Bootstrap union tests for unit roots in the presence of nonstationary volatility

We provide a joint treatment of three major issues that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not, and the possible presence of nonstationary volatility in the data. Harvey, Leybourne and Taylor (2010, Journal of Econometrics, forthcoming) propose decision rules based on a four-way union of rejections of QD and OLS detrended tests, both with and without allowing for a linear trend, to deal with the first two problems. However, in the presence of nonstationary volatility these test statistics have limit distributions which depend on the form of the volatility process, making tests based on the standard asymptotic critical values invalid. We construct bootstrap versions of the four-way union of rejections test, which, by employing the wild bootstrap, are shown to be asymptotically valid in the presence of nonstationary volatility. These bootstrap union tests therefore allow for a joint treatment of all three of the aforementioned problems.Unit root; local trend; initial condition; asymptotic power; union of rejections decision rule; nonstationary volatility; wild bootstrap

Testing against stochastic trend and seasonality in the presence of unattended breaks and unit roots

This paper considers the problem of testing against stochastic trend and seasonality in the presence of structural breaks and unit roots at frequencies other than those directly under test, which we term unattended breaks and unattended unit roots respectively. We show that under unattended breaks the true size of the Kwiatkowski et. al. (1992) [KPSS] test at frequency zero and the Canova and Hansen (1995) [CH] test at the seasonal frequencies fall well below the nominal level under the null with an associated, often very dramatic, loss of power under the alternative. We demonstrate that a simple modification of the statistics can recover the usual limiting distribution appropriate to the case where there are no breaks, provided unit roots do not exist at any of the unattended frequencies. Where unattended unit roots occur we show that the above statistics converge in probability to zero under the null. However, computing the KPSS and CH statistics after pre-filtering the data is simultaneously efficacious against both unattended breaks and unattended unit roots, in the sense that the statistics retain their usual pivotal limiting null distributions appropriate to the case where neither occurs. The case where breaks may potentially occur at all frequencies is also discussed. The practical relevance of the theoretical contribution of the paper is illustrated through a number of empirical examples.stationarity tests, structural breaks, pre-filtering, unattended unit roots

Testing for a change in persistence in the presence of non-stationary volatility

In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0) to I(1) behaviour or vice versa, of, inter alia, Kim (2000). We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical application to a variety of measures of U.S. price inflation data is provided.Persistence change; non-stationary volatility; wild bootstrap

Robust Inference on Seasonal Unit Roots via a Bootstrap Applied to OECD Macroeconomic Series

Recent experimental results presented in Burridge and Taylor (2001a,b, and 2003) show that, as usually implemented, the Hylleberg et al. (1990) seasonal unit root tests can be rather liberal, with true level often substantially higher than nominal level. This effect is due to the presence of any of three things: data-based lag selection in the implementation of the tests, and either or both periodic heteroscedasticity and serial correlation in the driving shocks. Burridge and Taylor (2003) demonstrate that under experimental conditions a carefully implemented bootstrap substantially corrects test level without loss of power. The present study applies their technique to a large number of publicly available series, and demonstrates conclusively that the bootstrap produces less liberal, and, given the experimental results cited above, more reliable inference. We report results for Sweden, the UK and the US, which are typical of the fifteen countries in our panel. Other results, the GAUSS code, and raw data are all available at: www.staff.city.ac.uk/p.burridge

Co-integration rank tests under conditional heteroskedasticity

In this paper we analyse the properties of the conventional Gaussian-based co-integrating rank tests of Johansen (1996) in the case where the vector of series under test is driven by possibly non-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 i.i.d. or stationary martingale difference innovations. We then propose wild bootstrap implementations of the co-integrating 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). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the re-sampled 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 also given.Co-integration; trace and maximum eigenvalue rank tests; conditional heteroskedasticity; IID bootstrap; wild bootstrap

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 nonstationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics of 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 remarkably well in practice.Cointegration; non-stationary volatility; trace and maximum eigenvalue tests; wild bootstrap

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

Many key macro-economic and ?nancial 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 identi?ed 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.co-integration; non-stationary volatility; trace and maximum eigenvalue tests; wild bootstrap

Bootstrap co-integration rank testing: the role of deterministic variables and initial values in the bootstrap recursion

In this paper we investigate the role of deterministic components and initial values in bootstrap likelihood ratio type tests of co-integration rank. A number of bootstrap procedures have been proposed in the recent literature some of which include estimated deterministic components and non-zero initial values in the bootstrap recursion while others do the opposite. To date, however, there has not been a study into the relative performance of these two alternative approaches. In this paper we fill this gap in the literature and consider the impact of these choices on both OLS and GLS de-trended tests, in the case of the latter proposing a new bootstrap algorithm as part of our analysis. Overall, for OLS de-trended tests our findings suggest that it is preferable to take the computationally simpler approach of not including estimated deterministic components in the bootstrap recursion and setting the initial values of the bootstrap recursion to zero. For GLS de-trended tests, we find that the approach of Trenkler (2009), who includes a restricted estimate of the deterministic component in the bootstrap recursion, can improve finite sample behaviour further.Co-integration; trace tests; i.i.d. bootstrap; OLS and GLS de-trending

Bootstrap Sequential Determination of the Co-integration Rank in VAR Models

Determining the co-integrating rank of a system of variables has become a fundamental aspect of applied research in macroeconomics and finance. It is wellknown that standard asymptotic likelihood ratio tests for co-integration rank of Johansen (1996) can be unreliable in small samples with empirical rejection frequencies often very much in excess of the nominal level. As a consequence, bootstrap versions of these tests have been developed. To be useful, however, sequential procedures for determining the co-integrating rank based on these bootstrap tests need to be consistent, in the sense that the probability of selecting a rank smaller than (equal to) the true co-integrating rank will converge to zero (one minus the marginal significance level), as the sample size diverges, for general I(1) processes. No such likelihood-based procedure is currently known to be available. In this paper we fill this gap in the literature by proposing a bootstrap sequential algorithm which we demonstrate delivers consistent cointegration rank estimation for general I(1) processes. Finite sample Monte Carlo simulations show the proposed procedure performs well in practice.co-integration; trace test; sequential rank determination; i.i.d. bootstrap; wild bootstrap