Rohit S. Deo

The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series

We establish some asymptotic properties of a log-periodogram regression estimator for the memory parameter of a long-memory time series. We consider the estimator originally proposed by Geweke and Porter-Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37). In particular, we do not omit any of the low frequency periodogram ordinates from the regression. We derive expressions for the estimators asymptotic bias, variance and mean squared error as functions of the number of periodogram ordinates, m, used in the regression. Consistency of the estimator is obtained as long as m∞ and n∞ with (m log m)/n 0, where n is the sample size. Under these and the additional conditions assumed in this paper, the optimal m, minimizing the mean squared error, is of order O(n4/5). We also establish the asymptotic normality of the estimator. In a simulation study, we assess the accuracy of our asymptotic theory on mean squared error for finite sample sizes. One finding is that the choice m = n1/2, originally suggested by Geweke and Porter-Hudak (1983), can lead to performance which is markedly inferior to that of the optimal choice, even in reasonably small samples.

Spectral tests of the martingale hypothesis under conditional heteroscedasticity

We study the asymptotic distribution of the sample standardized spectral distribution function when the observed series is a conditionally heteroscedastic martingale difference. We show that the asymptotic distribution is no longer a Brownian bridge but another Gaussian process. Furthermore, this limiting process depends on the covariance structure of the second moments of the series. We show that this causes test statistics based on the sample spectral distribution, such as the Cramer von-Mises statistic, to have heavily right skewed distributions, which will lead to over-rejection of the martingale hypothesis in favour of mean reversion. A non-parametric correction to the test statisticsis proposed to account for the conditional heteroscedasticity. We demonstrate that the corrected version of the Cramer von-Mises statistic has the usual limiting distribution which would be obtained in the absence of conditional heteroscedasticity. We also present Monte Carlo results on the finite sample distributions of uncorrected and corrected versions of the Cramer von-Mises statistic. Our simulation results show that this statistic can provide significant gains in power over the Box-Ljung-Pierce statistic against long-memory alternatives. An empirical application to stock returns is also provided.

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).

Linear Trend with Fractionally Integrated Errors

We consider the estimation of linear trend for a time series in the presence of additive long-memory noise with memory parameter d∈[0, 1.5). Although no parametric model is assumed for the noise, our assumptions include as special cases the random walk with drift as well as linear trend with stationary invertible autoregressive moving-average errors. Moreover, our assumptions include a wide variety of trend-stationary and difference-stationary situations. We consider three different trend estimators: the ordinary least squares estimator based on the original series, the sample mean of the first differences and a class of weighted (tapered) means of the first differences. We present expressions for the asymptotic variances of these estimators in the form of one-dimensional integrals. We also establish the asymptotic normality of the tapered means for d∈[0, 1.5) −{0.5} and of the ordinary least squares estimator for d∈ (0.5, 1.5). We point out connections with existing theory and present applications of the methodology.

On the integral of the squared periodogram

Let X1,X2,...,Xn be a sample from a stationary Gaussian time series and let I(·) be the sample periodogram. Some researchers have either proved heuristically or claimed that under general conditions, the asymptotic behaviour of is equivalent to that of the discrete version of the integral given by , where [lambda]i are the Fourier frequencies and [phi] and [eta] are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when [phi] is a non-linear function. We derive the exact finite sample variance of when {Xt} is Gaussian white noise and show that it is asymptotically different from that of . The asymptotic distribution of is also obtained in this case. The result is then extended to obtain the limiting distribution of when {Xt}is a stationary Gaussian series with spectral density f(·). From these results, the limiting distribution of the integral version of a goodness-of-fit statistic proposed in the literature is obtained.

Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter

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

d \in [0,1/2)

Asymptotic theory for certain regression models with long memory errors

to ensure that the corresponding counting process

On estimation and testing goodness of fit for m-dependent stable sequences

N(t)