Stationary Time Series

A time series is a sequence of random variables {y,}, t = 0, ± 1, ± 2,. . . . We assume Ey, = 0 for every t. (If Ey, Ф 0, we must subtract the mean before we subject it to the analysis of this chapter.) We say a sequence (y,) is strictly stationary if the joint distribution of any finite subset ytl, yh,. . . , yf, de­pends only on l2 — tx, t3 — tu. . . , tK— f, and not on r,. We say a sequence is weakly stationary if Ey, ys depends only on |t — $1 and not on t. If a process is strictly stationary and if the autocovariances exist, the process is weakly sta­tionary.

In Section 5.2 through 5.4 we shall be concerned only with strictly station­ary time series. The distributed-lag models discussed in Section 5.6 are gener­

ally not stationary in either sense. Time series with trends are not stationary, and economic time series often exhibit trends. However, this fact does not diminish the usefulness of Section 5.2 through 5.4 because a time series may be analyzed after a trend is removed. A trend may be removed either by direct subtraction or by differencing. The latter means considering first differences [y, — yt_i), second differences {(y, — y,_i) — (y,_, — y,_2)}> and so forth.

There are three fundamental ways to analyze a stationary time series. First, we can specify a model for it, such as an autoregressive model (which we shall study in Section 5.2) or a combined autoregressive, moving-average model (which we shall study in Section 5.3). Second, we can examine autocovar­iances Eytyt+h, h = 0, 1, 2,. . . . Third, we can examine the Fourier trans­form of autocovariances called spectral density. In Sections 5.1.2 and 5.1.3 we shall study autocovariances and spectral density.

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