Multi-Dimensional Sorting in the Data

When heterogeneous workers sort into heterogeneous jobs on the labor market, `how many' and especially `which' skills and job attributes matter for this choice? Based on our theory of multi-dimensional sorting under random search (Lindenlaub, Postel-Vinay 2017), this paper first develops an empirical test of how many heterogeneity dimensions matter for sorting: If the data is well-approximated by N-dimensional worker and job types, any two workers with the same type should face the same job ladder and therefore the same job acceptance sets. Conversely, if the assumption of N-dimensional heterogeneity is not justified but instead $M>N$ dimensions matter for sorting, then approximating different M-dimensional worker types by the same N-dimensional worker type will produce job ladder heterogeneity within those N-dimensional worker types, revealing misspecification of worker's attributes. To assess the accuracy of this test, we first implement it via simulations, where we simulate data from a large class of multi-dimensional models that complies with our theory. We then apply model selection methods to regressions of employment-to-employment indicators (as a proxy for job acceptance sets) on a large set of potential skills and job attributes in order to recover the true dimensionality of worker and job characteristics. We show that the model selection methods quite accurately reveal the `true' worker and job heterogeneity that matters for sorting. We then implement this test on US data at different points in time, which at each given point delivers a set of worker and job attributes that matters for labor market sorting and allows us to construct the multivariate distributions of skills and job attributes in the data. Second, we propose an application of multi-dimensional sorting to the observed slow-down in US labor market dynamics. We estimate the search model with multi-dimensional types developed in (Lindenlaub, Postel-Vinay 2017) at different points in time, using our constructed skill and job attribute distributions as inputs for the estimation. We then use the estimated model to decompose the slow-down in UE and EE flows in the part that is driven by (i) changes in the multivariate skill and job distributions, (ii) changes in technology, and (iii) changes in search frictions.