The two-sample version of the celebrated Pearson goodness-of-fit problem has been a topic of extensive research, and several tests like the Kolmogorov-Smirnov and Cramér-von Mises have been suggested. Although these tests perform fairly well as omnibus tests for comparing two probability density functions (PDFs), they may have poor power against specific departures such as in location, scale, skewness, and kurtosis. We propose a new test for the equality of two PDFs based on a modified version of the Neyman smooth test using empirical distribution functions minimizing size distortion in finite samples. The suggested test can detect the specific directions of departure from the null hypothesis. Specifically, it can identify deviations in the directions of mean, variance, skewness, or tail behavior. In a finite sample, the actual probability of type-I error depends on the relative sizes of the two samples. We propose two different approaches to deal with this problem and show that, under appropriate conditions, the proposed tests are asymptotically distributed as chi-squared. We also study the finite sample size and power properties of our proposed test. As an application of our procedure, we compare the age distributions of employees with small employers in New York and Pennsylvania with group insurance before and after the enactment of the “community rating” legislation in New York. It has been conventional wisdom that if community rating is enforced (where the group health insurance premium does not depend on age or any other physical characteristics of the insured), then the insurance market will collapse, since only older or less healthy patients would prefer group insurance. We find that there are significant changes in the age distribution in the population in New York owing mainly to a shift in location and scale.

In this paper we propose a new nonparametric test for conditional heteroskedasticity based on a measure of nonparametric goodness-of-fit (R2) that is obtained from the local polynomial regression of the residuals from a parametric regression on some covariates. We show that after being appropriately standardized, the nonparametric R2 is asymptotically normally distributed under the null hypothesis and a sequence of Pitman local alternatives. We also prove the consistency of the test and propose a bootstrap method to obtain the bootstrap p-values. We conduct a small set of simulations and compare our test with some popular parametric and nonparametric tests in the literature.

We propose a consistent kernel-based specification test for conditional density models when the dependent variable is categorical/discrete. The method is applicable to popular parametric binary choice models such as the logit and probit specification and their multinomial and ordered counterparts, along with parametric count models, among others. The test is valid when the conditional density function contains both categorical and real-valued covariates. Theoretical support for the test and for a bootstrap-based version of the test is provided. Monte Carlo simulations are conducted to assess the finite-sample performance of the proposed method.

It is often believed that without instruments, endogenous sample selection models are identified only if a covariate with a large support is available (see, e.g., Chamberlain, 1986, Journal of Econometrics 32, 189–218; Lewbel, 2007, Journal of Econometrics141, 777–806) . We propose a new identification strategy mainly based on the condition that the selection variable becomes independent of the covariates for large values of the outcome. No large support on the covariates is required. Moreover, we prove that this condition is testable. We finally show that our strategy can be applied to the identification of generalized Roy models.

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

Cai, Li, and Park (Journal of Econometrics, 2009) and Xiao (Journal of Econometrics, 2009) developed asymptotic theories for estimators of semiparametric varying coefficient models when regressors are integrated processes but the smooth coefficients are functionals of stationary processes. Using a recent result from Phillips (Econometric Theory, 2009), we extend this line of research by allowing for both the regressors and the covariates entering the smooth functionals to be integrated variables. We derive the asymptotic distribution for the proposed semiparametric estimator. An empirical application is presented to examine the purchasing power parity hypothesis between U.S. and Canadian dollars.