Giuseppe Cavaliere

Unit Root Tests under Time-Varying Variances

Abstract The paper provides a general framework for investigating the effects of permanent changes in the variance of the errors of an autoregressive process on unit root tests. Such a framework – which is based on a novel asymptotic theory for integrated and near integrated processes with heteroskedastic errors – allows to evaluate how the variance dynamics affect the size and the power function of unit root tests. Contrary to previous studies, it is shown that non-constant variances can both inflate and deflate the rejection frequency of the commonly used unit root tests, both under the null and under the alternative, with early negative and late positive variance changes having the strongest impact on size and power. It is also shown that shifts smoothed across the sample have smaller impacts than shifts occurring as a single abrupt jump, while periodic variances have a negligible effect even when a small number of cycles take place over a given sample. Finally, it is proved that the locally best invariant (LBI) test of a unit root against level stationarity is robust to heteroskedasticity of any form under the null hypothesis.

BOOTSTRAP UNIT ROOT TESTS FOR TIME SERIES WITH NONSTATIONARY VOLATILITY

The presence of permanent volatility shifts in key macroeconomic and financial variables in developed economies appears to be relatively common. Conventional unit root tests are unreliable in the presence of such behavior, having nonpivotal asymptotic null distributions. In this paper we propose a bootstrap approach to unit root testing that is valid in the presence of a wide class of permanent variance changes that includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. We make use of the so-called wild bootstrap principle, which preserves the heteroskedasticity present in the original shocks. Our proposed method does not require the practitioner to specify any parametric model for the volatility process. Numerical evidence suggests that the bootstrap tests perform well in finite samples against a range of nonstationary volatility processes.We thank two anonymous referees, Paulo Rodrigues, Peter Phillips, and seminar participants at the URCT conference held in Faro, Portugal, September 29 to October 1, 2005, for helpful comments on previous versions of this paper.

Testing for Co-Integration in Vector Autoregressions with Non-Stationary Volatility

Many key macro-economic and financial variables are characterised by permanent changes in unconditional volatility. In this paper we analyse vector autoregressions with non-stationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics computed as in Johansen (1988,1991) are potentially unreliable. In particular, their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co-integration tests, as we demonstrate numerically. A solution to the identified inference problem is provided by considering wild bootstrap-based implementations of the rank tests. These do not require the practitioner to specify a parametric model for volatility, nor to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. The bootstrap is shown to perform very well in practice.

COINTEGRATION RANK TESTING UNDER CONDITIONAL HETEROSKEDASTICITY

We analyze the properties of the conventional Gaussian-based cointegrating rank tests of Johansen (1996, Likelihood-Based Inference in Cointegrated Vector Autoregressive Models) in the case where the vector of series under test is driven by globally stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either independent and identically distributed (i.i.d.) or (strict and covariance) stationary martingale difference innovations. We then propose wild bootstrap implementations of the cointegrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first-order asymptotic null distributions of the rank statistics. We show that the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006, Econometrica 74, 1699–1714). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the resampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is given.

Bootstrap Determination of the Co-Integration Rank in Vector Autoregressive Models

This paper discusses a consistent bootstrap implementation of the likelihood ratio (LR) co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model that obtain under the reduced rank null hypothesis. A full asymptotic theory is provided that shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co-integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice.

LIMITED TIME SERIES WITH A UNIT ROOT

This paper develops an asymptotic theory for integrated and near-integrated time series whose range is constrained in some ways. Such a framework arises when integration and cointegration analysis are applied to persistent series which are bounded either by construction or because they are subject to control. The asymptotic properties of some commonly used integration tests are discussed; the bounded unit root distribution is introduced to describe the limiting distribution of the first-order autoregressive coefficient of a random walk under range constraints. The theoretical results show that the presence of such constraints can lead to drastically different asymptotics. Since deviations from the standard unit root theory are measured through noncentrality parameters, simple measures of the impact of range constraints on the asymptotic distributions are obtained. Finally, the proposed asymptotic framework provides an extremely adequate approximation of the finite sample properties of the unit root statistics under range constraints.

HETEROSKEDASTIC TIME SERIES WITH A UNIT ROOT

In this paper we provide a unified theory, and associated invariance principle, for the large-sample distributions of the Dickey–Fuller class of statistics when applied to unit root processes driven by innovations displaying nonstationary stochastic volatility of a very general form. These distributions are shown to depend on both the spot volatility and the integrated variation associated with the innovation process. We propose a partial solution (requiring any leverage effects to be asymptotically negligible) to the identified inference problem using a wild bootstrap–based approach. Results are initially presented in the context of martingale differences and are later generalized to allow for weak dependence. Monte Carlo evidence is also provided that suggests that our proposed bootstrap tests perform very well in finite samples in the presence of a range of innovation processes displaying nonstationary volatility and/or weak dependence.

STATIONARITY TESTS UNDER TIME-VARYING SECOND MOMENTS

In this paper we analyze the effects of a very general class of time-varying variances on well-known “stationarity†tests of the I(0) null hypothesis. Our setup allows, among other things, for both single and multiple breaks in variance, smooth transition variance breaks, and (piecewise-) linear trending variances. We derive representations for the limiting distributions of the test statistics under variance breaks in the errors of I(0), I(1), and near-I(1) data generating processes, demonstrating the dependence of these representations on the precise pattern followed by the variance processes. Monte Carlo methods are used to quantify the effects of fixed and smooth transition single breaks and trending variances on the size and power properties of the tests. Finally, bootstrap versions of the tests are proposed that provide a solution to the inference problem.We are grateful to Peter Phillips, a co-editor, and two anonymous referees whose comments on an earlier draft have led to a considerable improvement in the paper.

Bootstrap M Unit Root Tests

In this article we propose wild bootstrap implementations of the local generalized least squares (GLS) de-trended M and ADF unit root tests of Stock (1999), Ng and Perron (2001), and Elliott et al. (1996), respectively. The bootstrap statistics are shown to replicate the first-order asymptotic distributions of the original statistics, while numerical evidence suggests that the bootstrap tests perform well in small samples. A recolored version of our bootstrap is also proposed which can further improve upon the finite sample size properties of the procedure when the shocks are serially correlated, in particular ameliorating the significant under-size seen in the M tests against processes with autoregressive or moving average roots close to −1. The wild bootstrap is used because it has the desirable property of preserving in the resampled data the pattern of heteroskedasticity present in the original shocks, thereby allowing for cases where the series under test is driven by martingale difference innovations.

Time-Transformed Unit Root Tests for Models with Non-Stationary Volatility

Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non-stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near-) integrated time-series processes. Copyright 2007 The Authors