Andrew Tremayne

Estimation in conditional first order autoregression with discrete support

We consider estimation in the class of first order conditional linear autoregressive models with discrete support that are routinely used to model time series of counts. Various groups of estimators proposed in the literature are discussed: moment-based estimators; regression-based estimators; and likelihood-based estimators. Some of these have been used previously and others not. In particular, we address the performance of new types of generalized method of moments estimators and propose an exact maximum likelihood procedure valid for a Poisson marginal model using backcasting. The small sample properties of all estimators are comprehensively analyzed using simulation. Three situations are considered using data generated with: a fixed autoregressive parameter and equidispersed Poisson innovations; negative binomial innovations; and, additionally, a random autoregressive coefficient. The first set of experiments indicates that bias correction methods, not hitherto used in this context to our knowledge, are some-times needed and that likelihood-based estimators, as might be expected, perform well. The second two scenarios are representative of overdispersion. Methods designed specifically for the Poisson context now perform uniformly badly, but simple, bias-corrected, Yule-Walker and least squares estimators perform well in all cases.

Binomial thinning models for integer time series

This article considers some simple observation-driven time series models for counts. We provide a brief description of the class of integer-valued autoregressive (INAR) and integer-valued moving average (INMA) processes. These classes of models may be attractive when the data exhibit a significant serial dependence structure. We, therefore, briefly review various testing procedures useful for assessing the serial correlation in the data. Once it is established that the data are not serially independent, suitable INAR or INMA processes may be employed to model the data. In the important first order INAR model, we discuss various methods of estimating the structural parameters of the process. We also give a short account of the extension of some of these estimation procedures to second order INAR models. Moving average counterparts of both models are also entertained. Throughout, the models and methods are illustrated in the context of a famous data set from the branching process literature that turns out to be surprisingly difficult to model satisfactorily.

The wild bootstrap and heteroskedasticity-robust tests for serial correlation in dynamic regression models

Conditional heteroskedasticity is a common feature of financial and macroeconomic time series data. When such heteroskedasticity is present, standard checks for serial correlation in dynamic regression models are inappropriate. In such circumstances, it is obviously important to have asymptotically valid tests that are reliable in finite samples. Monte Carlo evidence reported in this paper indicates that asymptotic critical values fail to give good control of finite sample significance levels of heteroskedasticity-robust versions of the standard Lagrange multiplier test, a Hausman-type check, and a new procedure. The application of computer-intensive methods to removing size distortion is, therefore, examined. It is found that a particularly simple form of the wild bootstrap leads to well-behaved tests. Some simulation evidence on power is also given.

Testing for Serial Dependence in Time Series Models of Counts

In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first-order and second-order autoregressive as well as first-order moving average processes. Copyright 2003 Blackwell Publishing Ltd.

Convolution-Closed Models for Count Time Series with Applications

There has recently been an upsurge of interest in time series models for count data. Many papers focus on the model with first-order (Markov) dependence and Poisson innovations. Our paper considers practical models that can capture higher-order dependence based on the work of Joe (1996). In this framework we are able to model both equidispersed and overdispersed marginal distributions of data. The latter is approached using generalized Poisson innovations. Central to the models is the use of the property of closure under convolution of certain families of random variables. The models can be thought of as stationary Markov chains of finite order. Parameter estimation is undertaken by maximum likelihood, inference procedures are considered and means of assessing model adequacy employed. Applications to two new data sets are provided.

R-squared and prediction in regression with ordered quantitative response

Abstract This paper is concerned with the use of regression methods to predict values of a response variable when that variable is naturally ordered. An application to the prediction of student examination performance is provided and it is argued that, although individual scores are unlikely to be well predicted at the extremes of the range using the conditional mean, conditional on covariates, it is possible to usefully predict where an individual is likely to feature in the rank order of performance.

Exploratory data analysis and model criticism with posterior plots

The use of techniques of exploratory data analysis and model criticism represent important stages in many statistical investigations. One of the attractive features of a Bayesian analysis is that it can lend itself well to graphical summary. To produce this graphical summary it is generally necessary to restrict attention to a small number of key parameters. The graphical approach described can be adopted whenever an appropriate likelihood function can be specified. Solutions to some of the principal computational problems associated with implementing a graphical Bayesian analysis based on posterior plots are presented. Nuisance parameters are handled in two ways: by incorporating them directly into the computation of exact posterior distributions; and by integrating them out of a conditional analysis at an early stage when the former approach is infeasible. The latter proposal facilitates the handling of higher dimensional nuisance parameter vectors. Examples taken from the areas of time series and microeconomics are presented to illustrate the efficacy of the approach.