## Example

AVERAGE HOURLY EARNINGS IN RELATION TO EDUCATION, GENDER, AND RACE

Let us first present the regression results based on model (9.6.1). Using the data that were used to estimate regression (9.3.1), we obtained the following results:

Y, = —0.2610 — 2.3606D2, — 1.7327D3, + 0.8028X,-

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent.

The reader can check that the differential intercept coefficients are statistically significant, that they have the expected signs (why?), and that education has a strong positive effect on hourly wage, an unsurprising finding.

As (9.6.4) shows, ceteris paribus, the average hourly earnings of females are lower by about \$2.36, and the average hourly earnings of nonwhite non-Hispanic workers are also lower by about \$1.73.

We now consider the results of model (9.6.2), which includes the interaction dummy.

Y, = —0.26100 — 2.3606D2, — 1.7327D3, + 2.1289D2,D3, + 0.8028X, t = (—0.2357)** (—5.4873)* (—2.1803)* (1.7420)** (9.9095)** (9.6.5)

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent.

As you can see, the two additive dummies are still statistically significant, but the interactive dummy is not at the conventional 5 percent level; the actual p value of the interaction dummy is about the 8 percent level. If you think this is a low enough probability, then the results of (9.6.5) can be interpreted as follows: Holding the level of education constant, if you add the three dummy coefficients you will obtain: —1.964 ( = —2.3605 — 1.7327 + 2.1289), which means that mean hourly wages of nonwhite/non-Hispanic female workers is lower by about \$1.96, which is between the value of —2.3605 (gender difference alone) and —1.7327 (race difference alone).

312 PART ONE: SINGLE-EQUATION REGRESSION MODELS

The preceding example clearly reveals the role of interaction dummies when two or more qualitative regressors are included in the model. It is important to note that in the model (9.6.5) we are assuming that the rate of increase of hourly earnings with respect to education (of about 80 cents per additional year of schooling) remains constant across gender and race. But this may not be the case. If you want to test for this, you will have to introduce differential slope coefficients (see exercise 9.25)

Many economic time series based on monthly or quarterly data exhibit seasonal patterns (regular oscillatory movements). Examples are sales of department stores at Christmas and other major holiday times, demand for money (or cash balances) by households at holiday times, demand for ice cream and soft drinks during summer, prices of crops right after harvesting season, demand for air travel, etc. Often it is desirable to remove the seasonal factor, or component, from a time series so that one can concentrate on the other components, such as the trend.12 The process of removing the seasonal component from a time series is known as deseasonalization or seasonal adjustment, and the time series thus obtained is called the deseasonalized, or seasonally adjusted, time series. Important economic time series, such as the unemployment rate, the consumer price index (CPI), the producer's price index (PPI), and the index of industrial production, are usually published in seasonally adjusted form.

There are several methods of deseasonalizing a time series, but we will consider only one of these methods, namely, the method of dummy vari-ables.13 To illustrate how the dummy variables can be used to deseasonalize economic time series, consider the data given in Table 9.3. This table gives quarterly data for the years 1978-1995 on the sale of four major appliances, dishwashers, garbage disposers, refrigerators, and washing machines, all data in thousands of units. The table also gives data on durable goods expenditure in 1982 billions of dollars.

To illustrate the dummy technique, we will consider only the sales of refrigerators over the sample period. But first let us look at the data, which is shown in Figure 9.4. This figure suggests that perhaps there is a seasonal pattern in the data associated with the various quarters. To see if this is the case, consider the following model:

where Yt = sales of refrigerators (in thousands) and the D's are the dummies, taking a value of 1 in the relevant quarter and 0 otherwise. Note that

12A time series may contain four components: a seasonal, a cyclical, a trend, and one that is strictly random.

13For the various methods of seasonal adjustment, see, for instance, Francis X. Diebod, Elements of Forecasting, 2d ed., South-Western Publishers, 2001, Chap. 5.