3SLS obtains three stage least squares estimates of a set of nonlinear equations. It is a special case of LSQ with the options set for 3SLS estimation. The LSQ entry has a more complete description of the command.
Three stage least squares estimates are consistent and asymptotically normal, and, under some conditions, asymptotically more efficient than single equation estimates. In general, 3SLS is asymptotically less efficient than FIML, unless the model is linear in the parameters and endogenous variables.
3SLS (COVU=OWN or <residual covariance matrix>,DEBUG, FEI, HETERO,
INST=(<list of instrumental variables>),ITERU, MAXITW=0, ROBUST,
nonlinear options)
<list of equation names> ;
Usage
Three stage least squares is a combination of multivariate regression (SUR estimation) and two stage least squares. It obtains instrumental variable estimates, taking into account the covariances across equation disturbances as well. The objective function for three stage least squares is the sum of squared transformed fitted residuals.
Specification of the 3SLS command is the same as that of the LSQ command, except that the INST list is required. The variables in the INST list will be used to instrument all the equations, so that the actual instrumental variable matrix has the form given by Jorgenson and Laffont (1975), rather than that given by Amemiya (1977). In a simultaneous equations model, this means that a variable cannot be exogenous to one equation and endogenous to another. See the GMM command if you wish to relax this restriction.
Method
Three stage least squares estimates are obtained by estimating a set of nonlinear (or linear) equations with cross-equation constraints imposed, but with a diagonal covariance matrix of the disturbances across equations. This is the constrained two stage least squares estimator. The parameter estimates thus obtained are used to form a consistent estimate of the covariance matrix of the disturbances, which is then used as a weighting matrix when the model is reestimated to obtain new values of the parameters.
The actual method of parameter estimation is the Gauss-Newton method for nonlinear least squares described under LSQ. If the model is linear in the parameters and endogenous variables, only two iterations will be required, one to obtain the covariance matrix estimate, and one to obtain parameter estimates.
For further details on the properties of the linear three stage least squares estimator see the Theil text or Zellner and Theil (1962). For the nonlinear three stage least squares estimator, see Amemiya (1977) and Jorgenson and Laffont (1975). The method of estimation in TSP is described more fully in Berndt, Hall, Hall, and Hausman (1975), also available at this website.
COVU= residual covariance matrix (same as the old WNAME= option below).
DEBUG/NODEBUG specifies whether detailed computations of the model and its derivatives are to be printed out at every iteration. This option produces extremely voluminous output and is not recommended for use except by systems programmers maintaining TSP.
FEI/NOFEI specifies that models with additive individual fixed effects are to be estimated. The panel structure must have been defined previously with the FREQ (PANEL) command. The equations specified must be linear in the parameters (this will be checked) and variables. Individual-specific means will be removed from both variables and instruments.
INST= (list of instrumental variables). If this option is included, the LSQ estimator becomes nonlinear two stage least squares or nonlinear IV (if there is one equation) and nonlinear three stage least squares (if there is more than one equation). The list of instrumental variables supplied is used for all the equations. See the INST section of this manual and the references for further information on the choice of instruments.
ITERU/NOITERU specifies iteration on the COVU matrix; provides the same function as the old MAXITW= option.
MAXITW= the number of iterations to be performed on the parameters of the residual covariance matrix estimate. If MAXITW is zero the covariance matrix of the residuals is held fixed at the initial estimate (which is specified by WNAME). This option can be used to obtain estimates that are invariant to which equation is dropped in a shares model like translog.
HETERO/NOHETERO causes heteroskedastic-consistent standard errors to be used. See the GMM (NMA=) command for autocorrelation-consistent standard errors. Same as the old ROBUST option, or HCOV=R.
WNAME= the name of a matrix to be used as the starting value of the covariance matrix of the residuals.
WNAME=OWN specifies that the initial covariance matrix of the residuals is to be obtained from the residuals corresponding to the initial parameter values. If neither form of WNAME= is used, the initial covariance matrix is an identity matrix.
Nonlinear options control the iteration methods and printing. They are explained in the NONLINEAR section of this manual. Some of the common options are MAXIT, MAXSQZ, PRINT/NOPRINT, and SILENT/NOSILENT.
The only legal choice for HITER= is G (Gauss). HCOV=G is the default method for calculating standard errors; R (Robust) is the only other valid option.
Klein-I model:
FORM(VARPREF=C_) CONS CX C P P(-1) W;
FORM(VARPREF=I_) INV I C P P(-1) K(-1);
FORM(VARPREF=W_) WAGES W1 C E E(-1) TM;
3SLS(INST=(C,TM,W2,G,TX,P(-1),K(-1),E(-1))) CONS INV WAGES;
Amemiya, Takeshi, "The Maximum Likelihood and the Nonlinear Three-Stage Least Squares Estimator in the General Nonlinear Simultaneous Equation Model," Econometrica, May 1977, pp. 955-975.
Berndt, E. K., B. H. Hall, R. E. Hall, and J. A. Hausman, "Estimation and Inference in Nonlinear Structural Models," Annals of Economic and Social Measurement, October 1975, pp. 653-665.
Gallant, A. Ronald, and Dale W. Jorgenson, "Statistical Inference for a System of Simultaneous, Non-linear, Implicit Equations in the Context of Instrumental Variable Estimation," Journal of Econometrics 11, 1979, pp. 275-302.
Jorgenson, Dale W. and Jean-Jacques Laffont, "Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances," Annals of Economic and Social Measurement, October 1975, pp. 615-640.
Theil, Henri, Principles of Econometrics, John Wiley and Sons, New York, 1971, pp. 508-527.
Zellner, Arnold, and Henri Theil, "Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations," Econometrica 30 (1962), pp. 54-78. 1