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Traditional GARCH models describe volatility levels that evolve smoothly over time, generated by a single GARCH regime. However, nonstationary time series data may exhibit abrupt changes in volatility, suggesting changes in the underlying GARCH regimes. Further, the number and times of regime changes are not always obvious. This article outlines a nonparametric mixture of GARCH models that is able to estimate the number and time of volatility regime changes by mixing over the Poisson-Kingman process. The process is a generalisation of the Dirichlet process typically used in nonparametric models for time-dependent data provides a richer clustering structure, and its application to time series data is novel. Inference is Bayesian, and a Markov chain Monte Carlo algorithm to explore the posterior distribution is described. The methodology is illustrated on the Standard and Poor's 500 financial index.

Generalised autoregressive conditional heteroscedastic (GARCH) models estimate time-varying fluctuations around mean levels of a time series known as the volatility of a time series [

The Dirichlet process [

Theoretical developments and recent applications of the PKSS process are discussed in Lijoi et al. [

The methodology is illustrated through volatility and predictive density estimation of a GARCH(1,1) model applied to the Standard and Poor’s 500 financial index from 2003 to 2009. Results are compared between a no-mixture model and nonparametric mixtures over the Dirichlet, PD, and NGG processes. Under the criterion of marginal likelihood the NGG process performs the best. Also, the PD and NGG process outperforms the previously studied Dirichlet process which in turn outperforms the no-mixture model. The results suggest that alternatives to the Dirichlet process should be considered for applications of nonparametric mixture models to time-dependent data.

The paper proceeds as follows. Section

Let

We write

In Lau and So [

Understanding conditions for stationarity of a time series model is fundamental for statistical inference. Since our model is specified with zero mean over time, we provide a necessary and sufficient condition for the existence of a secondary order stationary solution for the infinite mixture of GARCH(1,1) models. The derivation closely follows Embrechts et al. [

Now consider the first two conditional moments from model (

The unconditional second moment could be derived according to this representation by marginalising over all the random variates. Also,

Finally, one might be also interested in the connection between models such as (

We now describe PKSS process and detail the Dirichlet, the PD, and NGG processes to illustrate how the more general PKSS process allows for richer clustering mechanisms. Let

All random probability measures within the class of PKSS processes feature an almost surely discrete probability measure represented as

A common characterization of (

Let

The

The PKSS process can be represented as either the Dirichlet, PD, or NGG processes in (

Taking

Setting that

The NGG process takes

In the above

Turning to the distribution of partitions, Pitman [

Notice that the joint distribution is dependent on the partition

We emphasise that we can always express

In time series analysis, we usually consider the first

Our Markov chain Monte Carlo (MCMC) sampling procedure generates distinct values and partitions alternatively from the posterior distribution,

Initialise

For

Generate

Generate

End.

To obtain our estimates we use the weighted Chinese restaurant Gibbs type process introduced in Lau and So [

The main idea of this algorithm is the “leave one out” principle that removes item

Define

To generate from

In (

We note that the special case of the above algorithm can be found for independent data of the normal mixture models from West et al. [

Usually, the parameters of interest are both the sequence

We now outline how the algorithms of Walker [

Furthermore, we can take the classification variables

The likelihood (

Papaspiliopoulos and Roberts [

The methodology is illustrated on the daily logarithm returns of the S&P500 (Standard & Poor’s 500) financial index, dated from 2006 Jan 03 to 2009 Dec 31. The data contains a total of 1007 trading days and is available from the Yahoo Finance (URL:

To compare the three different mixture models in Section

Figure

Volatility estimates.

Predictive density estimates.

Finally, we evaluate the goodness of fit in terms of the marginal likelihoods. The logarithm of the marginal likelihoods of the no mixture model, the Dirichlet process model, PD process model, and the NGG process model are −1578.085, −1492.086, −1446.275, and −1442.269, respectively. Under the marginal likelihood criterion all three mixture models outperform the GARCH(1,1). Further, the NGG process outperforms the PD process which in turn outperforms the model proposed in Lau and So [

In this paper we have extended nonparametric mixture modelling for GARCH models to the Kingman-Poisson process. The process includes the previously applied Dirichlet process and also includes the Poisson-Dirichlet and Normalised Generalised Gamma process. The Poisson-Dirichlet and Normalised Generalised Gamma process provide richer clustering structures than the Dirichlet process and have not been previously adapted to time series data. An application to the S&P500 financial index suggests that these more general random probability measures can outperform the Dirichlet process. Finally, we developed an MCMC algorithm that is easy to implement which we hope will facilitate further investigation into the application of nonparametric mixture modes to time series data.