"Frontmatter". In: Analysis of Financial Time Series

330 VECTOR TIME SERIESThe VARMA model of the transformed series y t can be obtained as follows:Thus, the model for y t is[y1ty 2t]−Lx t = LΦx t−1 + La t − LΘa t−1= LΦL −1 Lx t−1 + La t − LΘL −1 La t−1= LΦL −1 (Lx t−1 ) + b t − LΘL −1 b t−1 .[ ][ ] [ ]1.0 0 y1,t−1 b1t= −0 0 y 2,t−1 b 2t[ ][ ]0.4 0 b1,t−1. (8.30)0 0 b 2,t−1From the prior model, we see that (a) y 1t and y 2t are uncoupled series with concurrentcorrelation equal to that between the shocks b 1t and b 2t ,(b)y 1t follows aunivariate ARIMA(0,1,1) model, and (c) y 2t is a white noise series (i.e., y 2t = b 2t ).In particular, the model in Eq. (8.30) shows that there is only a single unit root in thesystem. Consequently, the unit roots of x 1t and x 2t are introduced by the unit rootof y 1t .The phenomenon that both x 1t and x 2t are unit-root nonstationary, but there isonly a single unit root in the vector series, is referred to as co-integration in theeconometric and time series literature. Another way to define co-integration is tofocus on linear transformations of unit-root nonstationary series. For the simulatedexample of model (8.29), the transformation shows that the linear combination y 2t =0.5x 1t + x 2t does not have a unit root. Consequently, x 1t and x 2t are co-integrated if(a) both of them are unit-root nonstationary, and (b) they have a linear combinationthat is unit-root stationary.Generally speaking, for a k-dimensional unit-root nonstationary time series, cointegrationexists if there are less than k unit roots in the system. Let h be the numberof unit roots in the k-dimensional series x t . Co-integration exists if 0 < h < k,and the quantity k − h is called the number of co-integrating factors. Alternatively,the number of co-integrating factors is the number of different linear combinationsthat are unit-root stationary. The linear combinations are called the co-integratingvectors. For the prior simulated example, y 2t = (0.5, 1)x t so that (0.5, 1) ′ is a cointegratingvector for the system. For more discussions on co-integration and cointegrationtests, see Box and Tiao (1977), Engle and Granger (1987), Stock andWatson (1988), and Johansen (1989).The concept of co-integration is interesting and has attracted a lot of attention inthe literature. However, there are difficulties in testing for co-integration in a realapplication. The main source of difficulties is that co-integration tests overlook thescaling effects of the component series. Interested readers are referred to Cochrane(1988) and Tiao, Tsay, and Wang (1993) for further discussion.While I have some misgivings on the practical value of co-integration tests, theidea of co-integration is highly relevant in financial study. For example, consider thestock of Finnish Nokia Corporation. Its price on the Helsinki Stock Market mustmove in unison with the price of its American Depositary Receipts on the New York