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Again, keep in mind that we are treating the first quarter as our base. As in (9.7.3), we see that the differential intercept coefficients for the second and third quarters are statistically different from that of the first quarter, but the intercepts of the fourth quarter and the first quarter are statistically about the same. The coefficient of X (durable goods expenditure) of about 2.77 tells us that, allowing for seasonal effects, if expenditure on durable goods goes up by a dollar, on average, sales of refrigerators go up by about 2.77 units, that is, approximately 3 units; bear in mind that refrigerators are in thousands of units and X is in (1982) billions of dollars.

CHAPTER NINE: DUMMY VARIABLE REGRESSION MODELS 317

EXAMPLE 9.6 (Continued)

An interesting question here is: Just as sales of refrigerators exhibit seasonal patterns, would not expenditure on durable goods also exhibit seasonal patterns? How then do we take into account seasonality in X? The interesting thing about (9.7.4) is that the dummy variables in that model not only remove the seasonality in Y but also the seasonality, if any, in X. (This follows from a well-known theorem in statistics, known as the Frisch-Waugh theorem.16) So to speak, we kill (deseasonalize) two birds (two series) with one stone (the dummy technique).

If you want an informal proof of the preceding statement, just follow these steps: (1) Run the regression of Y on the dummies as in (9.7.2) or (9.7.3) and save the residuals, say, S1; these residuals represent deseasonalized Y. (2) Run a similar regression for X and obtain the residuals from this regression, say, S2; these residuals represent deseasonalized X. (3) Regress S1 on S2. You will find that the slope coefficient in this regression is precisely the coefficient of X in the regression (9.7.4).

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