Bootstrap correcting the score test
The Lagrange multiplier test, or score test, suggested independently by Aitchison and Silvey (1958) and Rao (1948), tests for parametric restrictions. Although the score test is an intuitively appealing and often used procedure, the exact distribution of the score test statistic is generally unknown and is often approximated by its first-order asymptotic $\chi^2$ distribution. In problems of econometric inference, however, first-order asymptotic theory may be a poor guide, and this is also true for the score test, as demonstrated in different Monte Carlo studies. See e.g. Breusch and Pagan (1979), Bera and Jarque (1981), Davidson and MacKinnon (1983, 1984, 1992), Chesher and Spady (1991) and Horowitz (1994), among many others. One can use the bootstrap distribution of the score test statistic to obtain a critical value. This can give already satisfactory results in terms of ERP (error in rejection probability: the difference between nominal and actual rejection probability under the null hypothesis). However, the score test uses a quadratic form statistic. In the construction and implementation of such a quadratic form statistic two important aspects, which determine the performance of the test (both under the null and the alternative), are (i) the weighting matrix (the covariance matrix of the score vector) and (ii) the critical value. Since the score test statistic is asymptotically pivotal, the bootstrap critical value is second-order correct. However, one can achieve better performance, as well in terms of ERP as of power, by using a better estimate of the weighting matrix used in the quadratic form. In this paper we propose a bootstrap-based method to obtain both a second-order correct estimate of the covariance matrix of the score vector and a second-order correct critical value, using only one round of simulations (instead of B1 + B1 x B2). The method works as follows. Assume there exists a matrix A such that the score vector premultiplied by A is asymptotically pivotal. An obvious choice for A is the inverse of a square root of a covariance matrix estimate of the score vector, yielding a multivariate studentized score vector. This is not the only possible choice for A, though. Since then the transformed score vector is asymptotically pivotal, the bootstrap distribution is a second-order approximation to the exact finite sample distribution. As such, the bootstrap covariance matrix of the transformed score vector is also second-order correct. The next step is to construct a quadratic form statistic in the transformed score vector using its bootstrap covariance matrix as weighting matrix. This statistic is asymptotically (as both the sample size and the number of bootstrap simulations go to infinity) chi-squared distributed with q (the dimension of the score) degrees of freedom. In practice, however, the number of bootstrap simulations is fixed to, say, B simulations. In this case the statistic is asymptotically (for the sample size tending to infinity) Hotelling T-squared distributed with q and B-1 degrees of freedom. Using a finite B, the exact finite sample covariance matrix of the transformed score vector is estimated with some noise, but the T-squared critical values correct for this. When the T-squared critical values are used, one is still only first-order correct. But the distribution of the new statistic can also be approximated by the empirical distribution function of the quadratic forms in the bootstrap replications of the transformed score vector using the inverse of the bootstrap covariance matrix as weighting matrix. The appropriate quantile of this empirical distribution delivers a critical value which is second-order correct. In a Monte Carlo simulation study we look at the information matrix test (White, 1982) in the regression model. Chesher (1983) showed that the information matrix is a score test for parameter constancy. We correct the Chesher-Lancaster version (Chesher, 1983 and Lancaster, 1984) of the information matrix test with the method proposed above and look at the ERP under the null and the power under a heteroskedastic alternative. The corrected statistic outperforms the Chesher-Lancaster statistic both in terms of ERP (with asymptotic or bootstrap critical values) and power.
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