We show that the cumulated sum of squares statistic has a standard Brownian bridge–type asymptotic distribution in nonlinear regression models with (possibly) nonstationary regressors. This contrasts with cumulated sum statistics which have been previously studied and whose asymptotic distribution has been shown to depend on the functional form and the stochastic properties, such as persistence and stationarity, of the regressors. A recursive version of the test is also considered. A local power analysis is provided, and through simulations, we show that the test has good size and power properties across a variety of situations.

This study considers the estimation of spatial autoregressive models with censored dependent variables, where the spatial autocorrelation exists within the uncensored latent dependent variables. The estimator proposed in this paper is semiparametric, in the sense that the error distribution is not parametrically specified and can be heteroskedastic. Under a median restriction, we show that the proposed estimator is consistent and asymptotically normally distributed. As an empirical illustration, we investigate the determinants of the risk of assault and other violent crimes including injury in the Tokyo metropolitan area.

Long-run restrictions are a very popular method for identifying structural vector autoregressions, but they suffer from weak identification when the data is very persistent, i.e., when the highest autoregressive roots are near unity. Near unit roots introduce additional nuisance parameters and make standard weak-instrument-robust methods of inference inapplicable. We develop a method of inference that is robust to both weak identification and strong persistence. The method is based on a combination of the Anderson-Rubin test with instruments derived by filtering potentially nonstationary variables to make them near stationary using the IVX instrumentation method of Magdalinos and Phillips (2009). We apply our method to obtain robust confidence bands on impulse responses in two leading applications in the literature.

This article considers the problem of testing for an explosive bubble in financial data in the presence of time-varying volatility. We propose a sign-based variant of the Phillips, Shi, and Yu (2015, International Economic Review 56, 1043–1077) test. Unlike the original test, the sign-based test does not require bootstrap-type methods to control size in the presence of time-varying volatility. Under a locally explosive alternative, the sign-based test delivers higher power than the original test for many time-varying volatility and bubble specifications. However, since the original test can still outperform the sign-based one for some specifications, we also propose a union of rejections procedure that combines the original and sign-based tests, employing a wild bootstrap to control size. This is shown to capture most of the power available from the better performing of the two tests. We also show how a sign-based statistic can be used to date the bubble start and end points. An empirical illustration using Bitcoin price data is provided.

This paper derives the limit distribution of the rescaled sum of the reciprocal of the positive part of a random walk with continuously distributed innovations, and of the rescaled sum of the reciprocal of the absolute value of a random walk with continuously distributed innovations. It also considers this statistic for the case of a simple random walk, and shows that the limit distribution is different for this case.