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.

A Rescaled Range Statistics Approach to Unit Root Tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the sample path to lie within a closed interval is considered. To discuss this inferential problem, the concept of nearly-bounded integrated process is introduced, thus allowing to define formally the concept of boundary conditions within I(1) processes. When used to detect unknown boundaries, standard unit root tests do not maintain the usual power properties and new methods need developing. Therefore a new class of tests, which are based on the rescaled range of the process, are introduced. The limiting distribution of the proposed tests can be expressed in terms of the distribution of the range of particular Brownian functionals, while the power properties are obtained through the derivation of the limiting Brownian functional of a I(1) process with boundary conditions, which is done by referring to a new invariance principles for nonstationary time series with limited sample paths. Both theoretical and simulation exercises show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions.

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

Unit root tests under time-varyng variances

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 under permanent variance shifts, the conventional critical values can lead both to oversized and undersized tests. The paper concludes by showing that the power function of the unit root tests is affected by non-constant variances as well

Bootstrap determination of the co-integration rank in VAR 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 VAR model which obtain under the reduced rank null hypothesis. A full asymptotic theory is provided which 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.Bootstrap; Co-integration; Trace statistic; Rank determination Cointegrazione; Statistica “traccia”; determinazione del rango

Determining the number of cointegrating relations under rank constraints

This paper proposes likelihood-based procedures for determining the number of cointegrating vectors in the presence of constraints on the cointegration rank. The tests can be applied when a priori information suggests a lower bound on the number of common stochastic trends in the system. The likelihood ratio trace and and lambda max tests are obtained as special cases of the present setup. The tests are easy to implement and have comparable asymptotic power with respect to the trace test; it is also shown that the constrained tests are more powerful for some local alternatives.Cointegration rank, Likelihood ratio, Trace test, Asymptotic power

Wild bootstrap of the mean in the infinite variance case

It is well known that the standard i.i.d. bootstrap of the mean is inconsistent in a location model with infinite variance (?-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the i.i.d. bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986, Ann.Stat.) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the i.i.d. bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower confidence sets than other consistent methodsBootstrap, distribuzioni stabili, misure di probabilità stocastiche, convergenza debole Bootstrap, stable distributions, random probability measures, weak convergence

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

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