Cointegration Rank Research Articles

Local-Explosive Approximations to Null Distributions of the Johansen Cointegration Test, with an Application to Cyclical Concordance in the Euro Area

This paper considers approximating the nite sample null-distribution of a test statistic as its asymptotic distribution under a local alternative. We focus on the Likelihood Ratio test for the rank of cointegration and use nonlinearities that represent some nite sample distributional features. Reliable approximations are obtained using a class of locally explosive models. An empirical evaluation of the concordance of European business cycles through cointegration shows that some standard corrections lead to underestimating the number of cointegrating relations and induce volatile results.

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process Xt to be fractional of order d and cofractional of order d b; that is, there exist vectorsfor which � 0 Xt is fractional of order d b: The parameters d and b satisfy either db � 1=2, d = b � 1=2, or d = d0 � b � 1=2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1=2� bdd1 for any d1� d0. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator ofis asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also …nd the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

A bootstrap algorithm for testing cointegration rank in VAR models in the presence of stationary variables

In this paper, a bootstrap algorithm for a reduced rank vector autoregressive (VAR) model which also includes stationary regressors, is analyzed. It is shown that the bootstrap distribution for estimating the rank converges to the distribution derived from the usual asymptotic framework. Because the asymptotic distribution will typically depend on unknown parameters, bootstrap distributions are of considerable interest in this context. The result of an application and some Monte Carlo experiments are also presented.

An I(2) cointegration model with piecewise linear trends

Summary This paper presents likelihood analysis of the I(2) cointegrated vector autoregression which allows for piecewise linear deterministic terms. Limiting behaviour 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 Nielsen and Rahbek. The provided asymptotic theory extends also the results in Johansen et al. where asymptotic inference is discussed in detail for one of the cointegration parameters. 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.

Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach

This paper offers an evaluation of the output contribution of infrastructure. Drawing from a large data set of infrastructure stocks covering 88 countries and spanning the years 1960-2000, and using a panel time-series approach, the paper estimates a long-run aggregate production function relating GDP to human capital, physical capital, and a synthetic measure of infrastructure given by the first principal component of infrastructure endowments in transport, power and telecommunications. Tests of the cointegration rank allowing it to vary across countries reveal a common rank with a single cointegrating vector, which we interpret as the long-run production function. Estimation of its parameters is performed using the pooled mean group (PMG) estimator, which allows for unrestricted short-run parameter heterogeneity across countries while imposing the (testable) restriction of long-run parameter homogeneity. The long-run elasticity of output with respect to the synthetic infrastructure index ranges between 0.07 and 0.10. The estimates are highly significant, both statistically and economically, and robust to alternative dynamic specifications and infrastructure measures. There is little evidence of long-run parameter heterogeneity across countries, whether heterogeneity is unconditional, or conditional on their level of development, population size, or infrastructure endowments.

Do Long‐Run Theory Restrictions Help in Forecasting?

ABSTRACTDo long‐run equilibrium relations suggested by economic theory help to improve the forecasting performance of a cointegrated vector error correction model (VECM)? In this paper we try to answer this question in the context of a two‐country model developed for the Canadian and US economies. We compare the forecasting performance of the exactly identified cointegrated VECMs to the performance of the over‐identified VECMs with the long‐run theory restrictions imposed. We allow for model uncertainty and conduct this comparison for every possible combination of the cointegration ranks of the Canadian and US models. We show that the over‐identified structural cointegrated models generally outperform the exactly identified models in forecasting Canadian macroeconomic variables. We also show that the pooled forecasts generated from the over‐identified models beat most of the individual exactly identified and over‐identified models as well as the VARs in levels and in differences. Copyright © 2011 John Wiley & Sons, Ltd.

Panel Cointegration Rank Testing with Cross-Section Dependence

In this paper, we propose a test statistic to determine the cointegration rank of VAR processes in panel data allowing for cross-section dependence among the time series in the panel data. The cross-section dependence is accounted for through the specification of an approximate common factor model, which covers situations where there is cointegration among the cross-section dimension. Finite sample performance is investigated via a Monte Carlo experiment. We show that in some cases not accounting for common factors when they are present can lead to overestimating the cointegrating rank. We apply our proposed tests to two empirical applications and find strong evidence for panel cointegration once common factors are accounted for.

Tests for cointegration rank and the initial condition

Many economic events involve initial observations that substantially deviate from long-run steady state. Such initial conditions are known to affect the power of univariate unit root tests diversely, whereas their impact on multivariate tests is largely unknown. This paper investigates the impact of the initial condition on the power of tests for cointegration rank, such as Johansen’s widely used likelihood ratio test, tests with prior adjustment for deterministic terms, and a test based on the eigenvalues of the companion matrix. We find that the power of the likelihood ratio test is increasing in the magnitude of the initial condition, whereas the power of the other tests is generally decreasing. We exploit these findings in an application to price convergence.

Prepivoting Algorithm for Testing and Determining Cointegration Rank in VAR-Model

Prepivoting Algorithm for Testing and Determining Cointegration Rank in VAR-Model

Cointegration rank switching model: an application to forecasting interest rates

This paper proposes a new forecasting method in which the cointegration rank switches at unknown times. In this method, time series observations are divided into several segments, and a cointegrated vector autoregressive model is fitted to each segment. The goodness of fit of the global model, consisting of local models with different cointegration ranks, is evaluated using the information criterion (IC). The division that minimizes the IC defines the best model. The results of an empirical application to the US term structure of interest rates and a Monte Carlo simulation suggest the efficacy as well as the limitations of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

Residual-based block bootstrap for cointegration testing

We propose a new testing procedure to determine the rank of cointegration. This new method is based on the nonparametric resampling procedure, so-called Residual-Based Block Bootstrap (RBB), which is developed by Paparoditis and Politis (2003) in the context of unit root testing. Through Monte Carlo experiments we show that, in small samples, the RBB cointegration test has good power properties in relation to the other two well-known tests for cointegration, such as the Augmented Dickey–Fuller (ADF), applied to the residual of a cointegrating regression, and the Johansen's maximum eigenvalue tests. Likewise, this article looks at the influence played by the correlation of the ‘X’ variables with the errors of the cointegrating regression on the size and power properties of the above cointegration tests. In particular, we show that, when this correlation decreases, the RBB test for cointegration is the most powerful one.

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b; that is, there exist vectors β for which β′X_{t} is fractional of order d-b. The parameters d and b satisfy either d≥b≥1/2, d=b≥1/2, or d=d_{0}≥b≥1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2≤b≤d≤d_{1} for any d_{1}≥d_{0}. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

The Making of "Estimation of Common Long-Memory Components in Cointegrated Systems"

The eighties were very good years for music as well as econometrics. Intime-series econometrics, the first half of that decade was dominated by research on unit roots whilecointegration was the queen of the second half. Estimation and testing of a cointegrated system werethe key questions to answer. When I started my dissertation at the end of the eighties, under the supervision of CliveGranger and Robert Engle, you could sense that everyone was of the opinion that the testing problemof the cointegration rank had been solved by Johansen (1988). Johansen applied reduced rankregression techniques to the following error correction model (ECM) (Granger’s old notation is used tokeep the spirit of the eighties) and obtained the corresponding asymptotics. Nevertheless, estimation was still under close scrutiny,and new estimators appeared on a regular basis in the literature. It was also the case howeverthat different methods could lead to very different estimates of the cointegrating vector,and this was the reason Clive and Rob wanted me to start working on the problem. In Gonzalo (1994), I obtained the asymptotic distribution of the principal components (method proposed in Stock and Watson, 1988) and canonical correlations (between the vectorXt andXt−1proposed by Bossaerts, 1988) based estimators of cointegrating vectors and compared theirperformance to the most popular alternatives at the time: ordinary least square (Engle and Granger, 1987), nonlinear least square (Stock, 1987), and maximum likelihood (Johansen, 1988). Thelatter was the clear winner once the dispersion of the finite sample distributions was measured by theinterquartile range instead of the simple variance.

Model-based asymptotic inference on the effect of infrequent large shocks on cointegrated variables

Quasi-maximum-likelihood (QML) estimation of a model combining cointegration in the conditional mean and rare large shocks (outliers) with a factor structure in the innovations is studied. The goal is not only to robustify inference on the conditional-mean parameters, but also to find regularities and conduct inference on the instantaneous and long-run effect of the large shocks. Given the cointegration rank and the factor order, χ 2 asymptotic inference is obtained for the cointegration vectors, the short-run parameters, and the direction of each column of both the factor loading matrix and the matrix of long-run impacts of the large shocks. Large shocks, whose location is assumed unknown a priori, can be detected and classified consistently into the factor components.

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.

Extended generalized purchasing power parity and optimum currency area in East Asian countries

This article extends the theory of generalized purchasing power parity (G-PPP) by developing a model including foreign variables such as the real money supply, output and interest rate. A cointegration rank test with two structural breaks in the deterministic trend was adopted to selected Asian countries, with Japan as the base country. There were two cointegrating vectors and six common stochastic trends in the model. The cointegrating vectors were estimated by the same approach used by Pesaran et al. (2000). The first cointegrating vector is interpreted as G-PPP, and the second one is interpreted as the extended G-PPP. According to G-PPP, these countries could form a single currency area. The first stochastic trend is primarily driven by Japan's real money supply; the second, third and fourth by the real exchange rates of the Philippines, Singapore and Thailand, respectively; the fifth by Japan's output; and the sixth by Japan's government bond yields.

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.

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process X(t) to be fractional of order d and cofractional of order d-b; that is, there exist vectors β for which β′X(t) is fractional of order d-b. The parameters d and b satisfy either d≥b≥1/2, d=b≥1/2, or d=d₀≥b≥1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2≤b≤d≤d₁ for any d₁≥d₀. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.

Co-Integration Rank Testing under Conditional Heteroskedasticity

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 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 i.i.d. or (strict and covariance) 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 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 given.