Tests for cointegration with infinite variance errors

Abstract

This paper develops the asymptotic theory for residual-based tests and quasi-likelihood ratio tests for cointegration under the assumption of infinite variance errors. This article extends the results of Phillips and Ouliaris (1990) and Johansen (1988Johansen ( , 1991 which are derived under the assumption of square-integrable errors. Here the limit laws are expressed in terms of functionals of symmetric stable laws rather than Brownian motion. Critical values of the residual-based tests of Phillips and Ouliaris (1990) and likelihoodratio-based tests of Johansen (1991) are calculated and tabulated.We also investigate whether these tests are robust to infinite variance errors. We found that regardless of the index of stability , the residual-based tests are more robust to infinite variance errors than the likelihood-ratio-based tests.1998 Elsevier Science S.A. All rights reserved. JEL classification: C32

Unit Root Inference for Non-Stationary Linear Processes Driven by Infinite Variance Innovations

“…The asymptotic null distributions of conventional regression-based unit root statistics, such as the ordinary least squares (OLS) based statistics of Dickey and Fuller (1979), Said and Dickey (1987) and the semi-parametric statistic of Phillips (1987), di¤er from the …nite variance case when the innovations lie in the domain of attraction of a stable law, such that they have in…nite variance [IV]; see, in particular, Chan and Tran (1989), Phillips (1990), Samarakoon and Knight (2009), Rachev et al (1998) and Caner (1998). Indeed, in such cases these limiting null distributions are no longer pivotal, depending on the so-called tail index of the stable law and on the relative weights of the left and right tails of the stable distribution.…”

Purchasing power parity and uncovered interest parity: another look using stable law econometrics

“…The reciprocal of the published U.K. pound rate was utilised. Under the assumption of stable errors, we implement the Phillips-Perron (1988) procedures with Caner's (1998) critical values for the unit root tests in exchange rates and price indices. The estimated Phillips-Perron statistics along with Caner's critical values are reported in Table 5.1 and 5.2 for the levels and first differences of the exchange rates and price indices, respectively.…”

“…'' Stable parameters have been estiinated for stock returns by Fama (196S), Leitch and Paulson (197S), Arad (1980), McCulloch (1994), Buckle (1995), andManegna andStanley (1995); for interest rate movements by Roll (1970), andMcCulloch (1985); for foreign exchange rate changes by Bagshaw and Humpage (1987), So (1987a, b), Liu and Brorsen (1995), and Brousseau and Czarnecki (1993); for commodities price movements Caner (1998) derives the limit laws for the and Z, test statistics of Phillips and Ouliaris (1990) and the trace and maximum eigenvalue test statistics of Johansen (1988 under the assumption that the component processes are symmetric stable processes with identical stability indices. The limit laws consist of functionals of symmetric stable laws and, in the case of , involve the quadratic variation of a symmetric stable process.…”

“…The finding of the non-Gaussian stable errors and the unit root in those series provide the motivation for re-doing the cointegration tests for the PPP and UIP relationships. Under the assumption of stable non-Gaussian innovations, we implement the residual-based and multivariate likelihood-based cointegration tests of Caner (1998) In contrast, using residual-based cointegration tests for either PPP or UIP relationship in the data proves to be a disappointing exercise, since the null hypothesis of a unit root (i.e. no cointegration) cannot be rejected whether we assume normal errors or stable errors.…”

“…That is, it appears fi-om the evidence that a stable nonGaussian model may be more appropriate for these series in our data. Phillips-Perron unit root tests, along with the critical values in Caner (1998), implemented for determining the order of integration of those series generally cannot reject the null of a unit root. The finding of the non-Gaussian stable errors and the unit root in those series provide the motivation for re-doing the cointegration tests for the PPP and UIP relationships.…”