EMPIRICAL PERFORMANCE OF GARCH MODELS WITH HEAVY‐TAILED INNOVATIONS

We introduce a new type of heavy‐tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), to the GARCH and Glosten‐Jagannathan‐Runkle (1993) GARCH models, and compare its empirical performance with two other popular types of heavy‐tailed distribution, the Student's t distribution and the normal inverse Gaussian distribution (NIG), using a variety of asset return series. Our results illustrate that there is no overwhelmingly dominant distribution in fitting the data under the GARCH framework, although the NRIG distribution performs slightly better than the other two types of distribution. For market indexes series, it is important to introduce both GJR‐terms and the NRIG distribution to improve the models’ performance, but it is ambiguous for individual stock prices series. Our results also show the GJR‐GARCH NRIG model has practical advantages in quantitative risk management. Finally, the convergence of numerical solutions in maximum‐likelihood estimation of GARCH and GJR‐GARCH models with the three types of heavy‐tailed distribution is investigated.     

Bayesian model comparison by Markov chain simulation: Illustration using stock market data

The paper illustrates the computation of marginal likelihoods and Bayes factors when Markov Chain Monte Carlo has been used to produce draws from a model’s posterior distribution. The method is based on Raftery (1996) and does not require that Gibbs sampling is used or conditional posterior distributions are available in closed form. Models used include a normal finite mixture, a GARCH and a Student t -model as alternative models for the Standard and Poor’s stock returns.     

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.     

A note on the estimated GARCH coefficients from the S&P1500 universe

ABSTRACT

We employ 1440 stocks listed in the S&P Composite 1500 Index of the NYSE. Three benchmark GARCH models are estimated for the returns of each individual stock under three alternative distributions (Normal, t and GED). We provide summary statistics for all the GARCH coefficients derived from 11,520 regressions. The EGARCH model with GED errors emerges as the preferred choice for the individual stocks in the S&P 1500 universe when non-negativity and stationarity constraints in the conditional variance are imposed. 57% of the constraint’s violations are taking place in the S&P small cap stocks.     

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.     

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.     

German stock market dynamics

This article estimates generalized ARCH (GARCH) models for German stock market indices returns, using weekly and monthly data, various GARCH specifications and (non)normal error densities, and a variety of diagnostic checks. German stock return series exhibit significant levels of second-order dependence. Our results clearly demonstrate that for both weekly as well as monthly return series the Student-t distribution is superior to the standard normal distribution. In particular, the estimated GARCH-t models appear to be reasonably successful in accounting for both observed leptokurtosis and conditional heteroskedasticity from German stock return movements.     

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.     

Testing the Gaussian and Student's t copulas in a risk management framework

This paper introduces a semiparametric framework for selecting either a Gaussian or a Student's t copula in a d-dimensional setting. We compare the two models using four different approaches: (i) four goodness-of-fit graphical plots, (ii) a bootstrapped correlation matrix generated in each scenario with the empirical correlation matrix used as a benchmark, (iii) Value-at-Risk (VaR) and Expected Shortfall (ES) as risk measures, and (iv) co-Value-at-Risk (CoVaR) and Marginal Expected Shortfall (MES) as co-risk measures. We illustrate this four-step procedure using a portfolio of daily returns of six international stock indices. The VaR results confirm that the t-based copula model is an attractive alternative to the Gaussian. The ES analysis is less conclusive, and indicates that risk managers should jointly use the risk measure as well as the copula model. The results highlight the importance of promoting stress testing rather than ES in the risk management industry, particularly in the aftermath of a financial crisis.     

Long-Memory,Asymmetry and Fat-Tailed GARCH Models in Value-at-Risk Estimation: Empirical Evidence from the Global Real Estate Markets

In this paper, we address the question of whether long memory, asymmetry, and fat-tails in global real estate markets volatility matter when forecasting the two most popular measures of risk in financial markets, namely Value-at-risk (VaR) and Expected Shortfall (ESF), for both short and long trading positions. The computations of both VaR and ESF are conducted with three long memory GARCH-class models including the Fractionally Integrated GARCH (FIGARCH), Hyperbolic GARCH (HYGARCH), and Fractionally Integrated Asymmetric Power ARCH (FIAPARCH). These models are estimated under three alternative innovation’s distributions: normal, Student, and skewed Student. To test the efficacy of the forecast, we employ various backtesting methodologies. Our empirical findings show that considering for long memory, fat-tails, and asymmetry performs better in predicting a one-day-ahead VaR and ESF for both short and long trading positions. In particular, the forecasting ability analysis points out that the FIAPARCH model under skewed Student distribution turns out to improve substantially the VaR and ESF forecasts. These results may have several potential implications for the market participants, financial institutions, and the government.

     

Infinite-variance,alpha-stable shocks in monetary SVAR

This paper outlines a theory of what might be going wrong in the monetary SVAR (structural vector autoregression) literature and provides supporting empirical evidence. The theory is that macroeconomists may be attempting to identify structural forms that do not exist, given the true distribution of the innovations in the reduced-form VAR. This paper shows that this problem occurs whenever (1) some innovation in the VAR has an infinite-variance distribution and (2) the matrix of coefficients on the contemporaneous terms in the VAR's structural form is nonsingular. Since (2) is almost always required for SVAR analysis, it is germane to test hypothesis (1). Hence, in this paper, we fit α-stable distributions to the residuals from 3-lag and 12-lag monetary VARs, and, using a parametric-bootstrap method, we reject the null hypotheses of finite variance (or equivalently, α = 2) for all 12 error terms in the two VARs. These results are mostly robust to a sample break at the February 1984 observations. Moreover, ARCH tests suggest that the shocks from the subperiod VARs are homoskedastic in seven of 24 instances. Next, we compare the fits of the α-stable distributions with those of t distributions and a GARCH(1,1) shock model. This analysis suggests that the time-invariant α-stable distributions provide the best fits for two of six shocks in the VAR(12) specification and three of six shocks in the VAR(3). Finally, we use the GARCH model as a filter to obtain homoskedastic shocks, which also prove to have α < 2, according to ML estimates.     

A two-regime threshold model with conditional skewed Student t distributions for stock returns

This paper proposes a two-regime threshold model for the conditional distribution of stock returns in which returns follow a distinct skewed Student t distribution within each regime: the model allows capturing time variation in the conditional distribution of returns, as well as higher order moments. An application of the model to daily U.S. stock returns illustrates the advantages of the proposed model in comparison to alternative specifications: the model performs well in terms of in-sample fit; it more accurately estimates the conditional volatility; and it produces useful risk assessment as measured by the term structure of value at risk.     

Long range dependence in an emerging stock market’s sectors: volatility modelling and VaR forecasting

This study evaluates the sector risk of the Qatar Stock Exchange (QSE), a recently upgraded emerging stock market, using value-at-risk models for the 7 January 2007–18 October 2015 period. After providing evidence for true long memory in volatility using the log-likelihood profile test of Qu and splitting the sample and dth differentiation tests of Shimotsu, we compare the FIGARCH, HYGARCH and FIAPARCH models under normal, Student-t and skewed-t innovation distributions based on in and out-of-sample VaR forecasts. The empirical results show that the skewed Student-t FIGARCH model generates the most accurate prediction of one-day-VaR forecasts. The policy implications for portfolio managers are also discussed.     

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).     

ANALYTICAL EVALUATION OF VOLATILITY FORECASTS*

Estimation and forecasting for realistic continuous‐time stochastic volatility models is hampered by the lack of closed‐form expressions for the likelihood. In response, Andersen, Bollerslev, Diebold, and Labys (Econometrica, 71 (2003), 579–625) advocate forecasting integrated volatility via reduced‐form models for the realized volatility, constructed by summing high‐frequency squared returns. Building on the eigenfunction stochastic volatility models, we present analytical expressions for the forecast efficiency associated with this reduced‐form approach as a function of sampling frequency. For popular models like GARCH, multifactor affine, and lognormal diffusions, the reduced form procedures perform remarkably well relative to the optimal (infeasible) forecasts.     

PREDICTING STOCK RETURNS AND VOLATILITY WITH INVESTOR SENTIMENT INDICES: A RECONSIDERATION USING A NONPARAMETRIC CAUSALITY‐IN‐QUANTILES TEST

Evidence of monthly stock returns predictability based on popular investor sentiment indices, namely S^BW and S^PLS as introduced by Baker and Wurgler (2006, 2007) and Huang et al. (2015) respectively are mixed. While, linear predictive models show that only S^PLS can predict excess stock returns, nonparametric models (which accounts for misspecification of the linear frameworks due to nonlinearity and regime changes) finds no evidence of predictability based on either of these two indices for not only stock returns, but also its volatility. However, in this paper, we show that when we use a more general nonparametric causality‐in‐quantiles model of Balcilar et al., (forthcoming), in fact, both S^BW and S^PLS can predict stock returns and its volatility, with S^PLS being a relatively stronger predictor of excess returns during bear and bull regimes, and S^BW being a relatively powerful predictor of volatility of excess stock returns, barring the median of the conditional distribution.     

RISK PERCEPTION AND EQUITY RETURNS: EVIDENCE FROM THE SPX AND VIX

We use the semi‐nonparametric (SNP) model to study the relationship between the innovation of the Volatility Index (VIX) and the expected S&P 500 Index (SPX) returns. We estimate the one‐step‐ahead contemporaneous relation subject to leverage GARCH effect. Results agree with a body of newly established literature arguing non‐linearity, and asymmetries. In addition, the risk‐return behaviour depends on the signs as well as magnitudes of the perceived risk. We conclude that influence of fear or exuberance on the conditional market return is non‐monotonic and hump‐shaped. Very deep fear does not necessarily mean huge losses, instead, the loss may not be as bad as fears of normal levels. Results pass the robustness tests.     

The Efficiency of the Realized Range Measure of Daily Volatility: Evidence from Greece

In this study we estimate and compare the realized range volatility, a novel efficient volatility estimator computed by summing high–low ranges for intra‐day intervals, to the recently popularized realized variance estimator obtained by summing squared intra‐day returns. Our results, derived from a Greek equity high‐frequency data set, show that realized range‐based measures improve upon the corresponding realized variance‐based ones in most cases, especially for the most actively traded stocks. The usefulness of high‐frequency data in measuring and forecasting financial volatility is apparent throughout the paper.     

Unit Root Tests for Dependent Micropanels

This paper proposes a new test for the null hypothesis of panel unit roots for micropanels with short time dimensions (T) and large cross‐sections (N). There are several distinctive features of this test. First, the test is based on a panel AR(1) model allowing for cross‐sectional dependency, which is introduced by a factor structure of the initial condition. Second, the test employs the panel AR(1) model with AR(1) coefficients that are heterogeneous for finite N. Third, the test can be used both for the alternative hypothesis of stationarity and for that of explosive roots. Fourth, the test does not use the AR(1) coefficient estimator. The effectiveness of the test rests on the fact that the initial condition has permanent effects on the trajectory of a time series in the presence of a unit root. To measure the effects of the initial condition, the present paper employs cross‐sectional regressions using the first time‐series observations as a regressor and the last as a dependent variable. If there is a unit root in every individual time series, the coefficient of the regressor is equal to one. The t‐ratios for the coefficient are this paper's test statistics and have a standard normal distribution in the limit. The t‐ratios are based on the OLS estimator and the instrumental variables estimator that uses reshuffled regressors as instruments. The test proposed in this paper makes it possible to test for a unit root even at T = 2 as long as N is large. Simulation results show that test statistics have reasonable empirical size and power. The test is applied to college graduates' monthly real wage in South Korea. The number of time‐series observations for this data is only two. The null hypothesis of a unit root is rejected against the alternative of stationarity.     

A COMPARISON OF CONDITIONAL AND UNCONDITIONAL APPROACHES IN VALUE‐AT‐RISK ESTIMATION*

In this paper, we attempt to find the most important factor causing the differences in the performance of Value‐at‐Risk (VaR) estimation by comparing the performances of conditional and unconditional approaches. For each approach, we use various methods and models with different degrees of flexibility in their distributions including S_U‐normal distribution, which is one of the most flexible distribution functions. Our empirical results underscore the importance of the flexibility‐of‐distribution function in VaR estimation models. Even though it seems to be unclear which approach is better between conditional and unconditional approaches, it seems to be clear that the more flexible distribution we use, the better the performance, regardless of which approach we use.