Analysis of Financial Time Series, Third Edition

2.7 Unit-Root Nonstationarity

So far we have focused on return series that are stationary. In some studies, interest rates, foreign exchange rates, or the price series of an asset are of interest. These series tend to be nonstationary. For a price series, the nonstationarity is mainly due to the fact that there is no fixed level for the price. In the time series literature, such a nonstationary series is called unit-root nonstationary time series. The best known example of unit-root nonstationary time series is the random-walk model.

2.7.1 Random Walk

A time series {pt} is a random walk if it satisfies

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where p0 is a real number denoting the starting value of the process and {at} is a white noise series. If pt is the log price of a particular stock at date t, then p0 could be the log price of the stock at its initial public offering (IPO) (i.e., the logged IPO price). If at has a symmetric distribution around zero, then conditional on pt−1, pt has a 50–50 chance to go up or down, implying that pt would go up or down at random. If we treat the random-walk model as a special AR(1) model, then the coefficient of pt−1 is unity, which does not satisfy the weak stationarity condition of an AR(1) model. A random-walk series is, therefore, not weakly stationary, and we call it a unit-root nonstationary time series.

The random-walk model has widely been considered as a statistical model ...