So far, we have focused on measuring the efficiency of an individualproduction or decision-making unit (firm, country, etc.) relative to a frontier consistent with a behavior of this unit. In practice, researchers are often also interested in measuring the efficiency of a groupof similar units (entire industry of firms, region of countries) or particular types of these units (e.g., public firms vs. private firms, etc.) within such groups. Even when the focus is on the efficiency of individual units, at the end of the day, researchers might want to have just one or several aggregate numbers that summarize the results. This is especially important when the number of individual units is large and each of them cannot be published or easily comprehended. But, how can we aggregate? Can we just take an average? Which one: arithmetic, geometric, harmonic? Shall it be a weighted or a non-weighted average? The goal of this chapter is to outline the recently obtained and practically useful results of previous studies to answer these imperative questions.

THE AGGREGATION PROBLEM

The problem of constructing a group measure or a group score from individual analogues is an aggregation question, which has been recently studied in a number of works. The most important question here is the choice of aggregation weights. To illustrate the point, consider a hypothetical example (adapted from Simar and Zelenyuk, 2007) of an industry consisting of four firms, two firms in each of two types, whose efficiency and “an economic weight” (whatever that might be) are summarized in Table 5.1. Here, if a researcher were to use the simple (equally weighted) arithmetic average then group A and group Z are, on average, equally efficient. Note however that the efficiency scores are “standardized” so that they are between 0 and 1 and so they disregard the relative weights of the firms that attained these scores. If another researcher wanted to use a weighted arithmetic average, then a dramatically different conclusion might be reached – depending on the weighting scheme. For the example, in Table 5.1, group A has a higher-weighted average efficiency than that of group Z, yet the industry average could still be closer to the score of group Z if its group weight dominates the weight of group A (e.g., if their weight in the industry is 90 percent as in the table).