Fixed-b Analysis of LM Type Tests for a Shift in Mean

We analyze Lagrange Multiplier (LM) tests for a shift in mean of a univariate time series at an unknown date. We consider a class of LM statistics based on nonparametric kernel estimates of the long run variance and we develop a …xed-b asymptotic theory for the statistics. We provide results for the case of I(0) and I(1) errors. The …xed-b theory suggests that, for a given statistic, kernel, and signi…cance level, there exists a bandwidth such that the …xed-b asymptotic critical value is the same for both I(0) and I(1) errors. We compute these "robust" bandwidths for a selection of well known kernels. In the case of the supremum statistic, the robust bandwidth LM tests have good power that is monotonic and similar for all kernels. In the case of the mean and exponential mean statistics, power is non-monotonic when the robust bandwidths are used. Under the null hypothesis of no shift in mean, the Bartlett kernel supremum statistic has little, if any, over-rejection problems in …nite samples even when there is a moving average component with a negative coe¢ cient. In contrast the other supremum statistics tend to over-reject in this case. In practice we recommend the Bartlett kernel supremum LM statistic implemented with the robust bandwidth.