Determining the rank of cointegration with infinite variance

We study the issue of determining the rank of cointegration, R, in an N-variate time series yt, allowing for the possible presence of heavy tails. Our methodology does not require any estimation of nuisance parameters such as the tail index, and indeed even knowledge as to whether certain moments (such as the variance) exist or not is not required. Our estimator of the rank is based on a sequence of tests on the eigenvalues of the sample second moment matrix of yt. We derive the rates of such eigenvalues, showing that these do depend on the tail index, but also that there exists a gap in rates between the first N - R and the remaining eigenvalues. The former ones, in particular, diverge at a rate which is faster than the latter ones by a factor T (where T denotes the sample size), irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according as the relevant eigenvalue belongs in the set of first N - R eigenvalues or not. We then construct a randomised statistic based on this, using it as part of a sequential testing procedure. The resulting estimator of R is consistent, in that it picks the true value R with probability 1 as the sample size passes to infinity.