Unit Root Stochastic Process

This model resembles the Markov first-order autoregressive model that we discussed in the chapter on autocorrelation. If p = 1, (21.4.1) becomes a RWM (without drift). If p is in fact 1, we face what is known as the unit root problem, that is, a situation of nonstationarity; we already know that in this case the variance of Yt is not stationary. The name unit root is due to the fact that p = 1.11 Thus the terms nonstationarity, random walk, and unit root can be treated as synonymous.

If, however, |p| < 1, that is if the absolute value of p is less than one, then it can be shown that the time series Yt is stationary in the sense we have defined it.12

In practice, then, it is important to find out if a time series possesses a unit root.13 In Section 21.9 we will discuss several tests of unit root, that is, several tests of stationarity. In that section we will also determine whether the time series depicted in Figures 21.1 and 21.2 are stationary. Perhaps the reader might suspect that they are not. But we shall see.


The distinction between stationary and nonstationary stochastic processes (or time series) has a crucial bearing on whether the trend (the slow longrun evolution of the time series under consideration) observed in the constructed time series in Figures 21.3 and 21.4 or in the actual economic time series of Figures 21.1 and 21.2 is deterministic or stochastic. Broadly speaking, if the trend in a time series is completely predictable and not variable, we call it a deterministic trend, whereas if it is not predictable, we call it a stochastic trend. To make the definition more formal, consider the following model of the time series Yt .

"A technical point: If p = 1, we can write (21.4.1) as Yt — Yt-i = ut. Now using the lag operator L so that LYt = Yt—1, L2Yt = Yt—2, and so on, we can write (21.4.1) as (1 — L)Yt = ut. The term unit root refers to the root of the polynomial in the lag operator. If you set (1 — L) = 0, we obtain, L = 1, hence the name unit root.

12If in (21.4.1) it is assumed that the initial value of Y ( = Y0) is zero, |p| < 1, and ut is white noise and distributed normally with zero mean and unit variance, then it follows that E(Yt) = 0 and var (Yt) = 1/(1 — p2). Since both these are constants, by the definition of weak stationarity, Yt is stationary. On the other hand, as we saw before, if p = 1, Yt is a random walk or non-stationary.

13A time series may contain more than one unit root. But we will discuss this situation later in the chapter.


where ut is a white noise error term and where t is time measured chronologically. Now we have the following possibilities:

Pure random walk: If in (21.5.1) P = 0, fa = 0, fa = 1, we get

which is nothing but a RWM without drift and is therefore nonstationary. But note that, if we write (21.5.2) as

it becomes stationary, as noted before. Hence, a RWM without drift is a difference stationary process (DSP).

Random walk with drift: If in (21.5.1) fa = 0, = 0, fa = 1, we get

which is a random walk with drift and is therefore nonstationary. If we write it as

this means Yt will exhibit a positive > 0) or negative < 0) trend (see Figure 21.4). Such a trend is called a stochastic trend. Equation (21.5.3a) is a DSP process because the nonstationarity in Yt can be eliminated by taking first differences of the time series.

Deterministic trend: If in (21.5.1), fa = 0, fa = 0, p3 = 0, we obtain

which is called a trend stationary process (TSP). Although the mean of Yt is fa + fat, which is not constant, its variance (= a2) is. Once the values of fa and fa are known, the mean can be forecast perfectly. Therefore, if we subtract the mean of Yt from Yt, the resulting series will be stationary, hence the name trend stationary. This procedure of removing the (deterministic) trend is called detrending.

Random walk with drift and deterministic trend: If in (21.5.1), fa = 0, fa = 0, fa = 1, we obtain:

we have a random walk with drift and a deterministic trend, which can be seen if we write this equation as

which means that Yt is nonstationary.


FIGURE 21.5 Deterministic versus stochastic trend.

FIGURE 21.5 Deterministic versus stochastic trend.

Deterministic trend with stationary AR(1) component: If in (21.5.1)

which is stationary around the deterministic trend.

To see the difference between stochastic and deterministic trends, consider Figure 21.5.14 The series named stochastic in this figure is generated by an RWM: Yt = 0.5 + Yt-1 + ut, where 500 values of ut were generated from a standard normal distribution and where the initial value of Y was set at 1. The series named deterministic is generated as follows: Yt = 0.5t + ut, where ut were generated as above and where t is time measured chronologically.

As you can see from Figure 21.5, in the case of the deterministic trend, the deviations from the trend line (which represents nonstationary mean) are purely random and they die out quickly; they do not contribute to the long-run development of the time series, which is determined by the trend component 0.5t. In the case of the stochastic trend, on the other hand, the random component ut affects the long-run course of the series Yt.

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