In this paper we investigate methods for testing the existence of a cointegration relationship among the components of a nonstationary fractionally integrated (NFI) vector time series. Our framework generalizes previous studies restricted to unit root integrated processes and permits simultaneous analysis of spurious and cointegrated NFI vectors. We propose a modified F-statistic, based on a particular studentization, which converges weakly under both hypotheses, despite the fact that OLS estimates are only consistent under cointegration. This statistic leads to a Wald-type test of cointegration when combined with a narrow band GLS-type estimate. Our semiparametric methodology allows consistent testing of the spurious regression hypothesis against the alternative of fractional cointegration without prior knowledge on the memory of the original series, their short run properties, the cointegrating vector, or the degree of cointegration. This semiparametric aspect of the modelization does not lead to an asymptotic loss of power, permitting the Wald statistic to diverge faster under the alternative of cointegration than when testing for a hypothesized cointegration vector. In our simulations we show that the method has comparable power to customary procedures under the unit root cointegration setup, and maintains good properties in a general framework where other methods may fail. We illustrate our method testing the cointegration hypothesis of nominal GNP and simple-sum (M1, M2, M3) monetary aggregates. Copyright The Econometric Society 2004.

This paper develops an analytical study for the nonsense or spurious regressions that are generated by quite general integrated (of order d ) random processes. In doing this, we generalize the work of Phillips (Understanding spurious regressions in econometrics. J. Econ. 33 (1986), 311-40) who provided an analytical study of linear regressions involving only I( 1) stochastic processes. Our generalization of Phillips' work to the I(d) case is made employing fractional differencing techniques.

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We assume that some consistent estimator b b of an equilibrium relation between non-stationary series integrated of order d 2 ð0:5; 1:5Þ is used to compute residualsû t ¼ y t b bx t (or differences thereof). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence d of the equilibrium deviation u t : Provided b b converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of d: At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on d: This requires that d d40:5 for superconsistent b b; so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0pdo0:5; as well as for non-stationary but transitory equilibrium errors, 0:5odo1: In particular, if x t contains several series we consider the joint estimation of d and d: Wald statistics to test for parameter restrictions of the system have a limiting w 2 distribution. We also analyse the benefits of a pooled version of the estimate. The empirical applicability of our general cointegration test is investigated by means of Monte Carlo experiments and illustrated with a study of exchange rate dynamics. r 2005 Elsevier B.V. All rights reserved.JEL classification: C14; C22

In this paper we extend the well-known Sims, Stock and Watson (SSW, 1990)'s analysis on estimation and testing in VARs with integer unit roots and deterministic components to a more general set-up where nonstationary fractionally integrated (NFI ) processes are considered. To do so, we focus on partial VAR models where the conditioning variables are NFI processes since this is the only finite-lag VAR model compatible with such processes. We show how SSW 's conclusion remain valid in the sense that whenever a block of coefficients in the partial VAR can be written as coefficients on zero-mean I(0) regressors in models including a constant term, then they will have a joint asymptotic normal distribution. Monte Carlo simulations and an empirical application of our theoretical results are also provided.Keywords: Vector fractionally integrated processes; fractional cointegration; Granger causality; permanent income hypothesis.JEL classification: C12, C13, C32. * We are grateful to R. J. Smith and two anonymous referees for very useful comments which have led to considerable improvements of the paper.We also thank C. Alonso-Borrego, J. Davidson, J. Hidalgo, M. Jensen, D. Marinucci, P. M. Robinson, C. Velasco and participants at ESEM01, Laussane, for helpful comments and suggestions. Finally, we are especially grateful to Laura Mayoral for her help in designing the Monte Carlo simulations. This research has been partially financed with DGICYT grants PB98-0026 and SEC2001-0890. The second author thanks Instituto Flores de Lemus for financial support.

Herein we develop an analytical study of the asymptotic distributions obtained when we run linear regressions in the levels of stochastically independent integrated time series when the orders of integration of the dependent and independent variables are different. These theoretical findings largely explain the Monte Carlo results recently reported in Banerjee et al. (1993).

In this paper we propose an alternative characterization of the central notion of cointegration, exploiting the relationship between the autocovariance and the crosscovariance functions of the series+ This characterization leads us to propose a new estimator of the cointegrating parameter based on the instrumental variables~IV! methodology+ The instrument is a delayed regressor obtained from the conditional bivariate system of nonstationary fractionally integrated processes with a weakly stationary error correction term+ We prove the consistency of this estimator and derive its limiting distribution+ We also show that, in the I~1! case, with a semiparametric correction simpler than the one required for the fully modified ordinary least squares~FM-OLS!, our fully modified instrumental variables~FM-IV! estimator is median-unbiased, a mixture of normals, and asymptotically efficient+ As a consequence, standard inference can be conducted with this new FM-IV estimator of the cointegrating parameter+ We show by the use of Monte Carlo simulations that the small sample gains with the new IV estimator over OLS are remarkable+ INTRODUCTIONThe concept of cointegration was coined by Granger~1981, 1983! and Engle and Granger~1987! when the data generating process~DGP! was generated by integrated processes+ The time series x t is integrated of order d, denoted

The limit properties of the testing sequence underlying the Dickey-Pantula test for a double unit root in a time series are derived when the true data generating process is assumed to be nonstationary fractionally integrated. © 1997 Elsevier Science S.A.