Abstract. Generalizing the de®nition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non-stationary log-periodogram regression, Forthcoming J. Economet., 1997) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long-range-dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat. 23 (1995), 1630 61) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d P ( 3 4 ) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric ef®ciency of this estimate for stationary sequences also extends to the non-stationary framework.

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n 1=2 ; with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach. r 2005 Elsevier B.V. All rights reserved.JEL classification: C12

Whittle pseudo-maximum likelihood estimates of parameters for stationary time series have been found to be consistent and asumptotically normal in the presence of long-range dependence. Generalizing the definition of the memory parameter d, we extend these results to include possibly nonstationary (0.5 d < 1) or antipersistent (-0.5 < d < 0) observations. Using adequate data tapers we can apply this estimation technique to any degree of nonstationarity d ZLWKRXW SULRU NQRZOHGJH RI WKH PHPRU\ RI WKH VHULHV We analyse the performance of the estimates on simulated and real data.

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d* ) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the de"nition of the memory parameter d for non-stationary processes in terms of the (successively) di!erentiated series. We obtain that the log-periodogram estimate is asymptotically normal for d3 [ , ) and still consistent for d3[ , 1). We show that with adequate data tapers, a modi"ed estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation.1999 Elsevier Science S.A. All rights reserved.JEL classixcation: C11; C22

In this article we introduce efficient Wald tests for testing the null hypothesis of unit root against the alternative of fractional unit root. In a local alternative framework, the proposed tests are locally asymptotically equivalent to the optimal Robinson (1991Robinson ( , 1994a) Lagrange Multiplier tests. Our results contrast with the tests for fractional unit roots introduced by Dolado, Gonzalo and Mayoral (2002) which are inefficient. In the presence of short range serial correlation, we propose a simple and efficient two-step test that avoids the estimation of a nonlinear regression model. In addition, the first order asymptotic properties of the proposed tests are not affected by the pre-estimation of short or long memory parameters * Acknowledgements: We thank the co-editor and two referees for very useful comments, J. AbstractIn this article we introduce e¢ cient Wald tests for testing the null hypothesis of unit root against the alternative of fractional unit root. In a local alternative framework, the proposed tests are locally asymptotically equivalent to the optimal Robinson (1991Robinson ( , 1994a Lagrange Multiplier tests. Our results contrast with the tests for fractional unit roots introduced by Dolado, Gonzalo and Mayoral (2002) which are ine¢ cient. In the presence of short range serial correlation, we propose a simple and e¢ cient two-step test that avoids the estimation of a nonlinear regression model. In addition, the …rst order asymptotic properties of the proposed tests are not a¤ected by the pre-estimation of short or long memory parameters.We thank the co-editor and two referees for very useful comments,

We show the consistency of the log-periodogram regression estimate of the long memory parameter for long range dependent linear, not necessarily Gaussian, time series when we make a pooling of periodogram ordinates+ Then, we study the asymptotic behavior of the tapered periodogram of long range dependent time series for frequencies near the origin, and we obtain the asymptotic distribution of the log-periodogram estimate for possibly non-Gaussian observation when the tapered periodogram is used+ For these results we rely on higher order asymptotic properties of a vector of periodogram ordinates of the linear innovations+ Finally, we assess the validity of the asymptotic results for finite samples via Monte Carlo simulation+

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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.