Vnr0 U N[0 T

for some covariance matrix T that we have yet to estimate, it follows that the Wald statistic, nr'T-1 r x2(JX (17-59)

where the degrees of freedom J is the number of moment restrictions being tested and T is an estimate of T. Thus, the statistic can be referred to the chi-squared table.

It remains to determine the estimator of T. The full derivation of T is fairly complicated. [See Pagan and Vella (1989, pp. S32-S33).] But when the vector of parameter estimators is a maximum likelihood estimator, as it would be for the least squares estimator with normally distributed disturbances and for most of the other estimators we consider, a surprisingly simple estimator can be used. Suppose that the parameter vector used to compute the moments above is obtained by solving the equations a n n g(yi, xi, zi, 6) = g = 0, (17-60)

where 6 is the estimated parameter vector [e.g., (fi, a) in the linear model]. For the linear regression model, that would be the normal equations

Let the matrix G be the n x K matrix with i th row equal to g'. In a maximum likelihood problem, G is the matrix of derivatives of the individual terms in the log-likelihood function with respect to the parameters. This is the G used to compute the BHHH estimator of the information matrix. [See (17-18).] Let R be the n x J matrix whose i th row is r'. Pagan and Vella show that for maximum likelihood estimators, T can be estimated using

This equation looks like an involved matrix computation, but it is simple with any regression program. Each element of S is the mean square or cross-product of the least squares residuals in a linear regression of a column of R on the variables in G.22

Therefore, the operational version of the statistic is

C = nr'S-1r = -i'R[R'R - R'G(G'G)-1G'R]-1R'i, (17-62)

n where i is an n x 1 column of ones, which, once again, is referred to the appropriate critical value in the chi-squared table. This result provides a joint test that all the moment conditions are satisfied simultaneously. An individual test of just one of the moment

21It might be tempting just to use (1/n)R'R. This idea would be incorrect, because S accounts for R being a function of the estimated parameter vector that is converging to its probability limit at the same rate as the sample moments are converging to theirs.

22If the estimator is not an MLE, then estimation of T is more involved but also straightforward using basic matrix algebra. The advantage of (17-62) is that it involves simple sums of variables that have already been computed to obtain 6 and r. Note, as well, that if 6 has been estimated by maximum likelihood, then the term (G'G)-1 is the BHHH estimator of the asymptotic covariance matrix of 6. If it were more convenient, then this estimator could be replaced with any other appropriate estimator of Asy. Var[6].

restrictions in isolation can be computed even more easily than a joint test. For testing one of the L conditions, say the ¿th one, the test can be carried out by a simple t test of whether the constant term is zero in a linear regression of the ¿th column of R on a constant term and all the columns of G. In fact, the test statistic in (17-62) could also be obtained by stacking the J columns of R and treating the L equations as a seemingly unrelated regressions model with (i, G) as the (identical) regressors in each equation and then testing the joint hypothesis that all the constant terms are zero. (See Section 14.2.3.)

Example 17.8 Testing for Heteroscedasticity in the Linear Regression Model Suppose that the linear model is specified as y = & + fox; + fa; + ei •

To test whether

E [zf(sf - a2)] = 0, we linearly regress z2(e2 - s2) on a constant, ei,xiei, and ziei. A standard t test of whether the constant term in this regression is zero carries out the test. To test the joint hypothesis that there is no heteroscedasticity with respect to both x and z, we would regress both x?(e2 - s2) and z?(e2 - s2); on [1, ei, xiei, ziei] and collect the two columns of residuals in V. Then S = (1 /n)V'V. The moment vector would be n x)

The test statistic would now be

We will examine other conditional moment tests using this method in Section 22.3.4 where we study the specification of the censored regression model.

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