Monte Carlo Maximum Likelihood Estimation for Generalized Long-Memory Time Series Models

An exact maximum likelihood method is developed for the estimation of parameters in a non-Gaussian nonlinear density function that depends on a latent Gaussian dynamic process with long-memory properties. Our method relies on the method of importance sampling and on a linear Gaussian approximating model from which the latent process can be simulated. Given the presence of a latent long-memory process, we require a modification of the importance sampling technique. In particular, the long-memory process needs to be approximated by a finite dynamic linear process. Two possible approximations are discussed and are compared with each other. We show that an autoregression obtained from minimizing mean squared prediction errors leads to an effective and feasible method. In our empirical study, we analyze ten daily log-return series from the S&P 500 stock index by univariate and multivariate long-memory stochastic volatility models. We compare the in-sample and out-of-sample performance of a number of models within the class of long-memory stochastic volatility models.     

Long memory in intertrade durations,counts and realized volatility of NYSE stocks

We study the persistence of intertrade durations, counts (number of transactions in equally spaced intervals of clock time), squared returns and realized volatility in 10 stocks trading on the New York Stock Exchange. A semiparametric analysis reveals the presence of long memory in all of these series, with potentially the same memory parameter. We introduce a parametric latent-variable long-memory stochastic duration (LMSD) model which is shown to better fit the data than the autoregressive conditional duration model (ACD) in a variety of ways. The empirical evidence we present here is in agreement with theoretical results on the propagation of memory from durations to counts and realized volatility presented in Deo et al. (2009).     

Sieve bootstrapping the memory parameter in long-range dependent stationary functional time series

We consider a sieve bootstrap procedure to quantify the estimation uncertainty of long-memory parameters in stationary functional time series. We use a semiparametric local Whittle estimator to estimate the long-memory parameter. In the local Whittle estimator, discrete Fourier transform and periodogram are constructed from the first set of principal component scores via a functional principal component analysis. The sieve bootstrap procedure uses a general vector autoregressive representation of the estimated principal component scores. It generates bootstrap replicates that adequately mimic the dependence structure of the underlying stationary process. We first compute the estimated first set of principal component scores for each bootstrap replicate and then apply the semiparametric local Whittle estimator to estimate the memory parameter. By taking quantiles of the estimated memory parameters from these bootstrap replicates, we can nonparametrically construct confidence intervals of the long-memory parameter. As measured by coverage probability differences between the empirical and nominal coverage probabilities at three levels of significance, we demonstrate the advantage of using the sieve bootstrap compared to the asymptotic confidence intervals based on normality.

     

Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors

We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya-Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter α. We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.     

Tests for Trend: A Simulation Study

In this study, we use the wavelet analysis to construct a test statistic to test for the existence of a trend in the series. We also propose a new approach for testing the presence of trend based on the periodogram of the data. Since we are also interested in the presence of a long-memory process among the data, we study the properties of our test statistics under different degrees of dependency. We compare the results when using the band periodogram test and the wavelet test with results obtained by applying the ordinary least squares (OLS) method under the same conditions.     

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

In this paper we examine the relative increase in mean square forecast error fro fitting a weakly stationary process to the series of interest when 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 missspecification can lead to serious errors in terms of forecasting. We also show that corrections based on the AR(1) model can in some cases partially solve the problem. Received: March 15, 1999; revised version: February 14, 2000     

Semiparametric Whittle estimation of a cyclical long-memory time series based on generalised exponential models

This paper considers a semiparametric estimation of the memory parameter in a cyclical long-memory time series, which exhibits a strong dependence on cyclical behaviour, using the Whittle likelihood based on generalised exponential (GEXP) models. The proposed estimation is included in the so-called broadband or global method and uses information from the spectral density at all frequencies. We establish the consistency and the asymptotic normality of the estimated memory parameter for a linear process and thus do not require Gaussianity. A simulation study conducted using Monte Carlo experiments shows that the proposed estimation works well compared to other existing semiparametric estimations. Moreover, we provide an empirical application of the proposed estimation, applying it to the growth rate of Japan's industrial production index and detecting its cyclical persistence.     

Long memory versus structural breaks: An overview

We discuss the increasing literature on misspecifying structural breaks or more general trends as long-range dependence. We consider tests on structural breaks in the long-memory regression model as well as the behaviour of estimators of the memory parameter when structural breaks or trends are in the data but long memory is not. Methods for distinguishing both of these phenomena are proposed. The financial support of Volkswagenstiftung is gratefully acknowledged.     

Filtration of ASTA: a weak convergence approach

Consider a stochastic process (X,A), where X represents the evolution of a system over time, and A is an associated point process that has stationary independent increments. Suppose we are interested in estimating the time average frequency of the process X being in a set of states. Often it is more convenient to have a sampling procedure for estimating the time average based on averaging the observed values of X(T_n) (T_n being a point of A) over a long period of time: the event average of the process. In this paper we examine the situation when the two procedures—event averaging and time averaging—produce the same estimate (the ASTA property: Arrivals See Time Averages). We prove a result stronger than ASTA. Under a lack-of-anticipation assumption we prove that the point process, A, restricted to any set of states, has the same probabilistic structure as the original point process. In particular, if the original point process is Poisson the new point process is still Poisson with the same parameter as the original point process. We develop our results in the more general setting of a stochastic process (X,A), that is, a process with an imbedded cumulative process, A={A(t),t0}, which is assumed to be a Levy process with non-decreasing sample paths. This framework allows for modeling fluid processes, as well as compound Poisson processes with non-integer increments. First, we state the result in discrete time; the discrete-time result is then extended to the continuous-time case using limiting arguments and weak-convergence theory. As a corollary we give a proof of ASTA under weak conditions and a simple, intuitive proof of (Poisson Arrivals See Time Averages) under the standard conditions. The results are useful in queueing and statistical sampling theory.     

The Sensitivity of Detrended Long-Memory Processes

Long-memory tests are often complicated by the presence of deterministic trends. Hence, an additional step of detrending the data is necessary. The typical way to detrend a suspected long-memory series is to use OLS or BSP residuals. Applying the method of sensitivity analysis we address the of question of how robust these residuals are in presence of potential long memory components. Unlike short-memory ARMA process long-memory I(d) processes causes sensitivity to OLS/BSP residuals. Therefore, we develop a finite sample measure of the sensitivity of a detrended series based on the residuals. Based on our sensitivity measure we propose a “rule of thumb” for practitioners to choose between the two methods of detrending, has been provided in this article.     

Long Memory in Foreign-Exchange Rates

Using the Geweke–Porter-Hudak test, we find evidence of long memory in exchange-rate data. This implies that the empirical evidence of unit roots in exchange rates may not be robust to long-memory alternatives. Fractionally integrated autoregressive moving average (ARFIMA) models are estimated by both the time-domain exact maximum likelihood (ML) method and the frequency-domain approximate ML method. Impulse-response functions and forecasts based on these estimated ARFIMA models are evaluated to gain insight into the long-memory characteristics of exchange rates. Some tentative explanations of the long memory found in the exchange rates are discussed.     

A New Test for Short Memory in Long Memory Time Series

This article considers short memory characteristics in a long memory process. We derive new asymptotic results for the sample autocorrelation difference ratios. We used these results to develop a new portmanteau test that determines if short memory parameters are statistically significant. In simulations, the new test can detect short memory components more often than the Ljung-Box test when these short memory components are in fact within a long memory process. Interestingly, our test finds short memory autocorrelations in U.S. inflation rate data, whereas the Ljung-Box test fails to find these autocorrelations. Modeling these short memory autocorrelations of the inflation rate data leads to improved model accuracy and more precise prediction.     

Parameter identification for singular random fields arising in Burgers’ turbulence

The paper reports on a study of classical statistical inference problems for long-memory random fields arising as solutions of the nonlinear diffusion equation with random initial data (the Burgers’ turbulence problem).     

A bivariate integer-valued long-memory model for high-frequency financial count data

We propose a bivariate integer-valued fractional integrated (BINFIMA) model to account for the long-memory property and apply the model to high-frequency stock transaction data. The BINFIMA model allows for both positive and negative correlations between the counts. The unconditional and conditional first- and second-order moments are given. The model is capable of capturing the covariance between and within intra-day time series of high-frequency transaction data due to macroeconomic news and news related to a specific stock. Empirically, it is found that Ericsson B has mean recursive process while AstraZeneca has long-memory property.     

Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform.     

Orthogonal series density estimation in a disaggregation scheme

In this paper we consider long-memory processes obtained by aggregation of independent random parameter AR(1) processes. We propose an estimator of the density of the underlying random parameter. This estimator is based on the expansion of the density function on the basis of Gegenbauer polynomials. Rate of convergence to zero of the mean integrated square error (MISE) and of the uniform error are obtained. The results are illustrated by Monte-Carlo simulations.     

Parametric estimation for ARFIMA models via spectral methods

Given a fractional integrated, autoregressive, moving average,ARFIMA (p, d, q) process, the simultaneous estimation of the short and long memory parameters can be achieved by maximum likelihood estimators. In this paper, following a two-step algorithm, the coefficients are estimated combining the maximum likelihood estimators with the general orthogonal decomposition of stochastic processes. In particular, the principal component analysis of stochastic processes is exploited to estimate the short memory parameters, which are plugged into the maximum likelihood function to obtain the fractional differencingd.     

On the likelihood function of small time variance Gamma Lévy processes

We investigate the likelihood function of small generalized Laplace laws and variance gamma Lévy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Lévy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Lévy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.     

On quantile processes for m-dependent Rv's

We obtain strong and weak approximations for quantile processes for m-dependent random variables. We apply our results in the two-sample to obtain condidence bands for quantile plots and prove some CHERNOFF-SAVAGE theorems for m-dependent random variables     

Parameter Estimation for a Discretely Observed Integrated Diffusion Process

Abstract.  We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation is a discrete sampling of the integral of X at times i Δ , i  =  1 ,…, n . Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we first prove limit theorems for functionals associated with our observations. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent and asymptotically Gaussian with different rates for drift and diffusion coefficient parameters.