Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Asymptotics for stationary very nearly unit root processes

Abstract. This article considers a mean zero stationary first‐order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter ρ_n is very near to one in the sense that 1 ? ρ_n = o(n^?1).     

A mixed INAR(p) model

A mixed integer‐valued autoregressive model of order p is proposed. The existence of this unique, stationary and ergodic process is proved and its autocorrelation structure and some conditional stochastic characteristics are derived. Model parameters are estimated via Yule‐Walker, conditional least squares and conditional maximum likelihood methods. Finally, possible application of the model to real data sets is considered.     

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.     

Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004) , we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.     

Asymptotic Distribution of the Bias Corrected Least Squares Estimators in Measurement Error Linear Regression Models Under Long Memory

This article derives the consistency and asymptotic distribution of the bias corrected least squares estimators (LSEs) of the regression parameters in linear regression models when covariates have measurement error (ME) and errors and covariates form mutually independent long memory moving average processes. In the structural ME linear regression model, the nature of the asymptotic distribution of suitably standardized bias corrected LSEs depends on the range of the values of where d _X ,d _u , and d _ε are the LM parameters of the covariate, ME and regression error processes respectively. This limiting distribution is Gaussian when and non‐Gaussian in the case . In the former case some consistent estimators of the asymptotic variances of these estimators and a log(n)‐consistent estimator of an underlying LM parameter are also provided. They are useful in the construction of the large sample confidence intervals for regression parameters. The article also discusses the asymptotic distribution of these estimators in some functional ME linear regression models, where the unobservable covariate is non‐random. In these models, the limiting distribution of the bias corrected LSEs is always a Gaussian distribution determined by the range of the values of d _ε ? d _u .     

EXACT MAXIMUM LIKELIHOOD ESTIMATION IN AUTOREGRESSIVE PROCESSES

Abstract. The purpose of this paper is to complement the theory of exact maximum likelihood estimation in pure autoregressive processes by differentiating the exact Gaussian likelihood function with respect to the model parameters and obtaining a set of likelihood equations very similar in form to the Yule—Walker equations. The main contribution of this paper is a very simple expression for the derivatives and the resulting likelihood equations in terms of the components of a (p+ 1) x (p+ 1) function of the data, the model parameters (s?², φ) and the autocovariances at lags 0 through p. We propose an iterative algorithm for solving the likelihood equations by alternately solving two linear systems, first for (s?², φ) given current estimates of the autocovariances, then for updated estimates of the autocovariances given current estimates of (s?², φ). The number of operations per iteration is independent of the series length since the algorithm uses the data only through the value of the (p+ 1) x (p+ 1) sufficient statistic.     

Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

In this article, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous‐time asset return models with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps, are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality are provided through the conditional Lindeberg theorem as the time span T→∞ and the sample interval Δ _n →0. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns of stock index under 5‐minute high sampling frequency for real application.     

Estimation of Parameters in the NLAR(p) Model

Abstract. In this article, we study a new Laplace autoregressive model of order p– NLAR(p). Conditional least squares, weighted conditional least squares and maximum quasi‐likelihood are used to estimate the model parameters. Comparisons among these estimates of the NLAR(2) model are given via simulation studies.     

Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation

Abstract. We provide simulation and theoretical results concerning the finite‐sample theory of quasi‐maximum‐likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)–ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t‐tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean‐squared error of the maximum‐likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)‐ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1–10] invariance properties for the biases in these situations.     

Solutions of Yule‐Walker equations for singular AR processes

A study is presented on solutions of the Yule‐Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule‐Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.     

Inference for pth‐order random coefficient integer‐valued autoregressive processes

Abstract. A pth‐order random coefficient integer‐valued autoregressive [RCINAR(p)] model is proposed for count data. Stationarity and ergodicity properties are established. Maximum likelihood, conditional least squares, modified quasi‐likelihood and generalized method of moments are used to estimate the model parameters. Asymptotic properties of the estimators are derived. Simulation results on the comparison of the estimators are reported. The models are applied to two real data sets.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

Asymptotic theory for certain regression models with long memory errors

The asymptotic distribution of a weighted linear combination of a linear long memory series is shown to be normal for certain weights. This result can be used to derive the limiting distribution of the least squares estimators for polynomial trends and of the periodogram at fixed Fourier frequencies. A closed form expression for the asymptotic relative bias of the tapered periodogram at fixed Fourier frequencies is also obtained. A weighted least squares estimator, which is asymptotically efficient for polynomial trend regressors, is shown to be asymptotically normal.     

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.     

Asymptotic laws of successive least squares estimates for seasonal arima models and application

Abstract. In view of detecting the stochastic non-stationarity in time series, successive Yule–Walker estimates are considered for general seasonal ARIMA models and their asymptotic laws are obtained. This extends results known on least squares estimates for stable–unstable ARMA. Furthermore, these asymptotic laws are then compared with analogous results obtained under some additive seasonal model that corresponds to the case of deterministic seasonal behaviour. These results, combined with a simulation study, reveal that successive autoregressions provide a very useful tool both for identifying seasonal ARIMA processes and for distinguishing between stochastic and deterministic seasonal behaviours.     

REDUCTION OF THE ASYMPTOTIC BIAS OF AUTOREGRESSIVE AND SPECTRAL ESTIMATORS BY TAPERING

Abstract. The asymptotic bias to terms of order T ^-1, where T is the observed series length, is studied for estimators of the coefficients and disturbance variance in an AR( p ) model. Reduction of the asymptotic bias by tapering is established and, if the tapering function is defined appropriately to depend on T , not only is the asymptotic bias reduced, but the asymptotic distribution of the estimators is not altered. In addition, the asymptotic biases of other time series parameter estimators constructed from the sample covariance function, such as several types of spectral estimators, can also be reduced by tapering.     

Random environment integer‐valued autoregressive process

An r states random environment integer‐valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non‐stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the k‐step‐ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real‐life counting data.