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.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Length:  
Date of creation:  11 Aug 2004 
Date of revision:  
Handle:  RePEc:ecm:nasm04:228 
Contact details of provider:  Phone: 1 212 998 3820 Fax: 1 212 995 4487 Web page: http://www.econometricsociety.org/pastmeetings.asp Email: More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
 Chesher, Andrew D, 1984. "Testing for Neglected Heterogeneity," Econometrica, Econometric Society, vol. 52(4), pages 86572, July.
 Davidson, Russell & MacKinnon, James G., 1984.
"Convenient specification tests for logit and probit models,"
Journal of Econometrics,
Elsevier, vol. 25(3), pages 241262, July.
 Russell Davidson & James G. MacKinnon, 1982. "Convenient Specification Tests for Logit and Probit Models," Working Papers 514, Queen's University, Department of Economics.
 Chesher, Andrew, 1983. "The information matrix test : Simplified calculation via a score test interpretation," Economics Letters, Elsevier, vol. 13(1), pages 4548.
 Lancaster, Tony, 1984. "The Covariance Matrix of the Information Matrix Test," Econometrica, Econometric Society, vol. 52(4), pages 105153, July.
 Russell Davidson & James G. MacKinnon, 1994.
"Graphical Methods for Investigating the Size and Power of Hypothesis Tests,"
Working Papers
903, Queen's University, Department of Economics.
 Davidson, Russell & MacKinnon, James G, 1998. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," The Manchester School of Economic & Social Studies, University of Manchester, vol. 66(1), pages 126, January.
 Davidson, Russel & MacKinnon, James G., 1983.
"Small sample properties of alternative forms of the Lagrange Multiplier test,"
Economics Letters,
Elsevier, vol. 12(34), pages 269275.
 Russell Davidson & James G. MacKinnon, 1981. "Small Sample Properties of Alternative Forms of the Lagrange Multiplier Test," Working Papers 439, Queen's University, Department of Economics.
 Dhaene, Geert & Hoorelbeke, Dirk, 2004.
"The information matrix test with bootstrapbased covariance matrix estimation,"
Economics Letters,
Elsevier, vol. 82(3), pages 341347, March.
 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.
 Alastair Hall, 1987. "The Information Matrix Test for the Linear Model," Review of Economic Studies, Oxford University Press, vol. 54(2), pages 257263.
 Orme, Chris, 1990. "The smallsample performance of the informationmatrix test," Journal of Econometrics, Elsevier, vol. 46(3), pages 309331, December.
 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.
 Davidson, Russell & MacKinnon, James G, 1992.
"A New Form of the Information Matrix Test,"
Econometrica,
Econometric Society, vol. 60(1), pages 14557, January.
 Russell Davidson & James G. MacKinnon, 1988. "A New Form of the Information Matrix Test," Working Papers 724, Queen's University, Department of Economics.
 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.
 Horowitz, Joel L., 1994. "Bootstrapbased critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395411, April.
 Breusch, T S & Pagan, A R, 1979. "A Simple Test for Heteroscedasticity and Random Coefficient Variation," Econometrica, Econometric Society, vol. 47(5), pages 128794, September.
This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.
When requesting a correction, please mention this item's handle: RePEc:ecm:nasm04:228. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.