"Frontmatter". In: Analysis of Financial Time Series

60 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSDifferencingA time series y t is said to be an ARIMA(p, 1, q) process if the change series c t =y t − y t−1 = (1 − B)y t follows a stationary and invertible ARMA(p, q) model. Infinance, price series are commonly believed to be nonstationary, but the log returnseries, r t = ln(p t ) − ln(p t−1 ), is stationary. In this case, the log price series isunit-root nonstationary and hence can be treated as an ARIMA process. The idea oftransforming a nonstationary series into a stationary one by considering its changeseries is called differencing in the time series literature. More formally, c t = y t −y t−1is referred to as the first differenced series of y t . In some scientific fields, a timeseries y t may contain multiple unit roots and needs to be differenced multiple timesto become stationary. For example, if both y t and its first differenced series c t =y t −y t−1 are unit-root nonstationary, but s t = c t −c t−1 = y t −2y t−1 +y t−2 is weaklystationary, then y t has double unit roots, and s t is the second differenced series ofy t . In addition, if s t follows an ARMA(p, q) model, then y t is an ARIMA(p, 2, q)process. For such a time series, if s t has a nonzero mean, then y t has a quadratic timefunction and the quadratic time coefficient is related to the mean of s t . The seasonallyadjusted series of U.S. quarterly gross domestic product implicit price deflator mighthave double unit roots. However, the mean of the second differenced series is notsignificantly different from zero; see Exercises of the chapter. Box, Jenkins, andReinsel (1994) discuss many properties of general ARIMA models.2.7.4 Unit-Root TestTo test whether the log price p t of an asset follows a random walk or a random walkwith a drift, we employ the modelsp t = φ 1 p t−1 + e t (2.35)p t = φ 0 + φ 1 p t−1 + e t , (2.36)where e t denotes the error term, and consider the null hypothesis H o : φ 1 = 1versusthe alternative hypothesis H a : φ 1 < 1. This is the well-known unit-root testingproblem; see Dickey and Fuller (1979). A convenient test statistic is the t ratio of theleast squares (LS) estimate of φ 1 under the null hypothesis. For Eq. (2.35), the LSmethod givesˆφ 1 =∑ Tt=1p t−1 p t∑ Tt=1p 2 t−1∑ Tt=1, ˆσ e 2 = (p t − ˆφ 1 p t−1 ) 2,T − 1where p 0 = 0andT is the sample size. The t ratio isDF ≡ t-ratio = ˆφ 1 − 1std( ˆφ 1 ) =∑ Tt=1p t−1 e tˆσ e√ ∑Tt=1p 2 t−1,