Conditional Autoregressive Range Model Research Articles

Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models

This paper extends the conditional autoregressive range (CARR) model to the multivariate CARR (MCARR) model and further to the two-stage MCARR-return model to model and forecast volatilities, correlations and returns of multiple financial assets. The first stage model fits the scaled realised Parkinson volatility measures using individual series and their pairwise sums of indices to the MCARR model to obtain the fitted volatilities. Then covariances are calculated to construct the fitted variance–covariance matrix of returns which are imputed into the stage-two return model to capture the heteroskedasticity of assets’ returns. We investigate different choices of mean functions to describe the volatility dynamics. Empirical applications are based on the Standard and Poor 500, Dow Jones Industrial Average and Dow Jones United States Financial Service Indices. Results show that the stage-one MCARR models using asymmetric mean functions give better in-sample model fits than those based on symmetric mean functions. They also provide better out-of-sample volatility forecasts than those using CARR models based on two robust loss functions. We also find that the stage-two return models with constant means and multivariate Student-t errors give better in-sample fits than the Baba–Engle–Kraft–Kroner generalised autoregressive conditional heteroskedasticity models. The estimates and forecasts of value-at-risk (VaR) and conditional VaR based on the best MCARR-return models for each asset are provided and tested using backtests to confirm the accuracy of the VaR forecasts.

NEW LOSS RESERVE MODELS WITH PERSISTENCE EFFECTS TO FORECAST TRAPEZOIDAL LOSSES IN RUN-OFF TRIANGLES

AbstractModelling loss reserve data in run-off triangles is challenging due to the complex but unknown dynamics in the claim/loss process. Popular loss reserve models describe the mean process through development year, accident year, and calendar year effects using the analysis of variance and covariance (ANCOVA) models. We propose to include in the mean function the persistence terms in the conditional autoregressive range model for modelling the persistence of claim across development years. In the ANCOVA model, we adopt linear trends for the accident and calendar year effects and a quadratic trend for the development year effect. We investigate linear or log-transformed mean functions and four distributions, namely generalised beta type 2, generalised gamma, Weibull, and exponential extension, with positive support to enhance the model flexibility. The proposed models are implemented using the Bayesian user-friendly package Stan running in the R environment. Results show that the models with log-transformed mean function and persistence terms provide better model fits. Lastly, the best model is applied to forecast partial loss reserve and calendar year reserve for three years.

Range-based volatility forecasting: a multiplicative component conditional autoregressive range model

To capture the effect in volatility, a multiplicative component conditional autoregressive range (MCCARR) model is proposed. We show theoretically that the MCCARR model can capture the long-memory effect well. An empirical study is performed on the Standard & Poor's 500 index, and the results show that the MCCARR model outperforms both conditional autoregressive range and heterogeneous autoregressive models for in-sample and out-of-sample volatility forecasting.

Birnbaum‐Saunders autoregressive conditional range model applied to stock index data

AbstractThis article proposes a new approach to the conditional autoregressive range (CARR) model using the Birnbaum‐Saunders (BS) distribution. The model aims to develop volatility clustering, which incorporates extreme fluctuations, using a time‐varying evolution of the range process called the BSCARR model. Furthermore, diagnosis analysis tools for diagnosis analysis were developed to evaluate the goodness of fit, such as residual analysis, global influence measures based on Cook's distance, and local influence analysis. For illustrative purposes, three real financial market indices are analyzed. A comparison with classical CARR models was also carried out in these examples. The results indicated that the proposed model outperformed some existing models in the literature, especially a recent CARR model based on the gamma distribution even under the presence of atypical cases (observed values).

Forecasting volatility with component conditional autoregressive range model

In this paper, we propose a component conditional autoregressive range (CCARR) model for forecasting volatility. The proposed CCARR model assumes that the price range comprises both a long-run (trend) component and a short-run (transitory) component, which has the capacity to capture the long memory property of volatility. The model is intuitive and convenient to implement by using the maximum likelihood estimation method. Empirical analysis using six stock market indices highlights the value of incorporating a second component into range (volatility) modelling and forecasting. In particular, we find that the proposed CCARR model fits the data better than the CARR model, and that it generates more accurate out-of-sample volatility forecasts and contains more information content about the true volatility than the popular GARCH, component GARCH and CARR models.

Volatility forecasting using stochastic conditional range model with leverage effect

AbstractIn this paper, we propose a stochastic conditional range model with leverage effect (henceforth SCRL) for volatility forecasting. A maximum likelihood method based on the particle filters is developed to estimate the parameters of the SCRL model. Simulation results show that the proposed methodology performs well. We apply the proposed model and methodology to four stock market indices, the Shanghai Stock Exchange Composite Index of China, the Hang Seng Index of Hong Kong, the Nikkei 225 Index of Japan, and the S&P 500 Index of US. Empirical results highlight the value of incorporating leverage effect into range modeling and forecasting. In particular, the results show that our SCRL model outperforms the conditional autoregressive range model, the conditional autoregressive range model with leverage effect, and the stochastic conditional range model in both in‐sample fit and out‐of‐sample forecast.

Bayesian return forecasts using realised range and asymmetric CARR model with various distribution assumptions

A popular technique for measuring financial risk is to apply generalised autoregressive conditional heteroskedastic (GARCH)-type models to return-based time series. However recent studies are more focused on estimating volatility using the realised range calculated from high-frequency data. Making use of this efficient volatility measure, this paper analyses returns using a two-stage model: the first stage fits the realised range measures to the conditional autoregressive range (CARR) model whereas the second stage inputs the fitted values as observed volatilities in the return model. On modelling choices, we investigate how the model performance can be improved by different choices of error distributions and mean functions. We also study the effect of interval size on the realised range measures. A Bayesian Markov chain Monte Carlo approach via Rstan is used to estimate the parameters of these models. Empirical applications are based on three market indices. Results show that the CARR model with generalised beta type II distribution provides the most efficient modelling of volatility for all data. We also find that the realised range calculated using the most frequent 5 min intervals provides accurate estimates and forecasts of value-at-risk (VaR) and tail conditional VaR for both range and return than the daily range for all market indices.

Range-based volatility forecasting: an extended conditional autoregressive range model

This paper proposes an extended conditional autoregressive range (EXCARR) model to describe the range-based volatility dynamics of financial assets. Our EXCARR model not only takes the conditional autoregressive range (CARR) model as a special case but also considers the asymmetry between the upward range and the downward range. Empirical studies performed on a variety of stock indexes show that the EXCARR model outperforms not only the CARR model but also the asymmetric CARR (ACARR) model in both in-sample and out-of-sample forecasting. Hence, our EXCARR model provides a new benchmark for range-based volatility forecasting.

Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models

Abstract Risk measures such as value-at-risk (VaR) and expected shortfall (ES) may require the calculation of quantile functions from quantile regression models. In a parametric set-up, we propose to regress directly on the quantiles of a distribution and demonstrate a method through the conditional autoregressive range model which has increasing popularity in recent years. Two flexible distribution families: the generalised beta type two on positive support and the generalised-t on real support (which requires log transformation) are adopted for the range data. Then the models are extended to allow the volatility dynamic and compared in terms of goodness-of-fit. The models are implemented using the module fmincon in Matlab under the classical likelihood approach and applied to analyse the intra-day high-low price ranges from the All Ordinaries index for the Australian stock market. Quantiles and upper-tail conditional expectations evaluated via VaR and ES respectively are forecast using the proposed models.

Quantile range-based volatility measure for modelling and forecasting volatility using high frequency data

Volatility of asset prices in financial market is not directly observable. Return-based models have been proposed to estimate the volatility using daily closing prices. Recently, many new range-based volatility measures were proposed to estimate the financial volatility. This paper proposes quantile Parkinson (QPK) measure to estimate daily volatility and to show how it can robustify the Parkinson (PK) measure in the presence of intraday extreme returns. Results from extensive simulation studies show that the QPK measure is more efficient than intraday (open-to-close) squared returns and PK measures in the presence of intraday extreme returns. To demonstrate the applicability of QPK measure, we analyse the daily Standard and Poor 500 index by fitting the QPK measure to the conditional autoregressive range (CARR) models. Results show that choosing a suitable interquantile level width for the QPK measure will reduce its variance and hence improve its efficiency. In addition, the QPK measure using asymmetric CARR model gives the best in-sample model fit based on Akaike information criterion and provides the best out-of-sample forecast based on root mean squared forecast error and other measures. Mincer Zarnowitz test is carried out to confirm the unbiasedness of the forecasted volatility. Different levels of value-at-risk and conditional value-at-risk forecasts are also provided and tested.

Guilty speculators? Range-based conditional volatility in a cross-section of wheat futures

In response to the unusually high levels of price volatility during the world food price crisis of 2007/2008, US and EU regulators have introduced position limits that aim to protect commodity markets from exposure to excess speculation. Such regulatory initiatives presuppose that excess speculation is indeed responsible for excess volatility. Our results debunk this presupposition and show the opposite effect: speculative activity reduces price volatility, particularly during times of distress. Our findings are based on a cross-section of wheat futures contracts, traded at five different commodity exchanges with various degrees of speculative activity. Volatility is estimated based on a Conditional Autoregressive Range Model (CARR), which is further augmented with exogenous excess-speculation shocks (CARRX). These models capture herding, feedback and noise trading, and a threshold version (TCARRX) identifies regimes in which the anatomy of the volatility process changes according to the level of excess speculation. Our findings support Working’s hypothesis that a certain level of excess speculation is essential for a well-functioning market.

A conditional autoregressive range model with gamma distribution for financial volatility modelling

The commonly used conditional autoregressive range model with Weibull distribution (henceforth WCARR) suffers from serious inlier problem. We conjecture that this problem is due to a misspecified distribution to the disturbance, and propose a conditional autoregressive range model with gamma distribution (henceforth GCARR) to model the volatility of financial assets. In this paper, we first discuss the theoretical properties of the GCARR model and then compare its empirical performance with the WCARR. Empirical studies are performed on a broad set of stock indices in different countries over different time horizons. Consistent with the conjecture, we find that the GCARR model can reduce not only the inlier problem but also the outlier problem of the WCARR model. The results indicate that our GCARR model describes the dynamics of the range-based volatility better than the WCARR model and thus serves as a better benchmark.

Sudden changes in crude oil price volatility: an application of extreme value volatility estimator

This study provides a framework based on an extension of the Conditional Autoregressive Range (CARR) model which incorporates the impact of sudden changes in unconditional volatility. The results of the CARR model with and without volatility breaks are compared with the results of the GARCH model with and without volatility breaks to assess whether the forecasting ability of the CARR model is superior when endogenously determined structural breaks in volatility are accounted for. We undertake our analysis on WTI and Brent crude oil and find that the CARR model with volatility breaks effectively captures the dynamics in the crude oil volatility.

Sudden changes in extreme value volatility estimator: Modeling and forecasting with economic significance analysis

This study provides a framework based on an extension of the conditional autoregressive range (CARR) model which incorporates the impact of sudden changes in the unconditional volatility. This study proposes to use the RS estimator in the CARR model (called henceforth the CARRS model) instead of using the range. The results of the CARRS models with and without volatility breaks are compared with the results of the GARCH models with and without volatility breaks. We also compare the forecasting performance of CARRS models with the forecasting performance of EGARCH, TGARCH and FIGARCH models based on error statistics and regression approach. The findings indicate that the CARRS model with volatility breaks effectively captures the dynamics of volatility and provides better out-of-sample forecasts when compared with GARCH, EGARCH, TGARCH and FIGARCH models. We also devise a trading strategy to examine the economic significance of the proposed framework which indicates that the investor can make substantial gains (approximately 6%–10%) in return for most of cases based on volatility forecasts of CARRS model with volatility breaks. Results based on robustness check are consistent with our main findings.

Volatility spillovers between the oil market and the European Union carbon emission market

This paper examines the dynamics of volatility transmission between EU emission allowances (EUA) and oil markets using a range-based volatility measure. We propose a multivariate conditional autoregressive range model with bivariate lognormal distribution to capture volatility dynamics and volatility spillovers between oil and EUA markets. Our findings for Phase II of the European Union Emissions Trading Scheme point to the existence of volatility dynamics and leverage effects and to no significant volatility spillovers between these markets. These results remained robust to other volatility measures and model specifications.

Improving the CARR model using extreme range estimators

The aim of this article is to analyse the forecasting ability of the conditional autoregressive range (CARR) model proposed by Chou (2005) using the S&P 500. We extend the data sample, allowing for the analysis of different stock market circumstances and propose the use of various range estimators in order to analyse their forecasting performance. Additionally, we decide to divide the full sample into four sub-samples with the aim of analysing the forecasting ability of the different range estimators in various periods. Our results show that the original CARR model can be improved depending on three factors: the trend, the level of volatility in the analysis period and the error estimator that is used to analyse the forecasting ability of each model. The Parkinson model is better for upward trends and volatilities which are higher and lower than the mean while the CARR model is better for downward trends and mean volatilities.

Global and regional range-based volatility spillover effects

This study extends the univariate Weibull conditional autoregressive range (CARR) model to establish a bivariate Weibull CARR (BWCARR) model to investigate the range-based volatility spillover effect. The empirical results indicate that a conditional autoregressive range relationship exists on the US, Japan, mainland China, Hong Kong and Taiwan stock markets. The new BWCARR model is more credible and efficient than the CARR model. Moreover, the range-based volatility for the US and Japan has an impact on Taiwan, indicating that there exists a range-based global and regional stock market spillover effect that has an impact on the Taiwanese stock market.

Range-Based Analysis of Volatility Spillovers in European Financial Markets

We analyze multivariate time series of daily high-low ranges of national equity market indices to measure intra-daily volatility dynamics across four continental European markets. We use a dynamic linear model of expected daily range which is a variant of Chou’s conditional autoregressive range model. We find significant, but not uniform, range-based volatility spillovers between the European markets. The strongest spillover comes from the previous day’s realized range of the US market index. We also find that average daily range increased sharply during the European financial crisis, and the degree of autoregressive persistence also increased uniformly.

Forecasting volatility with asymmetric smooth transition dynamic range models

Abstract We propose a nonlinear smooth transition conditional autoregressive range (CARR) model for capturing smooth volatility asymmetries in international financial stock markets, building on recent work on smooth transition conditional duration modelling. An adaptive Markov chain Monte Carlo scheme is developed for Bayesian estimation, volatility forecasting and model comparison for the proposed model. The model can capture sign or size asymmetry and heteroskedasticity, such as that which is commonly observed in financial markets. A mixture proposal distribution is developed in order to improve the acceptance rate and the mixing issues which are common in random walk Metropolis-Hastings methods. Further, the logistic transition function is employed and its main properties are considered and discussed in the context of the proposed model, which motivates a suitable, weakly informative prior which ensures a proper posterior distribution and identification of the estimators. The methods are illustrated using simulated data, and an empirical study also provides evidence in favour of the proposed model when forecasting the volatility in two financial stock markets. In addition, the deviance information criterion is employed to compare the proposed models with their limiting classes, the nonlinear threshold CARR models and the symmetric CARR model.

Volatility contagion: A range-based volatility approach

This article proposes a new approach to evaluate volatility contagion in financial markets. A time-varying logarithmic conditional autoregressive range model with the lognormal distribution (TVLCARR) is proposed to capture the possible smooth transition in the range process. Additionally, a smooth transition copula function is employed to detect the volatility contagion between financial markets. The approach proposed is applied to the stock markets of the G7 countries to investigate the volatility contagion due to the subprime mortgage crisis. Empirical evidence shows that volatility is contagious from the US market to several markets examined.