# Time-Series-Based Econometrics: Unit Roots and Co-Integrations

## Excerpt

The most vigorous development in econometrics in recent years had undoubtedly been the unit-root field, including error correction and co-integration. I am most grateful to the editors of Advanced Texts in Econometrics, Professors C. W. J. Granger and G. E. Mizon, for allowing me an opportunity to survey these development. The survey has naturally suggested a number of research topics, and the results of one of them are included in the book. It is assumed that readers are acquainted with (a) fundamentals of the algebra of liner vector spaces, (b) standard time-series analysis of stationary process including the linear prediction theory, and (c) standard asymptotic theory of inference used in econometric theory. However, no other mathematics or statistics are presupposed nor is mathematical rigour sought in the writing. In particular unit-root problems are explained from their most elementary starting-points. Graduate students at an advanced level should be able to understand the book. The statistical procedures are explained in detail, and the results of applications are emphasized. The applications that I have in mind are primarily to macro-economic time series rather than to financial series, the analyses of the two kinds of series often requiring different concepts and tools. My survey does not trace the historical sequence of developments, but rather selects the topics worth noting as at the time of writing (the second half of 1992 to the end of 1993).

The book consists of two parts, Part I deals with the univariate unit root, i.e. to see if a stochastic trend is present when each time series is analysed separately. It is summarized in the last part of Chapter 1. Part II discusses co-integration, i.e. the empirical investigation of long-run relationships among a number of time series.

Critics of unit-root tests deny even their motivation. I disagree. The question that the tests try to answer is whether or not the economy is capable of restoring the trend line of a variable after it is forced by shocks to deviate from the line. If the answer is affirmative (negative) the variable is trend (difference) stationary. What the trend line is, and to what extent we can answer the question, is examined in Part I. As in the case of other problems in econometrics one should be warned against optimism about the precision of results, but I cannot imagine that answers to the question can be irrelevant to macroeconomic modelling.

If unit roots are involved in a set of variables we can investigate the longrun relationships among the variables, which is a new branch of econometrics. Moreover, unit roots influence the appropriate selection of inference methods to be adopted in all econometric studies. The new econometrics differs from the old even at the level of a simple regression model in its theoretical aspects, if not in most computations. This is explained in Part II.

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