  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, pointwise 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.
