Willa W. Chen

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 → δ >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.

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 averageestimator. Unlike current tests of goodness of fit, the asymptotic distribution of our test statisticallows the null hypothesis to be either a short or long range dependence model. Our test isin the frequency domain, is easy to compute and does not require the calculation of residualsfrom the fitted model. This is especially advantageous when the fitted model is not a finiteorder autoregressive model. The test statistic is a frequency domain analogue of the test byHong (1996) which is a generalization of the Box-Pierce (1970) test statistic. A simulation studyshows that our test has power comparable to that of Hongs test and superior to that of anotherfrequency domain test by Milhoj (1981).

Semiparametric Estimation of Multivariate Fractional Cointegration

We consider the semiparametric estimation of fractional cointegration in a multivariate process of cointegrating rank r > 0. We estimate the cointegrating relationships by the eigenvectors corresponding to the r smallest eigenvalues of an averaged periodogram matrix of tapered, differenced observations. The number of frequencies m used in the periodogram average is held fixed as the sample size grows. We first show that the averaged periodogram matrix converges in distribution to a singular matrix whose null eigenvectors span the space of cointegrating vectors. We then show that the angle between the estimated cointegrating vectors and the space of true cointegrating vectors is Op(ndu−d), where d and du are the memory parameters of the observations and cointegrating errors. The proposed estimator is invariant to the labeling of the component series and thus does not require that one of the variables be specified as a dependent variable. We determine the rate of convergence of the r smallest eigenvalues of the periodogram matrix and present a criterion that allows for consistent estimation of r. Finally, we apply our methodology to the analysis of fractional cointegration in interest rates.

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 suchthat 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 ofeigenvectors 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 estimatedcointegrating subspace is, with high probability, close to the k th true cointegrating subspace, in the sensethat the angle between the estimated cointegrating vector and the true cointegrating subspace convergesin probability to zero. This angle is Op(ni®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.

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 β 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.

On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems for Long-Memory Time Series

We show that for long-memory time series, the Toeplitz system Σn(f)x = b can be solved in O(nlog5/2n) operations using a well-known version of the preconditioned conjugate gradient method, where Σn(f) is the n × n covariance matrix, f is the spectral density, and b is a known vector. Solutions of such systems are needed for optimal linear prediction and interpolation. We establish connections between this preconditioning method and the frequency domain analysis of time series. Indeed, the running time of the algorithm is determined by the rate of increase in the condition number of the correlation matrix of the discrete Fourier transform (DFT) vector, as the sample size tends to ∞. We derive an upper bound for this condition number. The bound is of interest in its own right, because it sheds some light on the widely used but heuristic approximation that the standardized DFT coefficients are uncorrelated with equal variances. We present applications of the preconditioning methodology to the forecasting of volatility in a long-memory stochastic volatility model, and to the evaluation of the Gaussian likelihood function of a long-memory model.

Bias Reduction and Likelihood Based Almost-Exactly Sized Hypothesis Testing in Predictive Regressions Using the Restricted Likelihood

Difficulties with inference in predictive regressions are generally attributed to strong persistence in the predictor series. We show that the major source of the problem is actually the nuisance intercept parameter, and we propose basing inference on the restricted likelihood, which is free of such nuisance location parameters and also possesses small curvature, making it suitable for inference. The bias of the restricted maximum likelihood (REML) estimates is shown to be approximately 50% less than that of the ordinary least squares (OLS) estimates near the unit root, without loss of efficiency. The error in the chi-square approximation to the distribution of the REML-based likelihood ratio test (RLRT) for no predictability is shown to be null where | ρ | G (·) is the cumulative distribution function (c.d.f.) of a null random variable. This very small error, free of the autoregressive (AR) parameter, suggests that the RLRT for predictability has very good size properties even when the regressor has strong persistence. The Bartlett-corrected RLRT achieves an O ( n −2 ) error. Power under local alternatives is obtained, and extensions to more general univariate regressors and vector AR(1) regressors, where OLS may no longer be asymptotically efficient, are provided. In simulations the RLRT maintains size well, is robust to nonnormal errors, and has uniformly higher power than the Jansson and Moreira (2006, Econometrica 74, 681–714) test with gains that can be substantial. The Campbell and Yogo (2006, Journal of Financial Econometrics 81, 27–60) Bonferroni Q test is found to have size distortions and can be significantly oversized.

The Restricted Likelihood Ratio Test at the Boundary in Autoregressive Series

The restricted likelihood ratio test, RLRT, for the autoregressive coe-cient in autoregressive models has recently been shown to be second order pivotal when the autoregressive coe-cient is in the interior of the parameter space and so is very well approximated by the ´ 2 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 ´ 2 in the right tail. Together, the above two results imply that the ´ 2 distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution.

The Restricted Likelihood Ratio Test for Autoregressive Processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (−1, 1] × (−1, 1)p−1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.

Uniform Inference in Predictive Regression Models

The restricted likelihood has been found to provide a well-behaved likelihood ratio test in the predictive regression model even when the regressor variable exhibits almost unit root behavior. Using the weighted least squares approximation to the restricted likelihood obtained in Chen and Deo, we provide a quasi restricted likelihood ratio test (QRLRT), obtain its asymptotic distribution as the nuisance persistence parameter varies, and show that this distribution varies very slightly. Consequently, the resulting sup bound QRLRT is shown to maintain size uniformly over the parameter space without sacrificing power. In simulations, the QRLRT is found to deliver uniformly higher power than competing procedures with power gains that are substantial.