Double Generalized Threshold Models with constraint on the dispersion by the mean

doi:10.1016/j.csda.2014.08.003

Computational Statistics & Data Analysis

Volume 82, February 2015, Pages 59-73

https://doi.org/10.1016/j.csda.2014.08.003 Get rights and content

Abstract

Generalized Threshold Model (GTM) is a non-linear time series model which generalizes the Threshold Autoregressive Model (TAR) to implement the idea of the Generalized Linear Model under the threshold time series framework. However, the dispersion parameter is usually assumed as constant in the context of Generalized Linear Model which does not hold in general. In this paper, the GTM is extended to a Double Generalized Threshold Model (DGTM) where the dispersion parameter, defined as the expected deviance of the individual response about its mean, varies throughout the entire sample. The variation of the dispersion parameter can be predicted by another threshold type generalized linear model, which is interlinked with the threshold model for the mean and can be estimated simultaneously.

Introduction

The Threshold Autoregressive Model (TAR) proposed by Tong and Lim (1980) is one of the most popular proposed and discussed non-linear time series approaches. The Generalized Threshold Model (GTM) suggested by Samia et al. (2007) and Samia and Chan (2011) generalizes the TAR to responses of the entire exponential family and implements the idea of the Generalized Linear Model (Nelder and Wedderburn, 1972) under the threshold time series framework and they proved that the estimates of the model have desirable properties.

Nevertheless, the dispersion is often considered as a constant in the GLM framework and hence also for the GTM though varying dispersion is a common empirical phenomenon. Wedderburn (1974), McCullagh (1983), Nelder and Pregibon (1987), Smyth (1989), Smyth and Verbyla, 1996, Smyth and Verbyla, 1999 and Nelder et al. (1998) discussed modeling dispersion as the expected deviance of the individual response and its mean by regression models. Following these earlier works, the dispersion parameter in this paper is modeled by another threshold model which is interlinked with the threshold model for the mean and can be estimated simultaneously.

In this paper, a Double Generalized Threshold Model (DGTM) with heterogeneous dispersion will be introduced. In Section 2, the proposed approach will be motivated by explaining the application of a non-linear heteroscedastic model on a selected case study. Model formulation will be provided in Section 3, followed by a description of its estimation algorithm in Section 4. Furthermore, a score test for checking the occurrence of varying dispersion will be discussed in detail in Section 5. Simulations and case studies illustrate the advantage of dispersion modeling in Sections 6 Simulation study, 7 Case study, respectively. The paper will be rounded up with a conclusion and outlook of future research possibilities.

Section snippets

Motivation

In the framework of linear modeling, both time series and ordinary general linear models, the error or noise term $ϵ_{t}$ has been considered as a normally distributed random variable with mean 0 and variance $σ^{2}$ . Nevertheless, with the proposal of the Autoregressive Conditional Heteroscedastic (ARCH) model (Engle, 1982) and the Generalized ARCH model (Bollerslev, 1986), the residual $ϵ_{t}$ can be formulated as $ϵ_{t} = e_{t} σ_{t}$ , where $e_{t}$ is a standard normally distributed random variable and $σ_{t}^{2}$ is the

Mean submodel

The Generalized Threshold Model (GTM) relaxes the Gaussian error assumption of the Threshold AR model and allows a time series of length $T, {Y_{t}}$ , to switch between $n$ different generalized linear models depending on whether the threshold process, ${τ_{t}}$ at time $t - d$ falls into an interval $(r^{(j)}, r^{(j + 1)}]$ or not, where $d$ denotes the delay parameter and $r^{(j)}, r^{(j + 1)}$ are threshold values for the $j$ th regime.

Without loss of generality the formulation of a model with $n = 2$ regimes will be discussed in the

Parameter estimation and asymptotic properties

Let $D = {1, 2, \dots, D}$ be the parameter space of $d$ where $D < T$ is the highest possible time delay of the threshold process ${τ_{t}}$ . Because of its integral characteristic no function of $d$ is differentiable. Given known parameters $Θ = {\tilde{r}, \tilde{β}, \tilde{γ}, \tilde{λ}}$ and the data ${Y_{t}, τ_{t}}$ , the maximum likelihood estimate ${\hat{d}}_{M L} \in D$ is obtained as the argument such that: ${\hat{L}}_{Y} ({\hat{d}}_{M L}; Y, X) = max_{d} L_{Y} (d; Y, X) .$

Similarly, the maximum likelihood estimate of the threshold value $r$ can be found as follows: let $τ_{t}^{*}$ be the sorted version of the

Varying dispersion diagnostic

The discussion in the previous sections assumes that varying dispersion occurs and can be captured by a dispersion submodel. However, if the assumption is false and $ϕ$ is time-invariant, the dispersion submodel depends only on the intercept, $g_{ϕ} (ϕ_{t}) = γ_{0}$ , where $γ_{0}$ is a scalar. Let $γ = (γ_{0} | γ_{1}^{⊺})$ and $Z = (Z_{0} | Z_{1})$ denote the partitioned forms of $γ$ and $Z$ , respectively, where $Z_{0}$ and $γ_{0}$ represent the covariates and coefficients to which the model is restricted whereas $Z_{1}$ and $γ_{1}$ contain the remaining covariates

Simulation study

To check the asymptotic properties of the model, simulations of 1000 replicates each have been carried out for time series with 200, 400 and 800 Gamma distributed observations. Each series ${Y_{t}}$ was generated by the same mean submodel as defined in the varying dispersion diagnostic simulations in Section 5 except that the dispersion in each simulation is generated by the following two-regime model: $log (ϕ_{t}) = {\begin{matrix} - 1.5 - 0.6 z_{1 t} - 0 . 3_{2 t} - η_{t}, & τ_{t - 2} \leq 0.5, \\ 1 - 0.2 z_{1 t} + 0.8 z_{2 t} + 2 η_{t}, & τ_{t - 2} > 0.5 . \end{matrix}$

Furthermore, two scenarios

Case study

The maximum radiation levels in Tokyo of every three hours subsequent to the nuclear disaster in Fukushima were considered. The data are published in the website of the Tokyo Metropolitan Institution of Public Health (2011) (http://monitoring.tokyo-eiken.go.jp/monitoring/index-e.html) starting from 15th March 2011. To model the data by the DGTM, we assume that the process is piecewise linear autoregressive, highly depending on the level of nuclear leakage from Fukushima Daiichi with a certain

Conclusion

In this paper, we focus on the joint modeling of the mean and dispersion under the framework of threshold time series model. We are able to show that the goodness of fit improves and more efficient estimates of the model parameters of the mean component can be obtained if the varying dispersion assumption is adopted. Models with varying dispersion are no longer a luxurious add-on option to those classical approaches. In fact, by formulating and carrying out the score test on testing the

Acknowledgments

The authors thank the two anonymous referees for their very helpful comments and suggestions. W. K. Li’s research is partially supported by HKSAR Research Grants Council GRF grantHKU703711P.

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Double Generalized Threshold Models with constraint on the dispersion by the mean

Abstract

Introduction

Section snippets

Motivation

Mean submodel

Parameter estimation and asymptotic properties

Varying dispersion diagnostic

Simulation study

Case study

Conclusion

Acknowledgments

J. Econometrics

Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model

Ann. Statist.

Limiting properties of the least squares estimator of a continuous threshold autoregressive model

Biometrika

Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation

Econometrica

Sample splitting and threshold estimation

Econometrica

Exponential dispersion models

J. R. Stat. Soc. Ser. B Stat. Methodol.

Asymptotic theory on the least squares estimation of threshold moving-average models

Economic Theory

Estimation and testing stationarity for double-autoregressive models

J. R. Stat. Soc. Ser. B Stat. Methodol.

Quasi-likelihood functions

Ann. Statist.

Joint modelling of mean and dispersion

Technometrics