Stochastic volatility and GARCH: Do squared end-of-day returns provide similar information?

Allen, David E. Article Stochastic volatility and GARCH: Do squared end-of-day returns provide similar information? Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Allen, David E. (2020) : Stochastic volatility and GARCH: Do squared end-of-day returns provide similar information?, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 9, pp. 1-26, https://doi.org/10.3390/jrfm13090202 This Version is available at: https://hdl.handle.net/10419/239292 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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David Edmund Allen 1,2,3 1School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia; [email protected] 2Department of Finance, Asia University, Wufeng, Taichung 41354, Taiwan 3School of Business and Law, Edith Cowan University, Joondalup 6027, Australia Received: 19 June 2020; Accepted: 2 September 2020; Published: 7 September 2020   Abstract: The paper examines the relative performance of Stochastic Volatility (SV) and GARCH(1,1) models fitted to twenty plus years of daily data for three indices. As a benchmark, I use the realized volatility (RV) for the S&P 500, DOW JONES and STOXX50 indices, sampled at 5-minute intervals, taken from the Oxford Man Realised Library. Both models demonstrate comparable performance and are correlated to a similar extent with the RV estimates, when measured by OLS. However, a crude variant of Corsi’s (2009) Heterogenous Auto-Regressive (HAR) model, applied to squared demeaned daily returns on the indices, appears to predict the daily RV of the series, better than either of the two base models. The base SV model was then enhanced by adding a regression matrix including the first and second moments of the demeaned return series. Similarly, the GARCH(1,1) model was augmented by adding a vector of demeaned squared returns to the mean equation. The augmented SV model showed a marginal improvement in explanatory power. This leads to the question of whether we need either of the two standard volatility models, if the simple expedient of using lagged squared demeaned daily returns provides a better RV predictor, at least in the context of the indices in the sample. The paper thus explores whether simple rules of thumb match the volatility forecasting capabilities of more sophisticated models. Keywords: stochastic volatility; GARCH(1,1); S&P500; DOWJONES 50; RV 5 min; HAR model; demeaned daily squared returns JEL Classification: C22; G12 1. Introduction The paper explores the performance of Stochastic Volatility and GARCH(1,1) models as estimators of the volatility of the S&P 500, the DOW JONES and the STOXX50 indices. The volatilities estimated by these models are compared with realized volatility estimates for the three indices, obtained from the Oxford Man Realised Library, sampled at 5-minute intervals, as described in Heber et al. (2009). Their volatility forecasts are further compared with those derived from a simple historical volatility model. The data series features a 20 year sample window of daily prices taken from 3 January 2000 to 30 April 2020, initially comprising 5100 observations, also taken from the Oxford Man Realised Library. The study is a companion study to Allen and McAleer (2020). The paper is motivated by Poon and Granger (2003 , p. 507) who observed that: “as a rule of thumb, historical volatility methods work equally well compared with more sophisticated ARCH class and SV models.” The paper features a continued exploration of this observation in the context of basic GARCH and Stochastic Volatility models as compared with a simple historical volatility model based on lags of squared demeaned daily returns. J. Risk Financial Manag. 2020,13, 202; doi:10.3390/jrfm13090202 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 202 2 of 26 The R packages, stochvol and factorstochvol, are used, which employ Markov chain Monte Carlo (MCMC) samplers to conduct inference by obtaining draws from the posterior distribution of parameters and latent variables, which can then be used for predicting future volatilities. This is done within the context of a fully Bayesian implementation of heteroskedasticity modelling within the framework of stochastic volatility. For more information, see the discussion of the methods by Kastner and Frühwirth-Schnatter (2014), Kastner et al. (2017), and of the stochvol and factorstochvol packages by Kastner (2016,2019). Taylor (1982) developed modelling volatility probabilistically, through a state-space model where the logarithm of the squared volatilities—the latent states—follow an autoregressive process of order one, which became known as the stochastic volatility (SV) model. Jacquier et al. (1994), Ghysels et al. (1996), and Kim et al. (1998) provided evidence in support of the application of stochastic volatility models, but their practical use has been infrequent. The shortage of empirical applications of the SV models has been limited by two major factors: the variety (and potential incompatibility) of estimation methods for SV models, plus the lack of standard software packages (see Bos (2012)). The situation for multivariate SV was even more problematic until Hosszejni and Kastner (2019) developed the R package ’factorstochvol’. Taylor (1994) reviews the stochastic volatility and the ARCH/GARCH literature. Other reviews are by McAleer (2005) and Asai et al. (2006). Poon and Granger (2003, p. 485), noted the difficulties in the application of the SV model: “the SV model has no closed form, and hence cannot be estimated directly by maximum likelihood”. The advantage of the stochvol and factorstochvol R packages is that they incorporate an efficient MCMC estimation scheme for SV models, as discussed by Kastner and Frühwirth-Schnatter (2014) and Kastner et al. (2017). These two R library packages facilitate the analysis in the paper, which features a direct comparison of the volatility predictions of a SV model, a GARCH (1,1) model, and a simple application of a historical volatility based estimation method, as applied to the the three indices. The paper is divided into four sections: Section 2reviews the literature and econometric method employed. Section 3presents the results, and Section 4presents conclusions. 2. Previous Work and Econometric Models 2.1. Stochastic Volatility There have been numerous empirical studies of changes in volatility in various stock, currency, and commodities markets. The findings in volatility research have implications for option pricing, volatility estimation, and the degree to which volatility shocks persist. These research questions have been approached by means of different models and methodologies. Taylor (1982) suggested a novel SV approach, and Taylor (1994) developed the SV model as follows: if Pt denotes the prices of an asset at time t , and assuming no dividend payments, the returns on the asset can be defined, in the context of discrete time periods, as: Xt=ln[Pt/Pt−1]. (1) Volatility is customarily indicated by σ , and prices are described by a stochastic differential equation: d(lnP) = µdt +σdW, (2) with Wa standard Weiner process. If µand σare constants, Xthas a normal distribution, and: Xt=µ+σUt, (3) with Utindependent and identically distributed (i.i.d). J. Risk Financial Manag. 2020,13, 202 3 of 26 Equation (3) can be generalised by replacing σwith a positive random variable σt, to give: Xt=µ+σtUt, (4) where Ut∼N(0, 1). In circumstances where the returns process {Xt}can be presented by Equation (4), Taylor (1994) calls σt the stochastic volatility for period t . His definition assumes that (Xt−µ)/σt follows a normal distribution. The stochastic process {σt} generates realised volatilities {σ∗ t} which, in general, are not observable. For any realisation, σ∗ t: Xt|σt=σt∗(µ,σt∗2). (5) The mixture of these conditional normal distributions defines the unconditional distribution of Xt, which will have excess kurtosis whenever σthas positive variance and is independent of Ut. In the empirical section which follows, I use the RV of the S&P500, DOW JONES and STOXX50 indices, sampled at 5-minute intervals provided by Oxford Man, as a proxy for the true realised volatility. I then compare the estimates of volatility obtained from SV and GARCH(1,1) models, using the RV estimates as a benchmark. Taylor (1994, p. 3) suggests using “capital letters to represent random variables and lower case letters to represent outcomes”. I shall follow that convention. Given observed returns of It−1={x1,x2, .....xt−1}the conditional variance for period tis: ht=var(Xt|It−1). (6) Taylor (1994) notes that, in general, the random variable Ht , which generates the observed conditional variance ht is not, in general, equal to σ2 t . A convenient way to use economic theory to motivate changes in volatility is to assume that returns are generated by a number of intra-period price revisions, as in the manner of Clark (1973) and Tauchen and Pitts (1983). It is assumed that there are Nt price revisions during trading day t , each caused by unpredictable information. Let event ion day tchange the logarithmic price by ωit, with: Xt=µ+ Nt ∑ i=1 ωit. (7) If we assume that ωit ∼i . i . d . and is independent of the random variable Nt , with ωit ∼ N(0, σ2 ω), then: σ2 t=σ2 ωNtand Ht=σ2 ωE[Nt|It−1]. (8) The above model suggests that squared volatility is proportional to the amount of price information. The lack of standard software to estimate such a model is addressed by Kastner and Frühwirth-Schnatter (2014) who propose an efficient MCMC estimation scheme which is implemented in the R stochvol package, Kastner (2016). Kastner (2016, p. 2) proceeds by “letting y= (y1 , y2 , ..... yn)| be a vector of returns, with mean zero. An intrinsic feature of the SV model is that each observation yt is assumed to have its ’own’ contemporaneous variance, eht , which relaxes the usual assumption of homoscedasticity”. It is assumed that the logarithm of this variance follows an autoregressive process of order one. This assumption is fundamentally different to GARCH models, where the time-varying conditional volatility is assumed to follow a deterministic instead of a stochastic evolution. The centered parameterization of the SV model can be given as: yt|ht∼N(0, expht), (9) J. Risk Financial Manag. 2020,13, 202 4 of 26 ht|ht−1,µ,φ,σn∼N(µ,+φ(ht−1−µ),σ2 n), (10) h0|µ,φ,σn∼N(µ,σ2 n/(1−φ2), (11) where N(µ , σ2 n) denotes the normal distribution with mean µ and variance σ2 n . θ= (µ , φ , σn)| is the vector of parameters which consists of the level of the log variance µ , the persistence of log variance, φ , and the volatility of log variance, σn . The process h= (h0 , h1 , ......, hn)| which features in equations (10) and (11) is the unobserved or latent time-varying volatility process. Kastner (2016, p. 4) remarks that: “A novel and crucial feature of the algorithm implemented in stochvol is the usage of a variant of the “ancillarity-sufficiency interweaving strategy” (ASIS) which was suggested in the general context of state-space models by Yu and Meng (2011). ASIS exploits the fact that, for certain parameter constellations, sampling efficiency improves substantially when considering a non-centered version of a state-space model”. Another key feature of the algorithm used in stochvol is the joint sampling of all instantaneous volatilities “all without a loop” (AWOL), a technique with links to Rue (2001) and as discussed in McCausland et al. (2011). The combination of these features enables the R package stochvol to estimate SV models efficiently even when large datasets are involved. Kastner et al. (2017) suggest that the multivariate factor stochastic volatility (SV) model (Chib et al. 2006) provides a means of uniting simplicity with flexibility and robustness. It is simple in the sense that the potentially high-dimensional observation space is reduced to a lower-dimensional orthogonal latent factor space. It is flexible in the sense that these factors are allowed to exhibit volatility clustering, and it is robust in the sense that idiosyncratic deviations are themselves stochastic volatility processes, thereby allowing for the degree of volatility co-movement to be time-varying. Hosszejni and Kastner (2019) set up a factor SV model employed in the factorstochvol package in R, and the analysis also uses code from this package. 2.2. ARCH and GARCH Engle (1982) developed the Autoregressive Conditional Heteroskedasticity (ARCH) model that incorporates all past error terms. It was generalised to GARCH by Bollerslev (1986) to include lagged term conditional volatility. GARCH predicts that the best indicator of future variance is a weighted average of long-run variance, the predicted variance for the current period, and any new information in this period, as captured by the squared residuals. Consider a time series yt=Et−1(yt) + εt , where Et−1(yt) is the conditional expectation of yt at time t−1 and εtis the error term. The basic GARCH model has the following specification: εt=phtηtηt(0, 1)(12) ht=ω+ p ∑ j=1 αjε2 t−j+ q ∑ j=1 βjht−j(13) in which ω> 0, αj≥ 0 and βj≥ 0 (usually a positive fraction), to ensure a positive conditional variance, ht≥ 0 (see Tsay (1987)). The ARCH effect is captured by the parameter αj , which represents the short-run persistence of shocks to returns, βj captures the GARCH effect that contributes to long-run persistence, and αj+βj measures the persistence of the impact of shocks to returns to long-run persistence. A GARCH(1,1) process is weakly stationary if αj+βj≤ 1. (See the discussion in Allen et al. (2013). We contrast the estimates of volatility from the SV model with those from a GARCH(1,1) model, and assess which better explains the behaviour of the RV of FTSE sampled at 5-minute intervals. J. Risk Financial Manag. 2020,13, 202 5 of 26 2.3. Realised Volatility Use was made of the RV 5-min estimates from Oxford Man for the three indices as the RV benchmark (see: https://realized.oxford-man.ox.ac.uk/data). This database contains “daily (close to close) financial returns , and a corresponding sequence of daily realised measures rm1 , rm2 , ....., rmT . Realised measures are theoretically sound high frequency, nonparametric-based estimators of the variation of the price path of an asset during the times at which the asset trades frequently on an exchange. Realised measures ignore the variation of prices overnight and sometimes the variation in the first few minutes of the trading day when recorded prices may contain large errors”. The metrics were developed by Andersen et al. (2001), Andersen et al. (2003), and Barndorff-Nielsen and Shephard (2002). Shephard and Sheppard (2010) provide an account of the RV measures used in the Oxford Man Realised Library. The simplest realised metric is realised variance (RV): RVt=∑x2 j,t, (14) where xj,t=Xtj,t−Xtj−1,t . The tj,t are the times of trades or quotes on the t -th day. The theoretical justification of this measure is that, if prices are observed without noise then, as min j |tj,t−tj−1,t|↓ 0, it consistently estimates the quadratic variation of the price process on the t -th day. If the sampling is reduced to very small intervals of time, market microstructure noise may become a contaminant. In order to avoid this issue, we use RV estimates from Oxford Man, sampled at 5-minute intervals, hereafter RV5. 2.4. Historical Volatility Model Poon and Granger (2005) discuss various practical issues involved in forecasting volatility. They suggest that the HISVOL model has the following form: ˆ σt=φ1σt−1+φ2σt−2+.. +φτσt−τ, (15) where ˆ σt is the expected standard deviation at time t , φ is the weight parameter, and σ is the historical standard deviation for periods indicated by the subscripts. Poon and Granger (2005) suggest that this group of models include the random walk, historical averages, autoregressive (fractionally integrated) moving average, and various forms of exponential smoothing that depend on the weight parameter φ . We use a simple form of this model in which the estimate of σ is the previous day’s demeaned squared return. Poon and Granger (2005) review 66 previous studies, and suggest that implied standard deviations appear to perform best, followed by historical volatility and GARCH which have roughly equal performance. They also note that, at the time of writing, there were insufficient studies of SV models to come to any conclusions about this class of models. This observation provides the motivation for the current study which assesses the performance of all three classes of models. It also provides the motivation to use a crude rule of thumb in the form of 20 lags of daily squared demeaned returns. The choice of 20 lags is conditioned by Corsi’s (2009) “Heterogeneous Autoregressive model of Realized Volatility” (HAR) model, as discussed in the next section. A feature of this model is that it includes estimates of daily, weekly, and monthly ex-post realised volatility. The crude HISVOL model adopted in the paper takes 20 lags as an approximation for this, as it roughly represents a month of trading days. Barndorff-Nielsen and Shephard (2003) point out that taking the sums of squares of increments of log-prices has a long tradition in the financial economics literature. See, for example, Poterba and Summers (1986), Schwert (1989), Taylor and Xu (1997), Christensen and Prabhala (1998), Dacorogna et al. (1998) , and Andersen et al. (2001). (Shephard and Sheppard (2010), p 200, footnote 4) note that: “Of course, the most basic realised measure is the squared daily return”. We utilise this approach as the basis of our historical volatility model. J. Risk Financial Manag. 2020,13, 202 6 of 26 2.5. Heterogenous Autoregressive Model (HAR) Corsi (2009, p. 174) suggests “an additive cascade model of volatility components defined over different time periods. The volatility cascade leads to a simple AR-type model in the realized volatility with the feature of considering different volatility components realized over different time horizons and which he termed as a “Heterogeneous Autoregressive model of Realized Volatility”. Corsi (2009) suggests that the model successfully achieves the purpose of reproducing the main empirical features of financial returns (long memory, fat tails, and self-similarity) in a parsimonious way. He writes his model as: σ(d) t+1d=c+β(d)RV(d) t+β(w)RV(w) t+β(m)RV(m) t+˜ ω(d) t+1d, (16) where σ(d) is the daily integrated volatility, and RV(d) t , RV(w) t and RV(m) t are respectively the daily, weekly, and monthly (ex post) observed realized volatilities, and wt+1d=˜ ω(d) t+1d−ω(d) t+1d Corsi (2009) inspires the HISVOL model adopted in the paper, which uses lags of historical RV estimates, but, in the current case, lags of squared demeaned daily close-to-close returns are employed. A further justification for the approach adopted in the current study is provided by a recent publication by Perron and Shi (2020), who show that squared low-frequency daily returns can be expressed in terms of the temporal aggregation of a high-frequency series. They explore the links between the spectral density function of squared low-frequency and high-frequency returns. They analyze the properties of the spectral density function of realized volatility, constructed from squared returns with different frequencies under temporal aggregation. However, for the low frequency data on S&P 500 returns, they cannot infer whether the noise is stationary long memory but a long-memory process appears needed to explain the features related to high frequency S&P 500 futures. However, they caution that they cannot explain this difference and that it may also be related to the fact that they have used both spot and futures series for the S&P500 in the case of the high frequency data. Perron and Shi (2020, p. 14), suggest that: “that both the realized volatility and the squared daily returns contain the same information about long memory. However, the squared daily returns contain a larger noise component than does the realized volatility”. The current paper uses the long-memory component of low frequency squared daily demeaned returns to capture this long memory feature and to provide an approximation to a HAR model. The paper does not apply a full HAR model as the intention is to explore whether a simple HISVOL rule of thumb model, based on lagged squared demeaned returns, performs as well as standard GARCH or SV models. It is clear that a full HAR specification is likely to perform better, though Perron and Shi (2020) suggest that many of the HAR’s long memory features will be captured by the approach adopted in the paper. 3. Results of the Analysis 3.1. Preliminary Analysis The sample data set consists of 20 years and four months of daily data of adjusted continuously compounded close to close returns for S&P 500, DOW JONES and STOXX50 indices, taken from 3 January 2000 through to 30 April 2020. There are a matching set of daily RV5 estimates for the three indices obtained from the Oxford Man Realised library. These three indices are chosen because they constitute major components of global capital markets in the US and in Europe. The S&P500 reflects the performance of the top 500 stocks of leading companies in the USA, and is listed on the NYSE (New York Stock Exchange) and (Nasdaq Exchange). The companies included comprise roughly 80% coverage of the available US market capitalisation. The DOW JONES index is a much narrower index that measures the daily price movements of 30 large American companies on the Nasdaq and the New York Stock Exchange and is comprised of blue-chip stocks. The EURO STOXX 50 is a stock index of Eurozone stocks designed by STOXX, an index provider owned by Deutsche J. Risk Financial Manag. 2020,13, 202 7 of 26 Börse Group. STOXX suggests that the aim is “to provide a blue-chip representation of Supersector leaders in the Eurozone”. It is made up of fifty of the largest and most liquid stocks. These indices are representative of blue chip stocks in the USA and Eurozone, plus the broader US market. Thus, these indices are likely to be of great importance and appeal to investors given the nature of the markets that they cover. Summary statistics for the six series are provided in Table 1. The sample size varies for the three indices varies given a different incidence of holidays in the USA and in Europe. For example, in 2019, there were nine days of holiday related closures on the NYSE but only five in Europe. The total number of sample observations for the USA indices initially comprised 5100 data points whilst the total for the STOXX50 was 5180. Close to close returns were used because it was thought that these are likely to capture all the information released over a 24 h period and would provide more accurate measures of volatility for the demeaned squared return series used in the regression tests. The S&P500 has a mean daily return of 0.01361 per cent and a standard deviation of 1.248 per cent. Plots of the daily returns and RV5 estimates are provided in Figure 1. It had positive excess kurtosis and does not conform to a Gaussian distribution, as can be seen from the QQ plot in Figure 2. The S&P500 RV5 has a mean of 0.00011202 and a standard deviation of 0.00026873. However, Rv5 is measured as a variance, and, if we take the square root of its value and multiply it by 100, it will be on a common scale with the S&P500 returns. We undertake this transformation in some of the comparison plots in subsequent figures. It has very high skewness and kurtosis which is also evident in the QQ plots in Figure 2. The DOWJONES has a mean return of 0.01496 per cent and a standard deviation of 1.1946 per cent, while STOXX50 has a mean return of − 0.0098108 per cent and a standard deviation of 1.4439. DOWJONES RV5 has a mean of 0.00011386 and a standard deviation of 0.00028661, while RV5 of DOWJONES has a mean of 0.00011386 and a standard deviation of 0.00028661. STOXX50 RV5 has a mean of 0.00016066 and a standard deviation of 0.00033556. All three RV5 series are skewed and have high excess kurtosis. Table 1. Summary statistics. Mean Median Minimum Maximum Standard Deviation Skewness Excess Kurtosis SPRET 0.000136 0.000540 −0.126700 0.106420 0.012485 −0.360540 10.874 SPRV5 0.000112 0.000047 0.000001 0.007748 0.000269 10.6520 188.60 DJRET 0.000150 0.000488 −0.138070 0.107540 0.011946 −0.376640 13.380 DJRV5 0.000114 0.000049 0.000001 0.008624 0.000287 12.0870 238.02 STOXX50RET −0.000098 0.000236 −0.120050 0.105540 0.014399 −0.225470 5.8411 STOXX50RV5 0.000161 0.000081 0.000000 0.010827 0.000336 12.0260 256.24 KEY: SPRET Continuously compounded close to close return on the S&P500 Index SPRV5 Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man DJRET Continuously compounded close to close return on the DOWJONES Index DJRV5 Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man STOXX50RET Continuously compounded close to close return on the S&P500 Index STOXX50RV5 Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man J. Risk Financial Manag. 2020,13, 202 8 of 26 Figure 1. Series plots. J. Risk Financial Manag. 2020,13, 202 15 of 26 Table 4. Cont. DOWJONES OLS, using observations 2000-01-06–2020-04-30 (T= 5094) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const −0.000198625 5.37151 ×10−6−36.98 0.0000 SVDJ_1 0.000328020 4.75466 ×10−668.99 0.0000 Mean dependent var 0.000114 S.D. dependent var 0.000287 Sum squared resid 0.000216 S.E. of regression 0.000206 R20.483125 Adjusted R20.483024 F(1, 5092)4759.515 P-value(F) 0.000000 ˆ ρ0.331936 Durbin–Watson 1.336075 OLS, using observations 2000-01-06–2020-04-30 (T= 5094) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const −0.000170947 6.16017 ×10−6−27.75 0.0000 Djht_1 0.000280420 5.18265 ×10−654.11 0.0000 Mean dependent var 0.000114 S.D. dependent var 0.000287 Sum squared resid 0.000266 S.E. of regression 0.000228 R20.365056 Adjusted R20.364932 F(1, 5092)2927.610 p-value(F) 0.000000 ˆ ρ0.434841 Durbin–Watson 1.130290 OLS, using observations 2000-01-04–2020-04-30 (T= 5075) Dependent variable: lrv5 Coefficient Std. Error t-ratio p-value const 0.0405870 0.00329492 12.32 0.0000 SQDMDJRET_1 0.000165523 6.55973 ×10−625.23 0.0000 SQDMDJRET_2 0.000101750 6.61717 ×10−615.38 0.0000 SQDMDJRET_3 5.22487 ×10−56.99793 ×10−67.466 0.0000 SQDMDJRET_4 2.10587 ×10−56.99864 ×10−63.009 0.0026 SQDMDJRET_5 6.58804 ×10−57.00022 ×10−69.411 0.0000 SQDMDJRET_6 4.64120 ×10−57.00145 ×10−66.629 0.0000 SQDMDJRET_7 1.56715 ×10−67.13269 ×10−60.2197 0.8261 SQDMDJRET_8 1.54284 ×10−67.11737 ×10−60.2168 0.8284 SQDMDJRET_9 4.60489 ×10−57.11644 ×10−66.471 0.0000 SQDMDJRET_10 −2.98185 ×10−67.06670 ×10−6−0.4220 0.6731 SQDMDJRET_11 −1.35137 ×10−57.06710 ×10−6−1.912 0.0559 SQDMDJRET_12 −2.77575 ×10−67.11818 ×10−6−0.3900 0.6966 SQDMDJRET_13 5.94860 ×10−67.11930 ×10−60.8356 0.4034 SQDMDJRET_14 −8.20421 ×10−67.13518 ×10−6−1.150 0.2503 SQDMDJRET_15 −8.26171 ×10−67.00344 ×10−6−1.180 0.2382 SQDMDJRET_16 −4.30980 ×10−67.00422 ×10−6−0.6153 0.5384 SQDMDJRET_17 6.42163 ×10−67.00281 ×10−60.9170 0.3592 SQDMDJRET_18 2.34879 ×10−57.08619 ×10−63.315 0.0009 SQDMDJRET_19 2.02923 ×10−56.69703 ×10−63.030 0.0025 SQDMDJRET_20 −4.02896 ×10−66.63951 ×10−6−0.6068 0.5440 Mean dependent var 0.113763 S.D. dependent var 0.287164 Sum squared resid 230.9808 S.E. of regression 0.213782 R20.447966 Adjusted R20.445781 F(20, 5054)205.0615 p-value(F) 0.000000 ˆ ρ0.324610 Durbin–Watson 1.350773 J. Risk Financial Manag. 2020,13, 202 16 of 26 Table 4. Cont. STOXX50 OLS, using observations 2000-01-06–2020-04-30 (T= 5179) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const −0.000255606 7.97034 ×10−6−32.07 0.0000 STOXXSV_1 0.000249773 4.26226 ×10−658.60 0.0000 Mean dependent var 0.000161 S.D. dependent var 0.000336 Sum squared resid 0.000351 S.E. of regression 0.000260 R20.398798 Adjusted R20.398682 F(1, 5177)3434.088 p-value(F) 0.000000 ˆ ρ0.334086 Durbin–Watson 1.331768 OLS, using observations 2000-01-05–2020-04-30 (T= 5179) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const −3.19171 ×10−64.70679 ×10−6−0.6781 0.4977 h2STOXX50_1 0.783757 0.0140212 55.90 0.0000 Mean dependent var 0.000161 S.D. dependent var 0.000336 Sum squared resid 0.000364 S.E. of regression 0.000265 R20.376384 Adjusted R20.376264 F(1, 5177)3124.586 p-value(F) 0.000000 ˆ ρ0.342843 Durbin–Watson 1.314043 OLS, using observations 2000-02-02–2020-04-30 (T= 5160) Dependent variable: rv5 Coefficient Std. Error ct-ratio p-value const 2.05927 ×10−54.29557 ×10−64.794 0.0000 sq_DMSTOXXRET_1 0.118968 0.00666728 17.84 0.0000 sq_DMSTOXXRET_2 0.120534 0.00666684 18.08 0.0000 sq_DMSTOXXRET_3 0.0933684 0.00669582 13.94 0.0000 sq_DMSTOXXRET_4 0.0896290 0.00678649 13.21 0.0000 sq_DMSTOXXRET_5 0.0342222 0.00682410 5.015 0.0000 sq_DMSTOXXRET_6 0.0105941 0.00684945 1.547 0.1220 sq_DMSTOXXRET_7 0.0193777 0.00684880 2.829 0.0047 sq_DMSTOXXRET_8 0.0415985 0.00685618 6.067 0.0000 sq_DMSTOXXRET_9 0.0676568 0.00686950 9.849 0.0000 sq_DMSTOXXRET_10 0.0648663 0.00687561 9.434 0.0000 sq_DMSTOXXRET_11 0.00745897 0.00687600 1.085 0.2781 sq_DMSTOXXRET_12 0.00363238 0.00687086 0.5287 0.5971 sq_DMSTOXXRET_13 0.00415015 0.00685846 0.6051 0.5451 sq_DMSTOXXRET_14 0.00309319 0.00685311 0.4514 0.6518 sq_DMSTOXXRET_15 0.0459385 0.00685415 6.702 0.0000 sq_DMSTOXXRET_16 −0.0110292 0.00682975 −1.615 0.1064 sq_DMSTOXXRET_17 3.58184 ×10−50.00680192 0.005266 0.9958 sq_DMSTOXXRET_18 −0.0226702 0.00671209 −3.378 0.0007 sq_DMSTOXXRET_19 0.00604883 0.00668237 0.9052 0.3654 sq_DMSTOXXRET_20 −0.0196779 0.00668193 −2.945 0.0032 Mean dependent var 0.000161 S.D. dependent var 0.000336 Sum squared resid 0.000324 S.E. of regression 0.000251 R20.445202 Adjusted R20.443043 F(20, 5139)206.1915 p-value(F) 0.000000 ˆ ρ0.216381 Durbin–Watson 1.567234 Note: squared de-meaned return series used for the three index series. J. Risk Financial Manag. 2020,13, 202 17 of 26 3.3. Mincer–Zarnowitz Tests As an additional test of the accuracy of the forecasts, I used some Mincer and Zarnowitz (1969) regression tests. (I am grateful to a reviewer for this suggestion). The test involves regressing realised values on the forecasts: σt+1=β0+β1ˆ σt+1. The joint hypothesis tested is that β0= 0, and β1= 1. The results of the test on the full sample are shown in Table 5. Table 5. Mincer–Zarnowitz regressions. Method Test Statistic Probability S&P500 STOCHVOL 1.06838 ×1011 0.0 GARCH 1.01522 ×1011 0.0 sq_DMSPRET-1 2205.89 0.0 DOWJONES STOCHVOL 7.72384 ×1010 0.0 GARCH 7.33155 ×1010 0.0 sq_DMDJRET-1 1.84768 ×1016 0.0 STOXX50 STOCHVOL 626.635 0.0 GARCH 2.7139 ×1070.0 sq_DMSTOXRET-1 5449.58 0.0 The results in Table 5uniformly and significantly reject the accuracy of the forecasts. However, the forecasts involve long time series with in excess of 5000 observations. As a further check, I ran some rolling Mincer–Zarnowitz regressions using a 100 observation window to explore how frequently the slope coefficient had a value of 1 bounded within two standard deviation intervals. This was done to check the relative frequency of periods in which the forecasts could not reject the null hypothesis. The results for the S&P500 index are shown in Figure 5(I have omitted the results of the other series in the interests of brevity, but they are available from the author on request). It can be seen in Figure 5that there are extended periods of time in which the null hypothesis that the slope coefficient of the results of regressing the actual on the forecast is not significantly different from 1. This highlights the issues related to type one and type two errors. It also is consistent with the Mincer–Zarnowitz tests consistently rejecting the null hypothesis yet the adjusted Rsquares of the regressions of actuals on forecasts being consistently in the range of 40–50% J. Risk Financial Manag. 2020,13, 202 18 of 26 3.3 Mincer-Zarnowitz tests 20 Figure 5: Mincer-Zarnowitz rolling regression slope coecients bounded by 2 standard deviations. S&P500 SV S&P500 GARCH S&P500 sq_DMSPRET-1 3.3 Mincer-Zarnowitz tests 20 Figure 5: Mincer-Zarnowitz rolling regression slope coecients bounded by 2 standard deviations. S&P500 SV S&P500 GARCH S&P500 sq_DMSPRET-1 3.3 Mincer-Zarnowitz tests 20 Figure 5: Mincer-Zarnowitz rolling regression slope coecients bounded by 2 standard deviations. S&P500 SV S&P500 GARCH S&P500 sq_DMSPRET-1 Figure 5. Mincer–Zarnowitz rolling regression slope coefficients bounded by two standard deviations. J. Risk Financial Manag. 2020,13, 202 19 of 26 3.4. Further Analysis Kastner (2019) in the factorstochvol R package, together with the Hosszejni and Kastner (2019) vignette, on factorstochvol, demonstrated how to expand the capabilities of the original stochvol package. In particular, they explained how it is possible to construct a multiple regression model with an intercept, two regressors, and SVl residuals using the following construction: yt ht+1!|ht,ς, xt1 xt2!,β0, β1 β2!2 β0+β1xt1+β2xt2 u+ϕ(ht−µ)!,∑ρ!, (17) ∑ρ=  exp(ht)ρσexp(ht/2) ρσexp(ht/2 σ2 . This approach was adopted to include the second and third moments of the index series in the form of de-meaned squared returns and de-meaned cubed returns as the two regressors in the regression model. The intention was to check whether these simple enhancements would improve the performance of the basic stochastic volatility model, in regression tests which used the estimated volatility, as reported in the previous section, as the explanatory variable regressed on the benchmark values of RV5 for the three index series. Table 6presents the the stochastic volatility estimates for the augmented model and posterior density plots of parameters in θare shown in Figure 6. The values appear to be reasonably well-behaved and the density plots in Figure 6are acceptable, although the density estimates for the STOXX50 are the weakest of the three with some evidence of bimodality in the mu estimation plot. A similar simple enhancement was applied to the GARCH(1,1) conditional volatility model in which the mean equation featured the addition of a vector of squared de-meaned returns to assess whether this improved its explanatory performance. The results of the GARCH(1,1) models with enhanced mean equations are not reported in the paper in the interests of brevity, but are available on request. Suffice to say all the estimates appeared to be satisfactory. Table 7reports the results of the regression analyses featuring the volatility estimates from both the augmented stochastic volatility and GARCH models. The results are mixed. The benchmark in Table 4is provided by the ’rule of thumb’ historical volatility model in which 20 lags of de-meaned close-to-close returns were regressed on the Oxford Man RV5 estimates for the three indices. The adjusted R-squares in the cases of the S&P500, DOWJONES and STOXX50 were, respectively, 0.54, 0.54, and 0.44. The augmented SV model for the S&P500, as shown in the first regression in Table 6, has an adjusted R-square of 0.56, a marginal improvement. The original GARCH model for this index had an adjusted R-square of 0.36, which also improves marginally to 0.37. The DOWJONES historical volatility model’s adjusted R-square of 0.54 is not matched by that for the augmented SV model, which is 0.43 in Table 7. However, the adjusted R-square of the enhanced GARCH model has a value of 0.47, which is a considerable improvement over its previous value in Table 4of 0.37. Finally, in the case of the STOXX50, the original historical volatility model had an adjusted R-square of 0.44. This is surpassed by that of the augmented SV model which has a value of 0.47. However, the augmented GARCH model shows a marked deterioration with an adjusted R-square of 0.17. Thus, in two cases out of three, the augmented SV model does have higher explanatory power, when regressed on the RV5 estimates for these three indices. However, the improvement is by a couple of percent, and so it remains a moot point about whether it is of practical worth considering the difference in estimation complexity required to estimate an augmented SV model, as opposed to merely applying 20 lags of squared demeaned returns. J. Risk Financial Manag. 2020,13, 202 20 of 26 Table 6. Stochastic volatility estimates augmented SV model. Summary of 1000 MCMC draws after burn in of 1000 Prior Distributions µ∼normal mean = 0 S.D. = 100 (φ+1)/2 ∼βa0=5b0=1.5 σ2∼χ2(d f =1) S&P500 Posterior draws thinning = 1 Mean S.D. 5% 50% 95% µ−0.78 0.1562 −1.05 −0.78 −0.53 φ0.97 0.0048 0.96 0.97 0.97 σ0.36 0.0182 0.32 0.36 0.39 exp(µ/2)0.68 0.0529 0.59 0.68 0.77 σ20.13 0.0130 0.11 0.13 0.15 DOWJONES Posterior draws thinning = 1 Mean S.D. 5% 50% 95% µ−0.907 0.1447 −1.14 −0.913 −0.656 φ0.957 0.0058 0.95 0.957 0.966 σ0.437 0.0232 0.40 0.435 0.478 exp(µ/2)0.637 0.0462 0.56 0.634 0.720 σ20.191 0.0204 0.16 0.189 0.229 STOXX50 Posterior draws thinning = 1 Mean S.D. 5% 50% 95% µ−0.37 0.1197 −0.57 −0.37 −0.18 φ0.95 0.0062 0.94 0.95 0.96 σ0.41 0.0211 0.37 0.41 0.44 exp(µ/2)0.83 0.0494 0.75 0.83 0.92 σ20.17 0.0173 0.14 0.17 0.20 J. Risk Financial Manag. 2020,13, 202 21 of 26 S&P500 DOWJONES STOXX50 Figure 6. Posterior density plots of parameters in θ. J. Risk Financial Manag. 2020,13, 202 22 of 26 Table 7. Regression analysis of the three augmented stochastic volatility and GARCH models as explanators of RV5. S&P500 OLS, using observations 2000-01-05–2020-04-30 (T= 5098) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 4.80570 ×10−52.59906 ×10−618.49 0.0000 SVREGSPRET_1 5.61407 ×10−76.87701 ×10−981.64 0.0000 Mean dependent var 0.000112 S.D. dependent var 0.000269 Sum squared resid 0.000160 S.E. of regression 0.000177 R20.566679 Adjusted R20.566594 F(1, 5096)6664.347 p-value(F) 0.000000 ˆ ρ0.288688 Durbin–Watson 1.422414 OLS, using observations 2000-01-05–2020-04-30 (T= 5098) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 0.000136717 3.02817 ×10−645.15 0.0000 yhat3_1 −0.0810177 0.00148855 −54.43 0.0000 Mean dependent var 0.000112 S.D. dependent var 0.000269 Sum squared resid 0.000233 S.E. of regression 0.000214 R20.367610 Adjusted R20.367486 F(1, 5096)2962.324 p-value(F) 0.000000 ˆ ρ0.248686 Durbin–Watson 1.502451 DOWJONES OLS, using observations 2000-01-05–2020-04-30 (T= 5094) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 7.08759 ×10−53.11272 ×10−622.77 0.0000 DJSVREG_1 3.42022 ×10−75.52569 ×10−961.90 0.0000 Mean dependent var 0.000114 S.D. dependent var 0.000287 Sum squared resid 0.000239 S.E. of regression 0.000217 R20.429352 Adjusted R20.429240 F(1, 5092)3831.187 p-value(F) 0.000000 ˆ ρ0.382376 Durbin–Watson 1.235179 OLS, using observations 2000-01-06–2020-04-30 (T= 5093) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 0.000144841 2.96368 ×10−648.87 0.0000 yhat7_1 −0.00213366 3.18142 ×10−5−67.07 0.0000 Mean dependent var 0.000114 S.D. dependent var 0.000287 Sum squared resid 0.000222 S.E. of regression 0.000209 R20.469072 Adjusted R20.468968 F(1, 5091)4497.872 p-value(F) 0.000000 ˆ ρ0.347045 Durbin–Watson 1.305745 J. Risk Financial Manag. 2020,13, 202 23 of 26 Table 7. Cont. STOXX50 OLS, using observations 2000-01-05–2020-04-30 (T= 5179) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 3.92593 ×10−53.84434 ×10−610.21 0.0000 STOXXREGFAC_1 1.04919 ×10−61.55219 ×10−867.59 0.0000 Mean dependent var 0.000161 S.D. dependent var 0.000336 Sum squared resid 0.000310 S.E. of regression 0.000245 R20.468805 Adjusted R20.468702 F(1, 5177)4568.947 p-value(F) 0.000000 ˆ ρ0.212647 Durbin–Watson 1.574424 OLS, using observations 2000-01-05–2020-04-30 (T= 5179) Dependent variable: rv5 Coefficient Std. Error t-ratio p-value const 0.000191985 4.32834 ×10−644.36 0.0000 yhat2_1 −0.111512 0.00332039 −33.58 0.0000 Mean dependent var 0.000161 S.D. dependent var 0.000336 Sum squared resid 0.000479 S.E. of regression 0.000304 R20.178889 Adjusted R20.178731 F(1, 5177)1127.876 p-value(F) 7.0 ×10−224 ˆ ρ0.258795 Durbin–Watson 1.482403 4. Conclusions The paper featured a further examination of the effectiveness of SV and GARCH(1,1) models, as explanators of model-free estimates of the volatility of S&P500, DOWJONES and STOXX50, using RV samples at 5-minute intervals, as provided by Oxford Man Institute’s Realised Library, as a benchmark. In order to provide further contrast, 1 also used lags of squared demeaned daily returns on FTSE to provide a simple alternative estimate of daily volatility. The effectiveness of these three methods was explored via the application of ordinary least squares (OLS) regression analysis. Poon and Granger (2005) provided motivation in their analysis of 66 studies of this topic in which they noted that, at that time, there were an insufficient number of SV studies to provide a comparison between GARCH and HISVOL models. My intention in the paper was to further address this sparsity in the literature. The enhanced estimates of SV were obtained by means of the R package factorstockvol and the addition of a regression matrix which included vectors of the second and third moments of the demeaned return series for the three indices. The enhanced GARCH(1,1) model was obtained by adding the squared demeaned return series to the mean equation. The results were consistent with those of an earlier companion study by Allen and McAleer (2020) which featured the FTSE. In all the three base cases, the simple expedient of adopting 20 lags of squared demeaned returns in the regression model, out-performing the volatility estimates from those of the base SV model and the GARCH(1,1) models. The enhanced SV model did show higher explanatory power than the base models in the cases of both the S&P500 and the STOXX50. The results from the enhanced GARCH model were also variable, but in no case matched those of the simple HISVOL model. However, both performed relatively poorly as compared with the simple expedient of using squared demeaned daily returns on the three indices, in order to predict RV5 volatility. The results support Poon and Granger (2005), in that neither GARCH or SV models outperform a simple form of a HISVOL model in this sample when RV sampled at 5-minute intervals are used as a benchmark. In the case of the enhanced SV model, there was evidence of a marginal increase in adjusted R-squares in J. 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