Hausman Specification Test

The MODEL Procedure

Hausman Specification Test

Hausman's specification test, or m-statistic, can be used to test hypotheses in terms of bias or inconsistency of an estimator. This test was also proposed by Wu (1973). Hausman's m-statistic is as follows.

Given two estimators, ${\hat{{\beta}}_{0}}$ and ${\hat{{\beta}}_{1}}$ , where under the null hypothesis both estimators are consistent but only ${\hat{{\beta}}_{0}}$ is asymptotically efficient and under the alternative hypothesis only ${\hat{{\beta}}_{1}}$ is consistent, the m-statistic is

$m = {\hat{q}'} (\hat{V}_{1} - \hat{V}_{0})^{-}\hat{q}$

where ${\hat{V}_{1}}$ and ${\hat{V}_{0}}$ represent consistent estimates of the asymptotic covariance matrices of ${\hat{{\beta}}_{1}}$ and ${\hat{{\beta}}_{0}}$ ,and

$q = \hat{{\beta}}_{1} - \hat{{\beta}}_{0}$

The m-statistic is then distributed ${{\chi}^2}$ with k degrees of freedom, where k is the rank of the matrix ${(\hat{V}_{1} - \hat{V}_{0})}$ .A generalized inverse is used, as recommended by Hausman (1982).

In the MODEL procedure, Hausman's m-statistic can be used to determine if it is necessary to use an instrumental variables method rather than a more efficient OLS estimation. Hausman's m-statistic can also be used to compare 2SLS with 3SLS for a class of estimators for which 3SLS is asymptotically efficient (similarly for OLS and SUR).

Hausman's m-statistic can also be used, in principle, to test the null hypothesis of normality when comparing 3SLS to FIML. Because of the poor performance of this form of the test, it is not offered in the MODEL procedure. Refer to R.C. Fair (1984, pp. 246-247) for a discussion of why Hausman's test fails for common econometric models.

To perform a Hausman's specification test, specify the HAUSMAN option in the FIT statement. The selected estimation methods are compared using Hausman's m-statistic.

In the following example, OLS, SUR, 2SLS, 3SLS, and FIML are used to estimate a model, and Hausman's test is requested.

      proc model data=one out=fiml2;
         endogenous y1 y2;
   
         y1 = py2 * y2 + px1 * x1 + interc;
         y2 = py1* y1 + pz1 * z1 + d2;
   
         fit y1 y2 / ols sur 2sls 3sls fiml hausman;
         instruments x1 z1;
      run;

The output specified by the HAUSMAN option produces the following results.

The MODEL Procedure

Hausman's Specification Test Results
Comparing	To	DF	Statistic	Pr > ChiSq
OLS	SUR	6	32.47	<.0001
OLS	2SLS	6	13.86	0.0313
OLS	3SLS	6	-0.07	.
2SLS	3SLS	6	0.00	1.0000

Figure 14.45: Hausman's Specification Test Results

Figure 14.45 indicates that 2SLS, a system estimation method, is preferred over OLS. The model needs an IV estimator but not a full error covariance matrix. Note that the FIML estimation results are not compared.

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