Transition and limiting distributions when covariates are available

Kernel density techniques or finite-state Markov chains may be used to study the steady-state distribution of key variables of interest (like real GDP per capita) as well as the transition density. Such techniques are descriptive, in the sense that they do not allow us to examine the effect of certain covariates on the transition or limiting distribution of the variable of interest. We propose a bivariate mixture-of-normal-distributions to approximate accurately the joint distribution of the variable of interest at different points in time. The transition and the limiting distributions can be derived in a straightforward manner without solving integral equations. Moreover, the effect of covariates on the transition or limiting distribution can be examined in a principled way.