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 firstorder asymptotic $\chi^2$ distribution. In problems of econometric inference, however, firstorder 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 secondorder 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 bootstrapbased method to obtain both a secondorder correct estimate of the covariance matrix of the score vector and a secondorder 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 secondorder approximation to the exact finite sample distribution. As such, the bootstrap covariance matrix of the transformed score vector is also secondorder 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) chisquared 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 Tsquared distributed with q and B1 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 Tsquared critical values correct for this. When the Tsquared critical values are used, one is still only firstorder 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 secondorder 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 ChesherLancaster 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 ChesherLancaster statistic both in terms of ERP (with asymptotic or bootstrap critical values) and power.
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Date of creation:  11 Aug 2004 
Handle:  RePEc:ecm:nasm04:228 
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 Dhaene, Geert & Hoorelbeke, Dirk, 2004.
"The information matrix test with bootstrapbased covariance matrix estimation,"
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 Geert Dhaene & Dirk Hoorelbeke, 2002. "The Information Matrix Test with BootstrapBased Covariance Matrix Estimation," Working Papers Department of Economics ces0211, KU Leuven, Faculty of Economics and Business, Department of Economics.
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"Graphical Methods for Investigating the Size and Power of Hypothesis Tests,"
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 Horowitz, Joel L., 1994. "Bootstrapbased critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395411, April.
 Chesher, Andrew D, 1984. "Testing for Neglected Heterogeneity," Econometrica, Econometric Society, vol. 52(4), pages 865872, July.
 Teresa Aparicio & Inmaculada Villanua, 2001. "The asymptotically efficient version of the information matrix test in binary choice models. A study of size and power," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(2), pages 167182.
 Alastair Hall, 1987. "The Information Matrix Test for the Linear Model," Review of Economic Studies, Oxford University Press, vol. 54(2), pages 257263.
 Horowitz, Joel L. & Savin, N. E., 2000. "Empirically relevant critical values for hypothesis tests: A bootstrap approach," Journal of Econometrics, Elsevier, vol. 95(2), pages 375389, April.
 Breusch, T S & Pagan, A R, 1979. "A Simple Test for Heteroscedasticity and Random Coefficient Variation," Econometrica, Econometric Society, vol. 47(5), pages 12871294, September.
 Chesher, Andrew, 1983. "The information matrix test : Simplified calculation via a score test interpretation," Economics Letters, Elsevier, vol. 13(1), pages 4548.
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