It is well known that the conventional cumulative sum (CUSUM) test suffers from low power and large detection delay. In order to improve the power of the test, we propose two alternative statistics. The backward CUSUM detector considers the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector sequentially cumulates a triangular array of backwardly cumulated residuals. A multivariate invariance principle for partial sums of recursive residuals is given, and the limiting distributions of the test statistics are derived under local alternatives. In the retrospective context, the local power of the tests is shown to be substantially higher than that of the conventional CUSUM test if a break occurs in the middle or at the end of the sample. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is found to be much shorter than that of the conventional monitoring CUSUM procedure. Furthermore, we propose an estimator of the break date based on the backward CUSUM detector and show that in monitoring exercises this estimator tends to outperform the usual maximum likelihood estimator. Finally, an application of the methodology to COVID-19 data is presented.

In many nonlinear panel data models with fixed effects maximum likelihood estimators suffer from the incidental parameters problem, which often entails that point estimates are markedly biased. While the recent literature has mostly generated methods that yield a first-order bias reduction relative to maximum likelihood, we derive a first- and second-order bias correction of the profile likelihood based on “expected quantities” which differs from the corresponding correction based on “sample averages” derived in Dhaene and Sun (2021, Journal of Econometrics 220, 227–252). While consistency and asymptotic normality of our estimator are derived in a setting where both the number of individuals and the number of time periods grow to infinity, we illustrate in a simulation study that our second-order bias reduction indeed yields an estimator with substantially improved small sample properties relative to its first-order unbiased counterpart, especially when less than 10 time periods are available.

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

We develop theoretical finite-sample results concerning the size of wild bootstrap-based heteroskedasticity robust tests in linear regression models. In particular, these results provide an efficient diagnostic check, which can be used to weed out tests that are unreliable for a given testing problem in the sense that they overreject substantially. This allows us to assess the reliability of a large variety of wild bootstrap-based tests in an extensive numerical study.

We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient $\sigma $ is continuously differentiable in y, but with a jump discontinuity at some level y, say $y=0$. We first study sign-constrained kernel estimators of functions of the left and right limits of $\sigma $ at $0$. These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.