Trend stationarity versus long-range dependence in time series analysis

Empirically, it is difficult to offer unequivocal judgment as to whether many real economic variables are fractionally integrated or trend stationary. The objective of this paper is to study the effects of spurious detrending of a nonstationary fractionally integrated NFI(d), dE (1/2, 3/2). With respect to the performance of the traditional least squares estimators and tests we prove that the estimated time trend coefficient is consistent but that the corresponding t-Student test diverges. We also analyze a local version in the frequency domain of least squares. We are able to show the consistency of this estimator and that, after conveniently adjusting variance estimates, its t-ratio has a well-defined but nonstandard limiting distribution. Nonetheless, in this latter case it is possible to obtain a set of critical values giving rise to the correct size for any given dE (1/2, 3/2).Publicad

Spurius regression theory with nonstationary fractionally integrated processes

This paper develops an analytical study of the asymptotic distributions obtained when we run linear regressions in the levels of nonstationary fractionally integrated FI(d) processes, that are spuriously related in a multivariate single-equation setting which aIIows for the existence of co integrating relationships and quite general deterministic components. In doing this, the analytical studies of PhiIIips (1986), haldrup (1994) and Marmol (1995, 1996) are embedded in our results

Fractional integration versus trend stationary in time series analysis

The objective of this paper is to study the effects of spurious detrending of a nonstationary fractionally integrated process (NFI(d), d ~5) on the performance of the traditional least squares estimators and tests. We extend previous work on the subject undertaken by Durlauf and Phillips (1988) which considered only the leading difference stationary (d = 1) case. Moreover, we also consider the possibility of a double misspecification both in the stochastic and in the nonstochastic trends. Standard t-Student tests are shown to diverge in distribution invalidating any inference concerning the presence of time trends. On the other hand, we prove that, even under this double misspecification, the Durbin-Watson statistic remains to be a useful misspecification test

Residual log-periodogram inference for long-run relationships

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d(0.5,1.5) is used to compute residuals (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 δ of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of δ. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on δ. This requires that d-δ>0.5 for superconsistent , so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0δ<0.5, as well as for non-stationary but transitory equilibrium errors, 0.5<δ<1. In particular, if xt contains several series we consider the joint estimation of d and δ. Wald statistics to test for parameter restrictions of the system have a limiting χ2 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.Publicad

Residual Log-Periodogram Inference for Long-Run-Relationships

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d E (0:5; 1:5) is used to compute residuals ˆut = yt - xt (or differences there of). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence ± of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of ±. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on ±. This requires that d ¡ ± > 0:5 for superconsistent b¯, so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0 · ± < 0:5, as well as for non-stationary but transitory equilibrium errors, 0:5 < ± < 1. In particular, if xt contains several series we consider the joint estimation of d and ±. Wald statistics to test for parameter restrictions of the system have a limiting Â2 distribution. We also analyze 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

Residual Log-Periodogram Inference for Long-Run-Relationships.

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d E (0:5; 1:5) is used to compute residuals ˆut = yt - xt (or differences there of). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence ± of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of ±. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on ±. This requires that d ¡ ± > 0:5 for superconsistent b¯, so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0 · ±

Residual log-periodogram inference for long-run relationships.

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d(0.5,1.5) is used to compute residuals (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 δ of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of δ. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on δ. This requires that d-δ>0.5 for superconsistent , so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0δFractional cointegration; Semiparametric inference; Limiting normality; Long memory; Non-stationarity; Exchange rates;

Fractional cointegrating regressions in the presence of linear time trends

We consider regressions of nonstationary fractionally integrated variables dominated by linear time trends. The regression errors are short memory, long memory or even nonstationary, and hence allow for a very flexible cointegration model. In case of simple regressions, least squares estimation gives rise to limiting normal distribucions independently of the order of integration of the regressor, whereas the customary t-statistics diverge. We also investigate the possibility of testing for mean reverting equilibrium deviations by means of a residual-based log-periodogram regression. Asymptotic results become more complicated in the multivariate case

A beveridge-nelson decomposition for fractionally integrated time series

The purpose of this paper is to present a decomposition into trend or permanent component and cycle or transitory component of a time series that follows a nonstationary autoregressive fractionally integrated moving average (ARFlMA(p,d,q)) model. As a particular case, for d=l we obtain the well known BeveridgeNelson decomposition of a series. For d=2 we get the decomposition of an 1(2) series given by Newbold and Vougas (1996). The decomposition depends only on past data and is thus computable in real time. Computational issues are also discusse

Out-of-sample forecast errors in misspecified perturbed long memory processes

The correlogram is not a useful diagnosis tool in the presence of long-memory or long range depedent time series. The aim of this paper is to illustrate this claim by examining the relative increase in mean square forecast error from fitting a weakly stationary process to the series of interest hen in fact the true model is a so-called perturbed long-memory process recently introduced by Granger and Marmol (1997). This model has the property of being unidentifiable from a white noise process on the basis of the correlogram and the usual rule-of thumbs in the Box-Jenkins methodology. We prove that this kind of misspecification can lead to serious errors in terms of forecasting