This paper provides a framework for testing for a unit root in an observed time series against some alternatives considered previously by Anderson (1948). Some new tests for the unit root null hypothesis for the errors affecting a classical regression model against the non-stationary (including explosive) alternative hypothesis are developed. The previous results of Sargan and Bhargava (1983) and the new test statistics are then applied to test the simple random walk and the random walk with a constant drift null hypotheses against stationary and non-stationary one-sided alternatives. In each case, the test statistic is simplified in order that it could be viewed as a von Neumann type ratio and the exact significance points are tabulated. Finally, the unit root null hypotheses are tested using U.S. data on the velocity of money and the Michigan PSID.
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