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630 Chapter 14 Introduction to Time Series Regression and Forecasting

index of stock prices constructed by the Center for Research in Security Prices (CRSP). The
monthly percentage excess return is 100 * 5ln3(Pt + Divt) > Pt - 14 6 - ln(TBillt)6, where
Divt is the dividends paid on the stocks in the CRSP index and TBillt is the gross return
(1 plus the interest rate) on a 30-day Treasury bill during month t. The dividend–price ratio
is constructed as the dividends over the past 12 months, divided by the price in the current
month. We thank Motohiro Yogo for his help and for providing these data.

A p p e n d i x

14.2 Stationarity in the AR(1) Model

This appendix shows that if ͉ b1 ͉ 6 1 and ut is stationary, then Yt is stationary. Recall from
Key Concept 14.5 that the time series variable Yt is stationary if the joint distribution of
(Ys + 1, c, Ys + T) does not depend on s, regardless of the value of T. To streamline the
argument, we show this formally for T = 2 under the simplifying assumptions that
b0 = 0 and 5ut6 are i.i.d. N(0,s2u).

The first step is deriving an expression for Yt in terms of the ut’s. Because b0 = 0,
Equation (14.8) implies that Yt = b1Yt - 1 + ut. Substituting Yt - 1 = b1Yt - 2 + ut - 1 into this
expression yields Yt = b1(b1Yt - 2 + ut - 1) + ut = b21Yt - 2 + b1ut - 1 + ut. Continuing this
substitution another step yields Yt = b13Yt - 3 + b12ut - 2 + b1ut - 1 + ut, and continuing
indefinitely yields

∞

Yt = ut + b1ut - 1 + b21ut - 2 + b31ut - 3 + g = a b1i ut - i . (14.36)
i=0

Thus Yt is a weighted average of current and past ut’s. Because the ut’s are normally dis-
tributed and because the weighted average of normal random variables is normal (Section

2.4), Ys + 1 and Ys + 2 have a bivariate normal distribution. Recall from Section 2.4 that the
bivariate normal distribution is completely determined by the means of the two variables,

their variances, and their covariance. Thus, to show that Yt is stationary, we need to show

that the means, variances, and covariance of (Ys + 1, Ys + 2) do not depend on s. An extension
of the argument used below can be used to show that the distribution of

(Ys + 1, Ys + 2, c, Ys + T) does not depend on s.

The means and variances of Ys + 1 and Ys + 2 can be computed using Equation (14.36),

with the subscript s + 1 or s + 2 replacing t. First, because E(ut) = 0 for all t,
∞ ∞
E(Yt) = E( g i= 0bi1ut - i) = g i= 0b 1i E(ut - i) = 0, so the means of Ys + 1 and Ys + 2 are both

zero and in particular do not depend on s. Second, var(Yt) = var( g ∞ b1i ut - i) =
i=0