A conditionally heteroskedastic independent factor model with an application to financial stock returns

We propose a new conditionally heteroskedastic factor model, the GICA-GARCH model, which combines independent component analysis (ICA) and multivariate GARCH (MGARCH) models. This model assumes that the data are generated by a set of underlying independent components (ICs) that capture the co-movements among the observations, which are assumed to be conditionally heteroskedastic. The GICA-GARCH model separates the estimation of the ICs from their fitting with a univariate ARMA-GARCH model. Here, we will use two ICA approaches to find the ICs: the first estimates the components, maximizing their non-Gaussianity, while the second exploits the temporal structure of the data. After estimating and identifying the common ICs, we fit a univariate GARCH model to each of them in order to estimate their univariate conditional variances. The GICA-GARCH model then provides a new framework for modelling the multivariate conditional heteroskedasticity in which we can explain and forecast the conditional covariances of the observations by modelling the univariate conditional variances of a few common ICs. We report some simulation experiments to show the ability of ICA to discover leading factors in a multivariate vector of financial data. Finally, we present an empirical application to the Madrid stock market, where we evaluate the forecasting performances of the GICA-GARCH and two additional factor GARCH models: the orthogonal GARCH and the conditionally uncorrelated components GARCH.     

Using extreme value theory to measure value-at-risk for daily electricity spot prices

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets because the distinctive features of the former result in a highly unusual distribution of returns—electricity returns are highly volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of the return distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts. Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR in electricity markets.     

Indirect estimation of large conditionally heteroskedastic factor models,with an application to the Dow 30 stocks

We derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a Kalman-filter approximation with inequality constraints on the auxiliary model parameters. We also propose alternative indirect estimators for large-scale models, and explain how to apply our procedures to many other dynamic latent variable models. We analyse the small sample behaviour of our indirect estimators and several likelihood-based procedures through an extensive Monte Carlo experiment with empirically realistic designs. Finally, we apply our procedures to weekly returns on the Dow 30 stocks.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Temporal aggregation of volatility models

In this paper, we consider temporal aggregation of volatility models. We introduce semiparametric volatility models, termed square-root stochastic autoregressive volatility (SR-SARV), which are characterized by autoregressive dynamics of the stochastic variance. Our class encompasses the usual GARCH models and various asymmetric GARCH models. Moreover, our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables. The SR-SARV class is a natural extension of the class of weak GARCH models. This extension has four advantages: (i) we do not assume that fourth moments are finite; (ii) we allow for asymmetries (skewness, leverage effect) that are excluded from weak GARCH models; (iii) we derive conditional moment restrictions and (iv) our framework allows us to study temporal aggregation of IGARCH models.     

Time varying risk premia for real estate investment trusts: A GARCH-M model

This study employs the generalized autoregressive conditionally heteroskedastic in the mean (GARCH-M) methodology to investigate the return generating process of real estate investment trusts (REIT). The trade-off between excess returns and the conditional variance was positive for both equity and mortgage REITs but it was significant only for the latter. Changes in interest rates and their conditional variance were found to be inversely related to REIT excess returns. The 1986 tax law had a negative impact on the excess returns to both REIT sectors but the coefficient was significant only for mortgage REITs. The GARCH-M specification was determined to be more appropriate for the mortgage REIT portfolio than for the portfolio of equity REITs.     

Sequential estimation of shape parameters in multivariate dynamic models

Sequential maximum likelihood and GMM estimators of distributional parameters obtained from the standardised innovations of multivariate conditionally heteroskedastic dynamic regression models evaluated at Gaussian PML estimators preserve the consistency of mean and variance parameters while allowing for realistic distributions. We assess their efficiency, and obtain moment conditions leading to sequential estimators as efficient as their joint ML counterparts. We also obtain standard errors for VaR and CoVaR, and analyse the effects on these measures of distributional misspecification. Finally, we illustrate the small sample performance of these procedures through simulations and apply them to analyse the risk of large eurozone banks.     

Time-varying higher-order conditional moments and forecasting intraday VaR and Expected Shortfall

We estimate several GARCH- and Extreme Value Theory (EVT)-based models to forecast intraday Value-at-Risk (VaR) and Expected Shortfall (ES) for S&P 500 stock index futures returns for both long and short positions. Among the GARCH-based models we consider is the so-called Autoregressive Conditional Density (ARCD) model, which allows time-variation in higher-order conditional moments. ARCD model with time-varying conditional skewness parameter has the best in-sample fit among the GARCH-based models. The EVT-based model and the GARCH-based models which take conditional skewness and kurtosis (time-varying or otherwise) into account provide accurate VaR forecasts. ARCD model with time-varying conditional skewness parameter seems to provide the most accurate ES forecasts.     

No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects,jumps and i.i.d. noise: Theory and testable distributional implications

We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is utilized to alleviate microstructure frictions for realized volatility estimation. Size and power of the procedure are explored through Monte Carlo methods. Our empirical findings support the jump-diffusive representation for S&P500 futures returns but reveal it is critical to account for leverage effects and jumps to maintain the underlying semi-martingale assumption.     

Model‐free evaluation of directional predictability in foreign exchange markets

We examine directional predictability in foreign exchange markets using a model‐free statistical evaluation procedure. Based on a sample of foreign exchange spot rates and futures prices in six major currencies, we document strong evidence that the directions of foreign exchange returns are predictable not only by the past history of foreign exchange returns, but also the past history of interest rate differentials, suggesting that the latter can be a useful predictor of the directions of future foreign exchange rates. This evidence becomes stronger when the direction of larger changes is considered. We further document that despite the weak conditional mean dynamics of foreign exchange returns, directional predictability can be explained by strong dependence derived from higher‐order conditional moments such as the volatility, skewness and kurtosis of past foreign exchange returns. Moreover, the conditional mean dynamics of interest rate differentials contributes significantly to directional predictability. We also examine the co‐movements between two foreign exchange rates, particularly the co‐movements of joint large changes. There exists strong evidence that the directions of joint changes are predictable using past foreign exchange returns and interest rate differentials. Furthermore, both individual currency returns and interest rate differentials are also useful in predicting the directions of joint changes. Several sources can explain this directional predictability of joint changes, including the level and volatility of underlying currency returns. Copyright © 2007 John Wiley & Sons, Ltd.     

Model specification of conditional jump intensity: Evidence from S&P 500 returns and option prices

This study investigates the model specification of the conditional jump intensity under option pricing models having a generalized autoregressive conditional heteroskedastic with jumps (GARCH-jump). We compare three GARCH-jump models of Chang, Chang, Cheng, Peng, and Tseng (2018) to examine whether specifying asymmetric jumps in conditional jump intensity can improve the empirical performance. The empirical results from S&P 500 returns and options show that specifying the asymmetric jumps into the conditional jump intensity does improve the in-sample pricing errors and implied volatility errors. However, the out-of-sample results depend on the error measurement.     

Priced risk and asymmetric volatility in the cross section of skewness

We investigate the sources of skewness in aggregate risk factors and the cross section of stock returns. In an ICAPM setting with conditional volatility, we find theoretical time series predictions on the relationships among volatility, returns, and skewness for priced risk factors. Market returns resemble these predictions; however, size, book-to-market, and momentum factor returns are not always consistent with our predictions. We find evidence that size and book-to-market may be priced post-crisis but not in the decade before. Momentum does not appear priced by our test. We link aggregate risk and skewness to individual stocks and find empirically that the risk aversion effect manifests in individual stock skewness. Additionally, we find several firm characteristics that explain stock skewness. Smaller firms, value firms, highly levered firms, and firms with poor credit ratings have more positive skewness.     

GMM estimation of spatial autoregressive models with unknown heteroskedasticity

In the presence of heteroskedastic disturbances, the MLE for the SAR models without taking into account the heteroskedasticity is generally inconsistent. The 2SLS estimates can have large variances and biases for cases where regressors do not have strong effects. In contrast, GMM estimators obtained from certain moment conditions can be robust. Asymptotically valid inferences can be drawn with consistently estimated covariance matrices. Efficiency can be improved by constructing the optimal weighted estimation.     

Empirical modeling of high-income and emerging stock and Forex market return volatility using Markov-switching GARCH models

Using weekly data for stock and Forex market returns, a set of MS-GARCH models is estimated for a group of high-income (HI) countries and emerging market economies (EMEs) using algorithms proposed by Augustyniak (2014) and Ardia et al. (2018, 2019a,b), allowing for a variety of conditional variance and distribution specifications. The main results are: (i) the models selected using Ardia et al. (2018) have a better fit than those estimated by Augustyniak (2014), contain skewed distributions, and often require that the main coefficients be different in each regime; (ii) in Latam Forex markets, estimates of the heavy-tail parameter are smaller than in HI Forex and all stock markets; (iii) the persistence of the high-volatility regime is considerable and more evident in stock markets (especially in Latam EMEs); (iv) in (HI and Latam) stock markets, a single-regime GJR model (leverage effects) with skewed distributions is selected; but when using MS models, virtually no MS-GJR models are selected. However, this does not happen in Forex markets, where leverage effects are not found either in single-regime or MS-GARCH models.     

Modeling the conditional distribution of financial returns with asymmetric tails

This paper proposes a conditional density model that allows for differing left/right tail indices and time-varying volatility based on the dynamic conditional score (DCS) approach. The asymptotic properties of the maximum likelihood estimates are presented under verifiable conditions together with simulations showing effective estimation with practical sample sizes. It is shown that tail asymmetry is prevalent in global equity index returns and can be mistaken for skewness through the center of the distribution. The importance of tail asymmetry for asset allocation and risk premia is demonstrated in-sample. Application to portfolio construction out-of-sample is then considered, with a representative investor willing to pay economically and statistically significant management fees to use the new model instead of traditional skewed models to determine their asset allocation.     

The common and specific components of dynamic volatility

This paper develops a dynamic approximate factor model in which returns are time-series heteroskedastic. The heteroskedasticity has three components: a factor-related component, a common asset-specific component, and a purely asset-specific component. We develop a new multivariate GARCH model for the factor-related component. We develop a univariate stochastic volatility model linked to a cross-sectional series of individual GARCH models for the common asset-specific component and the purely asset-specific component. We apply the analysis to monthly US equity returns for the period January 1926 to December 2000. We find that all three components contribute to the heteroskedasticity of individual equity returns. Factor volatility and the common component in asset-specific volatility have long-term secular trends as well as short-term autocorrelation. Factor volatility has correlation with interest rates and the business cycle.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Maximum likelihood estimation of a GARCH-stable model

Maximum likelihood is used to estimate a generalized autoregressive conditional heteroskedastic (GARCH) process where the residuals have a conditional stable distribution (GARCH-stable). The scale parameter is modelled such that a GARCH process with normally distributed residuals is a special case. The usual methods of estimating the parameters of the stable distribution assume constant scale and will underestimate the characteristic exponent when the scale parameter follows a GARCH process. The parameters of the GARCH-stable model are estimated with daily foreign currency returns. Estimates of characteristic exponents are higher with the GARCH-stable than when independence is assumed. Monte Carlo hypothesis testing procedures, however, reject our GARCH-stable model at the 1% significance level in four out of five cases.     

Conditional correlations and volatility spillovers between crude oil and stock index returns

This paper investigates the conditional correlations and volatility spillovers between the crude oil and financial markets, based on crude oil returns and stock index returns. Daily returns from 2 January 1998 to 4 November 2009 of the crude oil spot, forward and futures prices from the WTI and Brent markets, and the FTSE100, NYSE, Dow Jones and S&P500 stock index returns, are analysed using the CCC model of Bollerslev (1990), VARMA-GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer, Hoti, and Chan (2008), and DCC model of Engle (2002). Based on the CCC model, the estimates of conditional correlations for returns across markets are very low, and some are not statistically significant, which means the conditional shocks are correlated only in the same market and not across markets. However, the DCC estimates of the conditional correlations are always significant. This result makes it clear that the assumption of constant conditional correlations is not supported empirically. Surprisingly, the empirical results from the VARMA-GARCH and VARMA-AGARCH models provide little evidence of volatility spillovers between the crude oil and financial markets. The evidence of asymmetric effects of negative and positive shocks of equal magnitude on the conditional variances suggests that VARMA-AGARCH is superior to VARMA-GARCH and CCC.     

Stock returns and volatility: Evidence from the Athens Stock market index

This paper investigates the volatility of the Athens Stock excess stock returns over the period 1990–1999 through the comparison of various conditional hetero-skedasticity models. The empirical results indicate that there is significant evidence for asymmetry in stock returns, which is captured by a quadratic GARCH specification model, while there is strong persistence of shocks into volatility.