Testing near or at the Boundary of the Parameter Space∗ (Job Market Paper)

Statistical inference about a scalar parameter is often performed using the two-sided t-test. In extremum problems, where the estimator satisfies the restrictions on the parameter space- such as the nonnegativity of a variance parameter-, the test suffers from size dis-tortions when the true parameter vector is near or at the boundary of the parameter space. Nevertheless, the two-sided t-test continues to be used when estimates are found to be close to the boundary. This can be attributed to a lack of inference procedures that appropriately account for boundary effects on the asymptotic distribution of the estimator. To address this issue, we propose an estimator that is asymptotically normally distributed, even when the true parameter vector is near or at the boundary, and the objective function is not defined outside the parameter space. The novel estimator allows the implementation of several ex-isting testing procedures and a new test based on the Conditional Likelihood Ratio statistic (CLR). Compared to the existing procedures, the new test is easy to implement and has good power properties. Moreover, it offers power advantages over the two-sided t-test, when the latter controls size. We also show the test to be admissible when inference is performed with respect to a scalar parameter. We apply the test to the random coefficients logit model using data on the European car market and find more evidence of heterogeneity in consumer preferences than suggested by the two-sided t-test