In dealing with the problem of estimating the parameters of a structural system of equations, we had not, in previous chapters, explicitly stated the form of the density of the random terms appearing in the system. Indeed, the estimation aspects of classical least squares techniques and their generalization to systems of equations are distribution free, so that no explicit assumption need be made with respect to the distribution of the error terms. On the other hand, in considering various tests of significance on 2SLS or 3SLS estimated parameters of a structural system, we have occasionally found it convenient to assert (joint) normality of the structural error terms. Under this assumption, the derivation of the asymptotic distribution of such estimators is simplified considerably.

Maximum Likelihood Methods

A method is given for estimating the coefficients of a single equation in a complete system of linear stochastic equations (see expression (2.1)), provided that a number of the coefficients of the selected equation are known to be zero. Under the assumption of the knowledge of all variables in the system and the assumption that the disturbances in the equations of the system are normally distributed, point estimates are derived from the regressions of the jointly dependent variables on the predetermined variables (Theorem 1). The vector of the estimates of the coefficients of the jointly dependent variables is the characteristic vector of a matrix involving the regression coefficients and the estimate of the covariance matrix of the residuals from the regression functions. The vector corresponding to the smallest characteristic root is taken. An efficient method of computing these estimates is given in section 7. The asymptotic theory of these estimates is given in a following paper [2]. When the predetermined variables can be considered as fixed, confidence regions for the coefficients can be obtained on the basis of small sample theory (Theorem 3). A statistical test for the hypothesis of over-identification of the single equation can be based on the characteristic root associated with the vector of point estimates (Theorem 2) or on the expression for the small sample confidence region (Theorem 4). This hypothesis is equivalent to the hypothesis that the coefficients assumed to be zero actually are zero. The asymptotic distribution of the criterion is shown in a following paper [2] to be that of $\chi^2$.

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Maximum Likelihood Methods

References

Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations

The Statistical Implications of a System of Simultaneous Equations

On the Statistical Treatment of Linear Stochastic Difference Equations

Statistical Methods for Research Workers

A textbook of econometrics

Related Papers (5)

Extended maximum likelihood

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Maximum likelihood in small samples: Estimation in the presence of nuisance parameters

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On Small-Sample Estimation