Felix Chan

Structure and Asymptotic Theory for Multivariate Asymmetric Conditional Volatility

Various univariate and multivariate models of volatility have been used to evaluate market risk, asymmetric shocks, thresholds, leverage effects, and Value-at-Risk in economics and finance. This article is concerned with market risk, and develops a constant conditional correlation vector ARMA–asymmetric GARCH (VARMA–AGARCH) model, as an extension of the widely used univariate asymmetric (or threshold) GJR model of Glosten et al. (1992), and establishes its underlying structure, including the unique, strictly stationary, and ergodic solution of the model, its causal expansion, and convenient sufficient conditions for the existence of moments. Alternative empirically verifiable sufficient conditions for the consistency and asymptotic normality of the quasi-maximum likelihood estimator are established under non-normality of the standardized shocks.

GENERALIZED AUTOREGRESSIVE CONDITIONAL CORRELATION

This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poors 500, Nikkei, and Hang Seng indexes.

Estimating Smooth Transition Autoregressive Models with GARCH Errors in the Presence of Extreme Observations and Outliers

The paper investigates several empirical issues regarding quasi-maximum likelihood estimation of smooth transition autoregressive (STAR) models with GARCH errors (STAR-GARCH) and STAR models with smooth transition GARCH errors (STAR-STGARCH). Empirical evidence is provided to show that different algorithms produce substantially different estimates for the same model. Consequently, the interpretation of the model can differ according to the choice of algorithm. Convergence, the choice of different algorithms for maximizing the likelihood function, and the sensitivity of the estimates to outliers and extreme observations, are examined using daily data for S&P 500, Hang Seng and Nikkei 225 for the period January 1986 to April 2000.

Modelling trends and volatility in atmospheric carbon dioxide concentrations

Abstract Atmospheric carbon dioxide concentration (ACDC) is a crucial variable for many environmental simulation models, and is regarded as an important factor for predicting temperature and climate changes. However, the conditional variance of ACDC levels has not previously been examined. This paper analyses the trends and volatility in ACDC levels using monthly data from January 1965 to December 2002. The data are a subset of the well known Mauna Loa atmosphere carbon dioxide record obtained through the Carbon Dioxide Information Analysis Center. The conditional variance of ACDC levels is modelled using the generalised autoregressive conditional heteroscedasticity (GARCH) model and its asymmetric variations, namely the GJR and EGARCH models. These models are shown to be able to capture the dynamics in the conditional variance in ACDC levels and to improve the out-of-sample forecast accuracy of ACDC.

Modelling thresholds and volatility in US ecological patents

Ecological patents have been increasing steadily over time. This paper analyses trends and volatility in ecological patents in the USA from 1975 to 1997. Germany contributed more than 10% of the total US ecological patents, and is by far the strongest foreign performer. This paper estimates a set of novel regime-switching models to investigate the time-varying nature of the conditional mean, as well as the conditional variance of the patent ratio, namely the ratio of US ecological patents to total US patents, using monthly data from January 1975 to December 1997. The regime-switching LSTAR-GARCH model is found to be optimal for modelling the ecological patent ratio.

An econometric analysis of hotel?motel room nights in New Zealand with stochastic seasonality

Seasonality has attracted considerable interest in empirical tourism research and forecasting. However, the analysis of such recurring phenomenon is sparse in hospitality research, with very few studies to date having analysed seasonal unit roots prior to forecasting guest nights for the tourist-lodging industry. While pricing and other strategies have been implemented to extract greater revenue in the hospitality and tourism industry, it is also essential to focus the application of revenue or yield management based on seasonal demand analysis and forecasting. This paper examines the seasonality of hotel–motel room night occupancy patterns in New Zealand using monthly time series from 1997 to 2007. The presence of seasonal unit roots is detected using the Hylleberg, Engel, Granger and Yoo (HEGY) procedures. When the Box–Jenkins model of the HEGY-based transformed series is used to forecast hotel–motel room nights, its forecast performance is worse than that of the 12 differenced SARMA(2, 2)(0, 2) 12 model and the ARMA(2, 2) model for the original series.

Modelling time-varying higher moments with maximum entropy density

Since the introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model of Engle [R. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50 (1982) 987-1007], the literature of modelling the conditional second moment has become increasingly popular in the last two decades. Many extensions and alternate models of the original ARCH have been proposed in the literature aiming to capture the dynamics of volatility more accurately. Interestingly, the Quasi Maximum Likelihood Estimator (QMLE) with normal density is typically used to estimate the parameters in these models. As such, the higher moments of the underlying distribution are assumed to be the same as those of the normal distribution. However, various studies reveal that the higher moments, such as skewness and kurtosis of the distribution of financial returns are not likely to be the same as the normal distribution, and in some cases, they are not even constant over time. These have significant implications in risk management, especially in the calculation of Value-at-Risk (VaR) which focuses on the negative quantile of the return distribution. Failed to accurately capture the shape of the negative quantile would produce inaccurate measure of risk, and subsequently lead to misleading decision in risk management. This paper proposes a solution to model the distribution of financial returns more accurately by introducing a general framework to model the distribution of financial returns using maximum entropy density (MED). The main advantage of MED is that it provides a general framework to estimate the distribution function directly based on a given set of data, and it provides a convenient framework to model higher order moments up to any arbitrary finite order k. However this flexibility comes with a high cost in computational time as k increases, therefore this paper proposes an alternative model that would reduce computation time substantially. Moreover, the sensitivity of the parameters in the MED with respect to the dynamic changes of moments is derived analytically. This result is important as it relates the dynamic structure of the moments to the parameters in the MED. The usefulness of this approach will be demonstrated using 5min intra-daily returns of the Euro/USD exchange rate.

Rolling regressions and conditional correlations of foreign patents in the USA

Patent registrations have often been used as a proxy of innovation as they reflect a countrys technological capability. Recently, some studies have found that the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model and an asymmetric extension, namely Glosten, Jagannathan and Runkles (GJR) model, are useful to model the time-varying volatility of the patent ratio, namely the ratio of foreign patents registered in the USA to total patents in the USA. However, this approach assumes that the conditional variance is independent across countries. Furthermore, the time series properties of the patent growth rate, namely the rate of change of foreign patents registered in the USA, have not previously been analysed. This paper examines the conditional variance of the patent growth rate from the leading four foreign countries, namely Canada, France, Germany and Japan, using the Constant Conditional Correlation - Multivariate GARCH (CCC-MGARCH), Vector Autoregressive Moving Average - GARCH (VARMA-GARCH) and VARMA - Asymmetric GARCH (VARMA-AGARCH) models. The results reveal the existence of cross-countries effects in the patent growth rate among the leading four countries, as well as asymmetric effects using monthly data from January 1975 to December 1998. Rolling estimates show that the restrictive assumption of constant conditional correlation is unlikely to hold, and models that accommodate dynamic conditional correlations may provide greater insights for investigating the effects of global factors on changes in innovation for the four leading foreign countries.

Trends and volatilities in foreign patents registered in the USA

This study analyses the patent trends and volatilities for the top 12 foreign patenting countries in the US market from 1975 to 1997. Japan is ranked first in terms of foreign patents registered in the USA, followed by Germany. Patent registrations from each of these countries have increased steadily over time, but at different rates. Using monthly time series data for 1975–1997, the time-varying volatility of Australian, Japanese and German patents registered in the USA is examined in detail. The asymmetric AR(1)-GJR(1,1) model is found to be suitable for Australia and Japan, while the best fitting model for Germany is the symmetric AR(1)-GARCH(1,1) model.

Modelling the volatility transmission and conditional correlations between A and B shares in forecasting value-at-risk

The aim of this paper is to investigate the effect of the Chinese B share market reform on the conditional correlation and information transmission between A and B Shares issued in the Shanghai and Shenzen stock exchanges. Daily returns for the Shanghai A share index (SHA), Shanghai B share index (SHB), Shenzen A share index (SZA) and Shenzen B share index (SZB) are used for the period 6 October 1992 to 8 February 2005. The impact of the reform on the volatility spillovers and volatility transmission were found to be significant. The results also suggest that all pairs of conditional correlations increase dramatically over the period analysed, but such increases began well before the reforms to the B share market. The importance of accommodating such an increase in conditional correlations and changes in the information transmission mechanism when estimating value-at-risk (VaR) thresholds is analysed. The results suggest that accommodating the B share market reform may not be particularly important in empirical analyses of volatility transmission.