## Applications of the FWL Theorem

A regression like (2.34), in which the regressors are broken up into two groups, can arise in many situations. In this section, we will study three of these. The first two, seasonal dummy variables and time trends, are obvious applications of the FWL Theorem. The third, measures of goodness of fit that take the constant term into account, is somewhat less obvious. In all cases, the FWL Theorem allows us to obtain explicit expressions based on (2.42) for subsets of the parameter estimates of a linear regression.

### Seasonal Dummy Variables

For a variety of reasons, it is sometimes desirable to include among the explanatory variables of a regression model variables that can take on only two possible values, which are usually 0 and 1. Such variables are called indicator variables, because they indicate a subset of the observations, namely, those for which the value of the variable is 1. Indicator variables are a special case of dummy variables, which can take on more than two possible values.

Seasonal variation provides a good reason to employ dummy variables. It is common for economic data that are indexed by time to take the form of quarterly data, where each year in the sample period is represented by four observations, one for each quarter, or season, of the year. Many economic activities are strongly affected by the season, for obvious reasons like Christmas shopping, or summer holidays, or the difficulty of doing outdoor work during very cold weather. This seasonal variation, or seasonality, in economic activity is likely to be reflected in the economic time series that are used in regression models. The term "time series" is used to refer to any variable the observations of which are indexed by the time. Of course, time-series data are sometimes annual, in which case there is no seasonal variation to worry about, and sometimes monthly, in which case there are twelve "seasons" instead of four. For simplicity, we consider only the case of quarterly data.

Since there are four seasons, there may be four seasonal dummy variables, each taking the value 1 for just one of the four seasons. Let us denote these variables as s1, s2, S3, and s4. If we consider a sample the first observation of which corresponds to the first quarter of some year, these variables look like