The following is the well known Klein's (1950) model I, along with an excerpt of the data. Where m n is first moment of the crossproduct. Table 1.1: Summary of PROC MODEL Estimation Methods Method The following table illustrates the relationship among some commonly used estimation methods for simultaneous equation models. Each block, corresponding to each single equation, is estimated using the residuals from the N2SLS estimates.
Note that the NH2SLS can be interpreted as a GMM estimator where the variance of the moment function V is taken to be block-diagonal. Where is a gk × gk block-diagonal matrix with each block j given by, for j = 1.
In such case, the nonlinear heteroscedastic 2SLS (NH2SLS) estimator is a single-equation estimator that attains a gain in efficency with respect to the N2SLS estimator by weighting observations with White's estimate of the error correlation matrix.įormally, the NH2SLS estimator is obtained by minimizing the following quantity: In the presence of heteroscedasticity, the 2SLS estimator is still consistent but no longer efficient. Such predicted values replace the right hand side endogenous variables in the model to obtain consistent estimates of the parameters.įormally, the N2SLS estimator is the value that minimizes the quantityĪ key assumption of N2SLS is that the disturbances are i.i.d. The nonlinear two stage least squares (N2SLS) is a commonly used single equation estimation method that consists of using instrumental variables (IV) that are uncorrelated with the disturbances to obtain predicted values for the endogenous variables.
is an unobservable disturbance vector with the following properties:
Where is a real valued vector function of, ,, g is the number of equations, l is the number of exogenous variables, p is the number of parameters and t ranges from 1 to n. DetailsĬonsider the general nonlinear simultaneous equations model One can use the MODEL procedure in SAS/ETS to compute the two-stage heteroscedastic estimates. Heteroscedastic two-stage least squares regression is a modification of the traditional two-stage least squares used to estimate simultaneous equation models when the disturbances are heteroscedastic.
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