Testing for a Unit Root with Near-Integrated Volatility

This paper considers tests for a unit root when the innovations follow a near-integrated GARCH process. We compare the asymptotic properties of the likelihood ratio statistic with that of the least-squares based Dickey-Fuller statistic. We first use asymptotics where the GARCH variance process is stationary with fixed parameters, and then consider parameter sequences such that the GARCH process converges to a diffusion process. In the fixed-parameter case, the asymptotic local power gain of the likelihood ratio test is only marginal for realistic parameter values. However, under near-integrated parameter sequences the difference in power is more pronounced.

Identifying, Estimating and Testing Restricted Cointegrated Systems: An Overview

The notion of cointegration has lead to a renewed interest in the identification and estimation of structural relations among economic time series, a field to which Henri Theil has made many pioneering contributions. This paper reviews the different approaches that have been put forward in the literature for identifying cointegrating relationships and imposing (possibly over-identifying) restrictions on them. Next, various algorithms to obtain (approximate) maximum likelihood estimates and likelihood ratio statistics are reviewed, with an emphasis on so-called switching algorithms. The implementation of these algorithms is discussed and illustrated using an empirical example.

Testing for periodic integration

A periodic autoregressive time-series model assumes that the autoregressive parameters vary with the season. This model can also be represented by a multivariate model for the annual vector containing the seasonal observations. When this multivariate model contains one unit root, a time-series is said to be periodically integrated of order 1. In this paper we propose tests for such a single unit root. These tests for periodic integration are applied to a periodic model for the quarterly German consumption series

Adaptive Testing for Cointegration With Nonstationary Volatility

This article develops a class of adaptive cointegration tests for multivariate time series with nonstationary volatility. Persistent changes in the innovation variance matrix of a vector autoregressive model lead to size distortions in conventional cointegration tests, which may be resolved using the wild bootstrap, as shown in recent work by Cavaliere, Rahbek, and Taylor. We show that it also leads to the possibility of constructing tests with higher power, by taking the time-varying volatilities and correlations into account in the formulation of the likelihood function and the resulting likelihood ratio test statistic. We find that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric volatility matrix estimation does not lead to a loss of asymptotic local power relative to the case where the volatilities are observed. The asymptotic null distribution of the test is nonstandard and depends on the volatility process; we show that various bootstrap implementations may be used to conduct asymptotically valid inference. Monte Carlo simulations show that the resulting test has good size properties, and higher power than existing tests. Empirical analyses of the U.S. term structure of interest rates and purchasing power parity illustrate the applicability of the tests

Adaptive wild bootstrap tests for a unit root with nonstationary volatility

Recent research has emphasised that permanent changes in the innovation variance (caused by structural shifts or an integrated volatility process) lead to size distortions in conventional unit root tests. Cavaliere and Taylor (2008) show how these size distortions may be resolved using the wild bootstrap. In this paper, we first derive the asymptotic power envelope for the unit root testing problem when the nonstationary volatility process is known. Next, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric estimation of the volatility process leads to the same asymptotic power envelope. Implementation of the resulting test involves cross-validation and the wild bootstrap. A Monte Carlo experiment shows that the asymptotic results are reflected in finite sample properties, and an empirical analysis of real exchange rates illustrates the applicability of the proposed procedures

Limiting distributions for explosive PAR(1) time series with strongly mixing innovation

This work deals with the limiting distribution of the least squares estimators of the coefficients a r of an explosive periodic autoregressive of order 1 (PAR(1)) time series X r = a r X r--1 +u r when the innovation {u k } is strongly mixing. More precisely {a r } is a periodic sequence of real numbers with period P \textgreater{} 0 and such that P r=1 |a r | \textgreater{} 1. The time series {u r } is periodically distributed with the same period P and satisfies the strong mixing property, so the random variables u r can be correlated

Adaptive information-based methods for determining the co-integration rank in heteroskedastic VAR models

Standard methods, such as sequential procedures based on Johansen’s (pseudo-)likelihood ratio (PLR) test, for determining the co-integration rank of a vector autoregressive (VAR) system of variables integrated of order one can be significantly affected, even asymptotically, by unconditional heteroskedasticity (non-stationary volatility) in the data. Known solutions to this problem include wild bootstrap implementations of the PLR test or the use of an information criterion, such as the BIC, to select the co-integration rank. Although asymptotically valid in the presence of heteroskedasticity, these methods can display very low finite sample power under some patterns of non-stationary volatility. In particular, they do not exploit potential efficiency gains that could be realised in the presence of non-stationary volatility by using adaptive inference methods. Under the assumption of a known autoregressive lag length, Boswijk and Zu (2022) develop adaptive PLR test based methods using a non-parameteric estimate of the covariance matrix process. It is well-known, however, that selecting an incorrect lag length can significantly impact on the efficacy of both information criteria and bootstrap PLR tests to determine co-integration rank in finite samples. We show that adaptive information criteria-based approaches can be used to estimate the autoregressive lag order to use in connection with bootstrap adaptive PLR tests, or to jointly determine the co-integration rank and the VAR lag length and that in both cases they are weakly consistent for these parameters in the presence of non-stationary volatility provided standard conditions hold on the penalty term. Monte Carlo simulations are used to demonstrate the potential gains from using adaptive methods and an empirical application to the U.S. term structure is provided