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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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8.5 Unit-Root Nonstationarity and Cointegration

When modeling several unit-root nonstationary time series jointly, one may encounter the case of cointegration. Consider the bivariate ARMA(1,1) model

8.31 8.31

where the covariance matrix Inline of the shock Inline is positive definite. This is not a weakly stationary model because the two eigenvalues of the AR coefficient matrix are 0 and 1. Figure 8.10 shows the time plots of a simulated series of the model with 200 data points and Inline = Inline, whereas Figure 8.11 shows the sample autocorrelations of the two component series xit. It is easy to see that the two series have high autocorrelations and exhibit features of unit-root nonstationarity. The two marginal models of Inline are indeed unit-root nonstationary. Rewrite the model as

Inline

Premultiplying the above equation by ...

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