A COMPANION TO Theoretical Econometrics

# Monte Carlo Tests: Econometric Applications

1.1 Pivotal statistics

In Dufour and Kiviet (1996, 1998), Kiviet and Dufour (1997), Dufour et al. (1998), Dufour and Khalaf (1998a, 1998c), Bernard, Dufour, Khalaf, and Genest (1998), Saphores, Khalaf, and Pelletier (1998), several applications of MC tests based on pivotal statistics are presented. The problems considered include: normality tests, heteroskedasticity tests including tests for (G)ARCH and tests for break in vari­ance at unknown points, independence tests and tests based on autocorrelations.8

The reader will find in the above papers simulation results which show clearly that the technique of Monte Carlo tests completely corrects often important size distortions due to poor large sample approximations; power studies are also reported on a case by case basis to assess the performance of MC size corrected tests.

Relevant results pertaining to the examples considered above are included in Tables 23.2 and 23.3. It is evident from Table 23.2 that the size of the JB and KS tests is perfectly controlled for all designs considered.9 Table 23.3 includes the empirical size of the MC LR test for linear restrictions. From (23.27), we see that under the distributional assumption (23.19), the simulated statistics may be obtained using draws from a nuisance-parameter free null distribution, namely the hypothesized distribution of the vector w. Consequently, application of the MC test procedure yields exact p-values. Indeed, it is shown in Table 23.3 that the MC LR test achieves perfect size control.10

Now to illustrate the feasibility of MMC tests and the usefulness of BMC tests, we will focus on examples involving nuisance parameters.

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## A COMPANION TO Theoretical Econometrics

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