Estimation of individual efficiency scores, correcting them for the bias and estimating their standard errors and confidence intervals are very useful when the focus is on particular DMUs. In most applications, however, the main focus is often on the big picture, i.e., on the overall tendencies of the population or its sub-populations of interest rather than (or in addition to) some individual DMUs. Such analysis can be done through various approaches, most popular of which are analysis of means, distributions or densities and a regression analysis, and the goal of this chapter is to outline these approaches and their caveats.
INFERENCE ON AGGREGATE OR GROUP EFFICIENCY
A common starting point of such aggregate analysis is to look at the aggregate efficiency and aggregate productivity measures. The simplest examples of such aggregate measures would be the sample means of the individual estimates. Such measures would be estimators of the population means. By the laws of large numbers, because the individual measures are consistent estimators of the true efficiency scores, the average of them is also a consistent estimator of the true mean of the population distribution of efficiency scores (under certain regularity conditions similar to those described in the previous chapter).
An important question, however, is whether the population mean is what should be of primary interest. As we discussed in Chapter 5, the equally weighted mean is a useful characteristic of a distribution, but just relying on that can misrepresent the situation dramatically. Indeed, when we look at the real world, one can see that many (if not most) industries are dominated by just a few firms, although they may also have many other small firms trying to have their place under the sun. For example, take the banking industry: in most countries just a few banks have a larger share of the industry (e.g., in terms of assets, loans, deposits, etc.) than the hundreds of remaining small banks. Imagine, hypothetically, if all those small banks have very high efficiency levels, say very close to 100 percent, while those few gigantic banks have low efficiency levels, say close to 50 percent, although they control most of the industry share, say at 90 percent.