Question

A chain of photography and electronics stores created a Web site to promote its photography lessons. The number of weekly visitors grew steadily, at a rate of about 300 new visitors each week. This timeplot shows the counts of unique visitors over the last 33 weeks.

To describe the growth, an analyst used a linear time trend and estimated the equation to be
Å·t = 666 + 294 t
with t = 1 denoting the first week, t = 2 the second, and so forth. The company hired a summer employee who had taken some statistics courses, and she suggested using a first-order auto regression [i.e., an AR(1) Model] instead of this time trend.
(a) What do you think the intercept and slope of the AR(1) equation are going to be?
(b) Do you think it€™s a good idea to use an auto regression in place of the linear time trend in this situation?

Answer 1

(a)        The intercept is about 300 and the slope is near 1. The time trend indicates an increase of about 300 per week. So the value this week is the same as the value last week, plus 300. That is

ŷ t = 300 + yt−1

 

 

 

(b)        Stick with the linear trend. Autoregressions are the natural choice for capturing meandering dependence, not linear growth such as in this case. Plus, the trend model is probably easier to explain to someone! Plus, as shown in the prior question, the autoregression may mean-revert, tending back to the average.

Sampling variation pushes the estimates from these ideal values, but both lie within the relevant confidence intervals.