Tests for qualitative features in the random coefficients model
The random coeﬃcients model is an extension of the linear regression model that allows for unobserved heterogeneity in the population by modeling the regression coeﬃcients as random variables. Given data from this model, the statistical challenge is to recover information about the joint density of the random coeﬃcients which is a multivariate and ill-posed problem. Because of the curse of dimensionality and the ill-posedness, nonparametric estimation of the joint density is diﬃcult and suﬀers from slow convergence rates. Larger features, such as an increase of the density along some direction or a well-accentuated mode can, however, be much easier detected from data by means of statistical tests. In this article, we follow this strategy and construct tests and conﬁdence statements for qualitative features of the joint density, such as increases, decreases and modes. We propose a multiple testing approach based on aggregating single tests which are designed to extract shape information on ﬁxed scales and directions. Using recent tools for Gaussian approximations of multivariate empirical processes, we derive expressions for the critical value. We apply our method to simulated and real data.