Whittle estimation of EGARCH and other exponential volatility models

The strong consistency and asymptotic normality of the Whittle estimate of the parameters in a class of exponential volatility processes are established. Our main focus here are the EGARCH model of [Nelson, D. 1991. Conditional heteroscedasticity in asset pricing: A new approach. Econometrica 59, 347–370] and other one-shock models such as the GJR model of [Glosten, L., Jaganathan, R., Runkle, D., 1993. On the relation between the expected value and the volatility of the nominal excess returns on stocks. Journal of Finance, 48, 1779–1801], but two-shock models, such as the SV model of [Taylor, S. 1986. Modelling Financial Time Series. Wiley, Chichester, UK], are also comprised by our assumptions. The variable of interest might not have finite fractional moment of any order and so, in particular, finite variance is not imposed. We allow for a wide range of degrees of persistence of shocks to conditional variance, allowing for both short and long memory.     

Extreme returns and intensity of trading

Asymmetric information models of market microstructure claim that variables such as trading intensity are proxies for latent information on the value of financial assets. We consider the interval‐valued time series (ITS) of low/high returns and explore the relationship between these extreme returns and the intensity of trading. We assume that the returns (or prices) are generated by a latent process with some unknown conditional density. At each period of time, from this density, we have some random draws (trades) and the lowest and highest returns are the realized extreme observations of the latent process over the sample of draws. In this context, we propose a semiparametric model of extreme returns that exploits the results provided by extreme value theory. If properly centered and standardized extremes have well‐defined limiting distributions, the conditional mean of extreme returns is a nonlinear function of the conditional moments of the latent process and of the conditional intensity of the process that governs the number of draws. We implement a two‐step estimation procedure. First, we estimate parametrically the regressors that will enter into the nonlinear function, and in a second step we estimate nonparametrically the conditional mean of extreme returns as a function of the generated regressors. Unlike current models for ITS, the proposed semiparametric model is robust to misspecification of the conditional density of the latent process. We fit several nonlinear and linear models to the 5‐minute and 1‐minute low/high returns to seven major banks and technology stocks, and find that the nonlinear specification is superior to the current linear models and that the conditional volatility of the latent process and the conditional intensity of the trading process are major drivers of the dynamics of extreme returns.     

Non-Gaussian models for CoVaR estimation

In this paper we show how to obtain estimates of CoVaR based on models that take into consideration some stylized facts about multivariate financial time series of equity log returns: heavy tails, negative skew, asymmetric dependence, and volatility clustering. While the volatility clustering effect is captured by AR-GARCH dynamics of the Glosten-Jagannathan-Runkle (GJR) type, the other stylized facts are explained by non-Gaussian multivariate models and copula functions. We compare the different models in the period from January 2007 to March 2020. Our empirical study conducted on a sample of listed banks in the euro area confirms that, in measuring CoVaR, it is important to capture the time-varying dynamics of the volatility. Additionally, a correct assessment of the heaviness of the tails and of the dependence structure is needed in the evaluation of this systemic risk measure.     

Semiparametric inference in a GARCH-in-mean model

A new semiparametric estimator for an empirical asset pricing model with general nonparametric risk-return tradeoff and GARCH-type underlying volatility is introduced. Based on the profile likelihood approach, it does not rely on any initial parametric estimator of the conditional mean function, and it is under stated conditions consistent, asymptotically normal, and efficient, i.e., it achieves the semiparametric lower bound. A sampling experiment provides finite sample comparisons with the parametric approach and the iterative semiparametric approach with parametric initial estimate of Conrad and Mammen (2008). An application to daily stock market returns suggests that the risk-return relation is indeed nonlinear.     

An econometric analysis of asymmetric volatility: Theory and application to patents

The purpose in registering patents is to protect the intellectual property of the rightful owners. Deterministic and stochastic trends in registered patents can be used to describe a country's technological capabilities and act as a proxy for innovation. This paper presents an econometric analysis of the symmetric and asymmetric volatility of the patent share, which is based on the number of registered patents for the top 12 foreign patenting countries in the USA. International rankings based on the number of foreign US patents, patent intensity (or patents per capita), patent share, the rate of assigned patents for commercial exploitation, and average rank scores, are given for the top 12 foreign countries. Monthly time series data from the United States Patent and Trademark Office for January 1975 to December 1998 are used to estimate symmetric and asymmetric models of the time-varying volatility of the patent share, namely US patents registered by each of the top 12 foreign countries relative to total US patents. A weak sufficient condition for the consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the univariate GJR(1,1) model is established under non-normality of the conditional shocks. The empirical results provide a diagnostic validation of the regularity conditions underlying the GJR(1,1) model, specifically the log-moment condition for consistency and asymptotic normality of the QMLE, and the computationally more straightforward but stronger second and fourth moment conditions. Of the symmetric and asymmetric models estimated, AR(1)–EGARCH(1,1) is found to be suitable for most countries, while AR(1)–GARCH(1,1) and AR(1)–GJR(1,1) also provide useful insights. Non-nested procedures are developed to test AR(1)–GARCH(1,1) versus AR(1)–EGARCH(1,1), and AR(1)–GJR(1,1) versus AR(1)–EGARCH(1,1).     

Long memory and structural breaks in modeling the return and volatility dynamics of precious metals

We investigate the potential of structural changes and long memory (LM) properties in returns and volatility of the four major precious metal commodities traded on the COMEX markets (gold, silver, platinum and palladium). Broadly speaking, a random variable is said to exhibit long memory behavior if its autocorrelation function is not integrable, while structural changes can induce sudden and significant shifts in the time-series behavior of that variable. The results from implementing several parametric and semiparametric methods indicate strong evidence of long range dependence in the daily conditional return and volatility processes for the precious metals. Moreover, for most of the precious metals considered, this dual long memory is found to be adequately captured by an ARFIMA–FIGARCH model, which also provides better out-of-sample forecast accuracy than several popular volatility models. Finally, evidence shows that conditional volatility of precious metals is better explained by long memory than by structural breaks.     

The stochastic volatility in mean model and automation: Evidence from TSE

This paper investigates the behavior of volatility in Canadian equity markets before and after automation. We employ a stochastic volatility in mean (SVM) model that incorporates the unobserved volatility as an explanatory variable in the mean equation. The volatility persistent estimates all increase post-automation, with the scaling parameter increasing as well. The parameter estimates which measure both the ex ante relationship between returns and volatility and the volatility feedback effect are found to be negative for all series, and to increase post-automation. Our results fall in line with those of (French, K.R., Schwert, G.W., & Stanbaugh, R.F. (1987). Expected stock returns and volatility. Journal of Financial Economics, 19, 3–29) who find similar relationship between unexpected volatility dynamics and returns and confirm the hypothesis that rational risk-averse investors require higher expected returns when unanticipated increase in future volatility are highly persistent. Finally, our findings are consistent since higher values of persistence are combined with larger negative values for the in-mean parameter.     

Measuring financial risk and portfolio reversion with time changed tempered stable Lévy processes

Given that underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper applies the stochastic volatility models driven by tempered stable Lévy processes to construct time changed tempered stable Lévy processes (TSSV) for financial risk measurement and portfolio reversion. The TSSV model framework permits infinite activity jump behaviors of returns dynamics and time varying volatility consistently observed in financial markets by introducing time changing volatility into tempered stable processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, capturing leptokurtosis, fat tailedness and asymmetry features of returns in addition to volatility clustering effect in stochastic volatility. Through employing the analytical characteristic function and fast Fourier transform (FFT) technique, the closed form formulas for probability density function (PDF) of returns, value at risk (VaR) and conditional value at risk (CVaR) can be derived. Finally, in order to forecast extreme events and volatile market, we perform empirical researches on Hangseng index to measure risks and construct portfolio based on risk adjusted reward risk stock selection criteria employing TSSV models, with the stochastic volatility normal tempered stable (NTSSV) model producing superior performances relative to others.     

Trading and non-trading period realized market volatility: Does it matter for forecasting the volatility of US stocks?

We study the potential merits of using trading and non-trading period market volatilities to model and forecast the stock volatility over the next one to 22 days. We demonstrate the role of overnight volatility information by estimating heterogeneous autoregressive (HAR) model specifications with and without a trading period market risk factor using ten years of high-frequency data for the 431 constituents of the S&P 500 index. The stocks’ own overnight squared returns perform poorly across stocks and forecast horizons, as well as in the asset allocation exercise. In contrast, we find overwhelming evidence that the market-level volatility, proxied by S&P Mini futures, matters significantly for improving the model fit and volatility forecasting accuracy. The greatest model fit and forecast improvements are found for short-term forecast horizons of up to five trading days, and for the non-trading period market-level volatility. The documented increase in forecast accuracy is found to be associated with the stocks’ sensitivity to the market risk factor. Finally, we show that both the trading and non-trading period market realized volatilities are relevant in an asset allocation context, as they increase the average returns, Sharpe ratios and certainty equivalent returns of a mean–variance investor.     

Sources of volatility persistence: A case study of the U.K. pound/U.S. dollar exchange rate returns

This paper utilizes a new approach to examine the inherent nonlinear dynamics of the exchange rate returns volatility. Specifically, we utilize a regime switching threshold (i) generalized autoregressive conditional heteroskedasticity (RS-TGARCH) and (ii) a fractional generalized autoregressive conditional heteroskedasticity (RS-TFIGARCH) model. The RS-TGARCH model is found to be adequate in analyzing the first two moments of the U.K. pound/U.S. dollar monthly exchange rate returns series. The RS-TFIGARCH is found to be adequate for the daily returns series. The volatility persistence and leverage effects associated with exchange rate returns series are jointly tested by means of a Wald Chi-square test.     

Forecasting volatility with time-varying leverage and volatility of volatility effects

Predicting volatility is of primary importance for business applications in risk management, asset allocation, and the pricing of derivative instruments. This paper proposes a measurement model that considers the possibly time-varying interaction of realized volatility and asset returns according to a bivariate model to capture its major characteristics: (i) the long-term memory of the volatility process, (ii) the heavy-tailedness of the distribution of returns, and (iii) the negative dependence of volatility and daily market returns. We assess the relevance of the effects of “the volatility of volatility” and time-varying “leverage” to the out-of-sample forecasting performance of the model, and evaluate the density of forecasts of market volatility. Empirical results show that our specification can outperform the benchmark HAR–GARCH model in terms of both point and density forecasts.     

Econometric analysis of jump-driven stochastic volatility models

This paper introduces and studies the econometric properties of a general new class of models, which I refer to as jump-driven stochastic volatility models, in which the volatility is a moving average of past jumps. I focus attention on two particular semiparametric classes of jump-driven stochastic volatility models. In the first, the price has a continuous component with time-varying volatility and time-homogeneous jumps. The second jump-driven stochastic volatility model analyzed here has only jumps in the price, which have time-varying size. In the empirical application I model the memory of the stochastic variance with a CARMA(2,1) kernel and set the jumps in the variance to be proportional to the squared price jumps. The estimation, which is based on matching moments of certain realized power variation statistics calculated from high-frequency foreign exchange data, shows that the jump-driven stochastic volatility model containing continuous component in the price performs best. It outperforms a standard two-factor affine jump–diffusion model, but also the pure-jump jump-driven stochastic volatility model for the particular jump specification.     

A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns

In this paper, we investigate the goodness-of-fit of three Lévy processes, namely Variance-Gamma (VG), Normal-Inverse Gaussian (NIG) and Generalized Hyperbolic (GH) distributions, and probability distribution of the Heston model to index returns of twenty developed and emerging stock markets. Furthermore, we extend our analysis by applying a Markov regime switching model to identify normal and turbulent periods. Our findings indicate that the probability distribution of the Heston model performs well for emerging markets under full sample estimation and retains goodness of fit for high volatility periods, as it explicitly accounts for the volatility process. On the other hand, the distributions of the Lévy processes, especially the VG and NIG distributions, generally improves upon the fit of the Heston model, particularly for developed markets and low volatility periods. Furthermore, some distributions yield to significantly large test statistics for some countries, even though they fit well to other markets, which suggest that properties of the stock markets are crucial in identifying the best distribution representing empirical returns.     

MULTIVARIATE VOLATILITY MODELING OF ELECTRICITY FUTURES

We model the dynamic volatility and correlation structure of electricity futures of the European Energy Exchange index. We use a new multiplicative dynamic conditional correlation (mDCC) model to separate long‐run from short‐run components. We allow for smooth changes in the unconditional volatilities and correlations through a multiplicative component that we estimate nonparametrically. For the short‐run dynamics, we use a GJR‐GARCH model for the conditional variances and augmented DCC models for the conditional correlations. We also introduce exogenous variables to account for congestion and delivery date effects in short‐term conditional variances. We find different correlation dynamics for long‐ and short‐term contracts and the new model achieves higher forecasting performance compared \to a standard DCC model. Copyright © 2012 John Wiley & Sons, Ltd.     

Forecasting risk measures using intraday data in a generalized autoregressive score framework

A new framework for the joint estimation and forecasting of dynamic value at risk (VaR) and expected shortfall (ES) is proposed by our incorporating intraday information into a generalized autoregressive score (GAS) model introduced by Patton et al., 2019 to estimate risk measures in a quantile regression set-up. We consider four intraday measures: the realized volatility at 5-min and 10-min sampling frequencies, and the overnight return incorporated into these two realized volatilities. In a forecasting study, the set of newly proposed semiparametric models are applied to four international stock market indices (S&P 500, Dow Jones Industrial Average, Nikkei 225 and FTSE 100) and are compared with a range of parametric, nonparametric and semiparametric models, including historical simulations, generalized autoregressive conditional heteroscedasticity (GARCH) models and the original GAS models. VaR and ES forecasts are backtested individually, and the joint loss function is used for comparisons. Our results show that GAS models, enhanced with the realized volatility measures, outperform the benchmark models consistently across all indices and various probability levels.     

Stochastic volatility models: conditional normality versus heavy‐tailed distributions

Most of the empirical applications of the stochastic volatility (SV) model are based on the assumption that the conditional distribution of returns, given the latent volatility process, is normal. In this paper, the SV model based on a conditional normal distribution is compared with SV specifications using conditional heavy‐tailed distributions, especially Student's t‐distribution and the generalized error distribution. To estimate the SV specifications, a simulated maximum likelihood approach is applied. The results based on daily data on exchange rates and stock returns reveal that the SV model with a conditional normal distribution does not adequately account for the two following empirical facts simultaneously: the leptokurtic distribution of the returns and the low but slowly decaying autocorrelation functions of the squared returns. It is shown that these empirical facts are more adequately captured by an SV model with a conditional heavy‐tailed distribution. It also turns out that the choice of the conditional distribution has systematic effects on the parameter estimates of the volatility process. Copyright © 2000 John Wiley & Sons, Ltd.     

The political uncertainty and stock market behavior in emerging democracy: the case of Taiwan

This paper examines the congressional effect between the pre- and post-democratization on the stock market by the asymmetric Generalized Autoregressive Conditional Heterosce desticity (GARCH) model in the period 1984–2004. The results found that the congressional effect is negative effect on stock returns but volatility is not significant. However, the democratic effect on stock returns is negative and increased of volatility. Moreover, the congressional effect on stock market returns following democratization significantly exceeds that before democratization, but have no significant effect for the volatility in the same circumstances. These results provide evidences consistent with the contention of liberalization (Hayek, Am. Econ. Rev. 35, 519–530, Individualism and Economic order, The university of Chicago press, Chicago, London, 1945, 1948; Popper, The open society and its Enemies, Princeton university, NJ, 1950).     

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.     

Using CARRX models to study factors affecting the volatilities of Asian equity markets

The range of daily asset prices is often used as a measure of volatility. Using a CARRX (conditional autoregressive range with exogenous variables) model, and the parsimony principle, the paper investigates the factors affecting the volatilities of Asian equity markets. Since the beginning of the new Century, emerging Asian markets such as Taiwan and Shanghai have been undergoing various stages of financial globalization. The volatility of the equity market may not be explained solely by its own dynamics. In this paper, we examine volatility using the following factors: (i) lagged returns; (ii) lagged absolute returns; (iii) own trading volume; (iv) U.S. factors; (v) European factors; and (vi) regional (Asian) factors. Points (i) and (iii) are by and large significant, while (ii) is not. Controlling for (i), (ii) and (iii), we find evidence that the volatility of European markets has spillovers on to both the Taiwan and Tokyo markets, mild evidence that the volatility of the U.S. market has spillovers on to the Hong Kong market, but there are no spillovers from the European or U.S. markets on to the Shanghai market.     

Estimating and predicting multivariate volatility thresholds in global stock markets

We propose a general double tree structured AR‐GARCH model for the analysis of global equity index returns. The model extends previous approaches by incorporating (i) several multivariate thresholds in conditional means and volatilities of index returns and (ii) a richer specification for the impact of lagged foreign (US) index returns in each threshold. We evaluate the out‐of‐sample forecasting power of our model for eight major equity indices in comparison to some existing volatility models in the literature. We find strong evidence for more than one multivariate threshold (more than two regimes) in conditional means and variances of global equity index returns. Such multivariate thresholds are affected by foreign (US) lagged index returns and yield a higher out‐of‐sample predictive power for our tree structured model setting. Copyright © 2006 John Wiley & Sons, Ltd.