In this paper we extend fixed-<italic>b</italic> asymptotic theory to the nonparametric Phillips–Perron (PP) unit root tests. We show that the fixed-<italic>b</italic> limits depend on nuisance parameters in a complicated way. These nonpivotal limits provide an alternative theoretical explanation for the well-known...

This paper analyzes tests for a shift in the trend function of a time series at an unknown date based on ordinary least squares (OLS) estimates of the trend function. Inference about the trend parameters depends on the serial correlation structure of the data through the long-run variance (zero...

This paper studies the properties of naive block bootstrap tests that are scaled by zero frequency spectral density estimators (long-run variance estimators). The naive bootstrap is a bootstrap where the formula used in the bootstrap world to compute standard errors is the same as the formula...

In this paper, test statistics for detecting a break at an unknown date in the trend function of a dynamic univariate time series are proposed. The tests are based on the mean and exponential statistics of Andrews and Ploberger (1994, <italic>Econometrica</italic> 62, 1383–1414) and the supremum statistic of...

In this paper we develop a canonical state space representation of autoregressive moving average (ARMA) processes with unit roots with integer integration orders at arbitrary unit root frequencies. The developed representation utilizes a state process with a particularly simple dynamic...