Dynamic conditional score models of degrees of freedom: filtering with score-driven heavy tails

This article extends the quasi-autoregressive (QAR) plus Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) dynamic conditional score (DCS) model. For the new DCS model, the degrees of freedom parameter is time varying and tail thickness of the error term is updated by the conditional score. We compare the performance of QAR plus Beta-t-EGARCH with constant degrees of freedom (benchmark model) and QAR plus Beta-t-EGARCH with time-varying degrees of freedom (extended model). We use data from the Standard and Poor’s 500 (S&P 500) index, and a random sample of its 150 components that are from different industries of the United States (US) economy. For the S&P 500, all likelihood-based model selection criteria support the extended model, which identifies extreme events with significant impact on the US stock market. We find that for 59% of the 150 firms, the extended model has a superior statistical performance. The results suggest that the extended model is superior for those industries, which produce products that people usually are unwilling to cut out of their budgets, regardless of their financial situation. We perform an application to compare the density forecast performance of both DCS models. We perform an application to Monte Carlo value-at-risk for both DCS models.     

Markov regime-switching Beta-t-EGARCH

We suggest a Markov regime-switching (MS) Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) model for U.S. stock returns. We compare the in-sample statistical performance of the MS Beta-t-EGARCH model with that of the single-regime Beta-t-EGARCH model. For both models we consider leverage effects for conditional volatility. We use data from the Standard Poor’s 500 (S&P 500) index and also a random sample that includes 50 components of the S&P 500. We study the outlier-discounting property of the single-regime Beta-t-EGARCH and MS Beta-t-EGARCH models. For the S&P 500, we show that for the MS Beta-t-EGARCH model extreme observations are discounted more for the low-volatility regime than for the high-volatility regime. The conditions of consistency and asymptotic normality of the maximum likelihood estimator are satisfied for both the single-regime and MS Beta-t-EGARCH models. All likelihood-based in-sample statistical performance metrics suggest that the MS Beta-t-EGARCH model is superior to the single-regime Beta-t-EGARCH model. We present an application to the out-of-sample density forecast performance of both models. The results show that the density forecast performance of the MS Beta-t-EGARCH model is superior to that of the single-regime Beta-t-EGARCH model.     

Is Beta-t-EGARCH(1,1) superior to GARCH(1,1)?

Statistical performance, in-sample point forecast precision and out-of-sample density forecast precision of GARCH(1,1) and Beta-t-EGARCH(1,1) models are compared. We study the volatility of nine global industry indices for period from April 2006 to July 2010. Competing models are estimated for periods before, during and after the United States (US) financial crisis of 2008. The results provide evidence of the superior out-of-sample predictive performance of Beta-t-EGARCH compared to GARCH after the US financial crisis.     

Dynamic conditional score models: a review of their applications

ABSTRACT

In this paper, applications of dynamic conditional score (DCS) models are reviewed and those models are discussed in relation to classical time series models from the literature. DCS models are robust to outliers, which improves their statistical performance compared to classical models. Three applications are presented in order to compare the statistical performances of DCS and classical models in three very different contexts: (i) The QAR (quasi-autoregressive) plus Beta-t-EGARCH (exponential autoregressive conditional heteroscedasticity) model is presented, which is a score-driven expected return plus volatility model. This model is used for daily returns on the DAX (Deutscher Aktienindex) equity index for the period of January 1988 to December 2017. (ii) The score-driven local level and seasonality plus Beta-t-EGARCH model is presented, which is used for daily AFN/USD (Afghan Afghani/United States Dollar) currency exchange rates for the period of March 2007 to July 2017. (iii) The Seasonal-t-QVAR (quasi-vector autoregressive) model is presented, which is a score-driven multivariate dynamic model of location. For this model, monthly US inflation rate and US unemployment rate are used for the period of January 1948 to December 2017. For all applications, the statistical performance of each DCS model is superior to that of a corresponding classical alternative.     

QARMA-Beta-t-EGARCH versus ARMA-GARCH: an application to S&P 500

Statistical performance and out-of-sample forecast precision of ARMA-GARCH and QARMA-Beta-t-EGARCH are compared. We study daily returns on the Standard and Poor’s 500 (S&P 500) index and a random sample of 50 stocks from the S&P 500 for period May 2006 to July 2010. Competing models are estimated for periods before and during the US financial crisis of 2008. Out-of-sample point and density forecasts are performed for periods during and after the US financial crisis. The results provide evidence of the superior in-sample statistical and out-of-sample predictive performance of QARMA-Beta-t-EGARCH.     

Score-driven Markov-switching EGARCH models: an application to systematic risk analysis

We introduce new Markov-switching (MS) dynamic conditional score (DCS) exponential generalized autoregressive conditional heteroscedasticity (EGARCH) models, to be used by practitioners for forecasting value-at-risk (VaR) and expected shortfall (ES) in systematic risk analysis. We use daily log-return data from the Standard & Poor’s 500 (S&P 500) index for the period 1950–2016. The analysis of the S&P 500 is useful, for example, for investors of (i) well-diversified US equity portfolios; (ii) S&P 500 futures and options traded at Chicago Mercantile Exchange Globex; (iii) exchange traded funds (ETFs) related to the S&P 500. The new MS DCS-EGARCH models are alternatives to of the recent MS Beta-t-EGARCH model that uses the symmetric Student’s t distribution for the error term. For the new models, we use more flexible asymmetric probability distributions for the error term: Skew-Gen-t (skewed generalized t), EGB2 (exponential generalized beta of the second kind) and NIG (normal-inverse Gaussian) distributions. For all MS DCS-EGARCH models, we identify high- and low-volatility periods for the S&P 500. We find that the statistical performance of the new MS DCS-EGARCH models is superior to that of the MS Beta-t-EGARCH model. As a practical application, we perform systematic risk analysis by forecasting VaR and ES.

Abbreviation Single regime (SR); Markov-switching (MS); dynamic conditional score (DCS); exponential generalized autoregressive conditional heteroscedasticity (EGARCH); value-at-risk (VaR); expected shortfall (ES); Standard & Poor's 500 (S&P 500); exchange traded funds (ETFs); Skew-Gen-t (skewed generalized t); EGB2 (exponential generalized beta of the second kind); NIG (normal-inverse Gaussian); log-likelihood (LL); standard deviation (SD); partial autocorrelation function (PACF); likelihood-ratio (LR); ordinary least squares (OLS); heteroscedasticity and autocorrelation consistent (HAC); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan-Quinn criterion (HQC).     

Regime-switching purchasing power parity in Latin America: Monte Carlo unit root tests with dynamic conditional score

We suggest a Monte Carlo simulation-based unit root test of the purchasing power parity theory for Latin American countries. Under the null hypothesis, we use a Markov regime-switching (MS) model with unit root in the conditional location and MS volatility dynamics. Under the alternative hypothesis, the proposed test incorporates Markov regime-switching autoregressive moving average (MS-ARMA) plus MS volatility dynamics. Under both the null and alternative hypotheses, one of the volatility models estimated is Beta-t-EGARCH, which is a recent dynamic conditional score volatility model. We use data on real effective exchange rate time series for 14 Latin American countries. For each country, we estimate by Monte Carlo simulation the critical values of the unit root test. We provide an economic discussion of the unit root test results and also study the robustness of MS-ARMA plus MS volatility with respect to smooth transition autoregressive models with Fourier function.     

Modeling and forecasting return jumps using realized variation measures

This paper proposes a simple HAR-RV-based model to predict return jumps through a conditional density of jump size with time-varying moments. We model jump occurrences based on a version of the autoregressive conditional hazard model that relies on past continuous realized volatilities. Applying our methodology to seven equity indices on the U.S. and Chinese stock markets, we reach the following key findings: (i) jump occurrence and size are dependent on past realized volatility, (ii) the proposed model yields superior in- and out-of-sample jump size density forecasts compared to an ARMA(1,1)-GARCH(1,1) model, (iii) and the occurrence and sign of return jumps are predictable to some extent.     

A new approach to modeling co-movement of international equity markets: evidence of unconditional copula-based simulation of tail dependence

Analyzing equity market co-movements is important for risk diversification of an international portfolio. Copulas have several advantages compared to the linear correlation measure in modeling co-movement. This paper introduces a copula ARMA-GARCH model for analyzing the co-movement of international equity markets. The model is implemented with an ARMA-GARCH model for the marginal distributions and a copula for the joint distribution. After goodness of fit testing, we find that the Student’s t copula ARMA(1,1)-GARCH(1,1) model with fractional Gaussian noise is superior to alternative models investigated in our study where we model the simultaneous co-movement of nine international equity market indexes. This model is also suitable for capturing the long-range dependence and tail dependence observed in international equity markets. Rachev’s research was supported by grants from Division of Mathematical, Life and Physical Science, College of Letters and Science, University of California, Santa Barbara, and the Deutschen Forschungsgemeinschaft (DFG). Sun’s research was supported by grants from the Deutschen Forschungsgemeinschaft (DFG) and Chinese Government Award for Outstanding Ph.D Students Abroad 2006, No. 2006-180. Kalev’s research was supported with a NCG grant from the Faculty of Business and Economics, Monash University. Data are supplied by Securities Industry Research Center of Asia-Pacific (SIRCA) on behalf of Reuters. The constructive comments of two anonymous referees, the Associate Editor, A.S. Wirjanto, and the Editor-in-charge, Baldev Raj, are gratefully acknowledged. The reviewers and editors are not responsible for any residual errors and omissions.     

Analysis of electricity prices for Central American countries using dynamic conditional score models

In this paper, we compare the performance of dynamic conditional score (DCS) and standard financial time-series models for Central American energy prices. We extend the Student’s t and the exponential generalised beta distribution of the second kind stochastic location and stochastic seasonal DCS models. We consider the generalised t distribution as an alternative for the error term and also consider dynamic specifications of volatility. We use a unique dataset of spot electricity prices for El Salvador, Guatemala and Panama. We consider two data windows for each country, which are defined with respect to the liberalisation and development process of the energy market in Central America. We study the identification of a wide range of DCS specifications, likelihood-based model performance, time-series components of energy prices, maximum likelihood parameter estimates, the discounting property of conditional score, and out-of-sample forecast performance. Our main results are the following. (i) We determine the most robust models of energy prices, with respect to parameter identification, from a wide range of DCS specifications. (ii) For most of the cases, the in-sample statistical performance of DCS is superior to that of the standard model. (iii) For El Salvador and Panama, the standard model provides better point forecasts than DCS, and for Guatemala the point forecast precision of standard and DCS models does not differ significantly. (iv) For El Salvador, the standard model provides better density forecasts than DCS, and for Guatemala and Panama, the density forecast precision of standard and DCS models does not differ significantly.     

Full versus quasi MLE for ARMA-GARCH models with infinitely divisible innovations

We compare the backtesting performance of ARMA-GARCH models with the most common types of infinitely divisible innovations, fit with both full maximum likelihood estimation (MLE) and quasi maximum likelihood estimation (QMLE). The innovation types considered are the Gaussian, Student’s t, α-stable, classical tempered stable (CTS), normal tempered stable (NTS) and generalized hyperbolic (GH) distributions. In calm periods of decreasing volatility, MLE and QMLE produce near identical performance in forecasting value-at-risk (VaR) and conditional value-at-risk (CVaR). In more volatile periods, QMLE can actually produce superior performance for CTS, NTS and α-stable innovations. While the t-ARMA-GARCH model has the fewest number of VaR violations, rejections by the Kupeic and Berkowitz tests suggest excessively large forecasted losses. The α-stable, CTS and NTS innovations compare favourably, with the latter two also allowing for option pricing under a single market model.     

Modelling the dynamics of Bitcoin and Litecoin: GARCH versus stochastic volatility models

We examine and compare a large number of generalized autoregressive conditional heteroskedastic (GARCH) and stochastic volatility (SV) models using series of Bitcoin and Litecoin price returns to assess the model fit for dynamics of these cryptocurrency price returns series. The various models examined include the standard GARCH(1,1) and SV with an AR(1) log-volatility process, as well as more flexible models with jumps, volatility in mean, leverage effects, t-distributed and moving average innovations. We report that the best model for Bitcoin is SV-t while it is GARCH-t for Litecoin. Overall, the t-class of models performs better than other classes for both cryptocurrencies. For Bitcoin, the SV models consistently outperform the GARCH models and the same holds true for Litecoin in most cases. Finally, the comparison of GARCH models with GARCH-GJR models reveals that the leverage effect is not significant for cryptocurrencies, suggesting that these do not behave like stock prices.     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.     

Do we need a global VAR model to forecast inflation and output in South Africa?

This study determines whether the global vector autoregressive (GVAR) approach provides better forecasts of key South African variables than a vector error correction model (VECM) and a Bayesian vector autoregressive (BVAR) model augmented with foreign variables. The article considers both a small GVAR model and a large GVAR model in determining the most appropriate model for forecasting South African variables. We compare the recursive out-of-sample forecasts for South African GDP and inflation from six types of models: a general 33 country (large) GVAR, a customized small GVAR for South Africa, a VECM for South Africa with weakly exogenous foreign variables, a BVAR model, autoregressive (AR) models and random walk models. The results show that the forecast performance of the large GVAR is generally superior to the performance of the customized small GVAR for South Africa. The forecasts of both the GVAR models tend to be better than the forecasts of the augmented VECM, especially at longer forecast horizons. Importantly, however, on average, the BVAR model performs the best when it comes to forecasting output, while the AR(1) model outperforms all the other models in predicting inflation. We also conduct ex ante forecasts from the BVAR and AR(1) models over 2010:Q1–2013:Q4 to highlight their ability to track turning points in output and inflation, respectively.     

QUANTILE FORECASTS OF FINANCIAL RETURNS USING REALIZED GARCH MODELS*

This article applies the realized generalized autoregressive conditional heteroskedasticity (GARCH) model, which incorporates the GARCH model with realized volatility, to quantile forecasts of financial returns, such as Value‐at‐Risk and expected shortfall. Student's t‐ and skewed Student's t‐distributions as well as normal distribution are used for the return distribution. The main results for the S&P 500 stock index are: (i) the realized GARCH model with the skewed Student's t‐distribution performs better than that with the normal and Student's t‐distributions and the exponential GARCH model using the daily returns only; and (ii) using the realized kernel to take account of microstructure noise does not improve the performance.     

Inflation Volatility and Forecast Accuracy

We examine the statistical properties of inflation in a sample of inflation‐targeting (IT) and non‐IT countries. It is hard to distinguish in which monetary regime inflation is less volatile. Inflation became easier to forecast in both groups of countries after the introduction of IT. The improvement was greater for IT countries, but forecast errors remain smaller for non‐IT countries. Our analysis is based on a stochastic volatility model proposed by Stock and Watson and its novel modification. Forecasts from the modified model are generally superior to both simple benchmarks and the original Stock and Watson model.     

On the regime-switching and asymmetric dynamics of economic growth in the OECD countries

This study estimates the SETAR and STAR models and examines the regime-switching and asymmetric dynamics of economic growth for a comprehensive set of 10 OECD countries. The SETAR models of both Tsay and Hansen consistently reject the null hypothesis of linearity against the alternative hypothesis of threshold nonlinearity for all the sample countries. The STAR model reinforces the evidence and rejects the null hypothesis of linearity against STAR nonlinearity for all the sample countries, except Italy. The sequential F tests for the nested nulls suggest LSTAR nonlinearity for Austria, Japan, Korea, Mexico, Netherlands and New Zealand, and ESTAR nonlinearity for Finland, Germany and Norway. The forecast evaluations suggest that the SETAR models of Tsay and Hansen perform better, as compared to the AR, ARMA and STAR models. The forecasting performance of the STAR model is approximately similar to the forecasting performance of the linear AR and ARMA rivals. The persistence of lower regimes (with negative-growth or moderate-expansions) necessitates the need for the adoption of expansionary economic policies. While the longer durations of upper regimes (with positive-growth or fast-expansions) support the sustainability of the expansionary economic policies, the adequate precautions need to be taken for the inflationary implications of these policies.     

Portfolio Diversification Strategy Via Tail‐Dependence Clustering and ARMA‐GARCH Vine Copula Approach

This study proposes a diversified portfolio construction method based on the tail dependence between the financial assets and adopting both market prior information and the exports’ subject views. In this paper, tail‐dependence clustering was applied to divide candidate assets into different groups according to their tail dependence during the crisis period and the ARMA‐GARCH vine copula‐opinion pooling approach was applied to select the minimum Conditional Value‐at‐Risk portfolio according to the clustering results. The daily closed prices of the components of DAX 20 from 3 January 2006 to 20 December 2014 were studied to illustrate the methodology. The results reveal that more than 90% of 450 possible portfolios are modelled by D‐vine structure and Student's t‐copula dominates almost all the cases for pair copula selection. As Student's t‐copula captures the symmetric tail dependence, the 450 possible portfolios do not show stronger lower tail dependence than upper tail dependence. This study contributes by combining cluster analysis with portfolios selection. It uses vine copula to capture the dependence structure among assets. Finally, it offers a flexible method to describe market and offers a strategy to construct diversified portfolios by adding the investors’ information into portfolio selection procedure at the 1‐day forecast horizon.     

Understanding forecast failure of ESTAR models of real exchange rates

The forecast performance of the empirical ESTAR model of Taylor et al. (2001) is examined for 4 bilateral real exchange rate series over an out-of-sample evaluation period of nearly 12?years. Point as well as density forecasts are constructed, considering forecast horizons of 1 to 22 steps head. The study finds that no forecast gains over a simple AR(1) specification exist at any of the forecast horizons that are considered, regardless of whether point or density forecasts are utilised in the evaluation. Non-parametric methods are used in conjunction with simulation techniques to learn about the models and their forecasts. It is shown graphically that the nonlinearity in the conditional means (or point forecasts) of the ESTAR model decreases as the forecast horizon increases. The non-parametric methods show also that the multiple steps ahead forecast densities are normal looking with no signs of bi-modality, skewness or kurtosis.