Detecting misspecifications in autoregressive conditional duration models and non‐negative time‐series processes

We develop a general theory to test correct specification of multiplicative error models of non‐negative time‐series processes, which include the popular autoregressive conditional duration (ACD) models. Both linear and nonlinear conditional expectation models are covered, and standardized innovations can have time‐varying conditional dispersion and higher‐order conditional moments of unknown form. No specific estimation method is required, and the tests have a convenient null asymptotic N(0,1) distribution. To reduce the impact of parameter estimation uncertainty in finite samples, we adopt Wooldridge's (1990a) device to our context and justify its validity. Simulation studies show that in the context of testing ACD models, finite sample correction gives better sizes in finite samples and are robust to parameter estimation uncertainty. And, it is important to take into account time‐varying conditional dispersion and higher‐order conditional moments in standardized innovations; failure to do so can cause strong overrejection of a correctly specified ACD model. The proposed tests have reasonable power against a variety of popular linear and nonlinear ACD alternatives.     

Analysis of accumulated rounding errors in autoregressive processes

In considering the rounding impact of an autoregressive (AR) process, there are two different models available to be considered. The first assumes that the dynamic system follows an underlying AR model and only the observations are rounded up to a certain precision. The second assumes that the updated observation is a rounded version of an autoregression on previous rounded observations. This article considers the second model and examines behaviour of rounding impacts to the statistical inferences. The conditional maximum‐likelihood estimates for the model are proposed and their asymptotic properties are established, including strong consistency and asymptotic normality. Furthermore, both the classical AR model and the ordinary rounded AR model are no longer reliable when dealing with accumulated rounding errors. The three models are also applied to fit the Ocean Wave data. It turns out that the estimates under distinct models are significantly different. Based on our findings, we strongly recommend that models for dealing with rounded data should be in accordance with the actions of rounding errors.     

On a Mixture GARCH Time‐Series Model

Abstract. Recently, there has been a lot of interest in modelling real data with a heavy‐tailed distribution. A popular candidate is the so‐called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy‐tailed features in real data. Some real examples illustrate the results.     

Estimation in Random Coefficient Autoregressive Models

Abstract. We propose the quasi‐maximum likelihood method to estimate the parameters of an RCA(1) process, i.e. a random coefficient autoregressive time series of order 1. The strong consistency and the asymptotic normality of the estimators are derived under optimal conditions.     

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.     

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.     

Functional Generalized Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic (GARCH) processes. More recently, multivariate variants of these processes have been the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Because of the need for estimation of many parameters, however, these models may not be suitable for modelling now prevalent high‐frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this article, theory is developed for a new functional version of the GARCH process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of strictly stationary solutions both in the space of square‐integrable and continuous functions. An estimation procedure is introduced, and its consistency and asymptotic normality are verified. A small empirical study highlights potential applications to intraday volatility estimation.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Rate of convergence in the central limit theorem for parameter estimation in a causal,invertible ARMA(p,q) model

In this study we consider the estimators of the parameters of a stable ARMA(p, q) process. The autoregressive parameters are estimated by the instrumental variable technique while the moving average parameters are estimated using a derived autoregressive process. The estimators are shown to be asymptotically normal and their rate of convergence to normality is derived.     

Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate

Abstract. We prove uniform convergence results for the integrated periodogram of a weakly dependent time series, namely a strong law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak‐dependence assumptions, the strong consistency and asymptotic normality of Whittle's estimate are established for a large class of models. For instance, the causal θ‐weak dependence property allows a new and unified proof of those results for autoregressive conditionally heteroscedastic (ARCH)(∞) and bilinear processes. Non‐causal η‐weak dependence yields the same limit theorems for two‐sided linear (with dependent inputs) or Volterra processes.     

A TEST FOR CONDITIONAL HETEROSKEDASTICITY IN TIME SERIES MODELS

Abstract. When testing for conditional heteroskedasticity and nonlinearity, the power of the test in general depends on the functional forms of conditional heteroskedasticity and nonlinearity that are allowed under the alternative hypothesis. We suggest a test for conditional heteroskedasticity and nonlinearity with the nonlinear autoregressive conditional heteroskedasticity model of Higgins and Bera as the alternative. Standard testing procedures are not applicable since our nonlinear autoregressive conditional heteroskedasticity (ARCH) parameter is not identified under the null hypothesis. To resolve this problem, we apply the procedure recently proposed by Davies. Power and size of the suggested test are investigated through simulation, and an empirical application of testing for ARCH in exchange rates is also discussed.     

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.     

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.     

PARAMETER ESTIMATION FOR PERIODIC ARMA MODELS

Abstract. In time series analysis of data sequences, the estimation of the parameters of an identified autoregressive moving-average (ARMA) model is a well-known and straightforward exercise. However, if the parameters of the model are periodic (i.e. a periodic ARMA (PARMA) model) then the estimation process becomes more difficult. This paper describes an on-line parameter estimation technique, based on methods from automatic control, which is demonstrated to provide consistent estimates of PARMA model parameters.     

Quasi-maximum likelihood estimation of periodic GARCH and periodic ARMA-GARCH processes

Abstract.  This article establishes the strong consistency and asymptotic normality (CAN) of the quasi-maximum likelihood estimator (QMLE) for generalized autoregressive conditionally heteroscedastic (GARCH) and autoregressive moving-average (ARMA)-GARCH processes with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution of the periodic GARCH (PGARCH) equation. As a result, it is shown that the moment of some positive order of the PGARCH solution is finite, under which we prove the strong consistency and asymptotic normality of the QMLE for a PGARCH process without any condition on its moments and for a periodic ARMA-GARCH (PARMA-PGARCH) under mild conditions.     

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ?_t = σ_tη_t and σ_t = f (σ_t?1, σ_t?2,…, σ_t?p, ?_t?1, ?_t?2,…, ?_t?q), when {?_t} is a process with just barely infinite variance, that is, {?_t} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.     

ROBUST FITTING OF INARCH MODELS

We discuss robust M‐estimation of INARCH models for count time series. These models assume the observation at each point in time to follow a Poisson distribution conditionally on the past, with the conditional mean being a linear function of previous observations. This simple linear structure allows us to transfer M‐estimators for autoregressive models to this situation, with some simplifications being possible because the conditional variance given the past equals the conditional mean. We investigate the performance of the resulting generalized M‐estimators using simulations. The usefulness of the proposed methods is illustrated by real data examples.     

Duration time‐series models with proportional hazard

Abstract. The analysis of liquidity in financial markets is generally performed by means of the dynamics of the observed intertrade durations (possibly weighted by price or volume). Various dynamic models for duration data have been considered in the literature, such as the Autoregressive Conditional Duration (ACD) model. These models are often excessively constrained, introducing, for example, a deterministic link between conditional expectation and variance in the case of the ACD model. Moreover, the stationarity properties and the patterns of the stationary distributions are often unknown. The aim of this article is to solve these difficulties by considering a duration time series satisfying the proportional hazard property. We describe in detail this class of dynamic models, discuss its various representations and provide the ergodicity conditions. The proportional hazard copula can be specified either parametrically, or nonparametrically. We discuss estimation methods in both contexts, and explain why they are efficient, that is, why they reach the parametric (respectively, nonparametric) efficiency bound.     

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.