Maximum quasi-likelihood estimation for the near(2) model

Abstract.  Maximum quasi-likelihood estimation is investigated for the NEAR(2) model, an autoregressive time series model with marginal exponential distributions. In certain regions of the parameter space, simulations indicate that maximum quasi-likelihood estimators perform better than two-stage conditional least squares estimators in terms of the per cent of estimates falling in the parameter space. The problem of out-of-range estimates is shown to be caused by the lack of information in the data rather than the characteristics of the method of estimation.     

Time‐series models with an EGB2 conditional distribution

A time‐series model in which the signal is buried in noise that is non‐Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation‐driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This specification produces a model that is not only easy to implement but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum‐likelihood (ML) estimator. Score‐driven models of this kind can also be based on conditional t distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An exponential general autoregressive conditional heteroscedastic (EGARCH) model based on the EGB2 distribution is also developed. This model complements the score‐driven EGARCH model with a conditional t distribution. Finally, dynamic location and scale models are combined and applied to data on the UK rate of inflation.     

Generalized autoregressive moving average models with GARCH errors

One of the important and widely used classes of models for non-Gaussian time series is the generalized autoregressive model average models (GARMA), which specifies an ARMA structure for the conditional mean process of the underlying time series. However, in many applications one often encounters conditional heteroskedasticity. In this article, we propose a new class of models, referred to as GARMA-GARCH models, that jointly specify both the conditional mean and conditional variance processes of a general non-Gaussian time series. Under the general modeling framework, we propose three specific models, as examples, for proportional time series, non-negative time series, and skewed and heavy-tailed financial time series. Maximum likelihood estimator (MLE) and quasi Gaussian MLE are used to estimate the parameters. Simulation studies and three applications are used to demonstrate the properties of the models and the estimation procedures.     

EXACT MAXIMUM LIKELIHOOD ESTIMATE AND LAGRANGE MULTIPLIER TEST STATISTIC FOR ARMA MODELS

Abstract. The paper provides a method for the computation of the derivatives of the exact log likelihood function of a Gaussian time series. Based on this result and using Fisher's scoring technique, an efficient method for computing the maximum likelihood estimates for an autoregressive moving average model has been obtained. Simulations suggest that the new procedure is as fast as the Box and Jenkins conditional least squares method. In a similar way, a procedure is derived to compute the Lagrange multiplier test statistics for testing the goodness of fit of the model.     

Local Gaussian Autocorrelation and Tests for Serial Independence

The traditional and most used measure for serial dependence in a time series is the autocorrelation function. This measure gives a complete characterization of dependence for a Gaussian time series, but it often fails for nonlinear time series models as, for instance, the generalized autoregressive conditional heteroskedasticity model (GARCH), where it is zero for all lags. The autocorrelation function is an example of a global measure of dependence. The purpose of this article is to apply to time series a well‐defined local measure of serial dependence called the local Gaussian autocorrelation. It generally works well also for nonlinear models, and it can distinguish between positive and negative dependence. We use this measure to construct a test of independence based on the bootstrap technique. This procedure requires the choice of a bandwidth parameter that is calculated using a cross validation algorithm. To ensure the validity of the test, asymptotic properties are derived for the test functional and for the bootstrap procedure, together with a study of its power for different models. We compare the proposed test with one based on the ordinary autocorrelation and with one based on the Brownian distance correlation. The new test performs well. Finally, there are also two empirical examples.     

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.     

Automatic identification of seasonal transfer function models by means of iterative stepwise and genetic algorithms

Abstract. In this article, we introduce an automatic identification procedure for transfer function models. These models are commonplace in time‐series analysis, but their identification can be complex. To tackle this problem, we propose to couple a nonlinear conditional least‐squares algorithm with a genetic search over the model space. We illustrate the performances of our proposal by examples on simulated and real data.     

Spatio‐temporal models for big multinomial data using the conditional multivariate logit‐beta distribution

We introduce a Bayesian approach for analyzing high‐dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio‐temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit‐beta distribution into the dependent multinomial data setting, which leads to conjugate full‐conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio‐temporal mixed effects model (MN‐STM). Additionally, we provide methodological developments including: the derivation of the associated full‐conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non‐stationary vector autoregressive model. We illustrate the MN‐STM through simulations and through a demonstration with public‐use quarterly workforce indicators data from the longitudinal employer household dynamics program of the US Census Bureau.     

Testing for parameter constancy in non‐Gaussian time series

This paper investigates testing for parameter constancy in models for non‐Gaussian time series. Models for discrete valued count time series are investigated as well as more general models with autoregressive conditional expectations. Both sup‐tests and CUSUM procedures are suggested depending on the complexity of the model being used. The asymptotic distribution of the CUSUM test is derived for a general class of conditional autoregressive models.     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.     

TESTING CHANGE-POINTS IN THE EXPLOSIVE GAUSSIAN AUTOREGRESSIVE PROCESSES

Abstract. A functional limit theorem with a particular function class and topology is derived for non-ergodic type time series. This limit theorem allows us to study the asymptotic law of the associated likelihood ratio test (LRT) statistic for testing the presence of a change in the covariance parameter in the explosive Gaussian autoregressive model. We show that the level of the LRT cannot be approximated without introducing appropriate normalization. The limit law of a particular weighted likelihood ratio test is examined through a simulation study and is compared with the well-known Kolmogorov distribution obtained in the stationary case; we conclude that for practical applications when the root is really close to unity one can use the same thresholds as in the stationary case. This procedure is applied to the study of three real time series known to be non-stationary.     

Bootstrap prediction intervals in state–space models

Abstract.  Prediction intervals in state–space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, with the true parameters substituted by consistent estimates. This approach has two limitations. First, it does not incorporate the uncertainty caused by parameter estimation. Second, the Gaussianity of future innovations assumption may be inaccurate. To overcome these drawbacks, Wall and Stoffer [ Journal of Time Series Analysis (2002) Vol. 23, pp. 733–751] propose a bootstrap procedure for evaluating conditional forecast errors that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. In this article, we propose a bootstrap procedure for constructing prediction intervals directly for the observations, which does not need the backward representation of the model. Consequently, its application is much simpler, without losing the good behaviour of bootstrap prediction intervals. We study its finite-sample properties and compare them with those of the standard and the Wall and Stoffer procedures for the local level model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series.     

ON BOOTSTRAP PREDICTIVE INFERENCE FOR AUTOREGRESSIVE PROCESSES

Abstract. In this paper we consider bootstrap-based predictive inference for autoregressive processes of order p. We consider both unconditional inference and inference conditional on the last p observed values. We make two contributions. Our first contribution is to point out the best way to apply the bootstrap to unconditional predictive inference when the process is Gaussian. Now, it may be argued that predictive inference for autoregressive processes of order p should be carried out conditional on the last p observed values. When the process is Gaussian, a bootstrap predictive inference conditional on the last p observed values is conveniently computed by 'running' the same autoregressive process backwards in time. This procedure is inappropriate for non-Gaussian autoregressive processes. Our second (and more important) contribution is to present a method (which is not computationally burdensome) for the computation of a bootstrap predictive inference for a non-Gaussian autoregressive process of order p conditional on the last p observed values.     

Conditional and Marginal Mutual Information in Gaussian and Hyperbolic Decay Time Series

We consider the amount of available information about an arbitrary future state of a Gaussian stochastic process. We derive an infinite series for the marginal mutual information in terms of the autocorrelation function. We derive an infinite series for the newly available information for prediction, the conditional mutual information, in terms of the moving average parameters, and directly characterize predictability in terms of sensitivity to random shocks. We apply our results to long memory, or more generally, hyperbolic decay models, and give information‐theoretic characterizations of the transition from persistence to anti‐persistence, stationary long memory to nonstationarity, and a stationary regime where the mutual information is not summable.     

A State space approach to bootstrapping conditional forecasts in arma models

A bootstrap approach to evaluating conditional forecast errors in ARMA models is presented. The key to this method is the derivation of a reverse-time state space model for generating conditional data sets that capture the salient stochastic properties of the observed data series. We demonstrate the utility of the method using several simulation experiments for the MA( q ) and ARMA( p, q ) models. Using the state space form, we are able to investigate conditional forecast errors in these models quite easily whereas the existing literature has only addressed conditional forecast error assessment in the pure AR( p ) form. Our experiments use short data sets and non-Gaussian, as well as Gaussian, disturbances. The bootstrap is found to provide useful information on error distributions in all cases and serves as a broadly applicable alternative to the asymptotic Gaussian theory.     

A CONDITIONAL LEAST SQUARES APPROACH TO BILINEAR TIME SERIES ESTIMATION

Abstract. In this paper a conditional least squares (CLS) procedure for estimating bilinear time series models is introduced. This method is applied to a special superdiagonal bilinear model which includes the classical linear autoregressive moving-average model as a particular case and it is proven that the limiting distribution of the CLS estimates is Gaussian and that the law of the iterated logarithm holds.     

Broadband Semiparametric Estimation of the Memory Parameter of a Long-Memory Time Series Using Fractional Exponential Models

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long-memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite-parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter-Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n )/ n , where n is the sample size. In a simulation study, we demonstrate the merits of using a finite-sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows' C_L criterion.     

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.     

Least absolute deviation estimation for general autoregressive moving average time‐series models

We study least absolute deviation (LAD) estimation for general autoregressive moving average time‐series models that may be noncausal, noninvertible or both. For ARMA models with Gaussian noise, causality and invertibility are assumed for the parameterization to be identifiable. The assumptions, however, are not required for models with non‐Gaussian noise, and hence are removed in our study. We derive a functional limit theorem for random processes based on an LAD objective function, and establish the consistency and asymptotic normality of the LAD estimator. The performance of the estimator is evaluated via simulation and compared with the asymptotic theory. Application to real data is also provided.     

Influence diagnostics for multivariate GARCH processes

This article presents diagnostics for identifying influential observations when estimating multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models. We derive influence diagnostics by introducing minor perturbations to the conditional variances and covariances. The derived diagnostics are applied to a bivariate GARCH model of daily returns of the S&P500 and IBM. We find that univariate diagnostic procedures may be unable to identify the influential observations in a multivariate model. Importantly, the proposed curvature‐based diagnostic identified influential observations where the correlation between the two series had a major change. These observations were not identified as influential using the univariate diagnostics for each asset separately. When estimating the bivariate GARCH model allowing for weights at influential observations, we found that the time‐varying correlations behaved differently from that implied by the model ignoring influential observations. The application therefore highlights the importance of extending univariate diagnostic procedures to multivariate settings.