Question

The models of this chapter emphasize forecasting the level of a series. For many business decisions, fore-casts of change are more important. If we€™re projecting an increase in sales, then we need to have more items in stock. This timeplot shows sales (in thou-sands of dollars) over a 25-week period (not including special holiday sales).

(a) The following output summarizes the ft of an AR(2) model to these data. Assuming that the model meets the usual conditions, is this a good description of the dependence in this series?

(b) If we use the same two predictors Yt -1 and Yt -2 to describe the changes in sales or use the differences Dt = Yt - Yt -1, what will be the estimated slopes for the lagged variables?
(c) Explain why se of the regression of Yt on these two lags is the same as the SD of the residuals when regressing the changes on these two lags.
(d) Although the models are equivalent (in the sense that you can get the slope for one from the other), most analysts prefer to ft models to Yt itself rather than to the differences. A partial explanation comes from a comparison of R2 for the two models. Which model do you think has a larger R2: the model with Yt as the response, or the model with the differences Dt?

Answer 1

(a)

Yes, with large (statistically significant) R2 and both slopes individually different from zero. 

 

(b)

The slope of yt−1 is smaller by one, and the intercept and slope of yt−2 are the same. The slope for the lag changes because we’ve moved most the lag into the response:

            ŷ t = 26.75 + 0.97yt−1 – 0.50yt−2 yt − yt−1 = 15 – 0.03yt−1 – 0.50yt−2

 

(c)

The SD of the residuals is the same because the residuals are the same. A residual in the model for shipments yt is

et = yt − ŷ t = yt − (26.75 + 0.97yt−1 – 0.50yt−2)

    = (yt − yt−1) − (26.75 − 0.03yt−1 0.50yt−2)

which is a residual in the model for the changes in shipments.

 

 

(d)

The residuals using differences are the same as those in levels, but the variation to start with is larger when working with yt itself. We’ve gotten down to the same residuals in both cases, but started with more variation when working with yt than with the differences. Hence, the R2 for the model in levels is larger.