## Info

Note: FRIG = refrigerator sales, thousands

DUR = durable goods expenditure, billions of 1992 dollars D2 = 1 in the second quarter, 0 otherwise D3 = 1 in the third quarter, 0 otherwise D4 = 1 in the fourth quarter, 0 otherwise Source: Business Statistics and Survey of Current Business, Department of Commerce (various issues).

Note: FRIG = refrigerator sales, thousands

DUR = durable goods expenditure, billions of 1992 dollars D2 = 1 in the second quarter, 0 otherwise D3 = 1 in the third quarter, 0 otherwise D4 = 1 in the fourth quarter, 0 otherwise Source: Business Statistics and Survey of Current Business, Department of Commerce (various issues).

CHAPTER NINE: DUMMY VARIABLE REGRESSION MODELS 315

EXAMPLE 9.6 (Continued)

Incidentally, instead of assigning a dummy for each quarter and suppressing the intercept term to avoid the dummy variable trap, we could assign only three dummies and include the intercept term. Suppose we treat the first quarter as the reference quarter and assign dummies to the second, third, and fourth quarters. This produces the following regression results (see Table 9.4 for the data setup):

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

Since we are treating the first quarter as the benchmark, the coefficients attached to the various dummies are now differential intercepts, showing by how much the average value of Y in the quarter that receives a dummy value of 1 differs from that of the benchmark quarter. Put differently, the coefficients on the seasonal dummies will give the seasonal increase or decrease in the average value of Y relative to the base season. If you add the various differential intercept values to the benchmark average value of 1222.125, you will get the average value for the various quarters. Doing so, you will reproduce exactly Eq. (9.7.2), except for the rounding errors.

But now you will see the value of treating one quarter as the benchmark quarter, for (9.7.3) shows that the average value of Y for the fourth quarter is not statistically different from the average value for the first quarter, as the dummy coefficient for the fourth quarter is not statistically significant. Of course, your answer will change, depending on which quarter you treat as the benchmark quarter, but the overall conclusion will not change.

How do we obtain the deseasonalized time series of refrigerator sales? This can be done easily. You estimate the values of Yfrom model (9.7.2) [or (9.7.3)] for each observation and subtract them from the actual values of Y, that is, you obtain (Yt - %) which are simply the residuals from the regression (9.7.2). We show them in Table 9.5.15

What do these residuals represent? They represent the remaining components of the refrigerator time series, namely, the trend, cycle, and random components (but see the caution given in footnote 15).

Since models (9.7.2) and (9.7.3) do not contain any covariates, will the picture change if we bring in a quantitative regressor in the model? Since expenditure on durable goods has an important factor influence on the demand for refrigerators, let us expand our model (9.7.3) by bringing in this variable. The data for durable goods expenditure in billions of 1982 dollars are already given in Table 9.3. This is our (quantitative) Xvariable in the model. The regression results are as follows

Yt = 456.2440 + 242.4976D2t + 325.2643D3t - 86.0804D4t + 2.7734Xt t = (2.5593)* (3.6951)* (4.9421)* (-1.3073)** (4.4496)* (9.7.4)

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

15Of course, this assumes that the dummy variables technique is an appropriate method of deseasonalizing a time series and that a time series (TS) can be represented as: TS = 5 + c + t + u, where 5 represents the seasonal, t the trend, c the cyclical, and u the random component. However, if the time series is of the form, TS = (s)(c)(t)(u), where the four components enter multiplicatively, the preceding method of deseasonalization is inappropriate, for that method assumes that the four components of a time series are additive. But we will have more to say about this topic in the chapters on time series econometrics.

Yt = 1222.1250 + 245.3750D2t + 347.6250D3( - 62.1250D4( t = (20.3720)* (2.8922)* (4.0974)* (-0.7322)**

(Continued)

316 PART ONE: SINGLE-EQUATION REGRESSION MODELS

EXAMPLE 9.6 (Continued) 