The determinants of equity risk and their forecasting implications: A quantile regression perspective

Bonaccolto, Giovanni; Caporin, Massimiliano Article The determinants of equity risk and their forecasting implications: A quantile regression perspective Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Bonaccolto, Giovanni; Caporin, Massimiliano (2016) : The determinants of equity risk and their forecasting implications: A quantile regression perspective, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 9, Iss. 3, pp. 1-25, https://doi.org/10.3390/jrfm9030008 This Version is available at: https://hdl.handle.net/10419/178575 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article The Determinants of Equity Risk and Their Forecasting Implications: A Quantile Regression Perspective Giovanni Bonaccolto 1,† and Massimiliano Caporin 2,*,† 1Department of Economics and Management “Marco Fanno”, University of Padova, Via del Santo 22, 35123 Padova, Italy; [email protected] 2Department of Statistical Sciences, University of Padova, Via Cesare Battisti 241, 35121 Padova, Italy *Correspondence: [email protected]; Tel.: +39-049-827-4199 † These authors contributed equally to this work. Academic Editor: Michael McAleer Received: 21 May 2016; Accepted: 29 June 2016; Published: 7 July 2016 Abstract: Several market and macro-level variables influence the evolution of equity risk in addition to the well-known volatility persistence. However, the impact of those covariates might change depending on the risk level, being different between low and high volatility states. By combining equity risk estimates, obtained from the Realized Range Volatility, corrected for microstructure noise and jumps, and quantile regression methods, we evaluate the forecasting implications of the equity risk determinants in different volatility states and, without distributional assumptions on the realized range innovations, we recover both the points and the conditional distribution forecasts. In addition, we analyse how the the relationships among the involved variables evolve over time, through a rolling window procedure. The results show evidence of the selected variables’ relevant impacts and, particularly during periods of market stress, highlight heterogeneous effects across quantiles. Keywords: realized range volatility; quantile regression; volatility quantiles and density forecasting; forecast assessment 1. Introduction Recent events, such as the subprime crisis, that originated in the United States and was marked by the Lehman Brothers’ default in September 2008 and the sovereign debt crisis, that hit the Eurozone in 2009, have highlighted the fundamental importance of risk measurement, monitoring and forecasting. The volatility of asset returns, a commonly used measure of risk, is a key variable in several areas of finance and investment, such as risk management, asset allocation, pricing and trading strategies. Therefore, estimating and forecasting the volatility point values and distribution play a critical role. The use of predicted volatility levels is central, for instance, in the pricing of equity derivatives, in the development of equity derivative trading strategies and in risk measurement when risk is associated with volatility. The volatility distribution is of interest when trading/pricing volatility derivatives, when designing volatility hedges for generic portfolios and when accounting for the uncertainty of volatility point forecasts. Many financial applications use constant volatility models [ 1 ], although empirical evidence suggests that variance changes over time. Several approaches have been developed with the purpose of achieving more accurate estimates, such as the class of autoregressive conditional heteroscedasticity (ARCH) [ 2 ] and generalized autoregressive heteroscedasticity (GARCH) [ 3 ] models and the stochastic volatility models [ 4 – 7 ]. Nevertheless, financial data are affected by several features, such as the so-called stylized facts [ 8 ], and standard GARCH and stochastic volatility models do not capture all of them [9]. J. Risk Financial Manag. 2016,9, 8; doi:10.3390/jrfm9030008 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2016,9, 8 2 of 25 We might also estimate volatility with non-parametric methods, such as by means of realized measures, which have been shown to perform better than traditional GARCH and stochastic volatility models when forecasting conditional second-order moments [ 10 ]. This approach has attracted considerable interest because of the availability of high-frequency financial data. In fact, as opposed to models that treat volatility as a latent (non-observable) element, realized measures use additional information from the intraday returns. We believe these methods provide more flexibility than standard GARCH-type models do, and therefore we focus on realized measures here. Among the various realized volatility measures, we choose a range-based measure for reasons of efficiency. Notably, the realized range estimator is five times more efficient than the corresponding realized variance [ 11 ]. This is because the realized range is built from two prices, the minimum and the maximum, for each intradaily interval, as opposed to using just the closing price in the realized volatility estimator. We therefore follow Christensen et al. [ 12 ] and adopt the realized range-based bias corrected bipower variation. Christensen et al. [ 12 ] demonstrated that their measure is a consistent estimator of the integrated variance in the presence of microstructure noise, arising from peculiar phenomena like non-continuous trading, infrequent trades and bid-ask bounce [13–15], and price jumps. Studies that focus on volatility forecasting have identified key macroeconomic and financial variables as important drivers of volatility, highlighting their power in improving forecast performances. For instance, Christiansen et al. [ 16 ] predicted the asset return volatility using macroeconomic and financial variables in a Bayesian model averaging framework. They considered several asset classes, such as equities, foreign exchange, bonds and commodities, over long time spans and found that economic variables provide information about future volatility from both an in-sample and an out-of-sample perspective. Paye [ 17 ] tested the power of financial and economic variables to forecast the volatility at monthly and quarterly horizons and rarely found a statistical difference between the performance of macroeconomic fundamentals and univariate benchmarks. Fernandes et al. [ 18 ] used parametric and semi-parametric heterogeneous auto regressive (HAR) processes to model and forecast the CBOE Volatility Index (VIX) and found significant results using financial and macroeconomic variables as additional regressors. Caporin and Velo [ 19 ] used an HAR model with asymmetric effects with respect to volatility and return and GARCH and Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) specifications for the variance equation. Caporin et al. [20] studied the relationship between the first principal component (FPC) of the volatility jumps, estimated using 36 stocks and a set of macroeconomic and financial variables, such as the VIX, the S&P 500 volume, the Credit Default Swap (CDS) and the Federal Fund rates, and found that CDS captures a large part of the moves of the expected jumps. Opschoor et al. [ 21 ] used the Bloomberg Financial Conditions Index, which comprises money, bond and equity markets, observing that worse financial conditions are associated with both higher volatility and higher correlations. Given the findings in the previous studies, we take into account several macroeconomic and financial variables commonly used in the literature; nevertheless, among them, we select just the VIX and the S&P 500 index, as all the other variables turned out to be non-significant in the model we propose. Our purpose is to generalize the previous contributions further. We model and forecast the conditional quantiles of the realized range-based bias corrected bipower variation using the quantile regression method introduced by Koenker and Bassett [22]. Focusing on realized volatility measures, we believe that the quantile regression approach can provide useful new evidence by allowing the entire conditional distribution of the realized volatility measure to be estimated, instead of restricting attention to the conditional mean. This approach could have a relevant advantage when the impact of covariates is changing, depending on market conditions (say, on low or high volatility states) or when the purpose is to recover density forecasts without making a distributional assumption. Other authors have applied quantile regression in a financial framework. For instance, Engle and Manganelli [ 23 ] proposed the Conditional Autoregressive Value at Risk (CAViaR) model to estimate the conditional value-at-risk, an important measure of risk that financial institutions and their regulators employ. In the present work, we do not consider the CAViaR model in order to J. Risk Financial Manag. 2016,9, 8 3 of 25 control the computational complexity of the analysis. White et al. [ 24 ] generalized the CAViaR to the multi-quantile CAViaR (MQ-CAViaR) model, studying the conditional skewness and kurtosis of S&P 500 daily returns. White et al. [ 25 ] extended the MQ-CAViaR model in the multivariate context to measure the systemic risk, a critical issue highlighted by the recent financial crises, taking into account the relationships among 230 financial institutions around the world. Li and Miu [ 26 ] proposed, on the basis of the binary quantile regression approach, a hybrid bankruptcy prediction model with dynamic loadings for both the accounting-ratio based and market-based information. Castro and Ferrari [ 27 ] used the ∆CoVaR model as a tool for identifying/ranking systemically important institutions. Finally, Caporin et al. [ 28 ] adopted quantile regressions in detecting financial contagion across bond spreads in Europe. Our work is closely related to the contribution of Zikes and Barunik [ 29 ]. However, our analysis differs in some important ways. First, Zikes and Barunik [ 29 ] estimated volatility using a realized measure that takes into account only the effects of jumps in the price process. In contrast, we use the realized range-based bias corrected bipower variation, which considers the impact of microstructure noise and that of jumps. To the best of our knowledge, quantile regression methods for the analysis of realized range volatility measures have never been used in the econometric and empirical financial literature, which provides a strong motivation for our study. Second, Zikes and Barunik [ 29 ] performed quantile forecasts, focusing on a few quantiles; we go farther by estimating a fine grid of quantiles in order to recover information about the entire realized volatility conditional distribution. Finally, on the basis of the obtained volatility quantiles, we build the entire conditional volatility density. Our work takes an empiric point of view and focuses on the high-frequency data of 16 stocks issued by large-cap companies that operate in different economic sectors. All companies are quoted on the U.S. market. In the first step, we analyse the FPC of the estimated volatility as a summary of the 16 series (a kind of market factor). Our first empirical contribution to the literature by showing that the VIX and the S&P 500 index have a significant impact on volatility quantiles, that their impact changes across quantiles, becoming irrelevant in some cases, and that the significance and strength of the relationship changes over time. This last finding is particularly evident when we focus on turbulent market phases, as in these periods we note an increase in the impact of both the VIX and the S&P 500 index. The last finding is particularly evident for high-volatility quantiles. The heterogeneity we observe across quantiles can also be interpreted as evidence against the location-shift hypothesis, that assumes homogeneous impacts of the covariates across the conditional quantiles of the response variable. Therefore, we highlight the benefits obtained through the quantile regression approach with respect to the classical least squares method, which focuses just on the conditional expected value of the volatility and is not able to capture the more interesting and significant relationships in its extreme conditional quantiles. This empirical evidence suggests that the uncertainty of volatility point forecasting, as measured by the influence of covariates on different volatility quantiles, changes depending on market states. Consequently, volatility forecast precision is ensured, as is the appropriateness of volatility density forecasting. Both of these elements have a central role in risk management and volatility hedging and trading, thus providing support for our analyses. In volatility hedging, or in the pricing of volatility derivatives, the most traditional measure is the realized volatility. However, as previously mentioned, the realized range estimator is more efficient and therefore provides more appropriate density forecasts as opposed to the realized variance density forecasts. Our second empirical contribution comes from the single-asset analyses. We develop a specific model for each asset to determine how the features of the 16 companies affect the relationships among the variables involved. We find some heterogeneity in the assets’ reactions to macro-level variables, such as the VIX and the S&P 500 index, which holds across both time and volatility quantiles. However, we find overall confirmation of the findings associated with the FPC. Therefore, the relevance of the quantile-based covariate impact on volatility is of interest both at the aggregated and at the single-asset level. J. Risk Financial Manag. 2016,9, 8 4 of 25 Our third empirical contribution stems from a forecasting exercise. We compare the quantile-based density forecasts to those of a benchmark model adapted to the realized range volatility mean and variance. The reference model combines an HAR structure of the realized volatility mean plus a GJR-GARCH [ 30 ] for the mean innovation variances. The HAR structure, inspired by the work of Corsi [ 9 ], captures the persistence of realized measures and is consistent with the presence of heterogeneous agents in the market, while the GJR-GARCH structure is coherent with the volatility of volatility hypothesis discussed in Corsi et al. [ 31 ]. We compare the benchmark model density forecasts and the quantile regression-based forecasts using the tests proposed by Berkowitz [ 32 ] and Amisano and Giacomini [ 33 ]. Moreover, by using a quantile-based loss function, we also consider the Diebold and Mariano [ 34 ] test. We stress that the Berkowitz [ 32 ] test allows for an absolute evaluation of the density forecasts provided by one model, while the Amisano and Giacomini [ 33 ] approach compares two competing models. The results confirm that our approach performs better, thus providing support for the use of quantile regression methods in all areas where volatility quantiles might have a role. Among the possible applications, we mention volatility trading and volatility hedging [ 35 , 36 ], as well as the evaluation of uncertainty around volatility point forecasts. The latter could be of crucial importance if we plan to combine volatility point and density forecasts derived from realized volatility measures with returns-based volatility models, as in the high-frequency-based volatility (HEAVY) models [37]. The remainder of the paper is structured as follows. Section 2includes the description of the data and of the conditioning variables. Section 3presents the model we propose to forecast the range bipower variation conditional quantiles. Section 4is devoted to the density forecast and predictive accuracy and provides the details about the tests we use. The results are analysed in Section 5, and Section 6consists of concluding remarks. 2. Realized Volatilities and Conditioning Variables The database we use includes stock prices recorded with a frequency of one minute, from 9:30 a.m. to 4:00 p.m. of every trading day between 2 January 2003 and 28 June 2013, inclusive. The equities analysed are those of large companies that operate in various economic sectors of the U.S. market: AT&T Inc. (ATT), Bank of America (BAC), Boeing (BOI), Caterpillar, Inc. (CAT), Citigroup, Inc. (CTG), FedEx Corporation (FDX), Honeywell International, Inc. (HON), Hewlett-Packard Company (HPQ), International Business Machines Corp. (IBM), JPMorgan Chase & Co. (JPM), Mondelez International, Inc. (MDZ), Pepsico, Inc. (PEP), The Procter & Gamble Company (PRG), Time Warner, Inc. (TWX), Texas Instruments, Inc. (TXN) and Wells Fargo & Company (WFC). The dataset is drawn from TickData. The prices are adjusted for extraordinary operations and filtered for errors, anomalies and outliers that arise from traders’ activities (for additional details see the company website http://www.tickdata.com). From the high-frequency data described above and after computing the assets returns, we estimate the daily volatilities of the 16 assets using the realized range-based bias corrected bipower variation ( RRVn,m BVBC ) introduced by Christensen et al. [ 12 ]. Notably, RRVn,m BVBC is a robust estimator of the integrated variance in the the presence of both price jumps and microstructure noise. The realized range volatility sequences show that the financial companies are the most volatile, while all the companies show sudden increases in the risk measure during the global financial crisis. Detailed descriptive analysis and plots of the estimated realized range are available upon request. A principal component analysis is carried out on the range-based bias corrected bipower variations of the 16 assets. In particular, the first principal component ( FPC ) explains 77% of the overall variance. The evolution of the assets’ volatilities have a strong common behaviour that we might interpret as market or systematic behaviour. Therefore, the analysis of FPC could produce useful results. Consequently, in the following, we model the conditional quantiles of both FPC and the single-asset realized volatilities. Previous studies in the literature, such as those by Caporin and Velo [ 19 ] and Caporin et al. [ 20 ], have used several indicators to analyse the realized variance and the range series. Therefore, in addition J. Risk Financial Manag. 2016,9, 8 5 of 25 to the data described so far, we have taken into account some macroeconomic and financial variables, such as the VIX and the S&P 500 indexes, the U.S. dollar–euro exchange rate, the crude oil price and the US Treasury interest rates, that convey important information about the overall market trend and risk. They will be considered exogenous variables that affect the conditional volatility quantiles. However, we included just two of them in our model: the daily return of the S&P 500 index ( sp 500) and the logarithm of the VIX index ( vix ), as all the others turned out to be non-significant in the model we propose. sp 500 reflects the trend of the U.S. stock market, and we expect that negative returns of the S&P 500 will have a positive impact on the market volatility, consistently to the well-known leverage effect. In contrast, high levels of vix reflect pessimism among the economic agents, so a positive relationship between vix and the volatility level is expected. The observations associated with sp500 and vix are recorded at a daily frequency and are recovered from Datastream. Many macroeconomic and financial time series are not stationary and are often characterized by unit-root non-stationarity [ 38 ]. Building on this evidence, we must determine whether the data-generating process of vix is affected by unit-root (we do not report the tests on sp 500, as they provided no useful results: the index level is non-stationary, while the index return is stationary). To this purpose, we consider two standard tests: the augmented Dickey–Fuller (ADF) test [ 39 ] and the Phillips–Perron (PP) test [ 40 ]. The ADF test rejects the null hypothesis at the 10% significance level, and the test statistic equals − 3.18 (with a p-value of 0.09). The PP test rejects the null hypothesis at the 5% level, and the test statistic is − 24.48 (with a p-value of 0.03). Therefore, we have a moderate amount of evidence against the presence of a unit root for the vix series. The rejection of the null is much clearer with the PP test. Beside vix and sp 500, we use other quantities to describe the evolution of realized range conditional quantiles. First, we follow Corsi [ 9 ] in introducing among the explanatory variables those commonly adopted in (HAR) models of realized volatility. Focusing on the FPC of realized ranges, whose observed value at t is denoted by f pct , a first explanatory variable is its lagged value, f pct−1 . This variable is usually accompanied by other quantities that are built from local averages of past elements: f pcm∗t=1 m∗ m∗−1 ∑ i=0 f pct−i. (1) Corsi’s [ 9 ] model used m∗= 5 and m∗= 21, representing the weekly and monthly horizons. These components allow the heterogeneous nature of the information arrivals [ 9 ] in the market to be considered. In fact, many operators have differing time horizons. For instance, intraday speculators have a short horizon, while insurance companies trade much less frequently. Therefore, agents whose time horizons differ perceive, react to and cause different types of volatility components. In our study, we use only m∗= 5 with the first lag. In fact, the longer horizon component, with m∗= 21, was not significant. We might interpret f pc5t−1 as reflecting the medium-term investors who typically rebalance their positions at a weekly frequency. We found that f pct−1 and f pc5t−1 are positively correlated with f pct , suggesting a positive impact at least on the mean. We will use similar variables (lagged and weekly elements) for each of the company-specific realized range sequences. The last explanatory variable we consider, denoted by JUMP , takes into account the impact of jumps in the price process. At the single-asset level, the jump intensity could be detected through the test statistic introduced by Christensen and Podolskij [ 41 ], here denoted as ZTP,t , by which we test the null hypothesis of no jumps at day t . For each asset, we test for the presence of jumps by computing the ratio between the number of days in which the null hypothesis of no jumps is rejected by the total number of days included in our dataset, with significance level set to 0.05 (the output of the test for jumps is available on request). We observed that the ratio ranges from 44.89% ( CTG ) to 91.97% ( MDZ ); therefore, the inferential procedure of Christensen and Podolskij [ 41 ] detects the presence of jumps, with clear heterogeneity across the 16 assets, and supports the inclusion of a component accounting for jumps in our model. In particular, we compute JUMP as the difference between the J. Risk Financial Manag. 2016,9, 8 6 of 25 realized range-based variance [ 11 , 42 ], denoted as RRVn,m , and the realized range-based bias corrected bipower variation [ 12 ], RRVn,m BVBC , which are, respectively, jump non-robust and jump robust estimators of the integrated variance. For the i -th stock, the observed value of JUMP at t is denoted by jumpi,t ; we checked that the jumpi,t series has evident peaks in periods of financial turmoil (the estimated jumps series are available on request). For reasons of consistency, when we analyse FPC , we compute jumptas the first principal component of jumpi,t, for i=1, ..., 16. To summarize, if we consider the FPC of the realized ranges, the variables of interest are: f pct (i.e., the dependent variable), f pct−1 , f pc5t−1 , vixt−1 , sp 500 t−1 and jumpt−1 (the explanatory variables). f pct−1 has the highest correlation with f pct (0.75), and the signs of the correlation coefficients are consistent with the expectations, as f pct−1 , f pc5t−1 , vixt−1 and jumpt−1 are positively correlated with f pct , while sp 500 t−1 has a negative correlation coefficient. We obtain similar results for each realized volatility sequence where the lagged values, the weekly lags and the jump variable are company-specific (the results are not reported for space constraints but are available upon request). 3. Modelling the Realized Range Conditional Quantiles We now introduce the model that we propose to study the conditional quantiles of volatility. As specified in the introduction, we focus on realized range quantiles, estimated using the quantile regression approach introduced by Koenker and Bassett [ 22 ]. Quantile regression overcomes some limitations of linear regression methods, including the sensitivity to outliers and the need to assume a linearity, and allows us to focus on the quantiles of the conditional distribution of a random variable without restricting the attention on the conditional mean. Let Y be a real-valued random variable with distribution function FY(y) = P(Y≤y) . For any 0 <τ< 1, the τ -th quantile of Y is equal to F−1 Y(τ) = inf{y:FY(y)≥τ} . In our work, Y coincides with FPC when we study the FPC of the stocks’ volatilities; at the single-asset level it is equal to the volatility of each stock. Let I(·) be the indicator function taking value 1 if the condition in (·) is true, 0 otherwise, and the approach introduced by Koenker and Bassett [22] makes use of the asymmetric loss function ρτ(e) = e[τ−I(e<0)], (2) showing that the minimizer ˜ yτ of the expected loss function E[ρτ(Y−˜ yτ)] satisfies FY(˜ yτ)− τ= 0. In particular, ˜ yτ is the conditional quantile function QY(τ|X1 , X2 , ..., Xδ) in the linear quantile regression: ˜ yτ=QY(τ|X1,X2, ..., Xδ) = β0(τ) + β1(τ)X1+... +βδ(τ)Xδ, (3) where X= (X1 , X2 , ..., Xδ) is the vector of δ explanatory variables. When we analyse the FPC quantiles, the covariates are f pct−1 , f pc5t−1 , vixt−1 , sp 500 t−1 and jumpt−1 ; in contrast, at the single-asset level, the explanatory variables are rrvi,t−1,rrv5i,t−1,vixt−1,sp500t−1and jumpi,t−1. Given the time index t= 1, ..., T , let yt and xj,t be, respectively, the realizations of Y and Xj for j= 1, ..., δ at t . Then, the parameter vector β β β(τ)=(β0(τ) , ..., βδ(τ)) is estimated as a solution of the quantile regression problem: min β β β∈Rδ+1 T ∑ t=1 ρτ(yt−β0−β1x1,t−... −βδxδ,t). (4) The specification adopted for the conditional quantiles of the FPC is the following: Qf pct(τ|xt−1) = x0 t−1β β β(τ) = β0(τ) + β1(τ)f pct−1+β2(τ)f pc5t−1+β3(τ)vixt−1 +β4(τ)sp500t−1+β5(τ)jumpt−1, (5) J. Risk Financial Manag. 2016,9, 8 7 of 25 where Qf pct(τ|xt−1) denotes the τ -th quantile of f pct , conditional to the information included in xt−1 . Although this approach is not novel, as conditional quantiles have already been used in a risk-management framework (e.g., Engle and Manganelli [ 23 ], White et al. [ 24 ] and White et al. [ 25 ]), we stress that, to the best of our knowledge, realized measures based on ranges have never been used. Therefore, even a simple estimation of Model (5) would provide useful results in terms of revealing the impact of covariates and the stability of the various coefficients across quantiles. Quantile regressions could also be used to forecast the conditional quantiles of the realized range volatility sequence, as we discuss below. Certain events, such as the subprime crisis that originated in the U.S. and that was marked by the Lehman Brothers’ default in September 2008 and the Eurozone sovereign debt crisis in 2010–2011, had considerable effects on the mechanisms that govern the international financial system. These extreme events could have affected the relationship between control variables and conditional quantiles, so it is necessary to determine whether the relationships that characterize Model (5) change over time before, during or after such periods of turmoil. For this purpose, we performed a rolling analysis with a step of one day and a window size of 500 observations. Thus, it is possible to determine how the coefficient values evolve over time and over τ . Further, Engle and Manganelli [ 23 ], within a risk management perspective, adopt quantile regression methods for the estimation of the value-at-risk. In their model, they introduce lagged quantiles among the conditioning variables. We do not follow their approach to control the computational complexity given the presence of a rolling method in the evaluation of conditional quantiles. We also built Model (5) to predict the conditional quantiles of the FPC computed on the realized range-based bias corrected bipower variations of the 16 assets. The forecasts, produced for a single step ahead, provide relevant details for the covariates’ prediction abilities. As the underlying companies have different features and operate in different economic sectors, it is also useful to build a model for each asset. These asset-specific models have the same structure of the one given in Equation (5), but the dependent variable is the conditional RRVn,m BVBC quantile of the one asset, and f pct−1 , f pc5t−1 and jumpt−1 are replaced with the analogous quantities computed for each asset. Therefore, the model built for the i-th asset, for i=1, ..., 16, is Qrrvi,t(τ|xi,t−1) = β0,i(τ) + β1,i(τ)rrvi,t−1+β2,i(τ)rrv5i,t−1+β3,i(τ)vixt−1 +β4,i(τ)sp500t−1+β5,i(τ)jumpi,t−1, (6) where rrvi,t is the observed RRVn,m BVBC related to the i -th company at day t , rrv5i,t is the mean of the rrvi,t values recorded in the last five days and jumpi,t−1 is the difference between RRVn,m and RRVn,m BVBC , computed for the i-th stock at day t−1. The models that we propose allow the conditional quantiles of a realized volatility measure to be estimated. We do not know the true, unobserved volatility of the assets’ returns (they are estimated), so measurement errors might play a role. However, we restrict our attention to the forecast of the realized measure at a given sampling frequency and not to a forecast of the future returns’ volatility. As a consequence, as Zikes and Barunik [ 29 ] discussed, the impact of measurement errors has limited importance. 4. Density Forecast and Predictive Accuracy In contrast to Section 3, we prefer providing more details on the forecast evaluation tools that we consider given the central role of volatility forecasting in our work. In Section 3, we described the models that we propose to estimate the dynamic governing of the conditional quantiles of the response variable Y; that is, the FPC of the assets’ volatilities in Model (5) or the volatilities of the single stocks in Model (6). Given the estimated model, we are able to forecast the conditional τ -th quantile of Y , which is Qyt(τ|xt−1). J. Risk Financial Manag. 2016,9, 8 8 of 25 The standard quantile regression approach allows estimating individual quantiles, but it does not guarantee their coherence, that is their increasing monotonicity in τ∈( 0, 1 ) . For instance, it might occur that the predicted 95th percentile of the response variable is lower than the 90th percentile. If quantiles cross, corrections must be applied to obtain a valid conditional distribution of volatility. For instance, to cope with the crossing problem, Koenker [ 43 ] applied parallel quantile planes, whereas Bondell et al. [ 44 ] estimated the quantile regression coefficients with a constrained optimization method. Here, we follow a different approach proposed by Zhao [ 45 ]. Given a collection of ϑ predicted conditional quantiles (Qyt(τ1|xt−1) , ..., Qyt(τϑ|xt−1) , for 0 <τj<τj+1< 1, j= 1, ..., ϑ− 1, we first rearrange them into ascending order, by making use of the quantile bootstrap method proposed by Chernozhukov et al. [ 46 ]. Then, starting from the rearranged quantiles, denoted by (Q∗ yt(τ1|xt−1) , ..., Q∗ yt(τϑ|xt−1)) , we estimate the entire conditional distribution with a nonparametric kernel method. The predicted density equals ˆ fYt(y∗|xt−1) = 1 ϑhϑ ϑ ∑ i=1 K y∗−Q∗ yt(τi|xt−1) hϑ!, (7) where y∗ are evenly interpolated points that generate the support of the estimated distribution, hϑ is the bandwidth, K(·) is the kernel function and ˆ fYt(y∗|xt−1) is the one-period ahead forecasted density of Y computed at y∗ , given the information set available in t− 1. Following Gaglianone and Lima [ 47 ], we use K(·)as the Epanechnikov kernel. With the solution above described, we are able to recover the entire volatility density, which could be of interest if one is dealing with volatility trading or volatility hedging applications. In this case, the analysis takes a density-forecasting perspective, where the assessment of a proposed approach’s predictive power (as compared to a benchmark model) and the evaluation of the potential benefits associated with introducing covariates are particularly important. Therefore, we apply three testing approaches: the tests proposed by Berkowitz [ 32 ] and Amisano and Giacomini [ 33 ] and a loss function-based forecast evaluation that builds on the Diebold and Mariano [34] testing approach. If, for simplicity, we denote the ex ante forecasted conditional density ˆ fYt(·|xt−1) that we estimate in t− 1 by ˆ ft−1(·) , the first step of the Berkowitz [ 32 ] test consists in computing, for all the available days, the variable νt=Zyt −∞ ˆ ft−1(u)du =ˆ Ft−1(yt), (8) where ˆ Ft−1(·)is the distribution function corresponding to the density ˆ ft−1(·). Under correct model specification, Rosenblatt [ 48 ] showed that νt is i.i.d. and uniformly distributed on ( 0, 1 ) , a result that holds regardless of the underlying distribution of yt , even when ˆ Ft−1(·)changes over time. Berkowitz [32] first pointed out that if νt∼ U(0, 1), then zt=Φ−1(νt)∼ N(0, 1), (9) where Φ−1(·)denotes the inverse of the standard normal distribution function. Given that under correct model specification zt should be independent and identically distributed as standard normal, an alternative hypothesis is that the mean and the variance differ from 0 and 1, respectively, with a first-order autoregressive structure. In particular, Berkowitz [ 32 ] considered the model zt−µb=ρb(zt−1−µb) + et(10) to test the null hypothesis H0:µb= 0, ρb= 0, var(et) = σ2 b= 1. The test builds on (10) and is based on the likelihood-ratio statistic LRb=−2[Lb(0, 1, 0; zt)−Lb(ˆ µb,ˆ σb,ˆ ρb;zt)], (11) J. Risk Financial Manag. 2016,9, 8 15 of 25 increase in the coefficient affects just the surface region in which τ takes high values. Unlike the HAR coefficients described above, the vixt−1 coefficient does not have a clear and stable increasing trend over τ . In addition, the relationship between the f pct conditional quantiles and vixt−1 is highly sensitive to the subprime crisis when pessimism among financial operators, reflected in the implied volatility of the S&P 500 index options, was acute. This result is again somewhat expected because we focus on U.S.-based data, and the subprime crisis had a high impact on the U.S. equity market. Our results are evidence that the perception of market risk has a great impact on the evolution of market volatility (as proxied by f pct ), particularly during financial turmoil. The impact is not so clear-cut during the European sovereign crisis, which had less effect on the U.S. equity market. Figure 4. Rolling analysis for vixt−1. Figure 5shows how the impact of sp 500 t−1 evolves over time and over τ . The surface given is almost always flat, the exception being the months between late 2008 and the end of 2010, when the effects of the subprime crisis were particularly acute; during this time, the coefficient values decrease as τ grows, mainly for values of τ above the median. The lagged value of the S&P 500 index return affects the entire conditional distribution of f pct and is statistically significant in almost all the quantiles considered. Moreover, Figure 5shows that the effect is negative and particularly pronounced during the subprime crisis when negative returns exacerbated market risk, increasing the upper quantiles’ volatility and increasing the likelihood of large and extreme volatility events. Figure 5. Rolling analysis for sp500t−1. J. Risk Financial Manag. 2016,9, 8 16 of 25 Finally, Figure 6shows the surface associated with the jump component. At the beginning of the sample, the jumpt−1 coefficient takes on small values over the τ levels; however, it starts to grow in 2007, reaching high peaks at high τ levels during the subprime crisis. Although the coefficient reaches considerable values in this region, their statistical significance is limited. After the second half of 2009, the surface flattens out again until the end of the sample, with the exception of some peaks of moderate size that were recorded in 2011 during the sovereign debt crises. Figure 6. Rolling analysis for jumpt−1. To summarize, using the rolling analysis, we show that two special and extreme market events (the U.S. subprime crisis and the European sovereign debt crisis) affected the relationships between the realized volatility quantiles and a set of covariates. Our results show that coefficients can reasonably vary, with a potential and relevant impact on the forecasts of both the mean (or median) volatility and the volatility distribution (starting from the quantiles). The effects differ across quantiles and change with respect to the volatility upper tails, as compared to the median and the lower tail. Therefore, when volatility quantiles are modeled, the impacts of covariates might differ over time and over quantiles, being crucial during certain market phases. This result further supports the need for quantile-specific estimations when there is an interest in single volatility quantiles. 5.3. Evaluation of the Predictive Power We evaluate the volatility density forecasts using the tests of Berkowitz [ 32 ], Amisano and Giacomini [ 33 ] and Diebold and Mariano [ 34 ]. We carry out the three tests by estimating a larger number of f pct conditional quantiles with respect to those analysed in Section 5.2. This allows recovering more precise volatility distribution functions. In particular, we consider 49 values of τ , ranging from 0.02 to 0.98, with steps of 0.02. Given the findings of the previous subsection, we must use a rolling procedure to build density forecasts. However, we modify the rolling scheme previously adopted to keep a balance between the reliability of the estimated coefficients and computational times. In particular, we recover the forecasts based on conditional quantiles estimated from subsamples of 100 observations, and we roll over the sample every 10 days. Within each 10-day window, we use fixed coefficients but produce one step-ahead forecasts updating the conditioning information set. With regard to the Berkowitz test, in the case of Model (5), we checked that zt is normally distributed, as the likelihood ratio test LRb equals 5.26 and the null hypothesis of the Berkowitz test, that is, zt∼ N( 0, 1 ) with no autocorrelation, is not rejected at the 5% significance level, thus validating the forecast goodness of Model (5). To determine whether the predictive power of our approach is affected by the U.S. subprime and the European sovereign debt crises, the series zt is divided into two parts of equal length: the first referring to a period of relative calm, from the beginning of 2003 to the first half of 2007, and the second referring to a period of market turmoil that was due to the two crises J. Risk Financial Manag. 2016,9, 8 17 of 25 between the second half of 2007 and the first half of 2013. In the first part, LRb equals 2.34, and in the second it equals 5.39. Nevertheless, the null hypothesis of the Berkowitz test is not rejected at the 5% level in both cases. Therefore, the conditional quantile model and the approach that we adopt to recover the conditional density forecasts are appropriate even during financial turbulence. An analysis of the results reveals that, as in the analysis of the full sample, f pc5t−1 , vixt−1 and jumpt−1 are not significant to explain the volatility quantiles at high values of τ in many of the subsamples. Therefore, we must determine whether this result affects the output of the Berkowitz test using a restricted model in which the regressors are only f pct−1 and sp 500 t−1 . LRb equals 2.60, a smaller value than in the previous cases. The last findings reported above suggest that the inclusion of non-significant explanatory variables penalizes the predictive power of Model (5). The restricted model gives the lowest value of LRb , but the predictive power could be improved by selecting only those variables that are significant for each value of τ and for each subsample to forecast the conditional distribution of the volatility. Thus, the structure of Model (5) would change over time and over τ . However, this approach is not applied in the present work because it would require using only the significant variables in 49 models, one for each specific value of τ , while it should be considered across all the rolling subsamples. So far, we have focused on an absolute assessment of our approach. Now, we compare it with a competing model that is fully parametric. We recover the predictive conditional distribution of the realized range volatility using a HARX model in which the mean dynamic is driven by a linear combination of the explanatory variables f pct−1 and f pc5t−1 (the HAR terms) and the exogenous variables vixt−1 , sp 500 t−1 and jumpt−1 (the X in the model’s acronym). In addition, to capture the volatility-of-volatility effect argued by Corsi et al. [ 31 ], a GJR-GARCH term [ 30 ] is introduced to the innovation (we name the model HARX-GJR). The error term is also assumed to follow a normal-inverse Gaussian ( NIG ) distribution with a mean of 0 and a variance of 1. Thus, the conditional variance is allowed to change over time, and the distribution of the error is flexible in relation to features like fat tails and skewness. We start by using the Berkowitz test to evaluate the density forecast performance of the HARX-GJR model. The likelihood ratio test LRb equals 113.88, a high value that suggests a clear rejection of the null hypothesis. We also determined whether the HARX-GJR model works better when we use the logarithm of the volatility as a response variable, following the evidence in Corsi et al. [ 52 ], and found that the likelihood ratio test LRb provides a much lower value (20.27). Even so, the test signals a rejection of the null hypothesis with a low p-value. As a first finding, our approach provides more flexibility than the parametric HARX-GJR model does and is better in terms of the Berkowitz test. To provide more accurate results, we move to a comparison of our approach to the HARX-GJR using the Amisano and Giacomini [ 33 ] and the Diebold and Mariano [ 34 ] tests. With regard to the Amisano and Giacomini [ 33 ] test, we use four weights to compute the quantity given in Equation (12): wCE f pcst t , wRT f pcst t , wLT f pcst t and wNW f pcst t . The associated likelihood ratio tests are denoted by AGCE , AGRT , AGLT and AGNW , respectively. Overall, our approach provides better results than the HARX-GJR because WLR is always positive; in fact, AGCE = 1.6521, AGRT = 2.2766, AGLT = 1.2961 and AGNW = 1.8810. Given that the critical values at the 5% level are equal to − 1.96 and 1.96 (we are dealing with a two-sided test), the null hypothesis of equal performance is rejected only when we give a higher weight to the right tail of the volatility conditional distribution. Similar results are obtained when we consider the single quantile loss function and the Diebold and Mariano [ 34 ] test statistic. Our approach provides lower losses because dDM,τ is negative for all the τ levels we considered (0.1, 0.5, 0.9). However, the differences are statistically significant only at τ= 0.9, given that DM0.1 =−1.7283 (0.0839), DM0.5 =−1.7332 (0.0831) and DM0.9 =−2.6884 (0.0072). To summarize, the results demonstrate the good performance of our approach, in particular when we focus on the right tail of the FPC (a kind of market risk factor) distribution. We note that the right tail assumes critical importance in our framework, as it represents periods of extreme risks. Our findings indicate another relevant contribution of this study, as the quantile regression approach that we propose can be used to recover density forecasts for a realized volatility measure. These forecasts J. Risk Financial Manag. 2016,9, 8 18 of 25 improve on those of a traditional approach because of the inclusion of quantile-specific coefficients. This feature of our approach might become particularly relevant in all empirical applications where predictive volatility density is required. 5.4. Single-Asset Results The results in Sections 5.1–5.3 were based on a summary of the 16 asset volatility movements, which was itself based on the FPC. Now, we search for confirmation of the main findings of Model (5) by running Model (6) at the single-asset level for all 16 assets. For simplicity, only the results associated with τ={ 0.1, 0.5, 0.9 } are shown (additional tables and figures relating to the single-asset estimates are available on request). Table 3provides the estimated parameters and their p-values. We first focus on the relationships between Qrrvi,t(τ|xi,t−1) and rrvi,t−1 , for i= 1, ..., 16. When τ equals 0.1, rrvi,t−1 is not significant at the 5% level to explain the conditional volatility quantiles of eight assets: ATT , CAT , HON , IBM , PEP , PRG , TWX and TXN . At τ= 0.5, rrvi,t−1 is not significant only for PRG , whereas, at τ= 0.9, rrvi,t−1 is not significant for PEP and PRG . Compared with the other regressors and in line with the results obtained for the FPC, rrvi,t−1 is one of the most significant explanatory variables at high levels of τ , so it assumes critical importance in the context of extreme events. The rrvi,t−1 coefficient takes a negative value only for PEP at τ= 0.1, but here it is not statistically significant. In all the other cases, it is always positive. Moreover, the magnitude of the impact that rrvi,t−1 has on Qrrvi,t(τ|xi,t−1)is a positive function of τ, for all 16 assets. The differences among the assets increase as τ grows, and at τ= 0.9 the financial companies ( BAC , CTG , JPM and WFC ) record the highest coefficient values, highlighting the crucial importance of the extreme events in the financial system. f pct−1 , the homologous regressor included in Model (5), has a weaker impact on the conditional volatility quantiles than rrvi,t−1 does only for the financial companies BAC ( τ= 0.1), JPM ( τ= 0.5) and CTG ( τ= 0.9). Therefore, the relationships between the conditional volatility quantiles and the lagged value of the response variable are stronger for the FPC than for the single assets. rrv5i,t−1 is not significant (significance level of 0.05) at τ= 0.1 for CTG , JPM and PRG , as the p -values of its coefficient indicate. At τ= 0.5 it is not significant only for PRG , whereas when τ equals 0.9 it is not significant for CTG , JPM and PRG . The rrv5i,t−1 coefficient is always positive and, with the exception of CTG and JPM , it is a positive function of τ . Moreover, the differences among the rrv5i,t−1 coefficient values are more marked at high τ levels for all 16 assets. As for the comparison with the results obtained from Model (5), at τ={ 0.1, 0.5 } , rrv5i,t−1 has a stronger impact on the conditional volatility quantiles with respect to f pc5t−1 for most of the 16 assets. When τ equals 0.9, it is pointless to compare the coefficients’ values because f pc5t−1 is not statistically significant at that level. Therefore, when we take into account the explanatory variables f pc5t−1 and rrv5i,t−1, we record stronger relationships for Model (6) than for Model (5). The p -values of the vixt−1 coefficient are less than 0.05 at τ={ 0.1, 0.5 } for all 16 assets. When τ equals 0.9, vixt−1 is significant in the cases of HON , HPQ , PRG , TWX and TXN . Unlike the coefficients previously mentioned, that of vixt−1 does not have a particular trend over τ ; it takes positive values for all 16 assets, therefore, as in the context of the FPC, it has a positive impact on the conditional volatility quantiles. In addition, at τ={ 0.1, 0.5 } , the coefficient of vixt−1 is larger in Model (5) than it is in Model (6) for all 16 companies. Therefore, vixt−1 has a more marked impact on the conditional volatility quantiles of the FPC. The comparisons made at τ= 0.9 are useless given the high p -values of the coefficients of interest. sp 500 t−1 is always significant ( α= 0.05) for the 16 assets, with the exception of BAC , CTG and JPM at τ= 0.1. Its coefficient is always negative, as expected, so sp 500 t−1 has a negative impact on the conditional volatilities’ quantiles. In addition, the magnitude of the impact is a negative function of τ for all the assets, and those relationships become more marked at high τ levels. With the exception of one case ( TXN at τ= 0.1), the impact of sp 500 t−1 at τ={ 0.1, 0.5, 0.9 } is more pronounced for the conditional volatility quantiles of the FPC than it is when the assets are considered individually, as comparing the absolute values of the related coefficients shows. J. Risk Financial Manag. 2016,9, 8 19 of 25 Table 3. Quantile regression results for each asset. Stocks Intercept rrvi,t−1rrv5i,t−1vixt−1sp500t−1jumpi,t−1 τ=0.1 ATT −0.002 (11.24) 4.589 (70.99) 28.691 (0.12) 0.002 (0.27) 0.055 (0.09) 3.515 (21.81) BAC −0.012 (0.02) 25.007 (0.00) 14.869 (1.09) 0.005 (0.01) −0.033 (18.18) −0.202 (99.31) BOI −0.008 (0.09) 15.852 (0.96) 21.132 (0.01) 0.004 (0.01) −0.081 (0.01) 4.509 (37.15) CAT −0.012 (0.00) 10.649 (19.31) 25.142 (0.00) 0.006 (0.00) −0.067 (0.695) −0.001 (99.97) CTG −0.030 (0.00) 18.252 (3.74) 12.645 (8.61) 0.012 (0.00) −0.081 (8.24) 95.128 (0.09) FDX −0.013 (0.00) 18.287 (0.46) 23.422 (0.00) 0.006 (0.00) −0.065 (0.34) −7.955 (11.99) HON −0.012 (0.00) 7.855 (36.51) 26.336 (0.01) 0.005 (0.00) −0.097 (0.05) 1.054 (70.67) HPQ −0.004 (1.75) 16.799 (0.85) 15.327 (0.33) 0.003 (0.00) −0.082 (0.21) −0.039 (99.12) IBM −0.003 (0.09) 13.709 (19.22) 22.922 (0.09) 0.002 (0.01) −0.050 (0.39) 23.369 (3.12) JPM −0.019 (0.02) 23.775 (2.59) 14.460 (10.22) 0.008 (0.01) −0.047 (13.31) 2.273 (89.03) MDZ −0.005 (0.00) 8.280 (1.17) 19.490 (0.00) 0.002 (0.00) −0.027 (0.49) −0.297 (82.09) PEP −0.005 (0.03) −2.564 (76.78) 26.404 (0.00) 0.003 (0.00) −0.036 (6.06) 6.751 (32.14) PRG −0.006 (0.08) 1.883 (86.89) 0.483 (91.48) 0.003 (0.00) −0.025 (0.72) 10.243 (49.12) TWX −0.005 (1.15) 13.005 (19.93) 29.034 (0.00) 0.003 (0.16) −0.072 (0.03) 18.752 (0.85) TXN −0.003 (19.32) 9.522 (16.87) 32.663 (0.00) 0.002 (0.38) −0.137 (0.00) 0.907 (82.14) WFC −0.012 (0.00) 20.878 (0.269) 20.321 (0.02) 0.005 (0.00) −0.064 (0.25) −0.044 (99.77) τ=0.5 ATT −0.002 (12.24) 30.163 (0.67) 52.921 (0.00) 0.001 (4.07) −0.079 (0.00) −0.400 (91.67) BAC −0.012 (0.19) 38.505 (0.00) 40.227 (0.00) 0.005 (0.14) −0.167 (0.00) 14.596 (66.46) BOI −0.011 (0.01) 36.842 (0.00) 36.566 (0.00) 0.005 (0.00) −0.112 (0.00) 7.570 (35.86) CAT −0.013 (0.00) 40.863 (0.00) 40.806 (0.00) 0.005 (0.00) −0.141 (0.00) −0.043 (99.25) CTG −0.031 (0.06) 44.363 (0.00) 25.602 (0.04) 0.013 (0.05) −0.109 (0.41) 71.026 (2.39) FDX −0.011 (0.00) 33.461 (0.00) 44.110 (0.00) 0.005 (0.00) −0.133 (0.00) −4.139 (11.64) HON −0.009 (0.01) 33.249 (0.02) 42.692 (0.00) 0.004 (0.00) −0.126 (0.00) −2.821 (34.55) HPQ −0.006 (0.07) 33.324 (0.00) 39.040 (0.00) 0.003 (0.00) −0.135 (0.00) −1.217 (69.97) IBM −0.004 (0.14) 33.790 (0.01) 42.321 (0.00) 0.002 (0.03) −0.084 (0.00) 39.259 (15.63) JPM −0.016 (0.01) 50.524 (0.00) 26.286 (0.00) 0.007 (0.01) −0.161 (0.00) −2.738 (92.41) MDZ −0.006 (0.00) 30.852 (0.00) 36.733 (0.00) 0.003 (0.00) −0.050 (0.00) −1.370 (70.27) PEP −0.004 (0.01) 31.618 (3.74) 39.554 (0.57) 0.002 (0.00) −0.047 (0.00) 3.569 (60.58) PRG −0.010 (0.22) 11.543 (65.76) 21.804 (20.76) 0.005 (0.13) −0.050 (0.55) 120.000 (10.11) TWX −0.007 (0.01) 37.437 (0.00) 39.468 (0.00) 0.003 (0.00) −0.098 (0.00) 18.763 (2.51) TXN −0.002 (24.69) 27.555 (0.00) 49.860 (0.00) 0.002 (0.63) −0.168 (0.00) −0.497 (91.84) WFC −0.011 (0.59) 37.207 (0.00) 43.713 (0.00) 0.005 (0.52) −0.133 (0.00) −0.109 (99.18) τ=0.9 ATT −0.003 (32.72) 70.584 (0.00) 69.582 (0.00) 0.002 (14.33) −0.179 (0.00) 0.282 (98.09) BAC −0.005 (60.02) 104.077 (0.00) 48.227 (0.39) 0.002 (60.64) −0.339 (0.02) 156.280 (21.58) BOI −0.008 (17.55) 60.555 (0.01) 79.933 (0.00) 0.004 (14.54) −0.254 (0.00) −0.471 (97.86) CAT −0.005 (40.02) 79.303 (0.00) 61.374 (0.00) 0.003 (22.98) −0.321 (0.00) −0.103 (99.14) CTG −0.003 (86.81) 158.940 (0.11) 19.051 (54.87) 0.001 (83.97) −0.376 (0.02) −73.284 (15.63) FDX −0.006 (16.06) 57.441 (0.00) 78.243 (0.00) 0.004 (6.12) −0.290 (0.00) −12.050 (44.62) HON −0.009 (9.87) 69.621 (0.00) 60.597 (0.00) 0.005 (3.83) −0.321 (0.00) −0.630 (92.34) HPQ −0.008 (12.83) 72.014 (0.00) 66.113 (0.00) 0.004 (4.62) −0.344 (0.00) −3.320 (47.36) IBM −0.001 (84.22) 55.602 (2.08) 101.283 (0.02) 0.000 (89.26) −0.218 (0.00) 41.659 (55.83) JPM −0.014 (6.25) 134.345 (0.00) 17.516 (5.77) 0.006 (6.81) −0.349 (0.00) 155.650 (8.94) MDZ 0.000 (97.15) 86.207 (0.00) 53.696 (0.18) 0.001 (50.69) −0.155 (0.00) -3.639 (23.02) PEP 0.000 (91.96) 36.885 (36.23) 117.046 (2.80) 0.000 (86.11) −0.107 (0.28) −0.578 (96.86) PRG −0.010 (2.21) 59.766 (16.98) 47.298 (13.89) 0.004 (1.35) −0.154 (0.01) 152.132 (37.35) TWX −0.010 (0.48) 86.695 (0.00) 52.144 (0.00) 0.004 (0.42) −0.232 (0.00) −1.217 (94.69) TXN −0.015 (0.22) 56.407 (0.00) 80.214 (0.00) 0.007 (0.03) −0.381 (0.00) 4.170 (60.83) WFC −0.004 (57.68) 94.470 (0.00) 64.679 (0.00) 0.002 (53.26) −0.309 (0.00) −0.292 (99.59) The table reports, for each stock, the coefficients (%) and the p-values (%) for Model (6), given τ={0.1, 0.5, 0.9}. J. Risk Financial Manag. 2016,9, 8 20 of 25 At τ= 0.1, jumpi,t−1 is significant, at the 5% level, for CTG , IBM and TWX . It is significant only for CTG and TWX at τ= 0.5, whereas it is never significant when τ equals 0.9. jumpi,t−1 ’s coefficient takes both negative and positive values and, with the exception of a few assets, it does not have a particular trend over τ . Comparing these results with those obtained for the f pct conditional quantiles, we find that, with the exception of CTG ( τ= 0.1) and PRG ( τ= 0.5), the lagged value of the component associated with jumps has more impact in Model (5) than in Model (6) at τ={ 0.1, 0.5 } . It is pointless to compare the coefficients at τ=0.9 given their high p-values. To summarize, the explanatory variables rrvi,t−1 , rrv5i,t−1 , vixt−1 and sp 500 t−1 are sufficient to explain the conditional volatility quantiles of the 16 assets in most of the cases studied. Their coefficients tend to take the same sign for the 16 assets: positive in the cases of rrvi,t−1 , rrv5i,t−1 and vixt−1 and negative in the case of sp 500 t−1 . Moreover, the coefficients of rrvi,t−1 , rrv5i,t−1 and sp 500 t−1 have a clear trend over τ , providing evidence against the location-shift hypothesis, which assumes homogeneous impacts of the regressors across quantiles. However, jumpi,t−1 is significant in only a few cases. Furthermore, with the exception of rrv5i,t−1 , we find that, in most of the studied cases, the relationships between the explanatory variables and the conditional volatility quantiles are more pronounced in the context of the FPC than when the assets are individually considered. This result shows that the FPC captures a kind of systematic effect, where the relationship between macro-level covariates, such as the VIX and the S&P 500 index, and volatility quantiles is clearer. At the single-asset level, the impact of covariates is more heterogeneous than for the FPC, perhaps suggesting the need for company- (or sector-) specific covariates. Table 4. Berkowitz test for the single asset analysis. Asset LRbLRb,HARX-GJR ATT 10.23 (0.0167) 150.26 (0.0000) BAC 3.00 (0.3916) 125.11 (0.0000) BOI 7.90 (0.04812) 93.09 (0.0000) CAT 9.31 (0.0254) 88.11 (0.0000) CTG 4.28 (0.2327) 142.99 (0.0000) FDX 12.62 (0.0055) 95.63 (0.0000) HON 5.95 (0.1140) 115.59 (0.0000) HPQ 5.33 (0.1491) 83.80 (0.0000) IBM 4.25 (0.2357) 157.06 (0.0000) JPM 3.26 (0.3532) 124.01 (0.0000) MDZ 5.49 (0.1392) 159.86 (0.0000) PEP 9.97 (0.0188) 164.45 (0.0000) PRG 2.25 (0.5222) 130.88 (0.0000) TWX 11.09 (0.0112) 99.10 (0.0000) TXN 1.90 (0.5934) 79.09 (0.0000) WFC 7.04 (0.0706) 143.04 (0.0000) The table reports for each stock (the ticker is given in the first column) the values of the likelihood ratio test (the p-values are given in brackets) proposed by Berkowitz [ 32 ], generated by Model (6), LRb, and the HARX-GJR model, LRb,HARX−GJR. The last point of our analysis refers to the assessment of the predictive power of Model (6), which we apply for each of the 16 assets. As in the case of the model for the FPC, we use the tests Berkowitz [ 32 ], Amisano and Giacomini [ 33 ] and Diebold and Mariano [ 34 ] proposed. With regard to the Berkowitz test, Table 4provides the values of the likelihood ratio defined by Equation (11) and the results generated by Model (6), showing that the null hypothesis of the test, that is, zt∼ N( 0, 1 ) with no autocorrelation, is not rejected for 10 assets: BAC , CTG , HON , HPQ , IBM , JPM , MDZ , PRG , TXN and WFC . The results from the other six cases stem from the fact that some variables, mainly jumpi,t−1 , are not significant in many subsamples for several τ levels. As indicated in Section 5.3, which focused on the FPC, the predictive power of our approach could be improved by selecting for each subsample and each τ only the regressors that are significant in order to explain the individually evaluated J. Risk Financial Manag. 2016,9, 8 21 of 25 conditional quantiles. Thus, the structure of Model (6) would change over time. Now, we compare our approach with the HARX-GJR model. The results that arise from using the HARX-GJR model are given in Table 4. As in the FPC context, the likelihood ratio test (11), denoted by LRb,HARX−GJR , takes high values for all 16 assets, suggesting that the null hypothesis is rejected with low p-values. Table 5reports the results of the Amisano and Giacomini [ 33 ] test and shows that overall our model provides better results: the test statistic values (13) are in most cases positive, and the null hypothesis of equal performance is almost always rejected at the 5% level. Similar results are obtained for the Diebold and Mariano [34] test, as we can see from Table 6. Table 5. Amisano–Giacomini test for the single asset analysis. Asset AGNW AGCE AGRT AGLT ATT 3.8957 (0.0001) 4.3336 (0.0000) 2.8878 (0.0039) 4.6406 (0.0000) BAC 4.0325 (0.0001) 4.3822 (0.0000) 3.0110 (0.0026) 5.4706 (0.0000) BOI 13.9648 (0.0000) 9.3975 (0.0000) 18.9700 (0.0000) 9.4658 (0.0000) CAT 4.3298 (0.0000) 4.5230 (0.0000) 2.9384 (0.0033) 5.1361 (0.0000) CTG 3.6318 (0.0003) 3.7586 (0.0002) 2.8238 (0.0047) 4.1108 (0.0000) FDX 3.0102 (0.0026) 2.9183 (0.0035) 3.9842 (0.0001) 2.0103 (0.0444) HON 2.9055 (0.0037) 3.0381 (0.0024) 2.4728 (0.0134) 2.8784 (0.0040) HPQ 16.6222 (0.0000) 7.4219 (0.0000) 2.7958 (0.0052) 5.3210 (0.0000) IBM −0.2210 (0.8251) 0.3105 (0.7562) −0.6036 (0.5461) 0.5044 (0.6140) JPM 0.7630 (0.4455) 0.6326 (0.5270) 1.0247 (0.3055) 0.4642 (0.6425) MDZ −0.3915 (0.6954) −0.3848 (0.7004) −0.3545 (0.7230) −0.4181 (0.6759) PEP 1.0614 (0.2885) 2.1311 (0.0331) 0.0232 (0.9815) 2.8849 (0.0039) PRG 3.2625 (0.0011) 3.2615 (0.0011) 3.3250 (0.0009) 3.2064 (0.0013) TWX 0.4999 (0.6171) 0.0756 (0.9397) 1.7995 (0.0719) 0.1434 (0.8860) TXN 3.1661 (0.0015) 3.1672 (0.0015) 2.4961 (0.0126) 3.0988 (0.0019) WFC 3.0723 (0.0021) 3.1620 (0.0016) 2.4755 (0.0133) 3.2976 (0.0010) The table reports, for each stock (the ticker is given in the first column), the values of the likelihood ratio test (the p-values are given in brackets) proposed by Amisano and Giacomini [ 33 ], for different weights. Each weight places greater emphasis on particular regions of the distribution: center ( AGCE ), right tail (AGRT) and left tail (AGLT). AGNW coincides with the unweighted likelihood ratio test. Table 6. Diebold–Mariano test for the single asset analysis. Asset DM0.1 DM0.5 DM0.9 ATT −1.9227 (0.0545) −2.9818 (0.0029) −3.0612 (0.0022) BAC −1.6373 (0.1016) −1.9346 (0.0530) −1.8357 (0.0664) BOI −1.8368 (0.0662) −3.2551 (0.0011) −3.3817 (0.0007) CAT −2.2598 (0.0238) −3.2398 (0.0012) −2.8698 (0.0041) CTG −1.8642 (0.0623) −1.9876 (0.0469) −1.7358 (0.0826) FDX −2.0123 (0.0442) −2.6834 (0.0073) −3.1223 (0.0018) HON −1.8906 (0.0587) −2.3123 (0.0208) −3.2590 (0.0011) HPQ −1.4229 (0.1548) −3.5429 (0.0004) −2.5960 (0.0094) IBM −2.6321 (0.0085) −4.2343 (0.0000) −5.7195 (0.0000) JPM −1.9990 (0.0456) −2.0265 (0.0427) −2.3821 (0.0172) MDZ −2.8359 (0.0046) −4.2537 (0.0000) −4.6886 (0.0000) PEP −2.4810 (0.0131) −4.7986 (0.0000) −4.1165 (0.0000) PRG −3.3287 (0.0009) −5.0239 (0.0000) −5.9189 (0.0000) TWX −1.6401 (0.1010) −2.7771 (0.0055) −2.0985 (0.0359) TXN −1.4331 (0.1518) −3.0835 (0.0020) −2.0368 (0.0417) WFC −1.6855 (0.0919) −1.7348 (0.0828) −2.3336 (0.0196) The table reports, for each stock (the ticker is given in the first column), the values of the likelihood ratio test (the p-values are given in brackets) proposed by Diebold and Mariano [ 34 ], at τ={0.1, 0.5, 0.9}. J. Risk Financial Manag. 2016,9, 8 22 of 25 The sign of the test statistic (15) is always negative, suggesting that our approach implies a lower loss. Moreover, the performances are in most cases statistically different, given that the null hypothesis is often rejected at the 5% level. Therefore, the three tests provide clear evidence of the better performance of our method with respect to the benchmark and also in the case of single-asset realized volatilities. To conclude, we found similar results for Models (5) and (6). In particular, for both models, the lagged value of the response variable and the lagged value of the S&P 500 return were fundamental explanatory variables at high τ levels, which are the most critical. In contrast, the lagged value of the jump component is significant in a few cases. We determined that the relationships between four explanatory variables ( rrvi,t−1 , vixt−1 , sp 500 t−1 and jumpi,t−1 ) and the conditional volatility quantiles are almost always stronger in Model (5) than in Model (6). However, in the case of rrv5i,t−1 , the relationships are stronger in Model (6) than in Model (5). Finally, even in the single-asset analysis, the goodness of the predicted power of our approach is validated using the three tests. 6. Conclusions We proposed a semi-parametric approach to model and forecast the conditional distribution of asset returns’ volatility. We used the quantile regression approach, considering as predictors, variables built from the lagged values of the estimated volatility, following the HAR structure Corsi [ 9 ] developed, and macro-level variables, such as the VIX and the S&P 500 index. We estimated volatility using the realized range-based bias corrected bipower variation introduced by Christensen et al. [ 12 ], which is a consistent estimator of the integrated variance in the presence of microstructure noise and jumps in the context of high-frequency data. Our analyses considered 16 companies that operate in a variety of sectors in the U.S. market, and the results provide evidence of relevant impacts by the explanatory variables. In particular, the lagged values of the estimated volatility and the S&P 500 return were critical indicators in the context of extreme events, where volatility can reach considerably high levels. These two regressors were highly significant in terms of their ability to explain the high quantiles of volatility. Moreover, the test Koenker and Bassett [ 54 ] introduced allowed us to reject the location-shift hypothesis, highlighting the heterogeneous impacts of the regressors across quantiles. To assess the evolution of the relationships among the variables over time, we carried out a rolling analysis with steps of one day and subsamples consisting of 500 observations. Thus, we verified that two special events, the U.S. subprime crisis and the European sovereign debt crisis, have affected those relationships. In particular, acute sensitivity was recorded at high levels of quantiles. Finally, the tests developed by Berkowitz [ 32 ], Amisano and Giacomini [ 33 ] and Diebold and Mariano [ 34 ] validated the forecast performances, even in periods of financial turmoil. We compared our approach with a HARX-GJR model, which combines a HAR structure of the realized volatility mean with additional exogenous variables, and a GJR-GARCH model [ 30 ] for the mean innovation variances. The results give evidence of the superior performance of our approach. Our findings provide supporting evidence for the use of quantile regression methods for the quantile forecasts and for the density forecast of the realized range volatility. The improvement over traditional methods is marked and will be relevant in all areas where volatility quantile values and volatility density forecasts play a role. The use of predicted volatility levels is central, for instance, in the pricing of equity derivatives, in the development of equity derivative trading strategies and in risk measurement when risk is associated with volatility. The volatility distribution is of interest when trading/pricing volatility derivatives, when designing volatility hedges for generic portfolios and when accounting for the uncertainty of volatility point forecasts. In the present paper, we worked with multiple assets’ volatilities inside a univariate framework. We left to future works the generalization of our approach to a multivariate version and its integration in returns-based volatility models where realized volatility measures represent further conditioning information sources. J. Risk Financial Manag. 2016,9, 8 23 of 25 Acknowledgments: The authors thank the Editor, two anonymous reviewers, Fabio Busetti, Sandra Paterlini and all the participants of the “International Conference on Computational and Financial Econometrics”, the “CIdE Workshop for PhD students in Econometrics and Empirical Economics”, the seminar at the European Business School of Wiesbaden (Germany), for the helpful comments and stimulating discussions. Massimiliano Caporin acknowledges financial support from the MIUR (Ministero dell’Istruzione, dell’Universitá e della Ricerca) PRIN (Progetti di Rilevante Interesse Nazionale) project MISURA—Multivariate Statistical Models for Risk Assessment. Both authors acknowledge financial support from the University of Padova. 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