Abstract
Estimation results obtained by parametric models may be seriously misleading when the model is misspecified or poorly approximates the true model. This study proposes a test that jointly tests the specifications of multiple response probabilities in unordered multinomial choice models. The test statistic is asymptotically chi-square distributed, consistent against a fixed alternative and able to detect a local alternative approaching to the null at a rate slower than the parametric rate. We show that rejection regions can be calculated by a simple parametric bootstrap procedure, when the sample size is small. The size and power of the tests are investigated by Monte Carlo experiments.
Highlights
Not infrequently, variables of interest in economic research are discrete and unordered, as we often find the variables that indicate the behavior or state of economic agents
The test statistic is asymptotically chi-square distributed with J −1 degrees of freedom, consistent against a fixed alternative and have nontrivial power against local alternatives approaching the null at the rate of 1/ nhq/2
The rejection region for the test statistic can be calculated through a simple parametric bootstrap procedure, when the sample size is small
Summary
Variables of interest in economic research are discrete and unordered, as we often find the variables that indicate the behavior or state of economic agents. This study proposes a new specification test that is directly applicable to any multinomial choice models with unordered outcome variables. These models set parametric assumptions on response probabilities that an option is chosen from multiple alternatives, and identical assumptions are often set for all response probabilities. A crucial point that makes parametric bootstrap work is that the orthogonality condition holds with bootstrap sampling under both the null and alternative hypotheses This is different from the specification test for the regression function that requires the wild bootstrapping procedure to calculate the rejection region proven by [7]. Extending empirical process-based tests and rate-optimal tests to unordered multinomial choice models is a task left for future research. The proofs of the lemmas and propositions are provided in the Appendix
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