Autoregressive conditional root model

In this paper we develop a time series model which allows long-term disequilibriums to have epochs of non-stationarity, giving the impression that long term relationships between economic variables have temporarily broken down, before they endogenously collapse back towards their long term relationship. This autoregressive root model is shown to be ergodic and covariance stationary under some rather general conditions. We study how this model can be estimated and tested, developing appropriate asymptotic theory for this task. Finally we apply the model to assess the purchasing power parity relationship.Cointegration; Equilibrium correction model; GARCH; Hidden Markov model; Likelihood; Regime switching; STAR model; Stochastic break; Stochastic unit root; Switching regression; Real Exchange Rate; PPP; Unit root hypothesis.

Likelihood Ratio Testing for Cointegration Ranks in I(2) Models.

This paper presents the likelihood ratio (LR) test for the number of cointegrating and multi-cointegrating relations in the I(2) vector autoregressive model. It is shown that the asymptotic distribution of the LR test for the (multi-) cointegration ranks is identical to the asymptotic distribution of the much applied test statistic based on the Two-Step procedure in Johansen (1995), Paruolo (1996), and Rahbek, Kongsted, and Jørgensen (1999). By construction the LR test statistic is smaller than the non-LR test statistic from the Two-Step procedure as the latter ignores some of the restrictions concerning the hypothesis of I(2), and application of the LR test may change rank selection in empirical work. Based on a study of existing empirical applications and related Monte Carlo simulations we conclude that the LR test has much better size properties when compared to the Two-Step based test. Overall, we propose to use of the LR test for rank determination in I(2) analysis as the Two-Step based statistic was developed as a feasible approximation to the then unobtainable LR test.vector autoregression; error correction model; cointegration; I(2); likelihood ratio test; Monte Carlo; reduced rank; rank testing

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

Poisson Autoregression

This paper considers geometric ergodicity and likelihood based inference for linear and nonlinear Poisson autoregressions. In the linear case the conditional mean is linked linearly to its past values as well as the observed values of the Poisson process. This also applies to the conditional variance, implying an interpretation as an integer valued GARCH process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and a nonlinear function of past observations. As a particular example an exponential autoregressive Poisson model for time series is considered. Under geometric ergodicity the maximum likelihood estimators of the parameters are shown to be asymptotically Gaussian in the linear model. In addition we provide a consistent estimator of the asymptotic covariance, which is used in the simulations and the analysis of some transaction data. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and non-perturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned.generalized linear models; non-canonical link function; count data; Poisson regression; likelihood; geometric ergodicity; integer GARCH; observation driven models; asymptotic theory

Purchasing power parity: A nonlinear multivariate perspective

The goal of this paper is to disentangle the respective contributions of the nominal exchange rate and the price differential to the adjustment towards the Purchasing Power Parity relation. To this end, we estimate a multivariate threshold vector equilibrium correction model, whose dynamics is consistent with the PPP in presence of trading costs. European data support the relevance of this model for Belgium, France and Italy, but this is not the case for the G7 data against the US Dollar. Furthermore, the adjustment in European countries seems to have been achieved only through nominal exchange rate changes.

An I(2) Cointegration Model With Piecewise Linear Trends: Likelihood Analysis And Application

This paper presents likelihood analysis of the I(2) cointegrated vector autoregression with piecewise linear deterministic terms. Limiting behavior of the maximum likelihood estimators are derived, which is used to further derive the limiting distribution of the likelihood ratio statistic for the cointegration ranks, extending the result for I(2) models with a linear trend in Nielsen and Rahbek (2007) and for I(1) models with piecewise linear trends in Johansen, Mosconi, and Nielsen (2000). The provided asymptotic theory extends also the results in Johansen, Juselius, Frydman, and Goldberg (2009) where asymptotic inference is discussed in detail for one of the cointegration parameters. To illustrate, an empirical analysis of US consumption, income and wealth, 1965 - 2008, is performed, emphasizing the importance of a change in nominal price trends after 1980.Cointegration, I(2); piecewise linear trends; likelihood analysis; US consumption

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

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