Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The monetary-policy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 JEL.tex Preliminary. Comments welcome. 2.1 Example 1: A simple backward-looking model of the transmission mechanism . . 7 2.2 Example 2: A simple forward-looking model of the transmission mechanism . . . 10 3 A direct optimal-control approach: Commitment to an optimal instrument rule . . . . 12 What Is Wrong with Taylor Rules? 3.1 The backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Using Judgment in Monetary Policy through Targeting Rules 3.2 The forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 A commitment to the optimal instrument rule is impracticable . . . . . . . . . . 16 4 Commitment to a simple instrument rule . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Lars E.O. Svensson¤ 4.1 Advantages of a commitment to a simple instrument rule . . . . . . . . . . . . . 20 Institute for International Economic Studies, Stockholm University; 4.2 Problems of a commitment to a simple instrument rule . . . . . . . . . . . . . . . 21 CEPR and NBER 5 Commitment to a targeting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 First draft: May 1999 5.1 Generalizing monetary-policy rules . . . . . . . . . . . . . . . . . . . . . . . . . . 27 This version: July 2001 5.2 Forecast targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 A commitment to a general forecast-targeting rule . . . . . . . . . . . . . . . . . 30 5.3.1 The backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3.2 The forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Abstract 5.3.3 A “commitment to continuity and predictability” . . . . . . . . . . . . . . 34 Most recent research on monetary-policy rules is restricted to consider a commitment to 5.3.4 Advantages and problems of a commitment to a general targeting rule . . 34 simple instrument rules, where the central-bank instrument is a simple function of available 5.4 A commitment to a speci…c forecast-targeting rule . . . . . . . . . . . . . . . . . 36 information about the economy, like the Taylor rule. However, a commitment to a simple 5.4.1 The backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . 36 instrument rule appears inadequate as a description of current goal-directed, forward-looking, 5.4.2 The forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . 39 systematic and therefore rule-like monetary policy, especially in‡ation targeting. The latter 5.4.3 Advantages and problems of a commitment to a speci…c targeting rule . . 41 can to a large extent instead be seen as in‡ation-forecast targeting, setting the instrument so 5.5 Some criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 that the corresponding conditional in‡ation forecast is consistent with the in‡ation target. It is argued, both from a descriptive and a prescriptive perspective, that in‡ation targeting 5.6 Interest-rate stabilization and smoothing . . . . . . . . . . . . . . . . . . . . . . . 43 is better understood as a commitment to a targeting rule, either a general targeting rule 5.7 Distribution forecast targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 in the form of clear objectives for monetary policy or a speci…c targeting rule in the form 6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 of a condition for (the forecasts of) the target variables, essentially the equality between A Review of the Operation of Monetary Policy in New Zealand . . . . . . . . . . . . . . 50 marginal rates of transformation and marginal rates of substitution for the target variables. B Terms of Reference for the Review of the Operation of Monetary Policy . . . . . . . . 51 Targeting rules allow the use of judgment and extra-model information, are more robust and C Policy Targets Agreement, December 1999 . . . . . . . . . . . . . . . . . . . . . . . . . 53 easier to verify than optimal instrument rules, but they can nevertheless bring the economy D The backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 close to the socially optimal equilibrium. These ideas are illustrated with the help of simple examples. Some recent defense of commitment to simple instrument rules and criticism of D.1 Equality of the MRT and MRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 forward-looking monetary policy and targeting rules by McCallum, Nelson and Woodford D.2 A constant-interest-rate in‡ation forecast . . . . . . . . . . . . . . . . . . . . . . 57 are also addressed. In the concluding section, robust and optimal rules for monetary policy E The forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 are suggested. E.1 Strict in‡ation targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 E.2 The discretion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 JEL Classi…cation: E42, E52, E58 E.3 Equality of the MRT and MRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Keywords: In‡ation targeting, forecast targeting E.4 Equality of the MRT and MRS for a more general aggregate-supply relation. . . 63 E.5 A constant-interest-rate in‡ation forecast . . . . . . . . . . . . . . . . . . . . . . 64 F An optimal reaction function with response to forecasts for an unchanged interest rate 65 F.1 The backward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 F.2 The forward-looking model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ¤ I thank Jon Faust, Dale Henderson, Bennett McCallum and Michael Woodford for many previous discussions on issues related to this paper, and Claes Berg, Hans Dillén, Giovanni Favara, Kai Leitemo, Bennett McCallum, Frank Smets, Michael Wickens, Michael Woodford, and participants in conferences at the Bank of England and Insead for comments on previous versions of this paper. I also thank Annika Andreasson for editorial and secretarial assistance. Remaining errors and expressed views are my own. 0 1 Introduction a class of simple instrument rules and a loss function for evaluation alternative simple instrument rules in the class. From a descriptive perspective, it has been examined to what extent simple What are the rules for good monetary policy? Here, “good monetary policy” is used in the instrument rules are a good empirical description of central-bank behavior (see, for instance, conventional meaning of successfully stabilizing in‡ation around a low average level with some Clarida, Galí and Gertler [23] and Judd and Rudebusch [39]). From a prescriptive perspective, concern for stabilizing output around potential output, what has been called “‡exible in‡ation it has been examined how simple instrument rules perform (in the sense of stabilizing in‡ation targeting” in the literature (see, for instance, the contributions to the in‡uential Jackson-Hole around an in‡ation target without causing unnecessary output-gap variability) in di¤erent macro symposium organized by Federal Reserve Bank of Kansas City [30]).1 What answer does the models. large literature on monetary-policy rules supply? Most of that literature uses a very narrow The research on instrument rules have contributed many important insights. These insights interpretation of “policy rule.” According to this interpretation, a policy rule expresses the include that stability of in‡ation and determinacy of equilibria in sticky-price models require the central bank’s instrument (usually a short interest rate, the instrument rate; the federal funds long-run response of the short interest rate to in‡ation to be larger than one-to-one, the so-called rate in the U.S., for instance) as an explicit function of information available to the central “Taylor principle” (see Taylor [87] and Woodford [100]),3 and that interest-rate smoothing may bank. Such a policy rule can be called an instrument rule. In particular, most of the literature be improve performance (see Rotemberg and Woodford [65] and [97]). Other insights are that focuses on simple instrument rules, where the instrument is a function of a small subset of the it is better that the instrument responds to the determinants of the target variables than to the information available to the central bank. The best-known simple instrument rule is the Taylor target variables themselves (for instance, even if in‡ation is the only target variable (the only rule [85], where the instrument rate responds only to the in‡ation and output gaps according to variable in the loss function), it is generally better to respond to both current in‡ation and the ¹ it = f + f¼ (¼t ¡ ¼¤ ) + fx xt ; (1.1) output gap, since both these are determinants of future in‡ation; see for instance, Svensson [72] ¹ and Rudebusch and Svensson [67]), and that the response coe¢cients in the optimal reaction where it is the instrument rate in period t, f is a constant, ¼t ¡ ¼¤ is the “in‡ation gap,” where function depend on the weights in the loss function on di¤erent target variables in sometimes ¼t is (the rate of) in‡ation and ¼¤ ¸ 0 is a given in‡ation target, xt ´ yt ¡ yt is the output ¤ ¤ nonintuitive and complex ways (see, for instance, Svensson [72]). One line of research has gap, where yt is (log) output and yt is (log) potential output, and the coe¢cients f¼ and fx ¹ examined to what extent a given simple instrument rule is “robust,” in the sense of performing are positive. The constant f equals the sum of the average short real interest rate and the reasonably well in di¤erent macro models. Given the uncertainty about which model is the best in‡ation target. In the original Taylor [85] formulation, the coe¢cients f¼ and fx are 1.5 and .5, representation of reality, little would then be lost if central banks would apply a robust simple respectively; the in‡ation target ¼¤ is 2%/year, the average short real interest rate is 2%/year, ¹ instrument rule. Results to date, although arguably from not too di¤erent models of closed and the coe¢cient f is hence 4%/year.2 economies, indicate that variants of the Taylor rule can be quite robust in this sense.4 Much research during the last two decades has examined simple instrument rules (mostly Thus, the answer from most of the literature on monetary-policy rules to the …rst question variants of the Taylor rule), both from a descriptive and a prescriptive perspective (see, for above is that central banks should commit to following a speci…c simple instrument rule. Thus, instance, McCallum [55] and the contributions in Bryant, Hooper and Mann [15] and Taylor [88]). central banks should announce their simple instrument rule and then mechanically follow it. The introduction by Taylor [86] gives a summary of the standard approach of specifying a model, 1 This has the further implication that once the decision about the instrument rule is made, A noncontroversial objective of monetary policy would be to contribute to the welfare of the representative citizen. This is not an operational objective, though. An increasing number of countries have instead announced the decision process of the bank is exceedingly simple and mechanical. For the Taylor rule, “price stability” (meaning low and stable in‡ation) as the primary objective for monetary policy, with some implicit or explicit concern also for the stability of the real economy, with the view that this is the best contribution 3 monetary policy can make to citizens’ welfare. Several recent papers, for instance, Benhabib, Schmitt-Grohe and Uribe [5], Carlstrom and Fuerst [17] and 2 Wicksell [94] and Henderson and McKibbin [35] have suggested other simple instrument rules with the Christiano and Gust [22], examine determinacy and multiplicity of equilibria under the assumption that the interest rate as the instrument. Meltzer [58] and McCallum [53] have suggested simple instrument rules with the central bank follows Taylor-type instrument rules. 4 monetary base as the instrument. The …rst empirical estimates of interest-rate reaction functions may have been McCallum has in several papers, for instance, [53], examined robustness properties of a simple instrument in the 1960s by Dewald and Johnson [27] and Christian [21]. Recent general discussions of Taylor rules include rule for the monetary base. Levin, Wieland and Williams [48] examine the robustness properties of Taylor-type Hetzel [37], Kozicki [44] and Woodford [100]. rules. 1 2 it just consists of regularly collecting data on in‡ation and output, collecting either external policy, the concept of monetary-policy rules has to be broadened and de…ned as “a prescribed estimates of potential output or doing constructing internal estimates, and then calculating the guide for monetary-policy conduct,” including “targeting rules” as well as “instrument rules.”6 output gap. (Estimating potential output is a nontrivial matter, though, and a major challenge Furthermore, it argues that the monetary-policy practice is better discussed in terms of targeting in practical monetary policy.) Once these inputs in the Taylor rule are available, calculating rules than instrument rules. A general targeting rule speci…es the objectives to be achieved, for the instrument-setting is completely mechanical. In particular, there is no room for judgment instance, by listing the target variables, the target levels for those variables, and the (explicit (except that judgment may enter in the estimation of potential output). As McCallum [56] has or implicit) loss function to be minimized. A speci…c targeting rule speci…es conditions for the expressed it, policy decisions could be turned over to “a clerk armed with a simple formula and target variables (or forecasts of the target variables), for instance, like the above rule of thumb of a hand calculator.” the Bank of England and the Riksbank. From a prescriptive perspective, this paper argues that The contrast between this answer from most of the literature on monetary-policy rules and a commitment to targeting rules has a number of advantages, for instance, in relying on more actual monetary-policy practice is striking. First, monetary-policy reform in a number of coun- information, in particular, allowing the use of judgment, being more robust to model variation tries during the 1990s has to a large extent focused on (1) formulating explicit and increasingly than instrument rules, and likely leading to better monetary-policy outcomes than instrument precise objectives for monetary policy and (2) creating an institutional setting where the central rules. Presumably, this is why real-world monetary policy and monetary-policy reform has bank is strongly committed to achieving those objectives (see, for instance, Bernanke, Laubach, shunned commitment to instrument rules. Mishkin and Posen [7]). Thus, there has been commitment to objectives rather than to simple Before the rational-expectations revolution in macroeconomics, behavior of …rms and house- instrument rules. Second, central banks have developed very elaborate and complex decision- holds was frequently represented by simple ad hoc reaction functions, for instance, consumption making processes, where large amounts of information is collected, processed and analyzed, and and investment functions. The rational-expectations revolution lead to an emphasis on opti- where considerable judgment is exercised (see, for instance, Brash [13]). Third, any simple rules mizing and forward-looking behavior by private agents, and their behavior being represented by of thumb used are conditions for target variables or forecasts of target variables, rather than …rst-order conditions, Euler conditions, derived from their objectives and constraints. Still, the explicit formulas for the instrument rate. This is the case, for instance, for the rule of thumb pioneers of the rational-expectations revolution continued to represent economic policy by me- expressed by the Bank of England and Sveriges Riksbank (the central bank of Sweden), that chanical reaction functions, missing that under optimizing policy those reaction functions would normally, the interest rate should be adjusted such that the resulting in‡ation forecast at an be as much subject to the essence of the Lucas critique (that reaction functions are endogenous) appropriate horizon (usually about two-years ahead) is on target.5 No central bank has so far as mechanical reaction functions for private agents. made a commitment to a simple instrument rule like the Taylor rule or variants thereof. Monetary policy by the world’s more advanced central banks is these days at least as optimiz- Thus, there appear to be a substantial gap between the research on instrument rules and the ing and forward-looking as the behavior of the most rational private agents. I …nd it strange that practice of monetary policy. This paper discusses and proposes a way to bridge that gap. From a large part of the literature on monetary policy still prefers to represent central bank behavior a descriptive perspective, it argues that, in order to be useful for discussing real-world monetary with the help of mechanical instrument rules. The concept of general and speci…c targeting 5 This rule furthermore refers to constant-interest-rate forecasts, since both the Bank of England and the rules is designed to provide a discussion of monetary policy rules that is fully consistent with Riksbank mainly rely on such forecasts. The rule has been stated by Charles Goodhart [32], former member of the Bank of England Monetary Policy the optimizing and forward-looking nature of modern monetary policy. From this point of view, Committee, as: “When I was a member of the MPC I thought that I was trying, at each forecast round, to set the level of interest rates, on each occasion, so that without the need for future rate changes prospective (forecast) general targeting rules essentially specify operational objectives for monetary policy and speci…c in‡ation would on average equal the target at the policy horizon. This was, I thought, what the exercise was supposed to be.” The rule has been stated by Lars Heikensten, First Deputy Governor of the Riksbank, as: “Monetary policy is targeting rules essentially specify operational Euler conditions for monetary policy. In particu- normally conducted so as to be on target, de…ned in terms of the CPI, one to two years ahead.” Furthermore, any departures from this general rule, due to transitory disturbances to in‡ation or real costs from a quick return lar, an optimal targeting rule expresses the equality of the marginal rates of transformation and of in‡ation to target, will be announced by the Riksbank in advance (Heikensten [36, p. 16]). Berg [6] gives an 6 extensive account and discussion of the Riksbank’s implementation of in‡ation targeting. Target(ing) rules have previously been discussed by Rogo¤ [64], Walsh [93], Svensson [72], Rudebusch and Svensson [67], Cecchetti [18] and [19], Clarida, Gali and Gertler [24] and Svensson and Woodford [81]. 3 4 the marginal rates of substitution between the target variables in an operational way. to some extent the previous two sections, also responds to recent discussion by McCallum [56], Much monetary-policy reform during the last decade can be interpreted in terms of achieving McCallum and Nelson [57] and Woodford [95] regarding the positive and normative role of a trinity of (1) a mandate in the form of clear objectives for monetary policy, (2) operational commitment to instrument rules and targeting rules. Section 6 summarizes and presents some independence for the central bank, and (3) accountability of the central bank for ful…lling the conclusions. Appendices A-F contain technical and other details. mandate. Operational independence (also called instrument-independence) protects the central bank from short-term political pressure to stray from its objectives and accountability structures 2 The monetary-policy problem strengthens the bank’s commitment to ful…lling the mandate. This trinity can be seen as directed In order to induce su¢cient precision and clarity in the discussion, and to avoid the confusion and towards making monetary policy goal-directed and therefore optimizing, systematic and rule- misunderstanding in some of the literature on monetary-policy rules, it is necessary to provide like. New Zealand since the passing of the Reserve Bank Act in 1989 provides a good example a bit of formal notation (no math or algebra will be used except in the appendix, though). and has been a source of inspiration for reform in many other countries. In May 2000, I was asked In‡ation targeting involves stabilizing in‡ation around an in‡ation target. In practice, as by the Minister of Finance of the New Zealand Government to conduct a review of monetary discussed in a number of recent contributions (see, for instance, Federal Reserve Bank of Kansas policy in New Zealand. The evaluation of the goal-directed and forward-looking monetary policy City [30]), in‡ation targeting is “‡exible” in‡ation targeting, in the sense that it also involves in New Zealand raised many interesting issues and is discussed brie‡y in appendix A and more some concern about the stability of the real economy.7 These objectives are conventionally extensively in Svensson [78]. and conveniently expressed as an intertemporal loss function to be minimized in each period t, The rest of the paper is outlined as follows. Section 2 presents the monetary-policy problem t = :::; ¡1; 0; 1; :::; consisting of the expected sum of discounted current and future losses, facing an in‡ation-targeting central bank, namely to stabilize in‡ation around an in‡ation target "1 # X ¿ with (under realistic “‡exible” in‡ation targeting) some weight also on stabilizing the output E ± Lt+¿ j It : (2.1) ¿ =0 gap. The central-bank objective is expressed as a conventional intertemporal loss function Here E[¢jIt ] denotes rational expectations conditional on the central bank’s information, It , in to be minimized, subject to the central bank’s information about the state of the economy period t about the state of the economy and the transmission mechanism of monetary policy, and its view of the transmission mechanism. For concreteness, two simple examples of models and ± (0 < ± < 1) is a discount factor. Furthermore, Lt denotes the period loss in period t, of the transmission mechanism are presented, one backward-looking and one forward-looking. given by a weighted sum of the squared in‡ation gap and the squared output gap, Section 3 discusses a direct optimal-control approach: to solve the optimization problem for the optimal reaction function once and for all and then make a commitment to follow that 1£ ¤ Lt = (¼t ¡ ¼¤ )2 + ¸x2 ; t (2.2) 2 reaction function. It is shown that even in the simple examples of the transmission mechanism used here, the optimal reaction functions are too complex to be practicable, not to speak of where ¸ > 0 is a given weight on output-gap stabilization. Since the implicit output target being veri…able. For this and a number of other reasons discussed, the direct optimal-control in (2.2) is not subject to choice but given by potential output, the output target is not “over- approach must be judged infeasible. Section 4 discusses commitment to simple instrument rules, ambitious”, so there is no conspicuous reason for an in‡ation bias (average in‡ation above the which, although in principle veri…able, are found to be inadequate as a positive description in‡ation target) as in the literature on the time-consistency problem following Kydland and of real-world in‡ation targeting and likely to be unsuitable as a normative recommendation Prescott [45] and Barro and Gordon [3].8 “Strict” in‡ation targeting would be the (unrealistic) 7 I thus here abstract from any separate objective to stabilize or smooth interest rates, which objective is for monetary policy. Section 5 de…nes targeting rules and argues that a commitment to a di¢cult to rationalize. Such objectives and their consequences are discussed separately in section 5.6. 8 Nevertheless, because the forward-looking model to be used has a positively sloped long-run Phillips curve, targeting rule is both an appropriate description of real-world in‡ation targeting and a suitable average in‡ation will in some equilibria di¤er from the in‡ation target (when the in‡ation target di¤ers from zero). normative recommendation for future monetary-policy developments. This section, as well as 5 6 special case of ¸ = 0. Thus, for ¸ > 0, we have ‡exible in‡ation targeting and both in‡ation and where the coe¢cient ®x is positive, zt+1 is an exogenous variable discussed below, and "t is an the output gap are target variables (target variables in the sense of entering the loss function).9 iid “cost-push” shock with zero mean and variance ¾2 . Let aggregate demand (in terms of the " Many papers assume a loss function of the form output gap) be given by 1 (Var[¼t ] + ¸Var[xt ]); (2.3) xt+1 = ¯ x xt + ¯ z zt+1 ¡ ¯ r (rt ¡ r) + ´ t+1 ; ¹ (2.5) 2 the weighted unconditional variances of in‡ation and the output gap. This loss function can be where the coe¢cients ¯ x and ¯ r are positive, rt is a short real interest rate given by seen as the limit of (2.1): Note that, if the intertemporal loss function (2.1) is scaled by 1 ¡ ± P rt ´ it ¡ ¼t+1jt ; (2.6) and written E [(1 ¡ ±) 1 ± ¿ Lt+¿ j It ], the limit when the discount factor approaches unity, ¿ =0 ± ! 1, can be shown to equal where it is a short nominal interest rate and the central bank’s instrument, r is the average ¹ 1© ª 1 real interest rate, and ´ t is an iid “excess demand” shock with zero mean and variance ¾ 2 .11 (E[¼t ] ¡ ¼¤ )2 + ¸E[xt ]2 + fVar[¼t ] + ¸Var[xt ]g: ´ 2 2 Furthermore, qt+¿ jt for any variable q denotes Et qt+¿ jt ´ E[qt+¿ jIt ], the rational expectation of Thus, if the unconditional means ful…ll E[¼ t ] = ¼¤ and E[xt ] = 0, the limit of the intertemporal qt+¿ conditional on the information available in period t, It . Under the assumption of symmetric loss function is (2.3). information, the private sector has the same information as the central bank, so ¼t+1jt is one- The monetary-policy problem for the central bank is then, in each period t, to set its period-ahead private-sector in‡ation expectations, and hence it ¡ ¼t+1jt is the short real interest monetary-policy instrument, it (usually a short interest rate), so as to minimize the intertem- ¤ rate. Potential output, yt , is assumed to be an exogenous stochastic process. poral loss function (2.1), subject to the central bank’s information It about the state of the Let zt+1 be an unobservable exogenous variable (it could easily be expanded to a vector of economy and its view of the transmission mechanism for monetary policy (how the instrument exogenous variables) called the deviation. The idea is that it represents additional determinants a¤ects the target variables). of future in‡ation and the output gap than current in‡ation and the output gap, or the deviation For concreteness, I will use two simple examples of standard models of the transmission of the true model of in‡ation and output-gap determination from the simple model with the mechanism, one “backward-looking” and one “forward-looking”. deviation equal to zero. Thus, the sequence of deviations, fzt+¿ g1 ¿ =¡1 can be interpreted as unobservable model perturbations, as in the literature on robust control.12 The central bank’s 2.1 Example 1: A simple backward-looking model of the transmission mechanism estimate in period t of zt+¿ is denoted by zt+¿ ;t . The sequence z t ´ fzt+¿ ;t g1 ¿ =¡1 of the bank’s This example of a simple backward-looking model of a closed economy is a variant of that in estimate of in period t of past and future deviations is identi…ed with the bank’s judgment in pe- Svensson [72].10 The model has a one-period control lag for the output gap, and a two-period riod t. It represents the unavoidable judgment (almost) always applied in monetary policy. Any control lag for in‡ation. For reasons explained below, it is practical (and not unrealistic) to let explicit model is always taken as, at best, an approximation of the true model of the economy, the period be some 3 quarters (the period in Svensson [72] is taken to be a year). The main and monetary-policy makers always …nd it necessary to make some judgmental adjustments simpli…cation of the backward-looking model is that private-sector expectations are implicitly to the results of any given model. The so-called “add factors” applied to model equations in treated as adaptive expectations, which simpli…es the discussion considerably. central-bank projections is one aspect of central-bank judgment, see Reifschneider, Stockton and Suppose aggregate supply (the Phillips curve) is given by Wilcox [63]. 11 A slightly more complex variant of the backward-looking model would replace the constant average real ¤ ¼t+1 = ¼t + ®x xt + ®z zt+1 + "t+1 ; (2.4) interest rate, r , with an exogenous stochastic time-varying Wicksellian real natural interest rate, rt , as in the ¹ forward-looking model below. 9 12 Note that, since the intertemporal loss function is the expected discounted future losses, this formulation See, for instance, Hansen and Sargent [34] and Onatski and Stock [60]. However, that literature deals with ¤ the more complex case when the model perturbations are endogenous and chosen by nature to correspond to a includes the realistic case when potential output, yt , is unobservable and has to be estimated. 10 Ball [2] has later used a similar model. worst-case scenario. 7 8 One information structure consistent with the deviation zt being unobservable, is when termined by previous decisions and current exogenous shocks (the shocks include the di¤erence ¤ in‡ation ¼t , output yt , potential output yt , and the short real rate rt are all observable in between the deviation and the previous private-sector judgment, zt+1 ¡ zt+1jt ). Current in‡ation period t, but the shocks "t and ´t are unobservable. Alternatively, we can interpret ¤ yt as expectations for the next period, ¼t+1jt , are also predetermined by the current in‡ation, output corresponding to an observable component of potential output and the deviation zt corresponding gap and the exogenous variable according to (2.8). Actual in‡ation in the next period, ¼t+1 , will to an unobservable time-variable component of potential output and/or an unobservable time- then equal these in‡ation expectations plus next period’s unobservable cost-push shock, "t+1 , variable component of the equilibrium real interest rate. and the e¤ect of any unanticipated shock to the deviation, zt+1 ¡ zt+1jt . Next period’s output Given this interpretation of the deviation zt+1 , it would be completely misleading to make a gap, xt+1 , will be determined by the current variables, current in‡ation expectations, the cur- simplifying assumption like it being an exogenous autoregressive process.13 Thus, I will refrain rent instrument setting, it , next period’s exogenous variable, zt+1 , and next period’s output-gap from such an assumption and instead leave the dynamic properties of zt+1 unspeci…ed—also, shock, ´ t+1 . Thus, the central bank can a¤ect the output gap in the next period, but it cannot there is no presumption that the deviation would have a zero unconditional mean. Instead, a¤ect in‡ation until two periods ahead. That is, the control lags for the output gap and in‡ation the focus will be on the central bank’s judgment z t in period t of the whole sequence of future are one and two periods (3 and 6 quarters), respectively. As shown in Rudebusch and Svensson deviations. For simplicity, I assume that the central bank’s judgment is exogenous in period t. [67], this simple backward-looking model …ts U.S. data quite well. For simplicity, I also assume that there is symmetric information in that the private sector has the same information about the economy and the transmission mechanism, and that the private 2.2 Example 2: A simple forward-looking model of the transmission mechanism sector’s rational expectation in period t of the sequence fzt+¿ g1 1 ¿ =¡1 , denoted zjt ´ fzt+¿ jt g¿ =¡1 As another example of a standard model of the transmission mechanism, consider the so-called (the private-sector judgment), coincides with the central-bank judgment, z t .14 New-Keynesian model with forward-looking aggregate-supply and aggregate demand relations, Thus, I assume that ¼t , yt and ¤ yt and hence xt are observable and known in the beginning of similar to the one used in Clarida, Galí and Gertler [24]. I use the variant in Svensson and period t. Furthermore, I assume that the central bank’s instrument it is then set for the duration Woodford [81], where current in‡ation and output gap is not forward-looking but predetermined of period t. Note that one-period-ahead in‡ation expectations, ¼t+1jt , are predetermined, one period (which is easily motivated as a minimum move towards realism). Instead, the one- period-ahead in‡ation and output-gap expectations (or “plans”, see below), ¼ t+1jt and xt+1jt , ¼t+1jt = ¼t + ®x xt + ®z zt+1jt ; (2.8) are forward-looking.16 Furthermore, I use a variant, as in Yun [101], which allow …rms to in the sense that they do not depend on the instrument setting in period t, it , and only de- index prices to the average in‡ation rate rather than, rather arbitrary, only allowing constant pend on ¼t , xt and zt+1jt which by (2.4), (2.5) and the assumption that zt is exogenous are prices between opportunities for price adjustment. The aggregate-supply and aggregate-demand predetermined.15 equations are Thus, the setup implies that in‡ation and the output gap in the current period t are prede- ¼t+1 ¡ ¼ = ±(¼t+2jt ¡ ¼) + ®x xt+1jt + ®z zt+1 + "t+1 ; (2.9) 13 For instance, if the deviation was assumed to follow, ¤ zt+1 = °zt + µt+1 ; (2.7) xt+1 = xt+2jt ¡ ¯ r (it+1jt ¡ ¼t+2jt ¡ rt+1 ) + ¯ z zt+1 + ´ t+1 ; (2.10) where 0 · ° < 1 and µ t is an iid shock with zero mean and variance ¾2 . µ 14 Thus, the central bank’s and the private sector estimates of the cost-push and excess-demand shocks are by where ¼ is the average in‡ation rate, "t+1 and ´ t+1 are iid “cost-push” and “excess-demand” (2.4) and (2.5) trivially given by ¤ shocks and where rt is an exogenous Wicksellian natural interest rate corresponding to a “neu- ¤ "t+1;t = "t+1jt = ¼ t+1 ¡ ¼t ¡ ®y (yt ¡ yt ) ¡ ®z zt+1;t tral” real interest rate consistent with a zero output gap in the absence of deviations (see and ¤ ¤ ´ t+1;t = ´ t+1jt = yt+1 ¡ yt+1 ¡ ¯ y (yt ¡ yt ) ¡ ¯ z zt+1;t + ¯ r (rt ¡ r); ¹ Woodford [99] for further discussion of the Wicksellian natural interest rate). (For simplicity, 15 Intuitively, a variable is predetermined if it only depends on lagged variables and current exogenous shocks. 16 Intuitively, a variable is forward-looking (non-predetermined, or a jump variable) if it depends on expectations Formally, a variable is predetermined if it has exogenous one-period-ahead forecast errors. of future variables. Formally, a variable is forward-looking if it has endogenous one-period-ahead forecast errors. 9 10 the private-sector discount factor ± in (2.9) is taken to be the same as in the monetary-policy 3 A direct optimal-control approach: Commitment to an optimal instrument loss function (2.1)). Again, the exogenous deviation zt+1 enter both equations (recall that the rule deviation could easily be expanded to be a vector, so as to correspond to separate judgemen- A direct optimal-control approach to the monetary-policy problem would be to solve the monetary- tal adjustments to the aggregate-supply and aggregate-demand equations), to emphasize the policy problem once-and-for-all for the optimal reaction function (for a given model, or, more approximative nature of the simple model and the unavoidability of central-bank judgment.17 generally, for a given probability distribution of models). This would result in an optimal reac- In this model, private-sector one-period-ahead “plans” for in‡ation and the output gap, tion function, where the instrument in period t would be a function of the information available ¼t+1jt and xt+1jt , are determined in period t by in period t; ¼t+1jt ¡ ¼ = ±(¼t+2jt ¡ ¼) + ®x xt+1jt + ®z zt+1jt ; it = F (It ): ¤ xt+1jt = xt+2jt ¡ ¯ r (it+1jt ¡ ¼t+2jt ¡ rt+1jt ) + ¯ z zt+1jt : The optimal reaction function referred to here is the “explicit” reaction function, in the sense Thus, the one-period-ahead in‡ation plan depend on expected future in‡ation, ¼t+2jt , the output- that the instrument is written as a function of current and lagged predetermined variables only. gap plan, xt+1jt , and the private-sector judgment, zt+1jt . The one-period-ahead output-gap plan In a linear model with predetermined and forward-looking variables and a quadratic loss function, depends on the expected future output gap, xt+2jt , the expected one-period-ahead real interest- there is a unique form of the explicit reaction function (see Currie and Levine [26], Söderlind rate gap, ¤ it+1jt ¡ ¼t+2jt ¡ rt+1jt ´ ¤ rt+1jt ¡ rt+1jt , and the private-sector judgment, zt+1jt . Actual [70] and Svensson [75]). (This is for a given minimum set of linearly independent predetermined in‡ation and output gap in period t will then di¤er from the plans because of the unanticipated variables; a model can of course trivially be expressed in terms of alternative sets of linearly shocks, independent predetermined variables). Since, in equilibrium, the forward-looking variables will be linear functions of the predetermined variables, the instrument can of course be written as ¼t+1 = ¼t+1jt + ®z (zt+1 ¡ zt+1jt ) + "t+1 ; ¤ ¤ a continuum of linear functions of both the forward-looking and the predetermined variables.19 xt+1 = xt+1jt + ¯ r (rt+1 ¡ rt+1jt ) + ¯ z (zt+1 ¡ zt+1jt ) + ´t+1 . In the literature, it is quite common to discuss such reaction functions where the instrument Thus, in this model, the period-t expectation of the instrument in period t + 1, it+1jt , is responds not to predetermined variables but to forward-looking variables. These can be called what a¤ects future in‡ation and the output gap. I will assume that the central bank in period “implicit” reaction functions, since they express a functional relation between the instrument t announces what interest rate it will set in period t + 1, it+1;t . Since the central bank has no and another endogenous non-predetermined variable. They are indeed equilibrium conditions, incentive to stray ex post from any such announcement (since the interest rate does not enter which need to be solved together with the rest of the model in order to determine the instrument the period loss function (2.2)), I assume that it will set the actual interest rate it+1 according setting. Thus, explicit and implicit reaction functions, and corresponding explicit and implicit to its previous announcement and that the announcement hence will be credible and equal to instrument rules, are conceptually distinct. In particular, implicit instrument rules are not the private-sector expectations,18 directly operational, since they involve an endogenous variable that depends on the instrument it+1 = it+1;t = it+1jt . setting. In any realistic model, current in‡ation and the output gap are predetermined, also in 17 The assumption that …rms can index prices to average in‡ation between price adjustment opportunities the intuitive sense that they cannot be a¤ected by current monetary-policy decisions. Then a has the advantage that the long-run Phillips curve becomes vertical rather than positively sloped (see also the appendix of the working-paper version of Benhabib, Schmitt-Grohe and Uribe [5]). In the common formulation, Taylor rule (with the current instrument rate responding to current in‡ation and the current used for instance in Clarida, Galí and Gertler [24], the Phillips curve is instead (without the assumption of prices 19 predetermined one period) Let Xt and Zt denote the column vectors of predetermined and forward-looking variables, respectively. Let ¼ t = ±¼ t+1jt + ®x xt ; it = F Xt be the unique explicit reaction function (for simplicity, under discretion; under commitment the optimal reaction function also involves lags of the predetermined variables, see below) where F is a unique row vector which implies that the long run Phillips curve (when ± < 1) ful…lls ¼ = ®x x=(1 ¡ ±), where ¼ and x is the average or a matrix, depending on whether there are one or several instruments. In equilibrium (under discretion), the in‡ation and output gap, respectively. forward-looking variables will be given by Zt = GXt (under discretion), where G is a unique matrix. For any 18 Formally, we could say that the central bank instrument in period t is really the announcement, it+1;t , of matrix K of appropriate dimension, the instrument ful…lls the implicit reaction function it = KZt + (F ¡ KG)Xt . the future interest rate, rather than the current interest rate, it . 11 12 output gap) is an explicit reaction function. In many (unrealistic) models, current in‡ation and even if the simple models presented above, once the optimal response to judgment is taken into the output gap are forward-looking variables. Then a Taylor rule is an implicit reaction function, account. an equilibrium condition. McCallum, for instance, in [55], has emphasized a related point; that 3.1 The backward-looking model a Taylor rule in a quarterly setting is not operational, since output and in‡ation in the current quarter is reported with a lag and therefore not known in the current quarter. However, in such For the backward-looking model, (2.4) and (2.5), appendix D shows that the optimal reaction a setting, as discussed in Rudebusch and Svensson [67], a Taylor rule can be reformulated in function is given by µ ¶ terms of responding to the current estimates of current, yet unreported, output and in‡ation. 1¡c ¯ it = r + ¼ ¤ + 1 + ¹ (¼t+1;t ¡ ¼¤ ) + x xt As long as these estimates rely on predetermined information, such a Taylor rule would still be ®x ¯ r ¯r ¯z ®z (1 ¡ c) an explicit instrument rule.20 Since, in practice, data on economic variables are revised several + zt+1;t + zt+2;t ; ~ (3.1) ¯r ®x ¯ r times, all published data, also of past economic variables, is imperfect estimates of underlying where it is practical to de…ne, for ¿ ¸ 0, variables. However, veri…cation of a particular instrument rule is of course easier if it relies on 1 X zt+¿ ;t ´ ~ (±c)s zt+¿ +s;t ; (3.2) published data. s=0 Once the above optimization problem is solved and the optimal (explicit) reaction function the discounted sum (with the discount factor ±c) of judgments of future deviations in period t, is determined, the central bank would then make a commitment to follow the optimal reaction starting ¿ periods ahead (in this case, ¿ = 2). Furthermore, the coe¢cient c depends on the function, and then follow it mechanically ever after. Thus, once the commitment is made, there parameters of the model,21 is an increasing function c(¸) of ¸; the relative weight on output- is no more optimizing. This can be called a commitment to the optimal instrument rule: gap variability in the loss function (2.2), and ful…lls 0 · c(¸) < 1, with c(0) = 0 and c(1) ´ There are many problems with this approach. For a model with forward-looking variables, lim¸!1 c(¸) = 1. Moreover, ¼t+1;t denotes the central bank’s one-period-ahead forecast of the optimal reaction function is generally not time-consistent, in the sense that the central bank in‡ation, given by has an incentive to depart from it in the future. This is the case even if the output target is ¼t+1;t = ¼t + ®x xt + ®z zt+1;t : (3.3) equal to potential output and not overambitious, as is the case in the period loss function (2.2). This in‡ation forecast is predetermined in period t, so it is independent of the instrument it in This is because, even if there is no problem with an average in‡ation bias, there is a problem period t. (Under the assumption of symmetric information, the central-bank in‡ation forecast, with “stabilization bias” and a lack of “history-dependence,” to be further discussed below. ¼t+1;t , coincides with private-sector in‡ation expectations, ¼t+1jt .) Therefore the commitment to the optimal instrument rule in a forward-looking variable requires The …rst row of (3.1) is not so complex but a simple Taylor-type rule, with intercept r + ¼¤ , ¹ a commitment mechanism, a mechanism by which the central bank can be bound to follow the a positive response to the one-period-ahead forecast of the in‡ation gap, ¼t+1;t ¡ ¼¤ , with a optimal reaction function in the future. This in turn requires the optimal instrument rule to be coe¢cient above unity (since 1 + (1 ¡ c)=®x ¯ r > 1), and a positive response to the output gap, veri…able, so that it can be objectively established whether the central bank is diverging from with the coe¢cient ¯ x =¯ r . Given (3.3), this can also be expressed as a response to the current the optimal reaction function or not. in‡ation gap, ¼t ¡ ¼ ¤ , with a coe¢cient above unity, in line with the Taylor principle mentioned However, in any realistic model, the technical problem of deriving the optimal reaction above. function is overwhelmingly di¢cult, and the resulting optimal reaction function overwhelmingly However, the second row of (3.1) shows the optimal response to judgment, which is quite complex. Thus, even if the optimal reaction function could be calculated, it would be far too complex even for this simple model. Of course, with the simplifying assumption that the devi- complex to ever be veri…able. In fact, the optimal reaction function is overwhelmingly complex ation is an AR(1) process, the second row would be simpler and consist of a response to zt;t , 20 Svensson and Woodford [82] discuss problems when these estimates depend on forward-looking observable 21 The coe¢cient c is the smaller root of the characteristic equation of the di¤erence equation resulting from variables. the model and the …rst-order conditions. 13 14 still with a rather complex coe¢cient. But, as argued above, such a simplifying assumption is Thus, (3.7) can be substituted into (3.5), in which case the optimal reaction function can be totally unwarranted for any realistic form of judgment.22 written as a complex response to current and previous judgment. With forward-looking variables, the optimal reaction function is not time-consistent, and the 3.2 The forward-looking model central bank each period has an incentive to temporarily depart from it. Thus, a commitment For the forward-looking model, appendix E shows that the optimal interest-rate decision and mechanism and corresponding veri…ability of the reaction function is necessary. However, even announcement in period t for the interest rate in period t + 1, it+1;t , is given by in this exceedingly simple model of the transmission mechanism, the optimal reaction function is quite complex, especially with regard to judgment, and certainly impossible to verify.24 ®x ¸ it+1;t = rt+1;t + ¼¤ + (1 ¡ ¤ ) c(1 ¡ c)xt;t¡1 ¸¯ r ®x ¯ ®x 3.3 A commitment to the optimal instrument rule is impracticable + z zt+1;t + (1 ¡ )®z cf[1 ¡ ±c(1 ¡ c)]~t+2;t ¡ (1 ¡ c)zt+1;t g: z (3.5) ¯r ¸¯ r For several reasons, the direct optimal-control approach with a once-and-for-all calculation of As shown in appendix E, the optimal policy has no average in‡ation bias, so average in‡ation the optimal reaction function and then a commitment to this reaction function is completely equals the in‡ation target, impracticable. Indeed, this conclusion seems to be part of the conventional wisdom, and a ¼ = ¼¤ : (3.6) commitment to an optimal instrument rule has no advocates, as far as I know. We have already Then the average output gap is equal to zero. The coe¢cient c in (3.5), although not identical seen that, even in these two very simple models, the optimal reaction functions are quite complex, to that in (3.1), is still also an increasing function of ¸, c(¸), that ful…lls 0 · c(¸) < 1, with as soon as there is a role for deviations from the simple model and judgment of those deviations. c(0) = 0 and c(1) ´ lim¸!1 c(¸) = 1. Furthermore, zt+2;t is de…ned as in (3.2) (for ¿ = 2), ~ This complexity makes veri…ability impossible, although veri…ability is necessary as soon as although with the new coe¢cient c(¸). there is a time-consistency problem and an incentive to temporarily depart from the optimal Thus, for the forward-looking model, the optimal reaction function is di¤erent from a Taylor reaction function. In more realistic models, the complexity further increases dramatically. The rule. Instead, it corresponds to a response to the central bank’s forecast of the Wicksellian real optimal reaction function indeed requires that every conceivable contingency can be anticipated, ¤ interest rate, rt+1;t , with a unit coe¢cient, a response to the lagged forecast of the output gap, which is clearly impossible. xt;t¡1 , and a quite complex response to judgment. The response to the lagged forecast of the There are also more fundamental problems with the idea of a once-and-for-all commitment. output gap illustrates that the optimal reaction function under commitment involves responses If this commitment is possible in a particular period, what is special with that period? Why to lagged states of the economy, what is called “history dependence” in Woodford [97].23 didn’t the commitment occur in a previous period, leaving no possibility to recommitment this Appendix E shows that the lagged forecast of the output gap ful…lls period? Woodford [95] has provided an ingenious solution to that problem, by proposing a more 1 ®x c X j sophisticated kind of commitment, “in a timeless perspective.” This involves a commitment to xt;t¡1 = ¡ ®z c zt¡j;t¡1¡j : ~ (3.7) ¸ recommit only to reaction functions to which one would have preferred to commit oneself far j=0 22 In the unrealistic case of strict in‡ation targeting, ¸ = 0, we have c(¸) = c(0) = 0, and the optimal reaction into the past. This is a commitment not to exploit the possibility of a one-time “surprise” at the function becomes µ ¶ time of the recommitment. It allows optimal recommitment, for instance when new information 1 ¯ ¯ ®z it = r + ¼ ¤ + 1 + ¹ (¼ t+1;t ¡ ¼ ¤ ) + x xt + z zt+1;t + zt+2;t (3.4) ®x ¯ r ¯r ¯r ®x ¯ r 24 In the unrealistic case of strict in‡ation targeting, ¸ = 0, the optimal reaction function is 0 (which corresponds to (3.1) with c(¸) = 0, when we use the convention that 0 = 1). Thus, we see how the ¤ ¯z ®z optimal instrument rate should respond to the one-period-ahead forecast of the in‡ation gap, ¼ t+1;t ¡ ¼ ¤ , the it+1;t = rt+1;t + zt+1;t ¡ (zt+2;t ¡ zt+1;t ): ¯r ®x ¯ r current output gap, xt , the one- and two-period-ahead forecasts of the exogenous variable, zt+1;t and zt+2;t . Even in the simple case of strict in‡ation targeting, judgment matters (in the form of the one- and two-period-ahead Under strict in‡ation targeting, there is no problem with time-consistency (because there is no tradeo¤ between forecasts of the deviation). in‡ation- and output-gap stability). The optimal reaction function still involves response to judgment for one and 23 We note that the response to the lagged output-gap forcast can be of either sign, depending on ¸ ? ®x =¯ r : two periods ahead. 15 16 about the transmission mechanism arrives, without the disadvantage of the negative e¤ect on Forecasting and Policy System (FPS) of the Reserve Bank of New Zealand, [9]. For a forecast- expectations that the possibility of surprises otherwise induces. horizon T su¢ciently long, the in‡ation forecast is no longer predetermined but depends on the But Woodford’s ingenious idea does not diminish the already overwhelming problem of en- instrument. For such a horizon, the forecast in most applications have been taken to be an equi- forcement and veri…ability of a commitment to a complex instrument rule. For practical pur- librium, or “rule-consistent,” forecast, meaning that it is an endogenous rational-expectations poses, the direct optimal-control approach with either a once-and-for-all commitment or contin- forecast conditional on an intertemporal equilibrium of the model.25 Thus, this reaction func- uous recommitment in a timeless perspective will only be a theoretical benchmark for evaluation tion is really an equilibrium condition that has to be satis…ed by simultaneously determined purposes. It is not a coincidence that no central bank has tried to implement this approach. For variables. This is called an implicit instrument rule in Rudebusch and Svensson [67], Svensson a practical monetary-policy rule, we have to look elsewhere. [75] and Svensson and Woodford [81]. The reaction function (4.2) is sometimes said to represent “forecast targeting”; a more precise 4 Commitment to a simple instrument rule and consistent terminology is “responding to forecasts.” In this paper, as in Rogo¤ [64], Walsh [93], Svensson [72] and [75], Rudebusch and Svensson [67], Cecchetti [18] and [19], Clarida, Galí Let me start by specifying the idea of a commitment to a simple instrument rule. The …rst step and Gertler [24] and Svensson and Woodford [81], “targeting variable Yt ” means minimizing is to consider a restricted class of reaction functions, namely where the instrument is a function ¹ a loss function that is increasing in the deviation between the variable and a target level. In of a particular small subset, It , of the central bank’s information, It , contrast, in some of the literature “targeting variable Yt ” refers to a reaction function where the ¹ it = f(It ): instrument responds to the same deviation. As discussed in Svensson [75, section 2.4], these two meanings of “targeting variable Yt ” are not equivalent. The reason is that it is generally better Typically, the instrument is restricted to be a linear function of the target variables (in‡ation (in the sense of minimizing the loss function) that the instrument responds to the determinants and (estimates of) the output gap) and the lagged instrument, which results in a Taylor-type of the target variables than to the target variables themselves (for instance, even if in‡ation is the rule with interest-rate smoothing. Then the reaction function is only target variable (the only variable in the loss function), it is generally better to respond to ¹ it = f + f¼ (¼t ¡ ¼¤ ) + fx xt + fi it¡1 ; (4.1) both current in‡ation and the output gap, when both these are determinants of future in‡ation; see, for instance, Svensson [72], Rudebusch and Svensson [67], and the optimal reaction function ¹ where the constant f and the coe¢cients f¼ , fx and fi remain to be determined. The Taylor under strict in‡ation targeting (3.4) in footnote 22). Note also that in the forward-looking model, rule, (1.1), is the best known special case. In the realistic situation when in‡ation and the the optimal reaction function (3.5) does not respond to any of the current target variables but output gap are predetermined, (4.1) makes the instrument a simple function of predetermined to their determinants. “Responding to variable Yt ” therefore seems to be a more appropriate variables, called an explicit instrument rule in Rudebusch and Svensson [67], Svensson [75] and description of the latter situation. Svensson and Woodford [81]. Once the class of reaction functions is determined, the second step is to determine numerical Another class of reaction functions is when the in‡ation and output gaps are replaced by a ¹ values of the constant f and the coe¢cients f¼ , fx and fi , as well as, for forecast-based instrument T -period ahead forecast of the in‡ation gap, ¼t+T;t ¡ ¼¤ , rules, the forecast horizon T and the nature of the forecast (equilibrium, unchanged interest-rate, ¹ it = f + f¼ (¼t+T;t ¡ ¼¤ ) + fi it¡1 : (4.2) or otherwise). The coe¢cients can either be chosen so that the reaction function minimizes the intertemporal loss function (2.1) for a particular model (is optimal for a given model and the This class of reaction functions has been referred to as forecast-based (instrument) rules and 25 Another possibility is to let the forecasts depend on an exogenous interest rate path, for instance a constant is promoted by, for instance, Batini and Haldane [4]. Variants of it are used in the Quarterly unchanged interest rate, see Jansson and Vredin [38], Sveriges Riksbank [83, p. 58–61], Rudebusch and Svensson [67] and appendix F below. Projection Model (QPM) of Bank of Canada, [25], Black, Macklem and Rose [10], and the 17 18 given class of reaction functions), or such that it performs reasonably well for a few alternative If the central bank has speci…ed the form (4.3) of the simple instrument rule, the decision- models (is “robust” over a class of models).26 making process is somewhat more elaborate after (2): (3) One alternative is to be sophisticated, For the backward-looking model above, one simple instrument rule is just to forget about form the judgment zt+1;t , and use this together with current in‡ation and the output gap to form the judgment part in (3.1) and instead follow the reaction function ¼t+1;t according to (3.3). Another alternative is to be unsophisticated, disregard any judgment, µ ¶ 1¡c ¯ and use only current in‡ation and the output gap in (3.3) to form ¼ t+1;t . In the former case, it = r + ¼ ¤ + 1 + ¹ (¼t+1;t ¡ ¼¤ ) + x xt : (4.3) ®x ¯ r ¯r there is still some partial role for judgment, in the latter not. (4) Use ¼t+1;t and xt in (4.3) to As noted, this is a variant of the Taylor rule, with the predetermined one-period ahead forecast calculate it . (5) Announce and implement it . (6) In the next period, start over again. of the in‡ation gap entering instead of the current in‡ation gap. Using the expression of the In the forward-looking model, the steps are the following: (1) Construct a one-period-ahead in‡ation forecast in (3.3) but disregarding the judgment part of (3.3) results in the corresponding ¤ forecast of the Wicksellian natural interest rate, rt+1;t . This is a nontrivial step, comparable simple reaction function of current in‡ation and output gaps, to estimating potential output. (2) Recall the one-period-ahead output-gap forecast from the µ ¶ µ ¶ ¤ 1¡c 1 ¡ c ¯x previous period, xt;t¡1 . (3) Use rt+1;t and xt+1;t in (4.5) to calculate it+1;t . (4) Construct the it = r + ¼ ¤ + 1 + ¹ (¼t ¡ ¼¤ ) + ®x + + xt : (4.4) ®x ¯ r ¯r ¯r one-period-ahead output-gap forecast, xt+1;t , to be used in the interest-rate decision next period. For the forward-looking model above, the obvious corresponding reaction function that disre- This is a nontrivial step; it involves combining (5.5), (5.6) and (4.5) to solve for the resulting gards judgment in (3.5) would be output-gap forecast. (5) Announce it+1;t . (6) In period t + 1, implement it+1 = it+1;t and start ®x ¸ over again. it+1;t = rt+1;t + ¼¤ + (1 ¡ ¤ ) c(1 ¡ c)xt;t¡1 : (4.5) ¸¯ r ®x The fact that the simple instrument rule (4.5) relies on the lagged one-period-ahead output- In this case, the obvious simple reaction function di¤ers from the Taylor rule. Also, arguably it gap forecast, and the fact that constructing this is not so easy, makes this simple instrument rule is not so simple, since it responds to the one-period-ahead forecast of the Wicksellian natural still somewhat complex. A simpler instrument rule would be to follow a Taylor-type rule with interest rate, which may be di¢cult to estimate in practice. Furthermore, it responds to the interest-rate smoothing, (4.1). As discussed in Svensson and Woodford [81], as long as f¼ > 1 so previous one-period-ahead output-gap forecast rather than the current output gap (estimate). the Taylor principle is upheld, this results in a unique equilibrium in the forward-looking model, All this makes it less simple and also less easy to verify.27 although the Taylor-type rule will result in a worse outcome (a larger value of the loss function The third step, …nally, is for the central bank to commit to the particular simple instrument (2.1)) than (4.5). In this case, the decision process would be the same simple one as for the rule chosen and then follow it ever after, or at least until there is a recommitment to a new backward-looking model with the instrument rule (4.3). instrument rule. More precisely, once the simple instrument rule has been speci…ed, the central bank’s decision process is exceedingly simple. 4.1 Advantages of a commitment to a simple instrument rule In the backward-looking model, if the central bank has speci…ed the form (4.4) of the simple The advantages of a commitment to a simple instrument rule are that (1) the simplicity of the instrument rule, the decision-making process thereafter can be described as follows: (1) Collect instrument rule makes commitment technically feasible, and (2) simple instrument rules may be data on current in‡ation and current output. (2) Estimate potential output (which as noted relatively robust. above is a nontrivial step) and then subtract from output to get the current output gap. (3) Use Su¢cient simplicity of the instrument rule, for instance, if it is restricted to be a Taylor- (4.4) to calculate it . (4) Announce and implement it . (5) In the next period, start over again. type rule with interest-rate smoothing like (4.1), implies that it is easily veri…able. Then, a 26 The contributions to the conference volume edited by Taylor, [88], provide many examples on commitment to alternative simple instrument rules; se especially the introduction by Taylor [86]. The dominance of this approach commitment is in principle feasible. The forecast-based instrument rule, (4.2) is less easy to in current research is indicated by the fact that, in this volume, only Rudebusch and Svensson [67] also consider targeting rules. 27 verify, since the forecasts are less easy to verify. Still, a good In‡ation Report with a published Also, recall that the response to the lagged output-gap forcast can be of either sign, cf. footnote 23. 19 20 transparently motivated forecast may allow veri…cation to a considerable extent, and perhaps Levin, Wieland and Williams [48]). For a smaller and more open economy, the real exchange also make a commitment to a forecast-based instrument rule feasible. rate, the terms of trade, foreign output and the foreign interest rate seem to be the minimum Some research indicates that a Taylor-type rule with interest-rate smoothing is relatively essential state variables that have to be added (see, for instance, Svensson [77]), increasing the robust, in the sense that it performs tolerably well for a variety of models. This idea of robust number of response coe¢cients that must be …xed. I am not aware of any agreed-upon levels of simple instrument rules has been promoted and examined in several papers by McCallum and the response coe¢cients for these variables. recently restated in McCallum [55]. Results of Levin, Williams and Wieland [48] for a set With forward-looking variables, the optimal reaction function is characterized by history- of models of the U.S. economy indicate that a Taylor-type reaction function with interest-rate dependence, as has been emphasized by Woodford [97] and as was demonstrated in (3.5) and smoothing may be relatively robust in this sense. Intuitively, in (almost) closed-economy models (3.7). The lack of history-dependence may seem to be a problem for a simple instrument rule. where future in‡ation and the output gap mainly depend linearly on current in‡ation, the output However, any response to the lagged instrument rate implies some history dependence, since gap and the instrument rate, a Taylor-type instrument rule with the right coe¢cients will be (4.1) can be written ¹ 1 X j f optimal or close to optimal (as is for instance the case in the backward-model above, in the it = + fi [f¼ (¼t¡j ¡ ¼¤ ) + fx xt¡j ]: (4.6) 1 ¡ fi j=0 absence of any deviations). The models examined by Levin, Williams and Wieland [48] are all Thus, a suitable choice of the coe¢cient fi may allow some approximation to the optimal history of this type, as is the model used in Rudebusch and Svensson [67]. On the other hand, even dependence and, as further discussed in Woodford [97], partly remedy this problem. for such a restricted class of reaction functions (where the Federal funds rate only depends on A second problem is that a commitment to an instrument rule does not leave any room the in‡ation gap, the output gap and the lagged Federal funds rate), there is still considerable for judgmental adjustments and extra-model information, made explicit by the inclusion of the variation in the suggested magnitudes for the three coe¢cients, as is apparent from the papers deviation zt and the central-bank judgment z t above. As I believe most students of practical in Taylor [88]. Thus, there is far from general agreement on what the precise coe¢cients should monetary policy would agree with, practical monetary policy cannot (at least not yet) rely on be. models only. As further discussed in Svensson [79], the use of judgmental adjustments and extra- 4.2 Problems of a commitment to a simple instrument rule model information is both desirable in principle and unavoidable in practice. For instance, when a rare event, like a stockmarket crash, an Asian crisis or the ‡oating of the Brazilian real occurs, The problems with the idea of a commitment to a simple instrument rule include that (1) the central bankers may have to use their judgments rather than their models in assessing its likely simple instrument rule may be far from optimal in some circumstances, (2) there is no room for e¤ect on future in‡ation and output. Given the lags in the e¤ects of monetary policy, it will be judgmental adjustments and extra-model information, (3) desired development of the instrument e¢cient to respond to such an event before it shows up in the variables that enter the simple rule due to learning and new information will con‡ict with the commitment, unless sophisticated instrument rule, like current GDP and in‡ation. Indeed, Taylor [85] to a large extent discusses and (arguably) unrealistic recommitment “in a timeless perspective”, as suggested by Woodford the Fed’s departures from the Taylor rule and their reasons. Put di¤erently, a commitment to [95], is allowed, and (4) in spite of all the academic work and promotion, no central bank has a simple instrument rule does not provide any rules for when discretionary departures from the actually chosen to do it, and prominent central bankers sco¤ at the idea. simple instrument rule are warranted. A …rst obvious problem for a Taylor-type rule, with or without interest-rate smoothing, is However, the forecast-based instrument rule of the form (4.2) seems less sensitive to criticism that, if there are other important state variables than in‡ation and the output gap, it will not on this point. The equilibrium forecasts that enter the rule can in principle incorporate all be optimal. For a large and not so open economy as the U.S., in‡ation and the output gap relevant information, in particular, the judgment, z t . Nevertheless, it is unsuitable for other may be the most important state variables, and the e¢ciency loss in not responding to other reasons. A …rst reason, as already noted, is that it is (when responding to equilibrium forecasts) variables may in many cases be moderate (as seem to be the case for the models examined by an equilibrium condition rather than an operational explicit reaction function of predetermined 21 22 variables. Still, it could be a desirable equilibrium condition. More precisely, it could be a A third problem with simple instrument rules would seem to be that a once-and-for-all desirable speci…c targeting rule, a reformulated …rst-order condition for optimal policy.28 A commitment to an instrument rule would not allow any improvement of the instrument rule second reason is then that, unfortunately, the forecast-based instrument rule of the form (4.2) is when new information about the transmission mechanism, the variability of shocks, or the not an optimal targeting rule for the conventional loss function (2.1) with (2.2). As explained in source of shocks arrives.30 A once-and-for-all commitment also faces the problem of an incentive some detail in Svensson [80], one can work backwards and …nd the loss function for which (4.2) to exploit the initial situation, for instance, by temporarily increase output by an initial surprise is an optimal …rst-order condition. This loss function is such that, in each period, the central in‡ation, and let the simple reaction function apply only in the future. As a solution to the bank puts weight on both instrument-rate stabilization and instrument-rate smoothing (that problem of once-and-for-all commitments, Woodford [95] has, as discussed above in section 3.3, is, the instrument rate has to be separate target variable and enter the loss function directly, suggested repeated recommitment “in a timeless perspective” to new revised instrument rules which, as noted in section 5.6, is di¢cult to rationalize). Furthermore, the central bank must when new information arrives. The timeless perspective is a self-imposed restriction to consider be concerned with stabilizing the in‡ation gap at a …xed horizon T only. Thus, in contrast with only long-run instrument rules that do not depend on the period when the commitment is made, the conventional intertemporal loss function (2.1) with (2.2), the central bank does not consider and by construction it eliminates any exploitation of the initial situation.31 Presumably, though, any tradeo¤ between in‡ation gaps at di¤erent horizons. Indeed, the implied loss function such recommitment to a new instrument rule would have to occur relatively infrequently and for the forecast-based instrument rule does not ful…ll the minimum requirement of being time- only after substantial accumulated information has arrived. Nevertheless, recommitment to new consistent in the classical sense of Strotz [71], and in any period the central bank will, with this long-run instrument rules is at least a logically possible (and theoretically elegant) solution to loss function, regret previous decisions made. In addition, the implied loss function does not the problem of once-and-for-all commitment. incorporate any concern for output-gap stability (except indirectly through the horizon T ).29 Of course, along the lines of Woodford’s recommitment in a timeless perspective, the central A third reason is that, in line with the above, as demonstrated by Levin, Wieland and bankers faced with an Asian crisis could ask: Suppose that in the past, when we designed Williams [49], in simulations on di¤erent macro models, the performance of this particular our current instrument rule, we had anticipated the possibility of a future Asian crisis. What forecast-based instrument rule, as long as it does not utilizing short forecast horizons, is infe- response to the crisis would we then have committed ourselves to? Answering this question rior and nonrobust, when evaluated according to (2.1) with (2.2) or the special case (2.3). All would amount to revising the simple instrument rule by incorporating this particular event. together, the forecast-based instrument rule of the form (4.2) quite problematic, in spite of its Undertaking a substantial revision of the instrument rule may not make much sense unless the entrenched position in the QPM of the Bank of Canada [25] and the FPS of the Reserve Bank same event is expected to occur reasonably frequently in the future, though. It would also seem of New Zealand [9]. (See also the discussion in section 5.5 below.) However, arguably a reaction to require that the event is somehow incorporated in the models used to derive the optimal function when the instrument responds the unchanged-interest-rate forecasts make more sense simple instrument rule. However, if the central bank tries to incorporate too many possible and can be seen as a …rst-order Taylor expansion of an optimal …rst-order condition/targeting events, the instrument rule would no longer be simple, and veri…cation of the bank’s adherence rule, even if the central bank does have a separate instrument-rate stabilization and/or smooth- to the rule becomes increasingly di¢cult. Furthermore, there could be times when a relatively ing objective, see appendix F below and Jansson and Vredin [38] and Rudebusch and Svensson swift response is called for, without leaving much time for a thorough revision of the instrument [67]. rule. It seems that we still lack rules for when departures from the simple instrument rule are 28 Indeed, when (4.2) is used in in the QPM of the Bank of Canada [25] and the FPS of the Reserve Bank of called for, without which the simple rule is either incomplete or ine¢cient. New Zealand [9], the interest rate is a 3-month interest rate, which is strictly speaking not an instrument rate but 30 a market interest rate over which the central bank has less than perfect control. In that case, strictly speaking, For a linear model of the transmission mechanism and a quadratic loss function, certainty-equivalence applies (4.2) is not an “instrument” rule or a “reaction function,” but a “targeting” rule. for the optimal reaction function. That is, the reaction function does not depend on the variance of the shocks. 29 Certainty-equivalence does not apply for simple reaction functions, so the coe¢cients of the optimal simple reaction Thus, we see that, a forecast-based instrument rule can be seen as a speci…c targeting rule for a loss function that involves instrument-rate stabilization and/or smoothing. It cannot, at this stage of research, be excluded function does depend on the variance of the shocks (see Currie and Levine [26]). 31 that other forms of forecast-based instrument rules than (4.2) could be a speci…c targeting rule for a reasonable The idea of commitment in a timeless perspective is worked out by Woodford [95] for optimal reaction loss function. functions rather than simple ones (the model used is so simple that the optimal reaction function is quite simple). 23 24 Suppose a central bank went ahead and wanted to implement a commitment to a simple lightened discretion is the rule. instrument rule? How would it actually commit itself to the instrument rule? One extreme As stated by King [42], possibility would be to have the Central Bank Act (or, in New Zealand, the Policy Targets Agreement (PTA), see appendix C) include the instrument rule in a veri…able way (and also Mechanical policy rules are not credible... No rule could be written down that describes how policy would be set in all possible outcomes. Some discretion is in- specify suitable sanctions for departures from the rule), with revisions of the law when new evitable. But that discretion must be constrained by a clear objective to which policy information calls for revisions of the instrument rule. A less extreme possibility would be an is directed... Instrument-Rule Report (rather than the In‡ation Reports issued by many in‡ation-targeting As expressed by Bernanke and Mishkin [8]: central banks), where the central bank presents its derivation and motivation of the current In‡ation targeting does not represent [a commitment to] an ironclad policy rule... instrument rule, solemnly commits itself to follow it, and invites external scrutiny of its adherence Instead, in‡ation targeting is better understood as a policy framework... to the rule (and criticism and embarrassment if it departs from the rule). The rule would then be in e¤ect until a new issue of the Instrument-Rule Report presents and motivates a new revision But do not simple instrument rules …t actual central-bank behavior well? Several researchers, of the rule. Revisions would probably have to be rather infrequent to limit the amount of for instance, Clarida, Galí and Gertler [23] and Judd and Rudebusch [39], have found that discretion. variants of Taylor-type rules with interest-rate smoothing …t U.S. data reasonably well. The The decision-making process inside the central bank would then be quite uneven. The interpretation of this …nding is not obvious, though. First, the similarity of the outcome of infrequent revisions of the rule would be highly active and demanding periods, using all the policy decisions with a simple instrument rule is completely consistent with the forward-looking bank’s intellectual capacity. During the presumably long periods in-between, monetary policy goal-directed behavior by the central bank, say in the form of discretionary period-by-period op- could be conducted by a clerk with a hand calculator, or even a pre-programmed computer. timization, in a situation where in‡ation and the output gap are important state variables. That Furthermore, the central bank would be forward-looking only at the time when it reconsiders is, the simple instrument rule is a reduced form rather than a primitive, the endogenous end and recommits to a new instrument rule; in-between it would not be forward-looking but behave point rather than the exogenous starting point of monetary policy. Second, even the best empiri- in a completely mechanical way. cal …ts leave one third or more of the variance of changes in the federal funds rate unexplained.32 Thus, an obvious fourth problem is that commitment to a simple instrument rule is far Thus, departures from the simple instrument rule are substantial and ask for an explanation. from an accurate description of current monetary policy, in‡ation targeting or other. Such a Indeed, as noted above, Taylor’s [85] …rst discussion of the Taylor rule to a considerable extent monetary-policy setup does not exist in the current in‡ation-targeting countries, nor has it ever emphasized and discussed Fed departures from the simple instrument rule.33 existed before. No central bank has (to my knowledge) announced and committed itself to Still, the fact that historical examples of successful policy are similar to variants of simple an explicit instrument rule. No central bank has issued anything similar to an Instrument-Rule instrument rules, the fact that variants of simple instrument rules perform reasonably well Report. Nor does there seem to be any attempt to construct a commitment mechanism, whereby in a variety of di¤erent models, together with the fact that they can be derived as optimal a central bank would be obliged to follow a mechanical instrument rule. In spite of the impressive in some circumstances (for instance, in the backward-looking model above in the absence of academic work on a commitment to a simple instrument rule, I doubt that we will ever see such any deviation) imply that these simple instrument rules can serve a very useful role as rough 32 an arrangement materialize. Certainly, prominent current and previous central bankers seem Judd and Rudebusch [39], for instance, estimate reaction functions for the Federal Reserve System during the terms of Arthur Burns, Paul Volcker and Alan Greenspan. The best …t is for Greenspan’s term (sample sceptic and maintain that some amount of discretion is inevitable. As Blinder [11, p. 49] puts 1987:1–1997:4) and the partial-adjustment form ¢it = °(i¤ ¡ it¡1 ) + ½¢it¡1 , where i¤ is given by a Taylor-type t t ¹ rule, i¤ = f + f¼ (¼ t ¡ ¼¤ ) + fy (yt ¡ y ¤ ). The best adjusted R2 is 0.67. t 33 it, Rudebusch [66] suggests that the high estimated inertia in estimated Fed interest-rate reaction functions (a high coe¢cient on the lagged interest rate) can be explained by the Fed reacting to persistent shocks, in other words, to serially correlated judgment, which is entirely consistent with the reasoning in the present paper. Rarely does society solve a time-consistency problem by rigid precommitment... En- 25 26 guidelines, in that large departures from them have better to have good explanations. But they is increasing in the distance of the target variables from prescribed “target levels.” “Targeting” are not more than rough guidelines. They are not su¢cient as rules for good monetary policy. is minimizing such a loss function. Thus, although alternative instrument rules can serve as informative guidelines (as empha- A “general targeting rule” is a high-level speci…cation of a monetary-policy rule that speci…es sized in Taylor [85]),34 and decisions ex post may sometimes be similar to those prescribed operational objectives, that is, the target variables, the target levels and the loss function to be by the simple instrument rules, a commitment to a simple instrument rule (even with Wood- minimized. A “speci…c targeting rule” is instead expressed directly as an operational condition ford’s recommitment in a timeless perspective) does not seem to be a realistic substitute for the for the target variables (or for forecasts of the target variables). Under certain circumstances, forward-looking decision framework applied by in‡ation-targeting central banks. Indeed, instead commitment to a “general targeting rule” may be directly related to a particular “speci…c of making infrequent forward-looking decisions at the time of the infrequent recommitment to targeting rule,” which describes conditions that the forecast paths must satisfy in order to a new simple instrument rule, it seems that central banks instead choose to be continuously minimize a particular loss function. Nonetheless, it may be important to distinguish between forward-looking and have a regular cycle of decision-making. To quote Greenspan [33, p. 244], the two ways of describing the policy commitment, on grounds either of di¤ering e¢ciency as Implicit in any monetary policy action or inaction, is an expectation of how the means of communicating with the public, or of di¤ering degrees of robustness to changes in the future will unfold, that is, a forecast. model of the economy used to implement them. Furthermore, as discussed in Svensson and The belief that some formal set of rules for policy implementation can e¤ectively Woodford [81], a speci…c targeting rule need not be equivalent to any obvious general targeting eliminate that problem is, in my judgment, an illusion. There is no way to avoid making a forecast, explicitly or implicitly.35 rule, and indeed one of the primary reasons for interest in such speci…cations here will be their greater ‡exibility. This ‡exibility makes it possible to avoid the stabilization bias and lack of Therefore, I now turn to an, in my mind, better way of describing current in‡ation targeting, history-dependence that results from discretionary optimization when there are forward-looking namely as a commitment to a targeting rule, “forecast targeting.” variables. Any policy rule implies a reaction function, and hence an instrument rule. That instrument 5 Commitment to a targeting rule rule should not, in general, be confused with the policy rule itself. For example, the implied 5.1 Generalizing monetary-policy rules instrument rule associated with a given targeting rule will generally depend on the model of the economy and hence change with the model. Thus, in the case of a commitment to a targeting Thus, I …nd that a commitment to a simple instrument rule is not a good description of current rule, the monetary-policy regime is de…ned by the targeting rule, and the instrument rule is in‡ation targeting, nor does the concept of instrument rules seem su¢cient to discuss monetary- implied. The targeting rule is exogenous and the instrument rule is endogenous. In contrast, policy rules. Instead, the concept of monetary-policy rules needs to be broadened. in the case of a commitment to an instrument rule, the monetary policy rule is de…ned by the In order to discuss alternative decision frameworks for monetary policy, it is practical to have instrument rule. In that case, there may exist a speci…c targeting rule implied by the instrument a consistent classi…cation of such decision frameworks. To repeat, as in Rudebusch and Svensson rule. That targeting rule will generally depend on the model. Then, the instrument rule is [67], Svensson [75] and Svensson and Woodford [81], a “monetary-policy rule” is interpreted exogenous and the targeting rule is endogenous. There may also be a general targeting rule, broadly as a “prescribed guide for monetary-policy conduct.”36 This allows not only the narrow that is, a loss function, that is implied by the instrument rule and a given model. “instrument rules” but also the broader, and arguably more relevant, “targeting rules”. “Target variables” are operational goal variables and variables that enter a loss function, a function that 5.2 Forecast targeting 34 See, for instance, the contributions in Taylor [88] and, with regard to the performance of a Taylor rule for the Eurosystem, Gerlach and Schnabel [31], Peersman and Smets [61] and Taylor [89]. “Forecast targeting” refers to using forecasts of the target variables e¤ectively as intermediate 35 Budd [16], which alerted me to this quote, contains an illuminating and detailed discussion of the advantages of explicitly considering forecasts rather than specifying reaction functions from observed variables to the instrument. 36 Indeed, the …rst de…nition of “rule” in Merriam-Webster [59] is “a prescribed guide for conduct or action.” target variables, as in King’s [40] early characterization of in‡ation targeting, and means mini- 27 28 mizing a loss function where forecasts enter as arguments. Monetary policy a¤ects the economy corresponding conditional forecasts minimize the intertemporal loss function, which, in practice, with considerable lags. Current in‡ation and output are, to a large extent, determined by previ- means that the in‡ation forecast returns to the in‡ation target and that the corresponding ous decisions of …rms and households. Normally, current monetary-policy actions can only a¤ect conditional output-gap forecast returns to zero, at an appropriate pace. If the in‡ation forecast the future levels of in‡ation and the output gap, in practice with substantial lags and with the is too high relative to the in‡ation target at the relevant horizon (but the output-gap forecast total e¤ects spread out over several quarters. This makes forecasts of the target variables crucial is acceptable), the instrument-rate path needs to be raised; if the conditional in‡ation forecast in practical monetary policy. is too low, the instrument-rate path needs to be lowered. The chosen instrument-rate path is Assume that the transmission mechanism is approximately linear, in the sense that the future then the basis for the current instrument setting.40 In regular decision cycles, the procedure is target variables depend linearly on the current state of the economy and the instrument (as in then repeated. If no new signi…cant information has arrived, the forecasts and the instrument- the above examples of the two simple backward- and forward-looking models). Furthermore, rate path are the same, and instrument-rate setting follows the same instrument-rate path. assume that any uncertainty about the transmission mechanism and the state of the economy (The time-consistency problem that arises when there are forward-looking variables is further shows up as “additive” uncertainty about future target variables, in the sense that the degree discussed below.) If new signi…cant information has arrived, the forecasts and the instrument- of uncertainty about future target variables only depends on the horizon but not on the current rate path are updated. This is essentially the procedure recommended by Blinder [11] and state of the economy and the instrument setting. (This is the way the deviation enters in the referred to as “dynamic programming” and “proper dynamic optimization.” Compared to many backward- and forward-looking models used above.) It is then a standard result in optimal- other intertemporal decision problems that households, …rms and investors solve one way or control theory that so-called certainty-equivalence applies, and that optimal policy need only another (usually without the assistance of a sizeable sta¤ of PhDs in economics), this particular focus on conditional mean forecasts of the future target variables, forecasts conditional on the decision problem is, in principle, not overly complicated or di¢cult. central bank’s current information and a particular future path for the instrument.37 Since this Forecast targeting requires that the central bank has a view of what the policy multipliers means treating the forecasts as (intermediate) target variables (that is, putting forecasts of the are, that is, how instrument-rate adjustments a¤ect the conditional in‡ation and output-gap target variables in the loss function), the procedure can be called “forecast targeting.” (As forecasts. But it does not imply that forecasts must be exclusively model-based. Instead, it noted above, forecast targeting does generally not imply that the instrument should respond to allows for extra-model information and judgmental adjustments, as well as very partial infor- forecasts in the manner of (4.2); instead, in equilibrium, the instrument will end up responding mation about the current state of the economy. It basically allows for any information that is to the determinants of the forecasts of the target variables.) relevant for the in‡ation and output-gap forecasts. This decision-making process in the central bank then involves making conditional forecasts of in‡ation and the output gap, conditional on di¤erent paths of the central bank’s instrument 5.3 A commitment to a general forecast-targeting rule rate, using all relevant information about the current and the future state of the economy and Let me be more speci…c. Let it = fit+¿ ;t g1 denote an instrument plan in period t. Con- ¿ =0 the transmission mechanism.38 39 Then, the instrument-rate path is chosen, for which the ditional on the central bank’s information in period t, It (including its view of the transmis- 37 For proof of the certainty-equivalence theorem for optimal-control theory, see Chow [20] for models with sion mechanism, etc.), and its judgment, z t , and conditional on alternative instrument plans predetermined variables only and Currie and Levin [26] for models with both predetermined and forward-looking variables. it , consider alternative (mean) forecasts for in‡ation, ¼t = f¼t+¿ ;t g1 , and the output gap, 38 See Brash [13] and Svensson [78] for a discussion of the decision-making process of the Reserve Bank of New ¿ =0 Zealand, which provides a prime example of forecast targeting. 39 Constructing conditional forecasts in a backward-looking model (that is, a model without forward-looking xt = fxt+¿ ;t g1 (consisting of the di¤erence between y t , the (mean) output forecast, and y¤t , ¿ =0 variables) is straightforward. Constructing such forecasts in a forward-looking model raises some speci…c dif- 40 …culties, which are explained and resolved in the appendix of the working-paper version of Svensson [75]. The The procedure results in an implicit reaction function, where the instrument is an implicit function of all information that goes into constructing the forecasts. To the extent that the current in‡ation and output gap are conditional forecasts for an arbitrary interest-rate path derived there assume that the interest-rate paths are “cred- important determinants of the conditional forecasts, they will be important arguments of this implicit reaction ible”, that is, anticipated and allowed to in‡uence the forward-looking variables. Leeper and Zha [46] present an function. Thus, forecast targeting is fully consistent with the instrument settings super…cially appearing to follow alternative way of constructing forecasts for arbitrary interest-rate paths, by assuming that these interest-rate a Taylor-type rule. Since variables other than current in‡ation and the output gap also a¤ect the forecasts paths result from unanticipated deviations from a normal reaction function. signi…cantly, further scrutiny will normally reveal that the instrument also depends on those other variables. 29 30 the (mean) potential-output forecast). That is, ¼t+¿ ;t = E[¼t+¿ j it ; It ; z t ], etc. Furthermore, the output gap, that is, implicitly minimizes (5.1). consider the intertemporal loss function in period t applied to the forecasts of the target vari- ables, that is, when the forecasts are substituted into the intertemporal loss function (2.1) with 5.3.1 The backward-looking model (2.2), In the backward-looking model, the central bank’s forecasting model in period t will be given by 1 X ¿1£ ¤ ± (¼t+¿ ;t ¡ ¼¤ )2 + ¸x2 ;t : t+¿ (5.1) (2.4)–(2.6), where the corresponding forecasts are substituted for actual values (and forecasts of 2 ¿ =0 the shocks are set equal to zero), By a commitment to a general forecast-targeting rule, I mean a commitment to minimize a loss function over forecasts of the target variables. For an intertemporal quadratic loss function ¼t+¿ +1;t = ¼t+¿ ;t + ®x xt+¿ ;t + ®z zt+¿ +1;t ; (5.2) like (5.1), in principle this requires that the in‡ation target, ¼¤ , the relative weight on output- xt+¿ +1;t = ¯ x xt+¿ ;t + ¯ z zt+¿ +1;t ¡ ¯ r (it+¿ ;t ¡ ¼t+¿ +1;t ¡ r); ¹ (5.3) gap stabilization, ¸, and the discount factor, ±, are speci…ed. In practice, the loss function is not speci…ed in this detail, and the central bank has some discretion over the translation of the stated for ¿ ¸ 0. Thus, the forecasts ful…lling (5.2) and (5.3) are conditional on the central bank’s objectives into a loss function, for instance, how the Reserve Bank of New Zealand interprets judgment, z t ´ fzt+¿ ;t g1 t ¿ =¡1 , and alternative instrument plans, i . As is shown in appendix the PTA (see appendix C). Each period t, conditional on the central bank’s forecasting model, D, a central bank minimizing the intertemporal loss function will implicitly be satisfying the information It and judgment z t , the bank then …nds the combination of forecasts and instrument …rst-order condition ¸ plan that minimizes (5.1), the optimal forecasts and instrument plan, denoted (^ ¼ t ; xt ; ^t ); ^ { and ¼t+¿ +2;t ¡ ¼¤ = (±xt+¿ +2;t ¡ xt+¿ +1;t ) (5.4) ±®x then makes the current instrument decision according to the current optimal instrument plan for ¿ ¸ 0. This is the implicit condition for forecasts of the in‡ation and output gaps “looking (the current instrument decision will be given by ^t;t in the backward-looking model and ^t+1;t { { good.” Combining (5.4) with (5.2) and (5.3) leads to the optimal forecasts and instrument path, in the forward-looking model). (^ t ; xt ; ^t ), and to the instrument-rate decision, ^t;t , each period t. This instrument-rate decision ¼ ^ { { As stated, this decision-making process implies discretionary minimization each period of will be consistent with (3.1), the complex optimal reaction function for this model. However, the a well-de…ned intertemporal loss function. The process will result in an endogenous reaction commitment to the general targeting rule means that the central bank never need to make this function for the current instrument decision, a function F (It ; z t ) of the central bank’s informa- reaction function explicit; instead it just repeatedly solves its optimization problem each period tion and judgment. This reaction function need not be speci…ed explicitly, and it need not be and implements its instrument-rate decision. In particular, the instrument setting incorporates followed mechanically. For a model without forward-looking variables, the resulting endogenous the central bank’s judgment in an optimal way. instrument-setting will follow the optimal reaction function derived under the direct optimal- control approach discussed above. For a model with forward-looking variables, this decision- 5.3.2 The forward-looking model making process will result in a di¤erent reaction function than the optimal one, to be further In the forward-looking model, the central bank’s forecasting model will be discussed below. More precisely, how does the central bank …nd the optimal forecasts and instrument plan? ¼t+¿ +1;t ¡ ¼ = ±(¼t+¿ +2;t ¡ ¼) + ®x xt+¿ +1;t + ®z zt+¿ +1;t ; (5.5) ¤ One possibility is that, conditional on the information It and the judgment zt , the central bank xt+¿ +1;t = xt+¿ +2;t ¡ ¯ r (it+¿ +1;t ¡ ¼t+¿ +2;t ¡ rt+¿ +1;t ) + ¯ z zt+¿ +1;t ; (5.6) sta¤ generates a set of alternative forecasts (¼t ; xt ) for a set of alternative instrument plans it . for ¿ ¸ 0, given the central bank’s forecast of the natural real interest rate, r¤t ´ frt+¿ ;t g1 , ¤ ¿ =0 This way, the sta¤ constructs the “feasible set” of forecasts and instrument plans. The decision- and its judgment z t . With forward-looking variables, straight-forward discretionary optimization making body of the central bank then selects the combination of forecasts that “looks best,” in each period t of the loss function (5.1) encounters the time-consistency problem: Even in the the sense of achieving the best compromise between stabilizing the in‡ation gap and stabilizing 31 32 absence of any new information in period t + 1, the optimal instrument-rate setting in period 5.3.3 A “commitment to continuity and predictability” t + 1 will deviate from the optimal instrument-rate plan in period t. As discussed in Svensson and Woodford [81], a commitment to a modi…ed general targeting Under the assumption that the central bank instead anticipates the result of future optimiza- rule can solve the time-consistency problem and avoid the loss from discretionary optimization. tion each period, time-consistency is assured. Appendix E then shows, that the central bank More precisely, let 't+1;t¡1 denote the shadow cost of increasing the two-period-ahead in‡ation will each period t set the instrument rate so as to implicitly achieve the …rst-order condition forecast in period t ¡ 1, ¼t+1;t¡1 , due to the impact on the one-period-ahead in‡ation forecast, ¸ ¼t+1;t ¡ ¼¤ = ¡ xt+1;t : (5.7) ¼t;t¡1 That is, 't+1;t¡1 is the marginal increase of the intertemporal loss in period t ¡ 1 from ®x a higher two-period-ahead in‡ation forecast when this is allowed to a¤ect the one-period-ahead Combining this …rst-order condition with (5.5) will result in optimal forecasts, (^ t ; xt ). Com- ¼ ^ forecast. Modify the general targeting rule by adding the term 't+1;t¡1 (¼t+1;t ¡ ¼¤ ) to the loss bining these forecasts with (5.6) will result in the optimal instrument path, ^t . The optimal { function (5.1) each period t, which results in instrument-rate decision, it+1;t , is then given by it+1;t = ^t+1;t , which follows from combining { 1 X ¤ (5.6) for ¿ = 0 with xt+1;t , xt+2;t , ¼t+2;t , rt+1;t and zt+1;t . The implied reaction function need ^ ^ ^ 1£ ¤ ±¿ (¼ t+¿ ;t ¡ ¼¤ )2 + ¸x2 ;t + 't+1;t¡1 (¼t+1;t ¡ ¼¤ ): t+¿ (5.10) 2 never be made explicit. It is shown in appendix E that the implied reaction function is given by ¿ =0 ®z c ~ ®x ®z c + ¯ z ¸ ~ This can be interpreted as a commitment to a general targeting rule that involves “continuity it+1;t = rt+1;t + ¼¤ + ¤ [¯ ¸ ¡ ®x (1 ¡ ±~)]~t+2;t + c z zt+1;t : (5.8) ¯r ¸ r ¯r ¸ and predictability,” in that the previous cost of adjusting the forecast is taken into account.41 This reaction function is clearly di¤erent from the optimal reaction function, (3.5), although It is very much in line with the transparency, predictability and continuity emphasized in actual there is still no average in‡ation bias, so average in‡ation equals the in‡ation target and (3.6) in‡ation targeting (see, for instance, King [41]). holds. The coe¢cient c is di¤erent from the coe¢cient c of the optimal reaction function. It is ~ As shown in appendix E, the central bank will then each period t choose the instrument-rate still a function of the relative weight ¸ on output gap stabilization, ful…lls 0 · c(¸) < 1, and is ~ plan it+1;t so as to implicitly achieve the …rst-order condition now given by ¸ c(¸) ´ ~ ¸ : (5.9) ¼t+1;t ¡ ¼¤ = ¡ (xt+1;t ¡ xt;t¡1 ): (5.11) ¸ + ®2 ®x x Moreover, zt+2;t is de…ned as in (3.2), but with the coe¢cient c substituted for c. ~ ~ According to this …rst-order condition, the one-period-ahead in‡ation-gap forecast shall be pro- Thus, under discretionary forecast-targeting in the forward-looking model, the resulting in- portional to the negative of the change in the current one-period-ahead output-gap forecast from strument setting described by (5.8) will di¤er from the optimal reaction function (3.5). This the previous period, with the proportionality factor ¸=®x . As shown in the appendix, this is illustrates that discretionary optimization results in stabilization bias (the response to shocks in the optimal …rst-order condition, and combining it with the forecasting model (5.5) and (5.6) (5.8) is di¤erent from that in (3.5)) and a lack of history-dependence (since there is no response will result in the optimal forecasts and instrument plan. Consequently, the implied instrument to previous shocks in (5.8)). decision will ful…ll (3.5) and be consistent with the optimal reaction function.42 The reason why discretionary optimization does not result in the optimal outcome is that, in the decision-period t ¡ 1, an increase in the two-period-ahead in‡ation forecast for t + 1, 5.3.4 Advantages and problems of a commitment to a general targeting rule ¼t+1;t¡1 , increases the one-period-ahead forecast, ¼ t;t¡1 , via (5.5) when t ¡ 1 is substituted for A commitment to a general targeting rule means specifying clear objectives for monetary policy. t and ¿ = 1. However, in the decision-period t, the in‡ation forecast for t + 1, ¼t+1;t , can Clear objectives in the form of a well-speci…ed loss function is often taken for granted in research be increased without any e¤ect on in‡ation in period t, since the latter is now predetermined. 41 Adding a linear term to the loss function is similar to the linear in‡ation contracts discussed in Walsh [92] and Persson and Tabellini [62]. Indeed, the term added in (5.10) corresponds to a state-contingent linear in‡ation Therefore, the tradeo¤ (the marginal rate of transformation) involved in adjusting the in‡ation contract, which, as discussed in Svensson [73], can remedy both stabilization bias and average-in‡ation bias. 42 The observant reader notes that the modi…ed loss function making the discretion equilibrium optimal is forecast for t + 1 is di¤erent between decision periods t ¡ 1 and t. related to the idea of recursive contracts by Marcet and Marimon [52]. 33 34 on monetary policy. Nevertheless, in practical monetary policy, specifying clear objectives is a which enforces the commitment equilibrium. As discussed in Faust and Svensson [28], increased substantial achievement. In practice, discretion in monetary policy has often meant discretion transparency may increase the reputational costs of deviating from announced goals and this also with respect to the objectives, as is still the case to some extent for the Federal Reserve way enforce a policy closer to the optimal commitment.44 System. Specifying explicit objectives, together with operational independence and e¤ective Remaining problems with a commitment to a general targeting rule can potentially be solved accountability structures is rightly considered essential in an e¤ective monetary-policy setup. by a commitment to a speci…c targeting rule, though. A major advantage with a commitment to a general targeting rule is also that the central bank is free to use all information deemed essential to achieve its objective. In particular, it 5.4 A commitment to a speci…c forecast-targeting rule allows the central bank to exercise its judgment and extra-model information, as demonstrated A speci…c targeting rule speci…es a condition for the forecasts of the target variables, which can in the backward- and forward-looking models used in the examples above. formally be written as What are the problems with a commitment to a general targeting rule? One problem is that G(¼t ; xt ) = 0: (5.12) the objectives may still not be su¢ciently well speci…ed not to be open to interpretation. For This condition may be an optimal …rst-order condition, or an approximate …rst-order condition. instance, the relative weight on output-gap stabilization in ‡exible in‡ation targeting, the ¸ in Indeed, the optimal speci…c targeting rule expresses the equality of the marginal rates of trans- (2.2), is not directly speci…ed by any in‡ation-targeting central bank. In practice, evaluation formation and the marginal rates of substitution between the forecasts of the target variables in of in‡ation-targeting monetary policy is left with examining reported forecasts of the in‡ation an operational way. Then, the monetary-policy problem consists of …nding the combination of and output gaps and assessing whether they “look good” and provide a reasonable compromise forecasts and instrument path, (¼t ; xt ; it ), that is consistent with the central bank’s forecasting between keeping in‡ation close to target and the output-gap movements necessary for this (as model and ful…lls the speci…c targeting rule, (5.12). Thus, in contrast to a commitment to a was the case for me in Svensson [78]). general targeting rule, once the condition (5.12) has been speci…ed, …nding the optimal forecasts A second potential problem, emphasized by Woodford [95], is the potential consequences of and instrument plan is not a matter of minimizing a loss function but …nding the solution to a the discretionary optimization under a commitment to a general targeting rule, more precisely system of di¤erence equations.45 that such discretionary optimization is not fully optimal in a situation with forward-looking variables. As we have seen, discretionary optimization results in stabilization bias and a lack 5.4.1 The backward-looking model of history dependence.43 The practical and empirical importance of the ine¢ciency caused by In the backward-looking model, appendix D shows that the …rst-order condition for the forecasts discretionary optimization is not obvious, though. It is perfectly possible that, in realistic models ¼t and xt in period t is with considerable inertia and strong backward-looking elements, this ine¢ciency is overwhelmed by bene…ts from both specifying clear objectives for monetary policy and allowing all relevant ¸ ¼t+¿ +2;t ¡ ¼¤ = (±xt+¿ +2;t ¡ xt+¿ +1;t ) (5.13) information and judgment to bear on monetary-policy decisions. Simulations by McCallum and ±®x Nelson [57] and Vestin [91] do not reject the hypotheses that the ine¢ciency is relatively small. for ¿ ¸ 0. This is the optimal speci…c targeting rule for this model. Appendix D shows Furthermore, the discretion involved in a commitment to a general targeting rule may be that it follows directly from the equality of the marginal rate transformation from the output constrained by a few more sophisticated mechanisms. The emphasis in in‡ation targeting on gap into in‡ation (following from the aggregate-supply relation, (5.2)) and the marginal rate 44 predictability and transparency may be interpreted as a commitment to not surprising the private Indeed, both these mechanisms arguably provide some foundations for McCallum’s [54] loosely speci…ed idea of “just do it.” 45 sector, e¤ectively similar to the “commitment to continuity and predictability” introduced above Alternatively, we can say that the ¢ ¡ central bank has a new (intermediate) intertemporal loss function to mimimize in period t, namely [G ¼t ; xt ]2 , the minimum of which occurs for (5.12). (Thus, to each speci…c 43 The remedies Woodford [95] suggests are actually commitments to alternative speci…c targeting rules— targeting rule, we can assing a trivial general targeting rule.) although they are not called so. 35 36 of substitution of in‡ation for the output gap (following from the intertemporal loss function, of substitution between in‡ation and the output gap (hence on the parameters ± and ¸), and (5.1)). When the discount factor is close to unity, ± ¼ 1, the speci…c targeting rule can be the aggregate-supply relation, the Phillips curve (5.2), via the marginal rate of transformation written approximately as between in‡ation and the output gap (hence on the parameter ®x ). Since the marginal rates ¸ of transformation only depends on the derivatives of the aggregate-supply relation with respect ¼t+¿ +2;t ¡ ¼¤ = (xt+¿ +2;t ¡ xt+¿+1;t ): (5.14) ®x to in‡ation and the output gap, the additive judgement (the “add factors”) do not appear in That is, the in‡ation-gap forecast should be proportional, with the factor ¸=®x , to the forecast the optimal targeting rule. This illustrates the relative robustness of targeting rules (relative to of the change in the output gap. reaction functions and instrument rules) suggested in Svensson [72] and further examined and The decision-making process of the central bank each period t is then to …nd in‡ation- and con…rmed in Svensson and Woodford [81]. output-gap forecasts that are consistent with the speci…c targeting rule. This means combining Strict in‡ation targeting, ¸ = 0. For the case of strict in‡ation targeting, the speci…c (5.13) (or its approximation (5.14)) with the forecasting model, (5.2) and (5.3) for the judgment targeting rule (5.13) simpli…es to the trivial z t , and …nding the appropriate in‡ation- and output-gap forecasts and the corresponding instru- ment path. In particular, this can be done in a two-step procedure. First, the speci…c targeting ¼t+2;t = ¼ ¤ : rule is combined with the Phillips curve, (5.2), and the optimal in‡ation- and output-gap fore- casts, (^ t ; xt ), are determined. Then these forecasts are used in the aggregate-demand relation, ¼ ^ That is, the two-period-ahead in‡ation should equal the in‡ation target. From (5.2) for ¿ = 1, (5.3), to infer the corresponding instrument path, ^t . { Again, the optimal instrument setting in it follows that the optimal one-period-ahead output-gap forecast, xt+1;t , must ful…ll ^ period t is then given by it = ^t;t . This instrument setting will be consistent with the optimal { 1 xt+1;t = ¡ ^ (¼t+1;t ¡ ¼¤ + ®z zt+2;t ): ®x reaction function, (3.1), but this reaction function need not be made explicit. Indeed, given (5.3) for ¿ = 0, the judgment z t and the optimal forecasts ¼t and xt , the optimal instrument-rate ^ ^ Using this in (5.15) will result in the desired instrument setting, ^t;t , which will be consistent { setting is given by with the reaction function (3.4). Again, this reaction function need never be made explicit. 1 ¯ ¯ ^t;t = r + ¼t+1;t ¡ xt+1;t + x xt + z zt+1;t : { ¹ ^ (5.15) ¯r ¯r ¯r A simple speci…c targeting rule As noted above and in footnote 5, the Bank of England The speci…c targeting rule can be formulated, using the approximation (5.14), as: “Select the and Sveriges Riksbank have formulated a simple speci…c targeting rule, “set the instrument instrument path so that the marginal rate of transformation of the output gap into in‡ation and rate so that a constant-interest-rate in‡ation forecast about two-years ahead equals the in‡ation the marginal rate of substitution of in‡ation for the output gap are equal, more speci…cally, that target.” This can be seen as an attempt to formulate an operational and simple targeting rule, the in‡ation-gap forecast equals the proportion ¸=®x of the change in the output-gap forecast.” not necessarily optimal but hopefully not far from being optimal. If the two-year horizon is seen The central bank then need not optimize but just solve di¤erence equations. as longer than the minimum horizon at which in‡ation can be a¤ected, it can be interpreted as We note that the targeting rule (5.13) only depends on the parameters ± (the discount corresponding to ‡exible rather than strict in‡ation targeting. Since in the simple model it takes factor), ¸ (the relative weight on output-gap stabilization) and ®x (the e¤ect of the output a minimum of two periods to a¤ect in‡ation, and since I have assumed above that the period gap on in‡ation, the slope of the short-run Phillips curve). In particular, the targeting rule is is 3 quarters, let me interpret the approximate two-year horizon as three periods (9 quarters). independent of the coe¢cients ®z and ¯ z . That is, it is independent of how the deviation a¤ects Thus, this simple rule can be interpreted as in‡ation and output, and hence also of the judgment, the forecast of the deviation. Furthermore, the targeting rule (5.13) is independent of all the parameters of the aggregate demand curve, ¼t+3;t = ¼ ¤ ; (5.16) (5.3). Indeed, the targeting rule only depends on the loss function, (5.1), via the marginal rate 37 38 where the 3-period-ahead in‡ation forecast, ¼t+3;t , is taken to be conditional on a constant We note that the optimal speci…c targeting rule for the forward-looking model, (5.17) and interest rate, in this case corresponding to it+1;t = it;t = it . The implied reaction function (5.18), is di¤erent from the optimal speci…c targeting rule for the backward-looking model, resulting from this simple targeting rule is derived in appendix D.2. (5.13). Comparing (5.17) to the approximation (5.14) for ± ¼ 1, we see that the right side is Clearly, the simple targeting rule (5.16) is generally di¤erent from the optimal speci…c tar- the same but have opposite signs. The is because the marginal rate of transformation between geting rule (5.13) or (5.14). Consequently, the implied reaction function derived in appendix D.2 in‡ation and the output gap, the dynamic tradeo¤ between the target variables, is di¤erent for is di¤erent from the optimal reaction function (3.1), corresponding to this simple targeting rule the two aggregate-supply relations. As has been observed in the literature, the dynamics of the not being optimal. In addition to not being optimal, there are a number of additional problems backward-looking and the forward-looking Phillips curves (5.2) and (5.5) are quite di¤erent. A with using constant-interest-rate forecasts, as discussed in Kohn [43]. steady increase in in‡ation corresponds to a positive output gap in the backward-looking Phillips curve but a negative output gap in the forward-looking one (see Ball [1] and Mankiw [51]). 5.4.2 The forward-looking model The simple speci…c targeting rule like (5.16) raises additional issues and problems in a For the forward-looking model, appendix E and Svensson and Woodford [81] show that the forward-looking model, as discussed in appendix E.5 and by Leitemo [47]. optimal speci…c targeting rule is ¸ A commitment to a speci…c price-level targeting rule Consider a commitment to an ¼t+¿ +1;t ¡ ¼¤ = ¡ (xt+¿ +1;t ¡ xt+¿ ;t ) (5.17) ®x alternative speci…c targeting rule, related to price-level targeting. First, let pt denote (the log of) for ¿ ¸ 0, where for ¿ = 0, as discussed in detail in [81], xt;t is interpreted as given by the price level in period t, and de…ne a (log) price-level target path, p¤t = fp¤ ;t g1 , according t+¿ ¿ =0 xt;t ´ xt;t¡1 ; (5.18) to ¸ but the one-period-ahead output-gap forecast in the previous period, xt;t¡1 (and not the current p¤ ´ pt + t;t xt;t¡1 ; (5.19) ®x output gap, xt ). Again, appendix E shows that this speci…c targeting rule follows directly from p¤ ;t t+¿ ¤ ¤ = pt;t + ¼ ¿ : (5.20) the equality of the marginal rate transformation from the output gap into in‡ation (following This price-level target path starts from p¤ and then increases at a rate equal to the in‡ation t;t from the aggregate-supply relation, (5.5)) and the marginal rate of substitution of in‡ation target. Since the starting point depends on the current price level which is subject to random for the output gap (following from the intertemporal loss function, (5.1)). Again, because the shocks ("t + ®z (zt ¡ ztjt¡1 )), some base drift occurs. Second, specify the speci…c price-level marginal rate of transformation depends on the derivatives of the aggregate-supply relation with targeting rule as respect to in‡ation and the output gap, the judgment part and the aggregate-demand relation ¸ pt+¿ +1;t ¡ p¤ +1;t = ¡ t+¿ xt+¿ +1;t (5.21) do not appear. ®x Thus, the central bank should …nd the in‡ation and output-gap forecasts that ful…ll the for ¿ ¸ 0. That is, the price-level-gap forecast should be proportional to the negative of the speci…c targeting rule (5.17) and (5.18). This is done by combining the speci…c targeting rule output-gap forecast. This speci…c targeting rule is equivalent to the optimal targeting rule t with the forward-looking Phillips curve, (5.5), which results in the optimal forecasts, ¼ and ^ xt . ^ (5.17) and (5.18). It illustrates the close relation between optimal in‡ation targeting under From the aggregate-demand relation, (5.6), for ¿ = 0, the optimal instrument decision is given commitment and price-level targeting under discretion previously discussed by Svensson [76], by Vestin [90], Svensson and Woodford [81] and Smets [69]. 1 ¯ ^t+1;t = rt+1;t + ¼¤ + (^ t+2;t ¡ ¼¤ ) + { ¤ ¼ (^t+2;t ¡ xt+1;t ) + z zt+1;t : x ^ ¯r ¯r The resulting instrument decision will be consistent with the optimal reaction function, (3.5). Again, the optimal reaction function need never be made explicit. 39 40 5.4.3 Advantages and problems of a commitment to a speci…c targeting rule such a loss function.46 47 Second, McCallum and Nelson have also argued that targeting rules can be replaced by A commitment to a speci…c targeting rule has the obvious advantage of providing a more speci…c, obvious instrument rules, thereby implying that targeting rules are redundant. Consider, for more operational, and more easily veri…able commitment than a commitment to a general target- instance, the targeting rule (5.17) and (5.18) with ¿ = 0 for the forward-looking model above, ing rule. This way it provides stronger accountability. The right speci…c targeting rule has also which can be written the potential to overcome the ine¢ciency caused by discretionary optimization, while retaining ¸ ¼t+1;t ¡ ¼¤ + (xt+1;t ¡ xt;t¡1 ) = 0: (5.22) the ‡exibility in allowing all relevant information and judgment to bear on the monetary-policy ®x decision. Compared to the benchmark of a commitment to the optimal instrument rule, it is The idea is that this can be replaced by an instrument rule of, for instance, the form more robust, in the sense of only depending on part of the model of the transmission mechanism, ½ · ¸¾ ¸ it = (1 ¡ fi ) r + ¼t + ° ¼t+1;t ¡ ¼¤ + ¹ (xt+1;t ¡ xt;t¡1 ) + fi it¡1 ; (5.23) namely the marginal rate of transformation between the target variables. ®x A potential disadvantage, however, is that a speci…c targeting rule, in order to be optimal, where the response coe¢cient, °, is very large (McCallum and Nelson [57] in simulations suggest depends on the precise marginal rate of transformation, the dynamic tradeo¤ between the target ° ¸ 50) and in the limit approaches in…nity. Such an instrument rule would ensure that the variables. Therefore, it is not robust to di¤erent models of the aggregate supply relation, as is term within the bracket is arbitrarily close to zero and hence ful…lls the targeting rule (5.22). apparent in the examples of the backward-looking and forward-looking models used above. Thus, However, as discussed in detail in Svensson and Woodford [81], this is a dangerous and com- it is clearly less robust than a commitment to a general targeting rule. pletely impracticable idea. It is completely inconceivable in practical monetary policy to have The simple speci…c targeting rule of the Bank of England and Sveriges Riksbank discussed reaction functions with very large response coe¢cients, since the slightest mistake in calculat- above is at most a rather preliminary attempt to formulate an operational speci…c targeting rule ing the argument of the reaction function would have grave consequences and result in extreme and raises a number of problems, as discussed in Kohn [43] and Leitemo [47]. instrument-rate volatility. (Such interest-rate volatility does not arise in McCallum and Nelson’s [57] simulations because no mistakes are allowed for.) That fact that McCallum is known for 5.5 Some criticism favoring robust instrument rules makes this idea even more surprising.48 McCallum, for instance in [56], and McCallum and Nelson, for instance in [57], have several Third, McCallum and Nelson have also argued that the Reserve Bank of New Zealand pro- times criticized various aspects of and defended alternatives to ideas presented in this paper. vides an example of a central bank that is committed to an instrument rule, as an argument in First, as a defense of instrument rules involving responses to target variables only, they have favor the practical relevance of instrument rules. It is true that the Reserve Bank uses a reaction suggested that there is a bene…t to discussing monetary policy without reference to explicit loss function of the form (4.2) in its Forecasting and Policy System (FPS) in order to generate an functions. At the same time, the results of these instrument rules have been evaluated in terms endogenous future interest-rate path. However, for the …rst few quarters of this interest-rate 46 of the resulting variances of in‡ation and the output gap. But this is equivalent to using a loss A common way of evaluating the outcome of alternative instrument rules is to plot the result in a graph with unconditional in‡ation variance on the horisontal axis and unconditional output-gap variance on the vertical function consisting of a weighted sum of the variances of in‡ation and the output gap, (2.3), for axis and then examine the result in relation to the “Taylor curve” (see Taylor [84]) of e¢cient combinations of the two variances. This is of course equivalent to using a loss function of the form (2.3), with di¤erent relative di¤erent values of the weight ¸. It seems more transparent to me, then, to be explicit about weights ¸ ¸ 0. Indeed, a common way to …nd the Taylor curve is to optimize over a class of reaction functions for values of ¸ from zero to in…nity. See, for instance, Rudebusch and Svensson [67] and several other papers in Taylor [88]. (Taylor [84] plotted the standard deviations along the axes; plotting the variances has the advantage that the (negative) slope at a preferred point on the Taylor curve can be interpreted as revealing 1=¸ in the loss function above.) 47 Another problem with restricting simple rules to respond to target variables only is that the general principle is that it is best to respond to the main determinants of the (forecasts) of the target variables. The set of these main determinants is likely to include more variables than the current target variables; indeed, the current target variables may not be among the determinants at all (which is the case when the target variables are forward-looking variables). 48 Furthermore, on a more technical note and as examined in Svensson and Woodford [81], stability properties of the model are not invariant between (5.22) and (5.23). 41 42 path, the interest rate is set by discretion and judgment, and the reaction function is only used commitment to an optimal speci…c targeting rule is a more direct way of achieving such history- further into the future. Hence, it is not the case that the current interest-rate decision or even dependence. The practical importance of history-dependence also remains to be established, as the …rst few quarters of the interest-rate path is given by the reaction function. Brash [13] and noted above. As also noted above, Rudebusch [66] suggests that the high coe¢cient on the lagged Svensson [78] provide some further details on the decision-making process of the Reserve Bank.49 federal funds rate in estimated Fed reaction functions can be explained by the Fed reacting to persistent shocks rather than to some separate interest-rate smoothing objective. (Even though 5.6 Interest-rate stabilization and smoothing the PTA for the Reserve Bank of New Zealand states that the Reserve Bank “shall seek to avoid The discussion of instrument rules and targeting rules here has, except brie‡y in the discussion unnecessary instability in output, interest rates and the exchange rate,” I must confess that I, of the forecast-based instrument rule (4.2) in section 4.2, been under the assumption of no in my evaluation [78], did not much consider stability of interest rates as a separate objective.) separate monetary-policy objectives of interest-rate stabilization and/or smoothing. That is, If interest-rate stabilization and/or smoothing nevertheless is a separate monetary policy only in‡ation and the output gap has been considered target variables and hence entered the objective, the interest-rate in question is also a target variable. Most of the above-mentioned loss function, and only (forecasts of) in‡ation and the output gap have entered the speci…c reason for such objectives would probably apply to something like a 3-month market interest targeting rules discussed. rate, like a 3-month money-market rate, rather than the instrument rate (which is typically a I …nd the case for explicit instrument-rate stabilization and/or smoothing objectives quite repo rate or an overnight rate). Then this market interest rate becomes an additional target weak (see Sack and Wieland [68] for further discussion and empirical evidence). Such objectives variable, separate from, but related to (via the expectations hypothesis, for instance) to the would correspond to adding the term ¸i (it ¡ ¹)2 { + ¸¢i (it ¡ it¡1 )2 , where ¸i and ¸¢i are positive instrument rate. weights and ¹ denotes the average instrument-rate level, in the period loss function (2.2). Possible { The more target variables, the more complex the speci…c targeting rules. In particular, if adverse consequences for …nancial markets of interest-rate volatility, beyond the real e¤ects an interest rate is a target variable, the targeting rules not only depend on the loss function represented by output-gap stabilization, are hardly convincing, except in special circumstances and the Phillips curve, but also on the aggregate-demand relation, since the latter involve the with an exceptionally weak …nancial sector. A desire to avoid too large interest-rate surprises tradeo¤s between output and interest rates. It would still be the case that additive judgement, would rather correspond to a term of the form ¸i (it ¡ itjt¡1 )2 , where itjt¡1 denotes previous as in the backward- and forward-looking example models used here, would not enter explicitly market expectations of the instrument rate, but with a systematic and transparent monetary in the targeting rules. In this respect, the optimal targeting rule would still be simpler than the policy as in current in‡ation targeting, instrument-rate surprises are small anyhow. In practical optimal instrument rule, since the judgment would enter the latter, in addition in a complex monetary policy, there are recent conspicuous deviations from instrument-rate smoothing, in way, as we have seen in the above example models. In case the instrument rate enters the loss Fed interest-rate reductions in the …rst half of 2001 and previously Willem Buiter’s voting function, so the instrument rate rather than a somewhat longer market rate is a target variable, in the Bank of England MPC. Deviations from Friedman’s optimal quantity of money could the targeting rule would be indistinguishable from an implicit forecast-based instrument rule motivate a quadratic interest-rate term (cf. Woodford [96]), but since most money pays some (although, for a reasonable loss function, of di¤erent form than (4.2)). interest these days, the distortion would seem to be minor, and it is di¢cult to see that such 5.7 Distribution forecast targeting costs could be signi…cant compared to the costs of variability of in‡ation and the output gap. Woodford [97] has shown that an instrument-smoothing objective under discretion can induce Under the above assumptions of a quadratic loss function and an essentially linear transmission a desirable history-dependence of monetary policy. In the perspective of this paper, though, a mechanism, together with additive uncertainty, the certainty-equivalence result implies that 49 In my review of the operation of monetary policy in New Zealand, [78], I actually criticize the Reserve Bank the mean forecasts are the relevant target variables, regardless of the degree of uncertainty. for its use of this form of a reaction function and suggest that it considers alternatives, with reference to the same problems as those reported in section 4.2. When the uncertainty about the transmission mechanism is “nonadditive,” that is, there is 43 44 uncertainty about the policy multipliers, or if the transmission mechanism is characterized by two di¤er. Hence, one interpretation of the adjustment of the mode because of the balance of signi…cant nonlinearities, certainty-equivalence no longer applies, and the mean forecasts of risk, is that it is just a way of constructing the mean, in which case the procedure is still one the target variables are not su¢cient. Instead, the “balance of risks” and indeed the whole of mean forecast targeting rather than true distribution forecast targeting. It would be more probability distribution of the target variables matter. As discussed in Svensson [79] and [74], transparent to always let the reported point forecasts be the mean forecasts rather than the forecast targeting can then be generalized from mean forecast targeting to distribution forecast mode forecasts, and then explicitly report whether the balance-of-risk considerations imply that targeting. the banks are deviating from mean forecast targeting. Distribution forecast targeting would then consist of constructing conditional probability distributions of the target variables instead of mean forecast only. Thus, for a given instrument- 6 Summary and conclusions rate path, the central bank would construct the joint conditional density function of the random This paper starts from the observation that most recent research on monetary-policy rules is path of in‡ation and the output gap, conditional upon all information available in period t and restricted to consider a commitment to a simple instrument rule, where the central-bank in- a given instrument-rate path. Then, the intertemporal loss function is evaluated with the help strument is a simple function of available information about the economy, like the Taylor rule. of this conditional probability distribution. First, this can be done informally, by the decision- However, the paper argues that a commitment to a simple instrument rule is inadequate as making body of the bank. In this case, the decision-making body could be presented with the a description of current monetary policy, especially in‡ation targeting. First, monetary-policy probability distributions of the target variables for a few alternative instrument-rate paths and reform in the last two decades is better described as the formulation of clear objectives for then decide which path and distribution provides the best compromise. This is in principle monetary policy and the creations of institutional commitment to those objectives. Second, the same problem that any economic agent is assumed to solve in countless applications of in‡ation-targeting central banks have developed elaborate decision-making processes, in which decision-making under uncertainty. Second, given a numerical representation of the probability huge amounts of data is collected and processed, conditional in‡ation- and output-gap forecasts distributions and a speci…cation of the parameters of the loss function, the loss function can are constructed with the exercise of considerable judgment and extra-model information, and easily be evaluated numerically. an instrument decision is reached with the help of those forecasts. This process can to a large In‡ation-targeting central banks already seem to consider the whole probability distribution extent be seen as in‡ation-forecast targeting, setting the instrument so that the corresponding of the forecast, by considering the “balance of risks.” Furthermore, the Bank of England and conditional in‡ation forecast, conditional on all relevant information and judgment, is consis- Sveriges Riksbank have developed sophisticated methods for constructing con…dence intervals tent with the in‡ation target and the output-gap forecast not indicating too much output-gap for the forecasts published in their In‡ation Reports (see Blix and Sellin [12] and Britton, Fisher variability. Third, no central bank has made an explicit commitment to a simple instrument and Whitley [14]). The Bank of England presents fan charts for both in‡ation and output, and rule. Instead, some prominent current and former central bankers seem highly sceptical about Sveriges Riksbank gives con…dence intervals for its in‡ation forecasts. Furthermore, scrutiny of the idea. the motivations for instrument-rate changes (including the minutes from the Bank of England’s The paper attempts to bridge the gap between the recent literature’s focus on simple in- Monetary Policy Committee and the Riksbank’s Executive Board) indicate that both banks strument rules and the actual monetary-policy practice by in‡ation-targeting central banks. It occasionally take properties of the whole distribution into account in their decisions, for instance, argues that, in order to be more useful, the concept of monetary-policy rules should be broad- when the risk is unbalanced and “downside risk” di¤er from “upside risk.” ened beyond the narrow instrument rules and also include targeting rules. It argues that, both However, the point forecasts (the center of the con…dence intervals) reported by the Bank from a descriptive and a prescriptive perspective, in‡ation targeting is better understood as a of England and Sveriges Riksbank are, by tradition, mode forecasts (that is, the most likely commitment to a targeting rule, either a general targeting rule in the form of clear objectives outcome), rather than mean forecasts. When the probability distribution is asymmetric, these for monetary policy or a speci…c targeting rule in the form of a condition for (the forecasts of) 45 46 the target variables. The optimal speci…c targeting rule is actually an operational speci…cation not appear in the derivatives. Still, the optimal speci…c targeting rules is fully consistent with of the equality of the marginal rates of transformation and the marginal rates of substitution the use of judgment and extra-model information, since these enter into the construction of the between the target variables. Targeting rules have the important advantage that they allow forecasts that have to ful…ll the speci…c targeting rule. In contrast, the optimal instrument rules the use of judgment and extra-model information. They are also more robust and easier to have to include judgment explicitly, making them overwhelmingly complex and, in practice, verify than optimal instrument rules, but they can nevertheless bring the economy close to the impossible to verify. socially optimal equilibrium. These ideas are illustrated with the help of two simple examples In conclusion, then, what are the rules for good monetary policy, the initial question posed of the transmission mechanism. Some recent defense of commitment to simple instrument rules in this paper? My suggestion is: (1) Specify operational objectives, the general targeting rule. and criticism of forward-looking monetary policy and targeting rules by McCallum, Nelson and That is, specify the target variables, the target levels, and the relative weight(s) on stabilizing Woodford are also addressed. the target variables around their target levels. (2) Estimate the dynamic tradeo¤s between the Whereas simple instrument rules, like variants of the Taylor rule, may serve as rough bench- target variables, the marginal rates of transformation. In the standard case when the target marks for good monetary policy, they are very partial rules, because they don’t specify when variables are in‡ation and the output gap, this means estimating a Phillips curve. (3) Given the central bank should or should not deviate from the simple instrument rule. Such deviations, these marginal rates of transformation and the marginal rates of substitution from the loss by discretion and judgment, have been and will be frequent, in a descriptive perspective (recall function, calculate a …rst-order condition for optimal policy, that is, a speci…c targeting rule. If that simple instrument rules at most explain two thirds of the empirical variance of interest-rate this speci…c targeting rule is too complicated to be operational, simplify. In most cases, this will changes), and they should be frequent, from a normative perspective (since the simple instru- result in an operational condition for the forecasts of the target variables. (4) Estimate the rest of ment rules are not optimal and do not take judgement into account). In contrast, targeting rules the transmission mechanism, that is, the dynamical impact of the instrument rate on the target should be much more complete rules, because there are few good reasons to deviate from them, variables. (5) Conditional on the estimated transmission mechanism and on current information since they allow the use of judgment and extra-model information. and judgment, construct a set of forecast paths for the target variables for a set of alternative Macroeconomics long ago stopped modeling private economic agents as following mechani- instrument-rate paths. Select the forecasts and the instrument paths that best ful…ll the speci…c cal rules for consumption, saving, production and investment decisions; instead, they are now targeting rule, and set the current instrument rate accordingly. (6) When estimates of the normally modeled as optimizing agents that achieve …rst-order conditions, Euler conditions. marginal rates of transformation between the target variables are updated, revise the speci…c It is long overdue to acknowledge that modern central banks are, at least when it comes to targeting rule correspondingly. (7) Explain all this in transparent monetary-policy reports. the in‡ation targeters, optimizing to at least the same extent time as private economic agents; These rules for good monetary policy acknowledge that the speci…c targeting rules, the Euler therefore their behavior can be better modeled with the help of targeting rules than with simple conditions of monetary policy, will depend on the transmission mechanism via the marginal rates instrument rules. of transformation between the target variables. Therefore, they allow for revisions of the speci…c As stated above (several times), optimal speci…c targeting rules simply state the equality of targeting rules when the estimate of these marginal rates of transformation change. This ways, the marginal rates of transformation and the marginal rates of substitution between the target the overall rules for good monetary policy are robust, but the speci…c targeting rule is allowed variables in an operational way. Since the marginal rates of transformation depend only on to change with the estimated marginal rates of transformation. the derivatives of the transmission mechanism with respect to the target variables, the optimal There may be cases when the dynamic tradeo¤s between the target variables are too complex speci…c targeting rules are inherently simpler and more robust than the optimal instrument rules, to result in a simple operational speci…c targeting rule. In such cases, the central bank may which depend on all aspects of the transmission mechanism. In particular, additive judgement have to abandon an attempt to …nd a speci…c targeting rule and instead have to rely on the and add factors do not enter in the formulation of the speci…c targeting rules, because they do general targeting rule, namely selecting the forecasts and the instrument path that best seem 47 48 to minimize the intertemporal loss function. Although this can be done in more informal and A Review of the Operation of Monetary Policy in New Zealand intuitive ways, given numerical representations of the alternative forecasts and a speci…ed loss Much monetary-policy reform during the last decade can be interpreted in terms of achieving function, the loss function can always be evaluated numerically for each forecast alternative, in a trinity of (1) a mandate in the form of clear objectives for monetary policy, (2) operational independence for the central bank, and (3) accountability of the central bank for ful…lling the order to assist the decision-making body of the bank in …nding the best alternative. mandate. Operational independence (also called instrument-independence) protects the central Further research on general and speci…c targeting rules should both lead to a better under- bank from short-term political pressure to stray from its objectives and accountability structures strengthens the bank’s commitment to ful…lling the mandate. New Zealand since the passing of standing of actual monetary-policy practice and also better contribute to the further improve- the Reserve Bank Act in 1989 provides a good example and has been a source of inspiration for ment of that practice: Regarding general targeting rules, how can central banks be more speci…c reform in many other countries. The objectives for monetary policy are speci…ed in the Policy Targets Agreement (PTA) between the Treasurer/Minister of Finance and the Governor of the about the loss function they (explicitly or implicitly) apply? Regarding ‡exible in‡ation tar- Reserve Bank of New Zealand. The most recent PTA is from December 1999 (see appendix C). geting, how can central banks specify the other objective(s) besides stabilizing in‡ation and the Section 2b states that “the policy target shall be 12-monthly increases in the CPI of between 0 and 3 percent.” Section 4a states that the Bank “shall constantly and diligently strive to relative weight(s) on this (these) objective(s)? Regarding speci…c targeting rules, is it possible meet the policy target established by this agreement.” Furthermore, section 4c states that “[i]n to provide more optimal but still operational targeting rules than the Bank of England’s and pursuing its price stability objective, the Bank shall implement monetary policy in a sustainable, consistent and transparent manner and shall seek to avoid unnecessary instability in output, the Riksbank’s “the constant-interest-rate in‡ation forecast about two years ahead should equal interest rates and the exchange rate.” Finally, section 4d states that “[t]he Bank shall be fully the in‡ation target?” For instance, given an empirical forward- and backward-looking Phillips accountable for its judgements and actions in implementing monetary policy.” In line with the best international practice of in‡ation targeting, the PTA is interpreted as curve, is there an operational close-to-optimal speci…c targeting rule involving in‡ation- and “‡exible” in‡ation targeting with a medium-term point in‡ation target of 1.5 percent. Flexible output-gap forecasts? How robust is such a speci…c targeting rule to realistic revisions of the in‡ation targeting in the literature means stabilizing in‡ation around a given in‡ation target with some weight on stabilizing the real economy, more precisely with some weight on stabilizing Phillips curve? If some fraction of the current research on simple instrument rules were directed the output gap. In principle, this can be implemented by choosing the instrument rate so towards the study of targeting rules, we would soon know the answers to these questions. that in‡ation and output-gap forecasts conditional on the instrument setting result in a good compromise between the speed at which the in‡ation forecast approaches the in‡ation target and the output-gap movements required for this. In practice, it means aiming at the in‡ation targeting at a longer horizon than the shortest possible and normally conducting policy in a gradual and measured way and by this measured policy avoid destabilizing output. The Reserve Bank has developed an elaborate decision-making process (Brash [13]). The process takes 8 weeks up to the release of the quarterly Monetary Policy Statement (MPS), where the new instrument rate (the O¢cial Cash Rate or OCR) decision is announced and motivated. Since monetary policy a¤ects output and in‡ation with considerable lags (a very rough rule of thumb is about a year for output and about two years for in‡ation), this process is forward-looking and relies to a large extent on constructing forecasts. During the process a huge amount of data and informal information is collected, processed and analyzed, an assessment of the current state of the economy is made, and forecasts of in‡ation, the output gap and other variables are constructed under alternative scenarios, assumptions and instrument-rate paths. Needless to say, considerable judgment is applied during the process. This leads up to an instrument-rate decision with the corresponding projections being in line with the medium-term in‡ation target and avoiding unnecessary variability of the real economy. Given the prominent role of forecasts, this process can be described as using forecasts as intermediate variables, forecast targeting. Similar processes are used by the Bank of England and Sveriges Riksbank. In May 2000, I was asked by the Minister of Finance of New Zealand to conduct a review of the operation of monetary policy there (see appendix B for the terms of reference for the review). The review was published in February 2001 (Svensson [78]). Had the PTA speci…ed a simple instrument rule, like the Taylor rule, for the Reserve Bank, conducting the review 49 50 would have been a very simple matter.50 Arguably, the Minister would not have needed to Terms of Reference appoint a foreign academic for the review. Anyone could have compared the Reserve Bank’s The review will consider: instrument settings to those prescribed by the PTA and reported on the degree of compliance. Instead, my main way of evaluating whether policy settings had been appropriate was with 1. The way in which monetary policy is managed in pursuit of the in‡ation target. The the help of Reserve Bank forecasts (which are published in the Bank’s regular Monetary Policy review will examine the way the Reserve Bank interprets and applies the in‡ation target Statements). Given evidence that the Bank’s forecasts have insigni…cant bias and good precision set out in the Policy Targets Agreement, with a view to ensuring that this approach to (relative to forecasts by external forecasters), I then examined the Bank’s forecasts at the time achieving medium-term price stability is consistent with avoiding undesirable instability of decisions (as reported in the MPSs) and assessed whether monetary-policy settings were in output, interest rates and the exchange rate. such that the in‡ation forecast met the in‡ation target at the appropriate horizon without unnecessarily destabilizing the output gap.51 2. The instruments of monetary policy. The review will assess whether the Reserve Bank has an adequate range of instruments and is using its current instruments e¤ectively in B Terms of Reference for the Review of the Operation of Monetary Policy altering monetary conditions in the desired direction. 3. The information used by the Reserve Bank in its decision-making. The review will consider Background the range of sources, availability, type and timeliness of data, and the impact of these Ten years have passed since the Reserve Bank of New Zealand Act 1989 (”the Act”) came into variables on forecasting and decision-making. force on 1 February 1990. Having come through a period of transition to sustained price stability, 4. The monetary policy decision-making process. The review will consider whether the it is now appropriate to review the way in which New Zealand’s monetary policy is conducted decision-making process and accountability structures promote the best outcomes pos- and its e¤ectiveness in contributing to broader social and economic objectives. sible. Goal 5. The coordination of monetary policy with other elements of the economic policy framework, including an evaluation of the relationship between monetary policy operations and other The goal of the review is to ensure that the monetary policy framework and the Reserve Bank’s Reserve Bank functions such as prudential oversight of …nancial institutions. operations within that framework are appropriate to the characteristics of the New Zealand economy and best international practice. 6. The communication of monetary policy. The Reserve Bank’s communication of monetary policy decisions will be reviewed to ensure that these decisions are explained to the public Context and …nancial markets in the simplest, clearest and most e¤ective way possible. The goal for monetary policy is set out in Section 8 of the Reserve Bank Act: The reviewer may, where appropriate, seek to identify lessons for the future from the last decade’s experience with the framework and the Reserve Bank’s conduct of monetary policy The primary function of the Bank is to formulate and implement monetary policy over that period. directed to the economic objective of achieving and maintaining stability in the general level of prices. Process Through the maintenance of medium-term price stability, monetary policy contributes to The reviewer will: the Government’s broader economic objectives. Accordingly, this section of the Act will not be reviewed, nor is the Government willing to lessen the accountability of the Reserve Bank for 1. Invite interested parties to submit their views on the operation of monetary policy in the in‡ation outcomes or alter the operational autonomy of the Bank. areas detailed in these terms of reference. The reviewer is not required to consult further 50 with parties making submissions or other parties but will consult where it is useful to the In that case, sections 2a, 2b, 4a and 4c in the PTA could be replaced by, for instance, the sentence: “The Bank shall set the OCR to equal 5 + 1:5(¼ ¡ 1:5) + 0:5x percent; here ¼ refers to 12-monthly increases in the review. CPI expressed in percent and x refers to the output gap expressed in percent, as published by Statistics New Zealand.” (In this example, the constant 5 percent would correspond to the sum of an assumed average short 2. Obtain such other relevant expertise, including external research services, as is desirable real interest rate of 3.5 percent and the in‡ation target of 1.5 percent. Furthermore, the estimation of potential output is assigned to Statistics New Zealand.) to assist in the examination of issues covered in the report. 51 The main conclusion of the review is that, both in an absolute sense and relative to New Zealand’s in‡ation history, the Reserve Bank has achieved a remarkable stabilization of in‡ation at a low level and successfully 3. Report to the Minister of Finance and make such recommendations pertaining to the anchored in‡ation expectations on the in‡ation target. Except for the period (mid 1997 to March 1999) when operation of monetary policy and legislation as is appropriate within the context speci…ed the Bank used a Monetary Conditions Index (MCI) to implement monetary policy, given the circumstances at the time of decisions, there is no evidence that policy has systematically caused unnecessary variability in output, above. interest rates and the exchange rate. In March 1999, the MCI implementation was replaced by the use of a conventional short interest rate (the O¢cial Cash Rate or OCR) as the monetary-policy instrument. The Reserve Bank’s current conduct of monetary policy is found to be entirely consistent with the best international practice Reporting Date of ‡exible in‡ation targeting (as represented by, for instance, the Bank of England and Sveriges Riksbank). The MCI period, however, represents a substantial departure from the best practice. The review report shall be with the Minister by 28 February 2001. 51 52 C Policy Targets Agreement, December 1999 or are projected to occur, and what measures it has taken, or proposes to take, to ensure that in‡ation comes back within that range. POLICY TARGETS AGREEMENT c) In pursuing its price stability objective, the Bank shall implement monetary policy in This agreement between the Treasurer and the Governor of the Reserve Bank of New Zealand a sustainable, consistent and transparent manner and shall seek to avoid unnecessary (the Bank) is made under sections 9(1) and 9(4) of the Reserve Bank of New Zealand Act 1989 instability in output, interest rates and the exchange rate. (the Act), and shall apply for the balance of the Governor’s present term, expiring on 31 August 2003. It replaces that signed on 15 December 1997. d) The Bank shall be fully accountable for its judgements and actions in implementing In terms of section 9 of the Act, the Treasurer and the Governor agree as follows: monetary policy. 1. Price stability Hon Michael Cullen Donald T Brash Consistent with section 8 of the Act and with the provisions of this agreement, the Bank shall Treasurer Governor formulate and implement monetary policy with the intention of maintaining a stable general level Reserve Bank of New Zealand of prices, so that monetary policy can make its maximum contribution to sustainable economic growth, employment and development opportunities within the New Zealand economy. Dated at Wellington, this 16th day of December 1999 2. Policy target D The backward-looking model a) In pursuing the objective of a stable general level of prices, the Bank shall monitor prices In period t, consider …nding the combination of forecasts and instrument plan, (¼t ; xt ; it ), that as measured by a range of price indices. The price stability target will be de…ned in terms minimizes (5.1) subject to (5.2), (5.3) and the judgment z t . of the All Groups Consumers Price Index (CPI), as published by Statistics New Zealand. Given that the only target variables are ¼t and xt , this minimization can be simpli…ed b) For the purpose of this agreement, the policy target shall be 12-monthly increases in the into two stages. The …rst stage is to minimize (5.1), conditional on ¼ t , ¼t+1;t , xt and z t and CPI of between 0 and 3 percent.52 including only the constraint (5.2). This results in optimal forecasts, ¼ t ´ f^ t+¿ ;t g1 and ^ ¼ ¿ =0 xt ´ f^t+¿ ;t g1 . The second stage is then to use these optimal forecasts in (5.3), which implies ^ x ¿ =0 3. Unusual events 1 ¯ ¯ it+¿ ;t = r + ¼t+¿ +1;t ¡ ¹ xt+¿ +1;t + x xt+¿ ;t + z zt+¿ +1;t ; (D.1) a) There is a range of events that can have a signi…cant temporary impact on in‡ation as ¯r ¯r ¯r measured by the CPI, and mask the underlying trend in prices which is the proper focus to infer the optimal instrument plan ^t = f^t+¿ ;t g1 . The optimal instrument setting in period { { ¿ =0 of monetary policy. These events may even lead to in‡ation outcomes outside the target t is then ^t = ^t;t , and ^t+¿ ;t can be seen as a forecast of future instrument setting conditional on { { { range. Such disturbances include, for example, shifts in the aggregate price level as a result current information (and current judgment). of exceptional movements in the prices of commodities traded in world markets, changes Consider the Lagrangian corresponding to stage 1, in indirect taxes, signi…cant government policy changes that directly a¤ect prices, or a natural disaster a¤ecting a major part of the economy. 1 X 1 Lt = ± ¿ f [(¼t+¿ ;t ¡¼¤ )2 +¸x2 ;t ]+±'¿ +1;t (¼t+¿ +1;t ¡¼t+¿ ;t ¡®x xt+¿ ;t ¡®z zt+¿ +1;t )g; (D.2) t+¿ b) When disturbances of the kind described in clause 3(a) arise, the Bank shall react in a 2 ¿ =0 manner which prevents general in‡ationary pressures emerging. where '¿ +1;t is the Lagrange multiplier of the constraint (5.2). Note that ¼t , xt , ¼t+1;t and zt+¿ ;t are predetermined for ¿ ¸ 1, and consider the …rst-order conditions for an optimum, with 4. Implementation and accountability respect to ¼t+¿ +1;t and xt+¿ ;t for ¿ ¸ 1. They are a) The Bank shall constantly and diligently strive to meet the policy target established by this agreement. ¼t+¿ +1;t ¡ ¼¤ + '¿ +1;t ¡ ±'¿ +2;t = 0 (D.3) b) It is acknowledged that, on occasions, there will be in‡ation outcomes outside the target with respect to ¼t+¿ +1;t , and range. On those occasions, or when such occasions are projected, the Bank shall explain ¸xt+¿ ;t ¡ ±®x '¿ +1;t = 0 (D.4) in Policy Statements made under section 15 of the Act why such outcomes have occurred, with respect to xt+¿ ;t . From (D.4), we have 52 Statistics New Zealand introduced a revised CPI regime from the September quarter, 1999. Until the June quarter 2000, 12-monthly increases in the CPI will be calculated by comparing the new CPI series with the old ¸ CPI series adjusted by removing the impact of changes in interest rates and section prices. This adjustment is '¿ +1;t = xt+¿ ;t : calculated by Statistics New Zealand. (Refer to the RBNZ’s November 1999 Monetary Policy Statement, page 8, ±®x for details.) 53 54 Using this in (D.3), we can write a consolidated …rst-order condition as For the case of strict in‡ation targeting, ¸ = 0, we have c(0) = 0, zt+¿ ;t ´ zt+¿ ;t and ~ wt+¿ +1;t = 0, so the (D.8) is replaced by ¸ ¼t+¿ +1;t ¡ ¼¤ + (xt+¿ ;t ¡ ±xt+¿ +1;t ) = 0: (D.5) ¼t+¿ +1;t ¡ ¼¤ = 0 ±®x In order to …nd the equilibrium, rewrite (5.2) as for ¿ ¸ 1. It follows from (D.6) and (D.8), that the corresponding output-gap forecast is 1 xt+¿ ;t = (¼t+¿ +1;t ¡ ¼t+¿ ;t ¡ ®z zt+¿ +1;t ) (D.6) 1 ®x xt+¿ ;t = [¡ (1 ¡ c)(¼t+¿ ;t ¡ ¼¤ ) + ®z (wt+¿ +1;t ¡ zt+¿ +1;t )] ®x and use this to eliminate xt+¿ ;t in (D.5). This results in a di¤erence equation for ¼t+¿ +1;t , 1¡c = ¡ [(¼t+¿ ;t ¡ ¼¤ ) + ®z zt+¿ +1;t ] : ~ ®x ¸ ¼t+¿ +1;t ¡ ¼¤ + [(¼t+¿ +1;t ¡ ¼t+¿ ;t ¡ ®z zt+¿ +1;t ) ¡ ±(¼t+¿ +2;t ¡ ¼t+¿ +1;t ¡ ®z zt+¿ +2;t )] = 0: By (D.1), the optimal interest setting in period t then follows ±®2x 1 ¯ ¯ For the case of ‡exible in‡ation targeting, ¸ > 0, rewrite the di¤erence equation as it = r + ¼ t+1;t ¡ ¹ xt+1;t + x xt + z zt+1;t ¯r ¯r ¯r 1 ®z 1¡c ¯ ¯ (¼t+¿ +2;t ¡ ¼¤ ) ¡ 2a(¼t+¿ +1;t ¡ ¼¤ ) + (¼t+¿ ;t ¡ ¼¤ ) = ¡ (zt+¿ +1;t ¡ ±zt+¿ +2;t ); = r + ¼ t+1;t + ¹ [(¼t+1;t ¡ ¼ ) + ®z zt+2;t ] + x xt + z zt+1;t ¤ ~ ± ± ®x ¯ r ¯r ¯r µ ¶ ¤ 1 ¡ c) ¤ ¯x where = r+¼ + 1+ ¹ (¼t+1;t ¡ ¼ ) + xt 1 ®2 ®x ¯ r ¯r + x 2a ´ 1 + (D.7) ¯ ®z (1 ¡ c) ± ¸ + z zt+1;t + zt+2;t : ~ (D.11) (since ¼t+1;t is given, it is natural to express the di¤erence equation in terms of the in‡ation ¯r ®x ¯ r forecasts). Under the special case (2.7), we have By standard methods, the solution to this di¤erence equation can be shown to ful…ll 1 ¡ ±° wt+¿ +1;t = ° ¿ +1 zt ; ¼t+¿ +1;t ¡ ¼¤ = c(¼t+¿ ;t ¡ ¼¤ ) + ®z wt+¿ +1;t (D.8) 1 ¡ ±°c(¸) 1 for ¿ ¸ 1 (recall that ¼t+1;t is predetermined). Here, the coe¢cient c ful…lls 0 < c < 1 and is zt+¿ ;t = ~ ° ¿ zt ; 1 ¡ ±°c(¸) the smaller root of the characteristic equation, 1 ¡ ±° ¿ +1 ¼t+¿ +1;t ¡ ¼¤ = c(¼ t+¿ ;t ¡ ¼¤ ) + ®z c° zt ; 1 1 ¡ ±°c 2 ¹ ¡ 2a¹ + = 0; (D.9) · ¸ ± 1¡c ®z xt+¿ ;t = ¡ (¼t+¿ ;t ¡ ¼¤ ) + ° ¿ +1 zt ; ®x 1 ¡ ±°c hence given by r · ¸ 1¡c ¯ ¯ ®z 1 ¡ c 1 it = r + ¼ t+1;t + ¹ (¼t+1;t ¡ ¼ ¤ ) + x xt + z + ° °zt : c´a¡ a2 ¡ : (D.10) ®x ¯ r ¯r ¯ r ®x ¯ r 1 ¡ ±°c ± Furthermore, c is an increasing function of ¸, c(¸), which ful…lls c(0) = lim¸!0 c(¸) = 0, D.1 Equality of the MRT and MRS c(1) ´ lim¸!1 c(¸) = 1. Finally, the variable wt+¿ +1;t is given by Given the aggregate-supply relation, (5.2), for ¿ ¸ 0, consider changes in xt+j;t , j ¸ 1, that 1 X result in d¼t+2;t 6= 0 and d¼t+i;t = 0 for all i 6= 2. This requires wt+¿ +1;t ´ c (±c)s (zt+¿ +1+s;t ¡ ±zt+¿ +2+s;t ) s=0 dxt+1;t 6= 0 1 X d¼t+2;t = ®x dxt+1;t = czt+¿ +1;t ¡ c±(1 ¡ c) (±c)s zt+¿ +2+s;t s=0 dxt+2;t = ¡ d¼t+2;t =®x = ¡ dxt+1;t (to make d¼t+3;t = 0) = zt+¿ +1;t ¡ (1 ¡ c)~t+¿ +1;t ; z dxt+j;t = 0; j ¸ 3: where Let MRT2;(1;¡1);t denote the marginal rate of transformation of the linear combination (xt+1;t ; xt+2;t ) = 1 X (1; ¡ 1)xt+1;t into ¼t+2;t . It ful…lls zt+¿ ;t ~ ´ (±c)s zt+¿ +s;t : ¯ s=0 d¼t+2;t ¯ ¯ MRT2;(1;¡1);t ´ = ®x : dxt+1;t ¯dxt+2;t =¡ dxt+1;t 55 56 Let Lt denote the value of the intertemporal loss function, (5.1), and let MRSi;j;t denote the Thus, the two-period-ahead output-gap forecasts depends on the one-period-ahead in‡ation fore- marginal rate of substitution of in‡ation in period t + i, ¼t+i;t , for the output gap in period t + j, cast, ¼t+1;t , the current output gap, xt , the one- and two-period-ahead forecast of the exogenous xt+j;t . It ful…lls ¯ variable, zt+1;t and zt+2;t , and the current and one-period-ahead forecasts of the interest rate, it d¼t+j;t ¯ ¯ ± j ¸xt+j;t and it+1;t : MRSi;j;t ´ =¡ i : (D.12) dxt+i;t ¯dLt =0 ± (¼t+i;t ¡ ¼ ¤ ) Construct the 3-period-ahead in‡ation forecast, Let MRS2;(1;¡1);t denote the marginal rate of substitution of ¼t+2;t for the linear combination ¼t+3;t = ¼t+2;t + ®x xt+2;t + ®z zt+3;t (xt+1;t ; xt+2;t ) = (1; ¡ 1)xt+1;t . It ful…lls ¯ = (1 + ®x ¯ r )¼t+1;t + ®x ¯ x xt + ®x ¯ z zt+1;t + ®z zt+2;t ¡ ®x ¯ r (it ¡ r) ¹ d¼t+2;t ¯¯ MRS2;(1;¡1);t ´ + ®x [(1 + ®x ¯ r + ¯ x )¯ r ¼t+1;t + (®x ¯ r + ¯ x )¯ x xt ] dxt+1;t ¯ dLt =0; dxt+2;t =¡ dxt+1;t + ®x [(®x ¯ r + ¯ x )¯ z zt+1;t + (®z ¯ r + ¯ z )zt+2;t ] + ®z zt+3;t = MRS2;1;t + MRS2;2;t (¡ 1) ¡ ®x [(®x ¯ r + ¯ x )¯ r (it ¡ r) + ¯ r (it+1;t ¡ r)] ¹ ¹ ¸xt+1;t ±¸xt+2;t = ¡ + = [1 + ®x (2 + ®x ¯ r + ¯ x )¯ r ]¼t+1;t + ®x (1 + ®x ¯ r + ¯ x )¯ x xt ±(¼t+2;t ¡ ¼¤ ) ±(¼t+2;t ¡ ¼¤ ) ¸(±xt+2;t ¡ xt+1;t ) + ®x (1 + ®x ¯ r + ¯ x )¯ z zt+1;t + [®x (®z ¯ r + ¯ z ) + ®z ]zt+2;t + ®z zt+3;t = : ¡ ®x (1 + ®x ¯ r + ¯ x )¯ r (it ¡ r) ¡ ®x ¯ r (it+1;t ¡ r): ¹ ¹ (D.17) ±(¼t+2;t ¡ ¼¤ ) Setting Consider the constant interest rate, it = it+1;t , for which ¼t+3;t = ¼ ¤ . This implies the MRT2;(1;¡1);t = MRS2;(1;¡1);t equation, gives (D.5) for ¿ = 1, after simpli…cation. Repeating the same argument for j ¸ 3 and d¼t+3;t , etc., results in (D.5) for ¿ ¸ 1. ¼¤ = [1 + ®x (2 + ®x ¯ r + ¯ x )¯ r ]¼t+1;t + ®x (1 + ®x ¯ r + ¯ x )¯ x xt + ®x (1 + ®x ¯ r + ¯ x )¯ z zt+1;t + [®x (®z ¯ r + ¯ z ) + ®z ]zt+2;t + ®z zt+3;t D.2 A constant-interest-rate in‡ation forecast ¡ ®x (2 + ®x ¯ r + ¯ x )¯ r (it ¡ r): ¹ By (5.2) we have that the one-period-ahead in‡ation forecast, ¼t+1;t is given by Solving for it gives the reaction function ¼t+1;t = ¼t + ®x xt + ®z zt+1;t (D.13) and cannot be a¤ected by the current instrument. In contrast, by (5.3), the one-period-ahead it = r + ¼ ¤ + f¼ (¼t+1;t ¡ ¼¤ ) + fx xt + fz1 zt+1;t + fz2 zt+2;t + fz3 zt+3;t ; ¹ (D.18) output-gap forecast, xt+1;t , is given by where xt+1;t = ¯ r ¼t+1;t + ¯ x xt + ¯ z zt+1;t ¡ ¯ r (it ¡ r) ¹ (D.14) 1 + ®x (2 + ®x ¯ r + ¯ x )¯ r f¼ ´ > 1; (D.19) and can be a¤ected by the current instrument. ®x (2 + ®x ¯ r + ¯ x )¯ r The two-period-ahead in‡ation forecast, ¼ t+2;t , will by (5.2) and (D.14) be given by (1 + ®x ¯ r + ¯ x )¯ x fx ´ > 0; (D.20) (2 + ®x ¯ r + ¯ x )¯ r ¼t+2;t = ¼ t+1;t + ®x xt+1;t + ®z zt+2;t (1 + ®x ¯ r + ¯ x )¯ z = ¼ t+1;t + ®x [¯ r ¼t+1;t + ¯ x xt + ¯ z zt+1;t ¡ ¯ r (it ¡ r)] + ®z zt+2;t ¹ fz1 ´ ; (D.21) (2 + ®x ¯ r + ¯ x )¯ r = (1 + ®x ¯ r )¼t+1;t + ®x ¯ x xt + ®x ¯ z zt+1;t + ®z zt+2;t ¡ ®x ¯ r (it ¡ r): ¹ (D.15) ®x (®z ¯ r + ¯ z ) + ®z fz2 = ; (D.22) Thus, we see that the two-period-ahead in‡ation forecast depends on the one-period-ahead ®x (2 + ®x ¯ r + ¯ x )¯ r in‡ation forecast, ¼t+1;t , the current output gap, xt , the one- and two-period-ahead forecasts of ®z fz3 = : (D.23) the exogenous variable, zt+1;t and zt+2;t , and the current interest rate relative to the average ®x (2 + ®x ¯ r + ¯ x )¯ r real interest rate, it ¡ r. ¹ Let us also note that the two-period-ahead output-gap forecast, xt+2;t , is given by E The forward-looking model xt+2;t = ¯ r ¼t+2;t + ¯ x xt+1;t + ¯ z zt+2;t ¡ ¯ r (it+1;t ¡ r) ¹ Consider the Lagrangian in period t for the problem of minimizing (5.1) subject to (5.5), = ¯ r [(1 + ®x ¯ r )¼t+1;t + ®x ¯ x xt + ®x ¯ z zt+1;t + ®z zt+2;t ¡ ®x ¯ r (it ¡ r)]¹ 1 X + ¯ x [¯ r ¼t+1;t + ¯ x xt + ¯ z zt+1;t ¡ ¯ r (it ¡ r)] + ¯ z zt+2;t ¡ ¯ r (it+1;t ¡ r) ¹ ¹ 1 Lt = ± ¿ f [(¼ t+¿ ;t ¡¼¤ )2 +¸x2 ;t ]+'t+¿ ;t [±(¼t+1+¿ ;t ¡¼)+®x xt+¿ ;t +®z zt+¿ ;t ¡(¼t+¿ ;t ¡¼)]g t+¿ = (1 + ®x ¯ r + ¯ x )¯ r ¼t+1;t + (®x ¯ r + ¯ x )¯ x xt 2 ¿ =0 + (®x ¯ r + ¯ x )¯ z zt+1;t + (®z ¯ r + ¯ z )zt+2;t (E.1) ¡ (®x ¯ r + ¯ x )¯ r (it ¡ r) ¡ ¯ r (it+1;t ¡ r) ¹ ¹ (D.16) 57 58 where 't+¿ ;t is the Lagrange multiplier for the constraint (5.5) for period t + ¿ , considered in where xt;t¡1 is the one-period-ahead output-gap forecast from the previous period. Equivalently, period t. Di¤erentiating with respect to ¼t+¿ ;t and xt+¿ ;t gives the …rst-order conditions we can let (E.5) apply for ¿ ¸ 1 instead of ¿ ¸ 2, with the initial condition ¼t+¿ ;t ¡ ¼¤ ¡ 't+¿ ;t + 't+¿ ¡1;t = 0; (E.2) xt;t = xt;t¡1 (E.8) ¸xt+¿ ;t + ®x 't+¿ ;t = 0 (E.3) imposed. That is, in the …rst-order condition (E.5), xt;t does not denote the current output gap, xt , but the forecast one period ago of the current output gap, xt;t¡1 . for ¿ ¸ 1; together with the initial condition Note that (E.5) implies that, in a steady state with ¼t+¿ +1;t = ¼, xt+¿ +1;t = x and zt+¿ +1;t = 't;t = 0: (E.4) 0 for ¿ ¸ 0, we will have (3.6), no average in‡ation bias. For the case of ‡exible in‡ation targeting (¸ > 0), substituting (E.5) into (5.5) leads to the Eliminating the Lagrange multipliers leads to the consolidated …rst-order condition di¤erence equation 1 ®x ¸ xt+¿ +2;t ¡ 2axt+¿ +1;t + xt+¿ ;t = ®z zt+¿ +1;t ¼t+¿ ;t ¡ ¼¤ + (xt+¿ ;t ¡ xt+¿ ¡1;t ) = 0 (E.5) ± ±¸ ®x for ¿ ¸ 0, where 1 ®2 for ¿ ¸ 2 and 2a = 1 + + x (E.9) ¸ ± ±¸ ¼t+1;t ¡ ¼¤ + xt+1;t = 0 (E.6) (note that (E.9) is similar to (D.7) except that the last term di¤ers). (Since xt;t is given (by ®x E.8), it is natural to express the di¤erence equation in terms of the output gap.) for ¿ = 1. Thus, …nding the optimal forecasts is reduced to the problem of …nding ¼t, xt and By standard methods, it can be shown that the solution to the di¤erence equation, the 't ´ f't+¿ ;t g1 that satisfy (5.5) and (E.2)–(E.4), or, equivalently, ¼t and xt that satisfy (5.5), ¿ =1 optimal output-gap forecast, ful…lls (E.5) and (E.6). 1 As noted in Woodford [95] and discussed in detail in Svensson and Woodford [81], these …rst- ®x c X order conditions de…ne a decision procedure that will not be time-consistent (under the case of xt+¿ +1;t = cxt+¿ ;t ¡ ®z (±c)s zt+¿ +1+s;t ¸ s=0 ‡exible in‡ation targeting, ¸ > 0). This can be seen from the fact that the initial condition (E.4) ®x c and the corresponding …rst-order condition for ¿ = 1, (E.6), are di¤erent from that for ¿ ¸ 2, = cxt+¿ ;t ¡ ®z zt+¿ +1;t ; ~ (E.10) ¸ (E.5). This results because, in deciding on ¼t+1;t , the central bank takes the previous period’s forecast ¼ t+1;t¡1 as given, and lets ¼t+1;t deviate from it without assigning any speci…c cost to where zt+¿ +1;t is de…ned as in (3.2) and c (0 · c < 1) is the smaller root of the characteristic ~ doing so. As a result, the forecasts in period t are not generally consistent with the forecasts equation (D.9) and hence again is given by (D.10) (but with a given by (E.9)), is an increasing made in period t ¡ 1, even if no new information is received in period t. function c(¸) of ¸, and ful…lls c(0) = 0, c(1) = 1:53 To see this, suppose that the forecasts ¼t¡1 and xt¡1 were constructed in period t¡ 1 so as to The optimal in‡ation forecast then ful…lls, by (E.5), minimize the intertemporal loss function (5.1) with t ¡ 1 substituted for t. The same procedure ¸ in period t¡1 as above then resulted in the same …rst-order conditions (E.5) and (E.6), although ¼t+¿ +1;t ¡ ¼¤ = (1 ¡ c)xt+¿ ;t + ®z c~t+¿ +1;t : z (E.11) ®x with t ¡ 1 substituted for t. Thus, in period t ¡ 1, the …rst-order condition for ¿ = 2 was The optimal interest-rate path will by (5.5) and (E.5) follow ¸ ¼t+1;t¡1 ¡ ¼¤ + (xt+1;t¡1 ¡ xt;t¡1 ) = 0: (E.7) 1 ¯ ®x it+¿ +1;t = rt+¿ +1;t + ¼¤ + (¼t+¿ +2;t ¡ ¼¤ ) + ¤ (xt+¿ +2;t ¡ xt+¿ +1;t ) + z zt+¿ +1;t ¯r ¯r Without any new information in period t relative to period t¡1, we should have ¼t+1;t = ¼t+1;t¡1 ®x ¯z ¤ ¤ ¤ and xt+1;t = xt+1;t¡1 for intertemporal consistency. From (E.6) and (E.7) it is apparent that = rt+¿ +1;t + ¼ + (1 ¡ )(¼ t+¿ +2;t ¡ ¼ ) + zt+¿ +1;t ¸¯ r ¯r this will not be the case, unless by chance xt;t¡1 = 0. ®x ¸ ®x ¯ As discussed in Svensson and Woodford [81]), time-consistency is ensured under optimization ¤ ¤ = rt+¿ +1;t + ¼ + (1 ¡ ) (1 ¡ c)xt+¿ +1;t + (1 ¡ )®z c~t+¿ +2;t + z zt+¿ +1;t : z “in a timeless perspective,” which corresponds to imposing the initial condition ¸¯ r ®x ¸¯ r ¯r (E.12) 't;t = 't;t¡1 The optimal interest-rate decision in period t for the interest rate in period t + 1 is then given for the Lagrange multiplier 't;t . That is, the multiplier 't;t is set equal to the shadow cost of by (note the loose relation to “forecast-based” instrument rules) the one-period-ahead in‡ation forecast from the previous period. ®x ¯ Equivalently, (E.6) is replaced by it+1;t = rt+1;t + ¼¤ + (1 ¡ ¤ )(¼t+2;t ¡ ¼¤ ) + z zt+1;t ¸¯ r ¯r ¸ 53 If the smaller root of (D.9) with (D.7) as a function of ¸ is denoted c(¸), the smaller root of (D.9) with (E.9) ^ ¼t+1;t ¡ ¼¤ + (xt+1;t ¡ xt;t¡1 ) = 0; is obviously c(±¸), for …xed ±. ^ ®x 59 60 ®x ¸ for ¿ ¸ 0 from (E.5). From (5.5) then follows = rt+1;t + ¼¤ + (1 ¡ ¤ ) c(1 ¡ c)xt;t ¸¯ r ®x ¯ ®x ®x xt+¿ +1;t + ®z zt+¿ +1;t = 0 + z zt+1;t + (1 ¡ )®z cf[1 ¡ ±c(1 ¡ c)]~t+2;t ¡ (1 ¡ c)zt+1;t g; z ¯r ¸¯ r and ®z where we use that the two-period-ahead in‡ation forecast is given by xt+¿ +1;t = ¡ zt+¿ +1;t : ®x ¸ The optimal instrument-rate decision in period t is then given by ¼t+2;t ¡ ¼¤ = (1 ¡ c)xt+1;t + ®z c~t+2;t z ®x h i 1 ¯ = ¸ (1 ¡ c) cxt;t ¡ ®z ®x c zt+1;t + ®z c~t+2;t ~ z it+1;t = rt+1;t + ¼¤ + ¤ (xt+2;t ¡ xt+1;t ) + z zt+1;t ®x ¸ ¯r ¯r ¸ ¯ ®z = (1 ¡ c)cxt;t ¡ ®z c(1 ¡ c)(zt+1;t + ±c~t+2;t ) + ®z c~t+2;t z z = rt+1;t + ¼¤ + z zt+1;t ¡ ¤ (zt+2;t ¡ zt+1;t ): ®x ¯r ®x ¯ r ¸ = c(1 ¡ c)xt;t + ®z cf[1 ¡ ±c(1 ¡ c)]~t+2;t ¡ (1 ¡ c)zt+1;t g; z E.2 The discretion case ®x As discussed in Svensson and Woodford [81], the …rst-order condition is where I have used that 1 X ¸ ¼t+¿ +1 ¡ ¼¤ = ¡ xt+¿ +1;t (E.13) zt+1;t ´ ~ (±c)s zt+1+s;t ´ ±c~t+2;t + zt+1;t : z ®x s=0 for ¿ ¸ 0. Combining (E.13) with (5.2) gives the di¤erence equation Note that, by (E.10), ¸ ±¸ ¡ xt+¿ +1;t = ¡ xt+¿ +2;t + ®x xt+¿ +1;t + ®z zt+¿ +1;t : xt;t = xt;t¡1 ®x ®x ®x c = cxt¡1;t¡2 ¡ ®z zt;t¡1 ~ The solution will ful…ll ¸ 1 ®x c X j ®x ®z c ~ = ¡ ®z c zt¡j;t¡1¡j : ~ xt+¿ +1;t = ±~xt+¿ +2;t ¡ c zt+¿ +1;t ¸ ¸ j=0 ®x ®z c ~ = ¡ zt+¿ +1;t ; ~ (E.14) In the special case (2.7), we have ¸ and 1 zt+¿ +1;t = ~ ° ¿ +1 zt ¼t+¿ +1;t ¡ ¼¤ = ®z czt+¿ +1;t ~~ 1 ¡ ±°c ®x c where zt+¿ +1;t is de…ned as in (3.2) (with c replacing c), and ~ ~ xt+¿ +1;t = cxt+¿ ;t ¡ ®z ° ¿ +1 zt ; ¸ 1 ¡ ±°c ¸ c ¸ ¼t+¿ +1;t ¡ ¼¤ = (1 ¡ c)xt+¿ ;t + ®z ° ¿ +1 zt ; 0 · c = c(¸) ´ ~ ~ < 1: ®x 1 ¡ ±°c ¸ + ®2 x · ¸ ®x ¸ ®x ° ¡ (1 ¡ c) ¯ z The corresponding reaction function is it+1;t = rt+1;t + ¼¤ + (1 ¡ ¤ ) c(1 ¡ c)xt;t + (1 ¡ )®z c + °zt ¸¯ r ®x ¸¯ r 1 ¡ ±°c ¯r 1 1 ¯ ®x X j ¤ it+1;t = rt+1;t + ¼t+2;t + (xt+2;t ¡ xt+1;t ) + z zt+1;t xt;t = ¡ ®z c c zt¡j;t¡1¡j ~ ¯r ¯r ¸ j=0 1 ¯ 1 = rt+1;t + ¼¤ + (¼t+2;t ¡ ¼¤ ) + (xt+2;t ¡ xt+1;t ) + z zt+1;t ¤ ®x °c X j ¯r ¯r = ¡ ®z c zt¡j;t¡1¡j : 1 ®x ®z c ~ ¯ ¸ 1 ¡ ±°c = rt+1;t + ¼¤ + ®z czt+2;t ¡ ¤ ~~ (~t+2;t ¡ zt+1;t ) + z zt+1;t z ~ j=0 ¯r ¸ ¯r 1 ®x ®z ¯ E.1 Strict in‡ation targeting = rt+1;t + ¼¤ + ®z czt+2;t ¡ ¤ ~~ c(~t+2;t ¡ ±~zt+2;t ¡ zt+1;t ) + z zt+1;t ~z c~ ¯r ¸ ¯r For ¸ = 0, we have ®z c ~ ®x ®z c + ¯ z ¸ ~ ¼t+¿ +1;t ¡ ¼¤ = 0 = rt+1;t + ¼¤ + ¤ [¯ r ¸ ¡ ®x (1 ¡ ±~)]~t+2;t + c z zt+1;t : ¯r ¸ ¯r ¸ 61 62 E.3 Equality of the MRT and MRS Let MRT2;(1;¡1=(1¡°)±;°=(1¡°)±);t denote the marginal rate of transformation of the linear combi- 1 ° nation (xt+1;t ; xt+2;t ; xt+3;t ) = (1; ¡ (1¡°)± ; (1¡°)± )xt+1;t into ¼t+2;t . It ful…lls Given the aggregate-supply relation, (5.5), for ¿ ¸ 0, consider changes in xt+j;t , j ¸ 1, that result in d¼t+2;t 6= 0 and d¼t+i;t = 0 for all i 6= 2. This requires ¯ d¼t+2;t ¯ ¯ ®x MRT2;(1;¡1=(1¡°)±;°=(1¡°)±);t ´ =¡ : dxt+1;t 6= 0 (to allow d¼t+1;t = 0) dxt+1;t ¯dxt+2;t =¡ dxt+1;t =(1¡°)±; dxt+3;t =°dxt+1;t =(1¡°)± (1 ¡ °)± ®x d¼t+2;t = ¡ dxt+1;t (to make d¼t+1;t = 0) Let MRS2;(1;¡1=(1¡°)±;°=(1¡°)±);t denote the marginal rate of substitution of ¼t+2;t for the ± 1 ° 1 1 linear combination (xt+1;t ; xt+2;t ; xt+3;t ) = (1; ¡ (1¡°)± ; (1¡°)± )xt+1;t . It will be given by dxt+2;t = d¼t+2;t = ¡ dxt+1;t (to make d¼t+3;t = 0) ®x ± ¯ dxt+j;t = 0; j ¸ 3: d¼t+2;t ¯¯ MRS2;(1;¡1=(1¡°)±;°=(1¡°)±);t ´ dxt+1;t ¯dLt =0; dxt+2;t =¡ dxt+1;t =(1¡°)±; dxt+3;t =°dxt+1;t =(1¡°)± Let MRT2;(1;¡1=±);t denote the marginal rate of transformation of the linear combination (xt+1;t ; xt+2;t ) = (1; ¡ 1=±)xt+1;t into ¼t+2;t . It ful…lls 1 ° = MRS2;1;t + MRS2;2;t (¡ ) + MRS2;3;t ¯ (1 ¡ °)± (1 ¡ °)± d¼t+2;t ¯ ¯ ®x MRT2;(1;¡1=±);t ´ =¡ : ¸xt+1;t ¸xt+2;t ¸±°xt+3;t dxt+1;t ¯ dxt+2;t =¡ dxt+1;t =± ± = ¡ + ¡ ±(¼t+2;t ¡ ¼¤ ) ±(¼t+2;t ¡ ¼¤ )(1 ¡ °) ±(¼t+2;t ¡ ¼¤ )(1 ¡ °) Let MRS2;(1;¡1=±);t denote the marginal rate of substitution of ¼t+2;t for the linear combina- ¸[±°xt+2;t ¡ xt+2;t + (1 ¡ °)xt+1;t ] tion (xt+1;t ; xt+2;t ) = (1; ¡ 1=±)xt+1;t . It ful…lls = ¡ : ¯ ±(¼t+2;t ¡ ¼¤ )(1 ¡ °) d¼t+2;t ¯ ¯ MRS2;(1;¡1=±);t ´ Setting dxt+1;t ¯ dLt =0; dxt+2;t =¡ dxt+1;t MRT2;(1;¡1=(1¡°)±;°=(1¡°)±);t = MRS2;(1;¡1=(1¡°)±;°=(1¡°)±);t 1 = MRS2;1;t + MRS2;2;t (¡ ) gives, after simpli…cation, ± ¸xt+1;t ¸xt+2;t ¸ = ¡ + ¼t+2;t ¡ ¼¤ ¡ [°(±xt+3;t ¡ xt+2;t ) ¡ (1 ¡ °)(xt+2;t ¡ xt+1;t )] ±(¼t+2;t ¡ ¼¤ ) ±(¼t+2;t ¡ ¼¤ ) ®x ¸(xt+2;t ¡ xt+1;t ) for ¿ = 2, etc. = : ±(¼t+2;t ¡ ¼¤ ) Setting E.5 A constant-interest-rate in‡ation forecast MRT2;(1;¡1=±);t = MRS2;(1;¡1=±);t As explained in the appendix of the working-paper version of Svensson [75], constructing constant- gives (E.5) for ¿ = 2, after simpli…cation. Repeating the same argument for j ¸ 3 and d¼t+3;t , interest-rate forecasts in a forward-looking model requires some special considerations (see also etc., results in (E.5) for ¿ ¸ 2. By the argument of the main text, optimality in a time-less Leitemo [47]). Basically, some assumptions must be made about future policy in order to con- perspective of the speci…c targeting rule then requires (E.5) to hold for ¿ = 1, with (5.18). struct determinate forecasts. The forecasts will not be rational-expectations forecasts, in that the constant-interest-rate path will not materialize even in the absence of new information or E.4 Equality of the MRT and MRS for a more general aggregate-supply relation. new judgment. It is easy to use the same method to establish the optimal speci…c targeting rule for a more Here is an example: general aggregate-supply relation. Suppose it is both backward- and forward-looking, as in ² In period t, impose the conditions that the interest rate is constant 3 periods ahead, ¼t+¿ +1;t = (1 ¡ °)±¼t+¿ +2;t + °¼t+¿ ;t + ®x xt+¿ +1;t + ®z zt+¿ +1;t it+1;t = it+2;t = it+3;t : (E.15) for ¿ ¸ 0 and 0 · ° · 1. Consider again changes in xt+j;t , j ¸ 1, that result in d¼t+2;t 6= 0 and d¼t+i;t = 0 for all i 6= 2. This now requires Furthermore, assume that we like to …nd the constant interest rate (for the next 3 periods) for which the corresponding 3-period-ahead in‡ation forecast is on target, dxt+1;t 6= 0 (to allow d¼t+1;t = 0) ®x ¼t+3;t = ¼¤ : (E.16) d¼t+2;t = ¡ dxt+1;t (to make d¼t+1;t = 0) (1 ¡ °)± 1 1 ² We must make some assumptions about the economy after period 3 in order to have a dxt+2;t = d¼t+2;t = ¡ dxt+1;t (to allow d¼t+3;t = 0) ®x (1 ¡ °)± determinate solution. Assume, for instance, that policy is optimal from (t + 4; t) onwards ° ° (where (t + ¿ ; t) denotes period t + ¿ seen from the forecasting done in period t). Then dxt+3;t = ¡ d¼ t+2;t = dxt+1;t (to make d¼t+3;t = 0) ®x (1 ¡ °)± ¼t+4;t , xt+4;t and it+4;t are given by (E.11), (E.10) and (E.12), respectively. In particular, dxt+j;t = 0; j ¸ 4: they depend on xt+3;t , which remains to be determined. 63 64 ² By (5.5) for (t + 3; t), given that ¼t+4;t is a function of xt+3;t , and given (E.16) and zt+3;t , The coe¢cients a, b and c are endogenously determined in the equilibrium, but once determined, we can solve for xt+3;t . Then ¼t+4;t and xt+4;t are determined. they are constant, due to the linearity of the model. Since ¼ t+2;t , xt+2;t and xt+1;t ful…ll (D.5) ^ ^ ^ for ¿ = 1, we have ² By (5.6) for (t+3; t), given xt+4;t , xt+3;t , ¼t+4;t , and zt+3;t , we get it+3;t and, by (E.15), also it+1;t and it+2;t . From now on, we can exploit the simple recursivity of the forward-looking ¸ [¹ t+2;t (it¡1 )¡¼¤ ¡a(^t;t ¡it¡1 )]+ ¼ { f[¹t+1;t (it¡1 )¡b(^t;t ¡it¡1 )]¡±[¹t+2;t (it¡1 )¡d(^t;t ¡it¡1 )]g = 0: x { x { model: ±®x ² By (5.6) for (t + 2; t), given xt+3;t , (E.16), zt+2;t , and it+2;t , we get xt+2;t . It follows that we can write ² By (5.5) for (t + 2; t), given (E.16), xt+3;t and zt+3;t we get ¼t+2;t . 1 ¸=®x ^t;t ¡it¡1 = { [¹ t+2;t (it¡1 )¡¼¤ ]¡ ¼ [±¹t+2;t (it¡1 )¡ xt+1;t (it¡1 )]: x ¹ a + ¸(b ¡ ±d)=±®x a + ¸(b ¡ ±d)=±®x ² By (5.6) for (t + 1; t), given xt+2;t , ¼t+2;t , zt+2;t , and it+1;t , we get xt+1;t . It follows that the optimal change in the interest rate from period t ¡ 1 to t, ^t;t ¡ it¡1 , can { ² By (5.5) for (t + 1; t), given ¼t+2;t , xt+1;t and zt+1;t we get ¼ t+1;t . be seen as a linear response to the deviation of a two-period-ahead forecast from the in‡ation ² Thus, we have found (¼ t ; xt ; it ) for which (E.16) and (E.15) holds. target, ¼t+2;t (it¡1 ) ¡ ¼¤ , and to the two-period-ahead forecast of the modi…ed change in the ¹ output gap, ±¹t+2;t (it¡1 ) ¡ xt+1;t (it¡1 ). x ¹ ² Suppose this procedure is followed each period t and it+1;t is implemented in each period t + 1. In period t + 1, even in the absence of any new information (any change in the F.2 The forward-looking model judgment), the resulting it+2;t+1 will di¤er from it+2;t , since in period t + 1 (E.16) is replaced by ¼t+4;t+1 = 0. Thus, we will have ¼t+2;t+1 and xt+2;t+1 di¤ering from ¼t+2;t For the forward-looking model, let (^ t ; xt ; ^t ) be the optimal equilibrium forecasts and instrument- ¼ ^ { and xt+2;t . In particular, rational plans by the private sector will incorporate rational rate path. Recall that in the forward-looking model, the relevant decision in period t concerns interest-rate expectations of the time-varying interest rate, so it+2jt = E[it+2;t+1 jIt ] 6= it+1;t . For a given i, let [¹ t (i); xt (i); ¹t (i)] correspond to an equilibrium where it+1;t = i but it+¿ ;t ¼ ¹ { it+1;t . Consequently, the private-sector plans ¼t+1jt and xt+1jt will di¤er from the constant- is optimal for ¿ ¸ 2 (conditional on it+1;t = i). Then, for ¹t (i), (E.5) will be ful…lled for ¿ ¸ 2, { interest-rate forecasts ¼t+1;t and xt+1;t . but not for ¿ = 1 (except if i = ^t+1;t ). Now, a decision in period t of an unchanged interest rate { corresponds to i = it . We realize that we have ² This will typically not ful…ll (E.5) and not be optimal. This may not even be close to optimal. ¼t+1;t = ¼t+1;t (it ) ¡ a(^t+1;t ¡ it ) ^ ¹ { xt+1;t = xt+1;t (it ) ¡ b(^t+1;t ¡ it ); ^ ¹ { ² See Leitemo [47] for more details. Also, see Kohn [43] for more general discussion of constant-interest-rate forecasts. where a ´ ¡ @¹ t+1;t (it )=@i and b ´ ¡ @ xt+1;t (it )=@i are the derivatives of ¼t+1;t and xt+1;t ¼ ¹ with respect to i at the equilibrium [¹ t (i); xt (i); ¹t (i)]. (Note that xt;t = xt;t¡1 is the previous ¼ ¹ { F An optimal reaction function with response to forecasts for an unchanged optimal forecast in period t ¡ 1 and is not a¤ected by i.) Again, the coe¢cients a, b and c interest rate are endogenously determined in the equilibrium but constant, once determined, because of the linearity of the model. Since ¼t+1;t and xt+1;t ful…ll (E.5) for ¿ = 1, we have ^ ^ This appendix shows how a forecast-based instrument rule involving precisely de…ned unchanged- interest-rate rather than equilibrium forecasts can be derived from the optimal targeting rule. ¸ [¹ t+1;t (it ) ¡ ¼ ¤ ¡ a(^t+1;t ¡ it )] + ¼ { [¹t+1;t (it ) ¡ b(^t+1;t ¡ it ) ¡ xt;t¡1 ]: x { ®x F.1 The backward-looking model It follows that we can write For the backward-looking forward-looking model, let (^ t ; xt ; ^t ) be the optimal equilibrium fore- ¼ ^ { 1 ¸=®x casts and instrument-rate path. For a given i, let [¹ t (i); xt (i); ¹t (i)] correspond to an equilibrium ¼ ¹ { ^t+1;t ¡ it = { [¹ t+1;t (it ) ¡ ¼¤ ] + ¼ [¹t+1;t (it ) ¡ xt;t¡1 ]: x a + ¸b=®x a + ¸b=®x where it;t = i but it+¿ ;t is optimal for ¿ ¸ 1 (conditional on it;t = i). Then, for ¹t (i), (D.5) will { be ful…lled for ¿ ¸ 1, but not for ¿ = 0 (except if i = ^t;t ). An unchanged interest rate in period { It follows that the optimal change in the interest rate from period t to t + 1, ^t+1;t ¡ it , can { t then corresponds to i = it¡1 . We realize that we have be seen as a linear response to the one-period-ahead forecast of the in‡ation gap, ¼t+1;t (it ) ¡ ¼¤ , ¹ and the change in the forecast of output gap, xt+1;t (it) ¡ xt¡1;t (relative not to the previous ¹ ¼t+2;t = ¼t+2;t (it¡1 ) ¡ a(^t;t ¡ it¡1 ); ^ ¹ { unchanged-interest-rate forecast xt;t¡1 (it¡1 ) but to the previous optimal forecast, xt;t¡1 ), where ¹ xt+1;t = xt+1;t (it¡1 ) ¡ b(^t;t ¡ it¡1 ); ^ ¹ { both forecasts for period t + 1 are conditional on an unchanged instrument rate, it+1;t = it . xt+2;t = xt+2;t (it¡1 ) ¡ d(^t;t ¡ it¡1 ); ^ ¹ { where a ´ ¡ @¹ t+2;t (it¡1 )=@i and b ´ ¡ @ xt+1;t (it¡1 )=@i and d ´ ¡ @ xt+2;t (it¡1 )=@i are the ¼ ¹ ¹ derivatives of ¼t+2;t and xt+1;t with respect to i at the equilibrium (¹ t (it¡1 ); xt (it¡1 ); ¹t (it¡1 )). ¼ ¹ { 65 66 References [18] Cecchetti, Stephen G. 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