Asymptotic Distributions of the Least-Squares Estimators and Test Statistics in the Near Unit Root Model with Non-Zero Initial Value and Local Drift and Trend

Seiji Nabeya · Bent E. Sørensen

26 citations0/0 files storedEconometric Theory

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Abstract

This paper considers the distribution of the Dickey-Fuller test in a model with non-zero initial value and drift and trend. We show how stochastic integral representations for the limiting distribution can be derived either from the local to unity approach with local drift and trend or from the continuous record asymptotic results of Sørensen [29]. We also show how the stochastic integral representations can be utilized as the basis for finding the corresponding characteristic functions via the Fredholm approach of Nabeya and Tanaka [16,17], This “link” between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic function. Using the same methods, we also find the characteristic function for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the various tests on the initial value of the time series.

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References

1979Distribution of the Estimators for Autoregressive Time Series with a Unit Root 2005Stochastic Integration and Differential Equations 1987Time Series Regression with a Unit Root 1990Inference in Linear Time Series Models with some Unit Roots 1990Stochastic Integration and Differential Equations 1987Towards a Unified Asymptotic Theory for Autoregression 1992LM TESTS FOR A UNIT ROOT IN THE PRESENCE OF DETERMINISTIC TRENDS*1951Multiple Wiener Integral

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