12.1 Stationary and integrated processes
Definition 12.1.1. A series is integrated of order \(d\), written \(I(d)\), if it must be differenced \(d\) times to become stationary (and invertible). A stationary series is \(I(0)\); a random walk is \(I(1)\), since \(\Delta Y_t = Y_t - Y_{t-1} = u_t\) is stationary.
Two models of a trending series are worth comparing. A trend-stationary series is \[ Y_t = \alpha + \delta t + u_t, \qquad u_t \text { stationary}, \] and a difference-stationary series, the random walk with drift, is \[ Y_t = \delta + Y_{t-1} + u_t, \qquad \text {so that} \qquad Y_t = Y_0 + \delta t + \sum _{s=1}^tu_s . \] Both grow on average by \(\delta \) a period. The difference is in the disturbance: in the first it is stationary; in the second it is the accumulated sum \(\sum u_s\), a stochastic trend.
Theorem 12.1.2. Let \(Y_t = \delta + Y_{t-1} + u_t\) with \(Y_0\) fixed and \(u_t\) independent with mean zero and variance \(\sigma ^2\). Then
- (i)
- \(E(Y_t) = Y_0 + \delta t\) and \(\Var (Y_t) = t\sigma ^2\);
- (ii)
- \(\Corr (Y_t, Y_{t+s}) = \sqrt {t/(t + s)}\);
- (iii)
- a shock \(u_t\) raises every later value \(Y_{t+s}\) by the same amount, whereas in the trend-stationary model with AR(1) disturbances \(u_t = \phi u_{t-1} + \varepsilon _t\) its effect on \(Y_{t+s}\) is \(\phi ^s\), which dies out;
- (iv)
- the variance of the error in forecasting \(Y_{t+s}\) from time \(t\) is \(s\sigma ^2\), which grows without bound, while for the trend-stationary model it tends to the finite \(\Var (u_t)\).
Proof. (i) \(Y_t - Y_0 - \delta t = \sum _{j=1}^tu_j\), a sum of \(t\) uncorrelated terms. (ii) \(\Cov (Y_t, Y_{t+s}) = \Cov \bigl (\sum _{j\leq t}u_j, \sum _{j\leq t+s}u_j\bigr ) = t\sigma ^2\), the variance of the common terms, so the correlation is \(t\sigma ^2/\sqrt {t\sigma ^2(t + s)\sigma ^2}\). (iii) \(Y_{t+s}\) contains \(u_t\) with coefficient one; in the other model \(u_{t+s} = \phi ^su_t + \dots \), as in Section 4.5. (iv) The forecast error is \(u_{t+1} + \dots + u_{t+s}\); for the trend-stationary model the error variance is that of Theorem 11.5.1, bounded by \(\sigma _u^2\). □
Part (ii) shows why a random walk looks so smooth: with \(t = 100\) and \(s = 10\) the correlation is \(0.95\), and the sample autocorrelations of a random walk decline very slowly, which is the classic visual symptom of a unit root. Part (iii) is the economic content of the distinction. If real output is difference-stationary, recessions permanently lower its path; if it is trend-stationary, output returns to the old trend line. The debate started by Nelson and Plosser (1982), who could not reject a unit root in most US macroeconomic series, is about exactly this.
The two models also require different treatment. Differencing a trend-stationary series gives \(\Delta Y_t = \delta + \Delta u_t\), whose moving average part has a unit root: it is over-differenced. Detrending a random walk, by regressing on time, leaves residuals that are still non-stationary, and the regression on time is itself spurious. Neither transformation is harmless, so one must first decide which model fits.
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