Unit root tests and dramatic shifts with infinite variance processes

Abstract

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.

Keywords:

• unit root
• stable processes
• partial sums
• limit distributions
• empirical size and power

JEL Classification :

• C31

Acknowledgements

Financial support under grant SFRH/BD/814/2000 from the Fundação para a Ciência e Tecnologia and Fundo Social Europeu is gratefully acknowledged. This paper is based on the fourth chapter of my Ph.D. thesis, Department of Economics, The Pennsylvania State University, USA. The author is grateful to Herman J. Bierens for his useful comments.

Notes

Phillips 29 does not provide the entire proof for this result. His main finding is

as . But because it follows easily that See [29, p. 46], for a definition of .

For a matter of comparison, we also report the standard critical values (α=2). The values in the bottom are those given in Table 10.A.2 of Fuller 12. With T=500, is approximated by , where and u _j is i.i.d. stable(α), is approximated by and by .

The model considered by Cavaliere is basically the same as in Kim et al. 18. Despite using a different statistic (KPSS, in Cavaliere's case), the conclusions are about the same.

For a fixed , each entry represents the percentage of values of the test statistic that are smaller than the asymptotic 5% critical value: , where X is the random variable that follows the empirical distribution.

Each value represents , where X is the random variable that follows the empirical distribution.

Here, dlmratio is the Mexico Dollarization ratio, oilprham is the nominal crude oil price index, irqoilpd is the Iraq crude oil production, and umcsent is the consumer sentiment index. * means significant at 1%, ** at 5%, and * * * at 10%.