  We study inference on the common stochastic trends in a non-stationary,
$N$-variate time series $y_{t}$, in the possible presence of heavy tails. We
propose a novel methodology which does not require any knowledge or estimation
of the tail index, or even knowledge as to whether certain moments (such as the
variance) exist or not, and develop an estimator of the number of stochastic
trends $m$ based on the eigenvalues of the sample second moment matrix of
$y_{t}$. We study the rates of such eigenvalues, showing that the first $m$
ones diverge, as the sample size $T$ passes to infinity, at a rate faster by
$O\left(T \right)$ than the remaining $N-m$ ones, 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
to whether the relevant eigenvalue belongs to the set of the first $m$
eigenvalues or not. We then construct a randomised statistic based on this,
using it as part of a sequential testing procedure, ensuring consistency of the
resulting estimator of $m$. We also discuss an estimator of the common trends
based on principal components and show that, up to a an invertible linear
transformation, such estimator is consistent in the sense that the estimation
error is of smaller order than the trend itself. Finally, we also consider the
case in which we relax the standard assumption of \textit{i.i.d.} innovations,
by allowing for heterogeneity of a very general form in the scale of the
innovations. A Monte Carlo study shows that the proposed estimator for $m$
performs particularly well, even in samples of small size. We complete the
paper by presenting four illustrative applications covering commodity prices,
interest rates data, long run PPP and cryptocurrency markets.
