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We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the break point and its form is that of the generalized Cramer–von Mises distribution, with degrees of freedom depending on the number of break points. The performance of this modified test is shown, via some simulation experiments, to be comparable with that of the LBI test. An unconditional test, based on the assumption that there is a single break at an unknown point, is also examined. The use of the tests is illustrated with data on the flow of the Nile and US gross national product.

A copula de…nes the probability that the observations from two time series are both below a given quantile. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with di¤erent quantiles may point to changes in di¤erent parts of the copula, with the lower quantiles being of particular interest in …nancial applications concerned with risk. Tests located at the median provide an overall test of a changing relationship. The properties of various tests are compared and it is shown that they are still e¤ective if pre-…ltering is carried out to correct for changing volatility or, more generally, changing quantiles. Applying the tests to data on IBM and General Motors stock returns indicates that the relationship is not constant over time.

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This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modi cations of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér-von Mises distributions. One such modi cation is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modi ed statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level.The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, in ation, and exchange rate data series.

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We consider how unit-root and stationarity tests can be used to study the convergence of prices and rates of inflation. We show how the joint use of these tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whether convergence occurs in relative prices or just in rates of inflation. Special attention is paid to the issue of whether a mean should be extracted in carrying out tests in first differences and whether there is an advantage to adopting a (Dickey-Fuller) unit-root test based on deviations from the last observation. The asymptotic distribution of this last test statistic is given and Monte Carlo simulation experiments show that the test yields considerable power gains for highly persistent autoregressive processes with 'relatively large' initial conditions. The tests are applied to the monthly series of the consumer price index in the Italian regional capitals over the period 1970-2003.

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