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Exercises 625

14.5 Prove the following results about conditional means, forecasts, and forecast
errors:

a. Let W be a random variable with mean mW and variance sw2 and let c
be a constant. Show that E3(W - c)24 = s2w + (mW - c)2.

b. Consider the problem of forecasting Yt, using data on Yt - 1, Yt - 2, c .
Let ft - 1 denote some forecast of Yt, where the subscript t - 1 on ft - 1
indicates that the forecast is a function of data through date t - 1. Let

E3(Yt - ft - 1)2 0 Yt - 1, Yt - 2, c4 be the conditional mean squared error

of the forecast ft - 1, conditional on values of Y observed through date
t - 1. Show that the conditional mean squared forecast error is mini-

mized when ft - 1 = Yt0t - 1, where Yt0t - 1 = E(Yt 0 Yt - 1,Yt - 2, c). (Hint:

Review Exercise 2.27.)

c. Let ut denote the error in Equation (14.13). Show that cov(ut, ut - j) = 0
for j 0. [Hint: Use Equation (2.27).]

14.6 In this exercise you will conduct a Monte Carlo experiment to study the
phenomenon of spurious regression discussed in Section 14.6. In a Monte
Carlo study, artificial data are generated using a computer, and then those
artificial data are used to calculate the statistics being studied. This makes
it possible to compute the distribution of statistics for known models when
mathematical expressions for those distributions are complicated (as they
are here) or even unknown. In this exercise, you will generate data so that
two series, Yt and Xt, are independently distributed random walks. The
specific steps are as follows:

i. Use your computer to generate a sequence of T = 100 i.i.d. standard
normal random variables. Call these variables e1, e2, c, e100. Set
Y1 = e1 and Yt = Yt - 1 + et for t = 2, 3, c, 100.

ii. Use your computer to generate a new sequence, a1, a2, c, a100, of
T = 100 i.i.d. standard normal random variables. Set X1 = a1 and
Xt = Xt - 1 + at for t = 2, 3, c, 100.

iii. Regress Yt onto a constant and Xt. Compute the OLS estimator, the
regression R2, and the (homoskedastic-only) t-statistic testing the null
hypothesis that b1 (the coefficient on Xt) is zero.

Use this algorithm to answer the following questions:

a. Run the algorithm (i) through (iii) once. Use the t-statistic from (iii)
to test the null hypothesis that b1 = 0, using the usual 5% critical
value of 1.96. What is the R2 of your regression?