Truncatednormal panel data estimation

The cross-sectional exponential formulation has served its purpose. Two major limitations preclude using it in empirical work. First, and most importantly, the model is actually under-identified. Second, because of its simplicity, the exponential distribution places undue restriction on the pattern of inefficiency, motivating search for more flexible formulations. A simple counting procedure motivates the intuition about why the cross-sectional model is under-identified. There are, in total, N + 3 parameters being estimated, namely the N inefficiency measures z1,z2 • ^,zN ; the two parameters characterizing the sampling distribution, namely a and (3 ; and the parameter X characterizing the inefficiency distribution. Yet, only i — 1, 2 • • • ,N sample points avail themselves for estimating these N + 3 unknowns. This point is taken up in Fernandez et al. (1997), where a simple remedy is also suggested. As long as we have panel data and the inefficiency terms are held constant across subsets of the data (usually its time dimension), the complete set of model parameters is fully identified. The second problem concerns the actual shape of the inefficiency distribution across the firms. The exponential distribution is a single-parameter distribution, offering considerable tractability, and, while highly useful for expositional purposes, it severely restricts the underlying inefficiencies. The form on which we focus attention in the remainder of the paper is the truncated-normal distribution. In the panel with j — 1, • • • ,Ni subunits (say, periods) corresponding to each of the i — 1, •••, N sample units (say, firms), we consider (firm) inefficiencies z1,z2 • • • ,zN ~ fTN (z^X, y), constant across time periods; and sampling errors Uj ~ fN(uj |0, a) for all j — 1,2 • • • ,Ni, i — 1,2 • • • ,N and the availability of panel data has two implications. It circumvents the under-identification issue and it affords great tractability permitting multiple linkages across sub-units and across time peri

ods. In the empirical application such flexibility, we will show, is extremely important. In short, panel data raises scope for nuanced empirical inquiry.

With a diffuse prior on the parameters 6 = (a, 3' ,X,Y)' the posterior has the form

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