Comment on ”Some recent theory for autoregressive count time series” by Dag Tjøstheim Konstantinos Fokianos Department of Mathematics & Statistics, University of Cyprus e-mail:
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April 26, 2012 The contribution by D. Tjøstheim reviews some probabilistic tools developed over the last years for the statistical analysis of count time series. As the author explains very vividly, large sample properties of maximum likelihood estimation for all type of models considered in the paper, are based upon proving a weak law of large numbers, a central limit theorem and existence of moments. All these facts and tools are standard and have been developed by numerous authors for several diverse models utilizing notions like mixing and Markov chain theory, for example. Unfortunately, such machinery was not available until recently for count time series models. These models fall under the broad framework of time series following generalized linear models as described in Fahrmeir and Tutz (2001, Ch.7) and Kedem and Fokianos (2002, Ch. 1–5). Chapter 4 of the last reference contains several models for count time series; however, as Kedem and Fokianos (2002) point out, the log-linear model is more natural since it corresponds to the canonical link function for the Poisson distribution. In particular, the last reference states Assumption A (see Kedem and Fokianos (2002, pp.16–17)) which in essence summarizes the needed conditions for asymptotic normality to hold. The same type of conditions have been used in early works by Slud and Kedem (1994) and Fokianos and Kedem (1998) for the study of binary and categorical time series, respectively. In a nutshell, this particular assumption calls for ergodicity of the covariates (see A.4). In addition, the covariates satisfy a condition which yields a positive definite information matrix and they are assumed to be bounded (see A.2). The other assumption items are standard in applications and such type of conditions are satisfied for most of the models considered in this article. These stipulations can be safely abandoned by the recent theory which is outlined in this paper, at least 1
in the context of modeling count time series. We point out that most regression models considered in Kedem and Fokianos (2002, Ch 1-4) do not contain a feedback mechanism like the models considered in this work. Hence, the results that we have been obtained so far and outlined by D. Tjøstheim are more general. However, the models discussed in Kedem and Fokianos (2002, Ch 1-4) always include covariates. As the author points out, there are several other models for the statistical analysis of count time series; integer autoregressive processes (INAR) being the most prominent ones; see Al-Osh and Alzaid (1987). I share though the same point of view with the author that models like the ones discussed in this article generalize naturally the standard ARMA theory (see Brockwell and Davis (1991), for instance) to the exponential family framework. This point has been made by Kedem and Fokianos (2002) and elaborated recently further by Fokianos (2012) in the context of count time series. In fact, combination of generalized linear models and likelihood theory provide a natural framework for the analysis of quantitative as well as qualitative time series data. Estimation, diagnostics, model assessment, and forecasting are implemented easily and computations are carried out by various available software. These issues are addressed in the list of desiderata suggested in Davis et al. (1999) and Zeger and Qaqish (1988). Model fitting for count time series is based on the autocorrelation function of the observed data. It is rather a stylistic fact of count time series that they usually exhibit a positive sample autocorrelation function. To be more specific, suppose that {Yt } denotes a count time series, which is conditionally distributed as a Poisson with mean {λt }. Then a model for the mean process {λt } includes either lagged variables of a function of {Yt } or/and lagged variables of a function of {λt }. Models that fall under the first case are usually entertained when a visual inspection of the sample autocorrelation function indicates a rapid decay of its values. Models which include past values of {λt } are employed when the sample autocorrelation function decays slowly. They are expected to be more parsimonious than models which include only lagged variables of a function of {Yt }’s; this is along the lines of GARCH modeling; Bollerslev (1986). The chosen model belongs to the class of observation driven models in the sense that given starting values for λt , then the mean process {λt } is completely recovered as a function of past values of {Yt } after repeated substitution; Cox (1981). As this article indicates, there are two main approaches so far, for developing the theory for autoregressive counts. The first one is based on Markov chain theory and a perturbation argument. We note here that the work by Meitz and Saikonnen (2008) is a basic tool. Note that Proposition 3.1 shows that the unobserved perturbed process {λm t } is geometrically ergodic Markov chain with finite moments under the assumed conditions on the parameter vector. This fact implies that the joint process {(Ytm , λm t , Ut )} is also geometrically ergodic Markov chain under the same conditions. For the linear (9), it is rather uncomplicated to show that when a + b < 1, then the perturbed and unperturbed processes are close to each other in the sense of Lemma 3.3. Furthermore, it is interesting to note that the essential assumption needed for the theory to be developed is based on a contraction condition used by Fokianos and Tjøstheim (2012),
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Doukhan et al. (2012), Neumann (2011) and Franke (2010). In particular, Doukhan et al. (2012) and Franke (2010) use the concept of weak dependence to study the probabilistic properties of count autoregressive models. Such an approach avoids the use of perturbation lemma and still provides the necessary tools for developing estimation and testing. The situation is quite different for the loglinear model (21). With an approach based on perturbation, the ergodicity conditions are quite different than those needed for the approximation lemma to hold. This is the same phenomenon reported in the papers by Davis et al. (2003) and Davis et al. (2005). Furthermore, the weak dependence concept might not be applicable because the logarithm does not possess the contraction property. However, in applications, and when positive correlation is observed, we anticipate that both linear and log-linear models will give similar results in terms of prediction. The application of the loglinear model, though natural for count time series, will be superior to the performance of linear models when there exists negative correlation among data. As the author indicates, there are several extensions of this work. I completely agree with the research guidelines that he puts forward and I complement those ideas by including a simple example of a binary time series. More specifically, suppose that a binary time series–that is a time series which can take only two values, say 0 and 1–is available. Let the transition probability given the past be denoted by pt ; that is pt = P (Yt = 1 | Ft−1 ). Because the resulting conditional distribution is Bernoulli, the canonical parameter
is defined by the log–odds ratio θt = log pt /(1 − pt ) . Therefore, in analogy to model (21) discussed in this article, we can study the properties of an auto–logistic model which takes the form θt = d + aθt−1 + bYt−1 . Another interesting model is obtained when we define θt = Φ−1 (pt ), where Φ(·) is the cdf of the standard normal distribution. With this specification, we obtain an auto–probit type of model. In fact, the last example, hints that we can replace the cumulative distribution function of the standard normal distribution with any other cumulative distribution function. Some other interesting examples for binary time series are given by the extreme value distributions. Models like those for binary data, can be developed for categorical time series. As a final remark, the existence of a universal condition which guarantees ergodicity and stationarity for time series following generalized linear models in general, seems hard to obtain. Each different data structure will pose specific problems and existence of such conditions might be true for very special cases of distributions and link functions.
References Al-Osh, M. A. and A. A. Alzaid (1987). First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8, 261–275. 3
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Brockwell, P. J. and R. A. Davis (1991). Time Series: Data Analysis and Theory (2nd ed.). New York: Springer. Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics 8, 93–115. Davis, R. A., W. T. M. Dunsmuir, and S. B. Streett (2003). Observation-driven models for Poisson counts. Biometrika 90, 777–790. Davis, R. A., W. T. M. Dunsmuir, and S. B. Streett (2005). Maximum likelihood estimation for an observation driven model for Poisson counts. Methodol. Comput. Appl. Probab. 7, 149–159. Davis, R. A., Y. Wang, and W. T. M. Dunsmuir (1999). Modelling time series of count data. In S. Ghosh (Ed.), Asymptotics, Nonparametric & Time Series, pp. 63–114. New York: Marcel Dekker. Doukhan, P., K. Fokianos, and D. Tjøstheim (2012). On weak dependence conditions for Poisson autoregressions. Statistics & Probability Letters 82, 942–948. Fahrmeir, L. and G. Tutz (2001). Multivariate Statistical Modelling Based on Generalized Linear Models (2nd ed.). New York: Springer. Fokianos, K. (2012). Count time series. In C. Rao and T. S. Rao (Eds.), Handbook of Statistics: Time Series Methods and Applications, Volume 30. Amserdam: Elsevier B. V. to appear. Fokianos, K. and B. Kedem (1998). Prediction and classification of non-stationary categorical time series. Journal of Multivariate Analysis 67, 277–296. Fokianos, K. and D. Tjøstheim (2012). Nonlinear Poisson autoregression. to appear in Annals of the Institute for Statistical Mathematics. Franke, J. (2010). Weak dependence of functional INGARCH processes. unpublished manuscript. Kedem, B. and K. Fokianos (2002). Regression Models for Time Series Analysis. Hoboken, NJ: Wiley. Meitz, M. and P. Saikonnen (2008). Ergodicity, mixing and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory 24, 1291–1320. Neumann, M. (2011). Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268– 1284.
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Slud, E. V. and B. Kedem (1994). Partial likelihood analysis of logistic regression and autoregression. Statistica Sinica 4, 89–106. Zeger, S. L. and B. Qaqish (1988). Markov regression models for time series: a quasi-likelihood approach. Biometrics 44, 1019–1031.
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