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

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

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

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

Determining the Cointegration Rank in Heteroskedastic VAR Models of Unknown Order

We investigate the asymptotic and finite sample properties of a number of methods for estimating the cointegration rank in integrated vector autoregressive systems of unknown autoregressive order driven by heteroskedastic shocks. We allow for both conditional and unconditional heteroskedasticity of a very general form. We establish the conditions required on the penalty functions such that standard information criterion-based methods, such as the Bayesian information criterion [BIC], when employed either sequentially or jointly, can be used to consistently estimate both the cointegration rank and the autoregressive lag order. In doing so we also correct errors which appear in the proofs provided for the consistency of information-based estimators in the homoskedastic case by Aznar and Salvador (2002). We also extend the corpus of available large sample theory for the conventional sequential approach of Johansen (1995) and the associated wild bootstrap implementation thereof of Cavaliere, Rahbek and Taylor (2014) to the case where the lag order is unknown. In particular, we show that these methods remain valid under heteroskedasticity and an unknown lag length provided the lag length is first chosen by a consistent method, again such as the BIC. The relative finite sample properties of the different methods discussed are investigated in a Monte Carlo simulation study. The two best performing methods in this study are a wild bootstrap implementation of the Johansen (1995) procedure implemented with BIC selection of the lag length and joint IC approach (cf. Phillips, 1996) which uses the BIC to jointly select the lag order and the cointegration rank

Testing and Inference in Nonlinear Cointegrating Vector Error Correction Models

We analyze estimators and tests for a general class of vector error correction models that allows for asymmetric and nonlinear error correction. For a given number of cointegration relationships, general hypothesis testing is considered, where testing for linearity is of particular interest. Under the null of linearity, parameters of nonlinear components vanish, leading to a nonstandard testing problem. We apply so-called sup-tests to resolve this issue, which requires development of new(uniform) functional central limit theory and results for convergence of stochastic integrals. We provide a full asymptotic theory for estimators and test statistics. The derived asymptotic results prove to be nonstandard compared to results found elsewhere in the literature due to the impact of the estimated cointegration relations. This complicates implementation of tests motivating the introduction of bootstrap versions that are simple to compute. A simulation study shows that the finite-sample properties of the bootstrapped tests are satisfactory with good size and power properties for reasonable sample sizes

A primer on bootstrap testing of hypotheses in time series models: with an application to double autoregressive models

This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this recordIn this article, we discuss the bootstrap as a tool for statistical inference in econometric time series models. Importantly, in the context of testing, properties of the bootstrap under the null (size) as well as under the alternative (power) are discussed. Although properties under the alternative are crucial to ensure the consistency of bootstrap-based tests, it is often the case in the literature that only validity under the null is discussed. We provide new results on bootstrap inference for the class of double-autoregressive (DAR) models. In addition, we review key examples from the bootstrap time series literature in order to emphasize the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics, while also providing an up-to-date review of existing approaches. DAR models are particularly interesting for bootstrap inference: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, even second-order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing nonstationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of the bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid. That is, it is able to replicate, under the null hypothesis, the correct limiting distribution. Importantly, we also show that the behavior of this bootstrap under the alternative hypothesis may be more involved because of possible lack of finite second-order moments of the bootstrap innovations. This feature makes for some parameter configurations, the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this possible drawback can be fixed by using a novel bootstrap in this framework. For this “hybrid bootstrap,” the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from unrestricted residuals. We show that the hybrid bootstrap mimics the correct asymptotic null distribution, irrespective of the null being true or false. Monte Carlo simulations illustrate the behavior of both the restricted and the hybrid bootstrap, and we find that both perform very well even for small sample sizes.Danish Council for Independent ResearchUniversity of Bologn