C. S. Wong

On a mixture autoregressive model

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.

On a Mixture Autoregressive Conditional Heteroscedastic Model

We propose a mixture autoregressive conditional heteroscedastic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture of K autoregressive components with autoregressive conditional heteroscedasticity; that is, the conditional mean of the process variable follows a mixture AR (MAR) process, whereas the conditional variance of the process variable follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MARARCH model allows a more flexible squared autocorrelation structure. The stationarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distributions is discussed. The estimation can be easily done through a simple EM algorithm, and the model selection problem is addressed. The shape-changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real datasets and compared to other competing models. The MAR-ARCH models appear to capture features of the data better than the competing models.

A note on the corrected Akaike information criterion for threshold autoregressive models

A bias-corrected Akaike information criterion AICC is derived for self-exciting threshold autoregressive (SETAR) models. The small sample properties of the Akaike information criteria (AIC, AICC) and the Bayesian information criterion (BIC) are studied using simulation experiments. It is suggested that AICC performs much better than AIC and BIC in small samples and should be put in routine usage.

Modelling Australian interest rate swap spreads by mixture autoregressive conditional heteroscedastic processes

The observed difference between the swap rate and the government bond yield of corresponding maturity is known as the swap spread. The swap spread reflects the risk premium that is involved in a swap transaction instead of holding risk-free government bonds. It is primarily composed of the liquidity risk premium and the credit risk premium. In recent years there has been growing interest in modelling swap spreads because the swap spread is the key pricing variable for the swap rate. The Australian interest rate swap market is the most important over-the-counter (OTC) derivative market in Australia. In this paper we apply the class of mixture autoregressive conditional heteroscedastic (MARCH) models to three (3-year, 5-year and 10-year) swap spread series in Australia. The MARCH model is able to capture both of the stylised characteristics of the observed changes of the swap spread series: volatility persistence and the dependence of volatility on the level of the data. The proposed MARCH model also allows for regime switches in the swap spreads.

Stress Testing Banks' Credit Risk Using Mixture Vector Autoregressive Models

This paper estimates macroeconomic credit risk of banksi¦ loan portfolio based on a class of mixture vector autoregressive models. Such class of models can differentiate distributions of default rates and macroeconomic conditions for different market situations and can capture their dynamics evolving over time, including the feedback effect from an increase in fragility back to the macroeconomy. These extensions can facilitate the evaluation of credit risks of loan portfolio based on different credit loss distributions.

Modeling Hong Kong's stock index with the Student t-mixture autoregressive model

It is well known that financial returns are usually not normally distributed, but rather exhibit excess kurtosis. This implies that there is greater probability mass at the tails of the marginal or conditional distribution. Mixture-type time series models are potentially useful for modeling financial returns. However, most of these models make the assumption that the return series in each component is conditionally Gaussian, which may result in underestimates of the occurrence of extreme financial events, such as market crashes. In this paper, we apply the class of Student t-mixture autoregressive (TMAR) models to the return series of the Hong Kong Hang Seng Index. A TMAR model consists of a mixture of g autoregressive components with Student t-error distributions. Several interesting properties make the TMAR process a promising candidate for financial time series modeling. These models are able to capture serial correlations, time-varying means and volatilities, and the shape of the conditional distributions can be time-varied from short- to long-tailed or from unimodal to multi-modal. The use of Student t-distributed errors in each component of the model allows for conditional leptokurtic distribution, which can account for the commonly observed unconditional kurtosis in financial data.

On a constrained mixture vector autoregressive model

A mixture vector autoregressive model has recently been introduced to the literature. Although this model is a promising candidate for nonlinear multiple time series modeling, high dimensionality of the parameters and lack of method for computing the standard errors of estimates limit its application to real data. The contribution of this paper is threefold. First, a form of parameter constraints is introduced with an efficient EM algorithm for estimation. Second, an accurate method for computing standard errors is presented for the model with and without parameter constraints. Lastly, a hypothesis-testing approach based on likelihood ratio tests is proposed, which aids in the selection of unnecessary parameters and leads to the greater efficiency at the estimation. A case study employing U.S. Treasury constant maturity rates illustrates the applicability of the mixture vector autoregressive model with parameter constraints, and the importance of using a reliable method to compute standard errors.

AN APPLICATION OF THE MIXTURE AUTOREGRESSIVE MODEL: A CASE STUDY OF MODELLING YEARLY SUNSPOT DATA

Many nonlinear time series models have been proposed in the literature in the past two decades. Numerous successful applications of these models are reported. These models usually specify a nonlinear conditional mean and/or variance function and assume a Gaussian conditional distribution. Under the normality assumption, the marginal and/or conditional distributions of the time series are unimodal and symmetric. The unimodal limitation persists even the Gaussian distribution is replaced by a heavy-tailed distribution such as the Student t distribution. In some real-life examples, a multimodal conditional distribution may seem more appropriate than a unimodal conditional distribution. The mixture distributions has a comparatively long history in the analysis of independent data. See, for example, Titterington et a2.l Le et aL2