linearmodels.iv.results.IVGMMResults.centered_j_stat¶

property IVGMMResults.centered_j_stat : InvalidTestStatistic | WaldTestStatistic¶

J-test of overidentifying restrictions using a centered moment covariance

Returns:¶

J statistic test of overidentifying restrictions that uses a covariance of the moment conditions that has been centered. An InvalidTestStatistic is returned if that covariance is singular, which can happen if there are too few clusters.

Return type:¶

linearmodels.shared.hypotheses.WaldTestStatistic or linearmodels.shared.hypotheses.InvalidTestStatistic

Notes

The statistic is defined as

\[n \bar{g}'\hat{S}_c^{-1}\bar{g} \sim \chi^2_q\]

where \(\bar{g} = n^{-1}\sum \hat{\epsilon}_i z_i\), \(z_i\) includes both the exogenous variables and instruments, \(\hat{\epsilon}_i\) are the model residuals and the degree of freedom is \(q = n_{instr} - n_{endog}\). \(\hat{S}_c\) is the estimator of the variance of \(\sqrt{n}\bar{g}\) selected by weight_type and weight_config, applied to the centered moment conditions \(g_i - \bar{g}\) and evaluated at the final parameter estimates. For weight_type="robust" it is

\[\hat{S}_c = n^{-1}\sum_{i=1}^{n}(g_i - \bar{g})(g_i - \bar{g})'\]

\(\hat{S}_c\) and the \(W\) of j_stat estimate the same quantity when the model is correctly specified, and so the statistics have the same asymptotic \(\chi^2_q\) null distribution. If the model is misspecified, \(E[g_i] \neq 0\), both statistics are consistent tests and grow at rate \(n\), but the uncentered estimator also contains \(\bar{g}\bar{g}'\). When it is evaluated using weights from the final parameter estimates, as in iterated GMM, this bounds the J statistic by \(n\), however large the violation of the moment conditions, whereas the centered statistic is not bounded. For weight_type="robust" and converged iterated GMM, the two are related by \(n \bar{g}'\hat{S}_c^{-1}\bar{g} = J / (1 - J / n)\), where \(J\) is the statistic in j_stat. This relationship is \(J / (1 - s J / n)\) if the estimator is scaled by \(s = n / (n - k)\) because debiased is True. The difference matters when \(J / n\) is not small.

Centering has no effect when weight_type is “unadjusted”, since that estimator uses the residual variance about its mean, and the statistic then coincides with j_stat up to the step of the estimation at which the weight matrix is evaluated. It also has no effect in IVGMMCUE models unless center=False was set.

In a weighted model the moment conditions are \(w_i z_i \hat{\epsilon}_i\), where \(w_i\) are the weights of the model, as in j_stat. The statistic and the test of whether the covariance is singular do not depend on the units in which the variables are measured.

References

Hall, A. R. (2000). Covariance matrix estimation and the power of the overidentifying restrictions test. Econometrica, 68(6), 1517-1528.

Hansen, B. E. and Lee, S. (2021). Inference for iterated GMM under misspecification. Econometrica, 89(3), 1419-1447. Theorem 1 and Example 1 discuss the centered weight matrix.