Let Xt be a linear process defined by
Xt = [sum ]k=0∞
ψkεt−k, where
{ψk, k ≥ 0} is a sequence of real
numbers and {εk, k = 0,±1,±2,...} is a
sequence of random variables. Two basic results, on the invariance
principle of the partial sum process of the Xt
converging to a standard Wiener process on [0,1], are presented
in this paper. In the first result, we assume that the innovations
εk are independent and identically distributed
random variables but do not restrict [sum ]k=0∞
|ψk| < ∞. We note that, for
the partial sum process of the Xt converging to a
standard Wiener process, the condition [sum ]k=0∞
|ψk| < ∞ or stronger conditions are
commonly used in previous research. The second result is for the situation
where the innovations εk form a martingale difference
sequence. For this result, the commonly used assumption of equal variance of
the innovations εk is weakened. We apply these general
results to unit root testing. It turns out that the limit distributions of the
Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS) test statistic still hold for the more general models under very
weak conditions.
A note on joint functional convergence of partial sum and maxima for linear processes
