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Multiple time series regression with integrated processes. (English) Zbl 0599.62103

An invariance principle for multiple time series is formulated and proved as the starting point for investigation of integrated models. A multiple time series \(y_ t=Ay_{t-1}+u_ t\) is analyzed, where \(u_ t\) is a weakly stationary sequence.
The authors develop an asymptotic theory for sample moments of this integrated process and construct a test of \(H_ 0: A=I\). Then they examine the multiple regression equation \(y_ t=Ax_ t+u_ t\), where \(x_ t=x_{t-1}+v_ t\). The elements of the matrix A are unknown parameters and \((u_ t,v_ t)\) are joint innovations. A test of \(H_ 0: R vec A=r\) is proposed, where R and r are known. Asymptotical properties of this test are quite different from those of classical regression tests. The results are extended to models with fitted drift vectors.
The research is motivated by some evidence about the behaviour of macroeconomic time series.
Reviewer: J.Anděl

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles
91B84 Economic time series analysis
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