Multiple Time Series Regression with Integrated Processes

Abstract

This paper develops a general asymptotic theory of regression for processes which are integrated of order one. The theory includes vector autoregressions and multivariate regressions amongst integrated processes that are driven by innovation sequences which allow for a wide class of weak dependence and heterogeneity. The models studied cover cointegrated systems such as those advanced recently by Granger and Engle and quite general linear simultaneous equations systems with contemporaneous regressor error correlation and serially correlated errors. Problems of statistical testing in vector autoregressions and multivariate regressions with integrated processes are also studied. It is shown that the asympotic theory for conventional tests involves major departures from classical theory and raises new and important issues of the presence of nuisance parameters in the limiting distribution theory. Unlike many of the time series encountered in the natural sciences, economic time series frequently exhibit characteristics that are widely believed to be intrinsically nonstationary. For example, real macroeconomic variables such as output and consumption typically display a strong secular or growth component as well as cyclical behaviour; and many financial series like common stock prices behave in general as if they had no fixed mean. Recognizing these typical characteristics of economic time series, econometricians have devoted attention to the problem of describing and modelling nonstationarity. In the 1960's important contributions in the area were made by Granger, Hatanaka and their associates in Granger and Hatanaka (1964), Granger and Morgenstern (1963), Brillinger and Hatanaka (1968) and Hatanaka and Suzuki (1967). Later, following the influential work of Box and Jenkins (1976), attention shifted to the role of integrated processes in modelling economic time series. While undoubtedly restricting the class of nonstationary models, integrated processes of the ARIMA type have been found to produce highly satisfactory representations of many observed time series in economics. Quite recently, Nelson and Plosser (1982) have published a detailed empirical study of historical economic time series for the U.S.A. These authors provide some convincing evidence that macroeconomic time series normally thought to be stationary about a time trend are better described as integrated processes with drift. Amongst the latest research in this field have been the studies of cointegration by Granger and Weiss (1983) and Granger and Engle (1985). Two time series are said to be cointegrated if some linear combination of the series has a lower order of integration than the individual series. These authors argue that the notion of (steady state) equilibrium in economics implies the existence of such relationships. Thus, a classical economist's view of the interaction of money growth and price movements would require these series