The Variance Ratio Statistic at large Horizons

We make three contributions to using the variance ratio statistic at large horizons. Allowing for general heteroscedasticity in the data, we obtain the asymptotic distribution of the statistic when the horizon k is increasing with the sample size n but at a slower rate so that k/n¨0. The test is shown to be consistent against a variety of relevant mean reverting alternatives when k/n¨0. This is in contrast to the case when k/n¨ƒÂ>0, where the statistic has been recently shown to be inconsistent against such alternatives. Secondly, we provide and justify a simple power transformation of the statistic which yields almost perfectly normally distributed statistics in finite samples, solving the well known right skewness problem. Thirdly, we provide a more powerful way of pooling information from different horizons to test for mean reverting alternatives. Monte Carlo simulations illustrate the theoretical improvements provided.Mean reversion, frequency domain, power transformations

Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s>=1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by d_k for k=1,...,s. We estimate each cointegrating subsspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k'th estimated coingetraging subspace is, with high probability, close to the k'th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. The angle is O_p(n^{- \alpha_k}), where n is the sample size and \alpha_k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to \infty more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Fractional Cointegration; Long Memory; Tapering; Periodogram

GMM Estimation for Long Memory Latent Variable Volatility and Duration Models

We study the rate of convergence of moment conditions that have been commonly used in the literature for Generalised Method of Moments (GMM) estimation of short memory latent variable volatility models. We show that when the latent variable possesses long memory, these moment conditions have an n^{1/2-d} rate of convergence where 0GMM, long memory, stochastic volatility and durations

The Variance Ratio Statistic at Large Horizons

We make three contributions to using the variance ratio statistic at large horizons. Allowing for general heteroscedasticity in the data, we obtain the asymptotic distribution of the statistic when the horizon k is increasing with the sample size n but at a slower rate so that k=n ! 0. The test is shown to be consistent against a variety of relevant mean reverting alternatives when k=n ! 0. This is in contrast to the case when k=n ! – > 0; where the statistic has been recently shown to be inconsistent against such alternatives. Secondly, we provide and justify a simple power transformation of the statistic which yields almost perfectly normally distributed statistics in finite samples, solving the well known right skewness problem. Thirdly, we provide a more powerful way of pooling information from different horizons to test for mean reverting alternatives. Monte Carlo simulations illustrate the theoretical improvements provided. --Mean reversion,Frequency domain,Power transformation

Estimation of Mis-Specified Long Memory Models

We study the asymptotic behaviour of frequency domain maximum likelihood estimators of mis-specified models of long memory Gaussian series. We show that even if the long memory structure of the time series is correctly specified, mis-specification of the short memory dynamics may result in parameter estimators which are slower than pn consistent. The conditions under which this happens are provided and the asymptotic distribution of the estimators is shown to be non-Gaussian. Conditions under which estimators of the parameters of the mis-specified model have the standard pn consistent and asymptotically normal behaviour are also provided. --

A Small Sample Study of Goodness-of-fit Tests for Time Series Models

We study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box- Ljung-Pierce portmanteau test, one in the time domain and the other in the frequency domain. The tests are found to be oversized under the null of white noise but undersized under other null hypotheses. The cause for this effect is investigated and a finite sample correction proposed which ameliorates this effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant in certain alternatives. Furthermore, the corrected tests also have uniformly better power than the Box-Ljung-Pierce portmanteau test, unlike the uncorrected versions.Statistics Working Papers Serie

The Restricted Likelihood Ratio Test at the Boundary in Autoregressive Series

The restricted likelihood ratio test, RLRT, for the autoregressive coefficient in autoregressive models has recently been shown to be second order pivotal when the autoregressive coefficient is in the interior of the parameter space and so is very well approximated by the chi-square distribution. In this paper, the non-standard asymptotic distribution of the RLRT for the unit root boundary value is obtained and is found to be almost identical to that of the chi-square in the right tail. Together, the above two results imply that the chi-square distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution.Department of Statistics, Texas A&M University; Stern School of Business, New York UniversityStatistics Working Papers Serie

A Generalized Portmanteau Goodness-of-fit Test for Time Series Models

We present a goodness of fit test for time series models based on the discrete spectral average estimator. Unlike current tests of goodness of fit, the asymptotic distribution of our test statistic allows the null hypothesis to be either a short or long range dependence model. Our test is in the frequency domain, is easy to compute and does not require the calculation of residuals from the fitted model. This is especially advantageous when the fitted model is not a finite order autoregressive model. The test statistic is a frequency domain analogue of the test by Hong (1996) which is a generalization of the Box-Pierce (1970) test statistic. A simulation study shows that our test has power comparable to that of Hong's test and superior to that of another frequency domain test by Milhoj (1981).Statistics Working Papers Serie

Estimating fractional cointegration in the presence of polynomial trends

We propose and derive the asymptotic distribution of a tapered narrow-band least squares estimator (NBLSE) of the cointegration parameter ÃÂò in the framework of fractional cointegration. This tapered estimator is invariant to deterministic polynomial trends. In particular, we allow for arbitrary linear time trends that often occur in practice. Our simulations show that, in the case of no deterministic trends, the estimator is superior to ordinary least squares (OLS) and the nontapered NBLSE proposed by P.M. Robinson when the levels have a unit root and the cointegrating relationship between the series is weak. In terms of rate of convergence, our estimator converges faster under certain circumstances, and never slower, than either OLS or the nontapered NBLSE. In a data analysis of interest rates, we find stronger evidence of cointegration if the tapered NBLSE is used for the cointegration parameter than if OLS is used.Statistics Working Papers Serie

Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s ¸ 1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by dk, for k = 1; : : : ; s. We estimate each cointegrating subspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k’th estimated cointegrating subspace is, with high probability, close to the k’th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. This angle is Op(n¡®k ), where n is the sample size and ®k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to 1 more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Statistics Working Papers Serie