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626 Chapter 14 Introduction to Time Series Regression and Forecasting

b. Repeat (a) 1000 times, saving each value of R2 and the t-statistic.
Construct a histogram of the R2 and t-statistic. What are the 5%,
50%, and 95% percentiles of the distributions of the R2 and the
t-statistic? In what fraction of your 1000 simulated data sets does the
t-statistic exceed 1.96 in absolute value?

c. Repeat (b) for different numbers of observations, such as T = 50 and
T = 200. As the sample size increases, does the fraction of times that
you reject the null hypothesis approach 5%, as it should because you have
generated Y and X to be independently distributed? Does this fraction
seem to approach some other limit as T gets large? What is that limit?

14.7 Suppose that Yt follows the stationary AR(1) model Yt = 2 .5 + 0 .7Yt - 1 + ut,
where ut is i.i.d. with E1ut2 = 0 and var1ut2 = 9.

a. Compute the mean and variance of Yt. (Hint: See Exercise 14.1.)
b. Compute the first two autocovariances of Yt. (Hint: Read Appendix 14.2.)
c. Compute the first two autocorrelations of Yt.

d. Suppose that YT = 102 .3. Compute YT + 10T = E(YT + 1 0 YT, Yt - 1, c).

14.8 Suppose that Yt is the monthly value of the number of new home construc-
tion projects started in the United States. Because of the weather, Yt has a
pronounced seasonal pattern; for example, housing starts are low in Janu-
ary and high in June. Let mJan denote the average value of housing starts in
January and let mFeb, mMar, c, mDec denote the average values in the other
months. Show that the values of mJan, mFeb, c, mDec can be estimated from
the OLS regression Yt = b0 + b1Febt + b2Mart + g + b11Dect + ut, where
Febt is a binary variable equal to 1 if t is February, Mart is a binary variable
equal to 1 if t is March, and so forth. (Hint: Show that b0 + b2 = mMar, and
so forth.)

14.9 The moving average model of order q has the form

Yt = b0 + et + b1et - 1 + b2et - 2 + g + bqet - q,

where et is a serially uncorrelated random variable with mean 0 and vari-
ance s2e.

a. Show that E1Yt2 = b0.
b. Show that the variance of Yt is var(Yt) = s2e(1 + b21 + b22 + g+ bq2).
c. Show that rj = 0 for j > q.
d. Suppose that q = 1. Derive the autocovariances for Y.