  We revisit classical asymptotics when testing for a structural break in
linear regression models by obtaining the limit theory of residual-based and
Wald-type processes. First, we establish the Brownian bridge limiting
distribution of these test statistics. Second, we study the asymptotic
behaviour of the partial-sum processes in nonstationary (linear) time series
regression models. Although, the particular comparisons of these two different
modelling environments is done from the perspective of the partial-sum
processes, it emphasizes that the presence of nuisance parameters can change
the asymptotic behaviour of the functionals under consideration. Simulation
experiments verify size distortions when testing for a break in nonstationary
time series regressions which indicates that the Brownian bridge limit cannot
provide a suitable asymptotic approximation in this case. Further research is
required to establish the cause of size distortions under the null hypothesis
of parameter stability.
