## Several Groupings

The case in which several sets of dummy variables are needed is much the same as those we have already considered, with one important exception. Consider a model of statewide per capita expenditure on education y as a function of statewide per capita income x. Suppose that we have observations on all n = 50 states for T = 10 years. A regression model that allows the expected expenditure to change over time as well as across states would be yit = a + 3 xit + 5i + 6t + eu. (7-2)

As before, it is necessary to drop one of the variables in each set of dummy variables to avoid the dummy variable trap. For our example, if a total of 50 state dummies and 10 time dummies is retained, a problem of "perfect multicollinearity" remains; the sums of the 50 state dummies and the 10 time dummies are the same, that is, 1. One of the variables in each of the sets (or the overall constant term and one of the variables in one of the sets) must be omitted.

### Example 7.2 Analysis of Covariance

The data in Appendix Table F7.1 were used in a study of efficiency in production of airline services in Greene (1997b). The airline industry has been a favorite subject of study [e.g., Schmidt and Sickles (1984); Sickles, Good, and Johnson (1986)], partly because of interest in this rapidly changing market in a period of deregulation and partly because of an abundance of large, high-quality data sets collected by the (no longer existent) Civil Aeronautics Board. The original data set consisted of 25 firms observed yearly for 15 years (1970 to 1984), a "balanced panel." Several of the firms merged during this period and several others experienced strikes, which reduced the number of complete observations substantially. Omitting these and others because of missing data on some of the variables left a group of 10 full

1See Suits (1984) and Greene and Seaks (1991).