The sampling distribution of Directional Mobility Indices applied to the income of Italian families

In economics, transition matrices are often used to describe the dynamics of individuals among a discrete set of states, defined on the basis of an economically relevant variable. Typical examples are the analysis of flows of workers in the labor market or of householders among various income levels. We consider here the problem of comparing different matrices through a specific mobility index called directional index. Generally speaking, mobility indices are functions of a given transition matrix P. The directional index is a particular function able to indicate both the level of mobility and the prevailing direction (left/right) in the dynamics under study. Here we focus on the comparison between two different sampling matrices: consequently, in order to rigorously determine if the level of mobility has significantly changed, we provide the analysis of its asymptotic sampling distribution. Empirical applications regard the analysis of sampling transition matrices about the income of Italian families. Such matrices cover four consecutive two-years periods, from 2004 to 2012. We make statistical inference to analyze changes of mobility with respect of time, and among families with different income levels. Starting from 2006, results show a prevailing negative mobility, that is the tendency of Italian families to move towards lower income classes. In particular, we observe negative peaks in the mobility values in the period 2010-2012, for middle/ high income classes, indicating that Italian families are still suffering the 2008 downturn.