VOLATILITY FORECASTS WITH THE HAR RNG CJ.

(PRELIMINARY, COMMENTS WELCOMED.)

LOUIS SCOTT, KIEMA ADVISORS 1 AND DAN DIBARTOLOMEO, NORTHFIELDINFORMATION SERVICES 2

Overview

Our interest is in forecasting the one day ahead volatility, and explore the use of arange model based on Corsi 2003 and Andersen, Bollerslev and Diebold 2005 HAR-RVmodels. We use Range based estimators and models that allow the introduction of pastvolatility, innovations, volume and news in our forecasts. We then use Patton’s Volatilityforecast comparison using imperfect volatility proxies to compare our designs ability toconditionally forecast the volatilities. The designs are an alternative to the RV whenmicrostructure, illiquidity and noise limit the use of RV based models.

Literature Review

ABCD: Aggregate versus asset level, conditional versus unconditional andhigh frequency versus noisy dirty data. Anderson, Bollerslev, Christoffersen andDiebold 2004 review the econometrics of volatility and correlation modeling stress threepoints. First, the importance of distinguishing aggregate portfolio level and asset levelmodeling. Noting that risk measurement generally only requires a portfolio level model,whereas risk management requires asset level modeling. Second, they are quite directin stating - ”for most financial risk management purposes, the conditional perspective isexclusively relevant ...” Their final emphasis is on the use of higher frequency data inmeasuring volatility. This final point, the idea of intra-day volatility forecasts was intro-duced in Anderson, Bollerslev 1998 and more recently, Liu Patton and Sheppard 2012compare a number of estimators and find that the 5 minute realized volatility calculationperforms quite well. The use of intra-day data would provide the most precise estim-ates, but as Hansen and Lunde 2006 point out, the data suffers from bad prices and iscomplicated by micro-structure issues. Barndorff-Nielson, Hansen, Lunde and Shephard2008 suggest the use of smoothing kernels to measure intra-day volatility. The discus-sion of ABCD 2004 directly question the viability of noisy high frequency data, a pointChristoffersen concedes may be viable only for the most liquid instruments.

Date: December 2013.1

2LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

Range volatility estimators and robustness to noise. Others (CITATIONS) suggestthat range estimators are capable of providing estimates without the complications ofmicro-structure noise.

For our purposes, the cleanest and highest frequency data set available is daily, theliterature suggest range based estimators provide the best estimates.

Latent variable forecasting is just harder. When the forecast variable of interestis unobserved perhaps even ex post, the researcher’s task is more complicated. Stand-ard measures like Mincer-Zarnowitz or Diebold-Mariano tests have been found to stillhold for noisy proxies when they are conditionally unbiased as Hansen and Lunde (2006)demonstrate. Andersen and Bollerslev 1998 and Andersen et al 2005 demonstrate thatthe results using unbiased proxies may not always lead to the same outcomes as if thetrue latent variable were used. Patton 2010 reviews the problem and offers generalizedloss functions that preserve rankings of proxies to latent variables under noise.

Range Estimators. Parkinson and Garman-Klass introduced range estimators thatprovide more efficient estimates than close to close variances. While the literature isover thirty years old, Molnar notes that these are not well understood, and points outsome flaws in the literature of range estimators. For instance, Molnar 2010 notes that

Bali and Weinbaum 2005 wrongly assume that σestimated =√σ2, Bollen and Inder 2002

correctly adjust

√σ2simple but not the other range estimators.1 See Molnar for a deriva-

tion of the bias adjustments, and monte carlo confirmation of the results. We remind thereader of what Anderson, Bollerslev, Christofferson and Diebold stated as the prize:

The near log-normality of realized volatility, together with the near-normalityof returns standardized by realized volatility, holds promise for relativelysimple-to-implement lognormal-normal mixture models in financial risk man-agement.

Molnar also examines the normalized return zt = rt/σt for all rescaled estimators under theaddition of noise and finds the Garman Klass is the only estimator suitable for examiningnormalized returns. Under monte carlo simulations of the various estimators with noise,all others exhibit excess kurtosis, other tail distortions under noise.

Data and Methods

A total of 402 assets traded on the NYSE, NYSE-ARCA and NASDAQ exchangesincluding ETFs for equities, bonds, and currencies. The vast majority of assets areequities and a number of ADRs representing the largest market cap names. The ADRsinclude some thinly traded and pink sheet names to help address the diversity in liquidity,

1E[σ]2 is different than E[σ2] consider the random variable that takes values 0,1,2 with equal probability.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.) 3

efficiency count mean stdev kurtosis signifmin-.01 1314.113 4039 0.077 1.094 -0.106 -1.374.01-.05 211.295 11443 0.036 0.991 -0.094 -2.061.05-.10 243.866 18426 0.023 0.966 -0.123 -3.412.10-.25 188.618 50419 0.021 0.960 -0.232 -10.626.25-.50 43.469 96511 0.021 0.981 -0.218 -13.834.50-.75 92.228 111334 0.036 0.995 -0.203 -13.795.75-.90 75.328 63442 0.043 1.024 -0.177 -9.096.90-.95 287.950 22052 0.055 1.040 -0.128 -3.868.95-.99 36.149 19241 0.052 1.046 -0.078 -2.202.99-max 107.998 5286 0.065 1.040 -0.130 -1.926

Table 1. statistics for Garman Klass estimator in-sample by illiquidity tiles

micro-structure noise that would be evident in any wider set of assets. Among the ETFsare the VIX, the S&P 500, and the Barclay Agg Bond index.

The Usual Suspects. Assuming a geometric Brownian motion for the price P, ln(p) isa Brownian process with zero drift and diffusion σ. Next consider a time series data setof the Open, High, Low, Close of the day. Let

ct = ln(Ct) − ln(Ot)

ht = ln(Ht) − ln(Ot)

lt = ln(Lt) − ln(Ot)

The return ct is a normal process with zero drift, and c2 is an unbiased estimate of σ2

ct ∼ N(0, σ2t )

The intuition behind the range estimators is that the intra-day high and low are mean-ingful additional information about volatility. We state the following list of estimators,the motivated reader is directed to Shu and Zhang for a review of the estimators.

Simple: σ2Simple = c2t

Range: σ2Range = (h− l)2

Parkinson’s : σ2Park = (h−l)2

4 ln 2

Garman Klass : σ2GK = 0.5(h− l)2 − (2 ln 2 − 1)c2

Rogers Satchell : σ2RS = h(h− c) + l(l − c)

Meijilson:

4LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

Modeling anticipated quarterly earnings announcements. One known propertyof earnings announcement dates is that volatility and volume is lower the day prior to theannouncement and larger on the announcement as trading absorbs the new information.During our in-sample period we create bins for each of the (on average) 63 days of thequarter and look for a signature seasonal jump in volatility. Out of sample, we look forthe date within two days on either side of the prior announcement date with the highestvolatility. In practice, one would subscribe to a service that provides the anticipatedand actual announcement dates. The method correctly identifies a small sample of datesexamined by hand including MCD.

Figure 1. The seasonalilty of earnings announcements.

The Jumps. The realized volatility literature suggest that the log price is composed ofa slow diffusion process and a less predictable jump component. 2 Barndorff-Nielson andShephard 2004 introduce a generalization of quadratic variation called bipower variation(BPV). Rather than summing a squared return series, this employs a rescaled product ofadjacent high frequency absolute returns. They derive a jump ratio test based on BPVand apply it to bivariate USD/DM, USD/JPY exchange series over a 10 year period end-ing November 1996. The data is the most recent mid-quote sampled every 5 minutes.They find that the null of a continuous process is frequently rejected and that often thejumps coincide with macroeconomic announcements. Andersen, Bollerslev and Diebold2005 (ABD2005) apply the BPV results to obtain a variant of a HAR-RV model theHAR-RV-CJ. The HAR-RV is a heterogeneous AR model using realized volatility overthree horizons - the 1,5 and 22 day where the latter two are defined as the average oflagged one day RV over the appropriate horizon.

2See Merton (1976), Ball and Torous (1983), Beckers (1981), Jarrow and Rosenfeld (1984) for earlywork on jump-diffusion process in finance.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.) 5

The HAR-RV model of Corsi (2003).

(1) RVt+1 = β0 + βDRVt + βWRVt−5,t + βWRVt−22,t + εt+1

Longer horizon versions alter the left hand side and require Newey-West standard errorcorrections for the resulting serial correlation induced by the overlap of daily estimationon a longer horizon. A HAR-RV-J version adds a jump to the above design. A nonlinearversion forecast the RV 1/2 rather than RV, or a log version. Unsurprisingly, the βDis most important for a one day forecast, the βW for a weekly forecast and βM for themonthly. The βJ are significantly negative suggesting that the jumps are short lived burstof volatility.

The HAR-RV-CJ model of Andersen, Bollerslev and Diebold (2005).

RVt+h = β0+βCDCt + βCWCt−5,t + βCMCt−22,t + . . .(2)

βJDJt + βJWJt−5,t + βJMJt−22,t + εt,t+h(3)

This nests the Corsi model but allows for differing behavior for the continuous diffusion(which we expect is persistent) and the jumps (which we expect are not). Note that thediscussion is with respect to realized volatility (RV) measures using very liquid instru-ments. The S&P 500 futures, USD/DM spot, and the 30 year T-bond futures. We goback to Christoffersen’s admission in ABCD 2002 that RV designs may be viable onlyfor the most liquid instruments. (LIT REVIEW INDICATING THE USE OFEXTREMELY LIQID INSTRUMENT IN RV MODELS.) This paper proposesreplacing the RV measure with range estimates, specifically we consider the intra-dayGarman Klass as the Ct diffusion term and the overnight jump as our Jt term in theabove - our HAR-RNG-CJ. This of course limits the jumps, they are no longer associatedwith intra-day macro data announcements. But we argue that this is largely a systemicrisk effect, and that its application is thus appropriate for instruments like those used inABD 2005. Our interest is in portfolio management, hence at the asset level. Here thedata is largely to noisy to measure without imposing restrictions based on intra-day jumpson the aggregate index like the S&P 500. One can further decompose the HAR-RNG-CJto include a jump component due to the intra-day index moves and an idiosyncratic stocklevel overnight jump. The latter will capture firm level news as these are most oftenannounced after hours.3

3I am of course, arguing a constraint imposed by the available data. I also believe that outside themost liquid names, the range version makes the most sense. Paper after paper raise the concern of noisydata, and the tremendous influence it has on our volatility estimation problem. If you wish to knowthe conclusions in the literature, read the methods and data section. Realized volatility focuses on veryliquid instruments.

6LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

The HARMA Model.

RVt+h = β0+βCDCt + βCWCt−5,t + βCMCt−22,t + . . .(4)

βJDJt + βJWJt−5,t + βJMJt−22,t + βMA(1)εt−1,t+h−1 + εt,t+h(5)

Our expectation is that the innovations are more important as the assets become lessliquid. We solve the model using iterative least squares.

Augmenting models with additive slope designs. Suppose we wish to add an asym-metry to our designs. We could alter the model

yt = β0 + βxt−1 + γzt−1 to

= β0 + βUPxt−1 + βDNxt−1 + γzt−1

The distinction is a joint test of significance, where what we actually want is a testif the nested model hypothesis H0 : βUP = βDN . We now define a dummy indicator1(x < 0), and change the model to

yt = βxt−1 + βDNxt−11(x < 0) + γzt−1

Now the nested model hypothesis can be rejected given the t-stat on βDN , the nestedmodel implies βDN is zero. The forecast for when 1(x < 0) is an additive slope β + βDN .

The HARMA model with earnings, asymmetries. The nested models for the hypo-thesis, ”Do asymmetries matter?” or ”are announcement dates different?” can be readilyexamined by placing the above designs. 1(x < 0) would be applied to all terms to see thefull decomposition. Similarly on announcement events and quantified news.

The pooled panel. The proposed designs quickly explode he number of estimated para-meters if estimated individually. The plan of attack is to first aggregate using the restric-tion that the coefficients are constant across assets, a pooled panel model. At this stagewe are asking if there is an effect on average to a decomposition. Bayesian model aver-aging is then used across a draws of the variables to establish a posterior probability thateach model factor is part of the likeliest model design.

Adding Amihud illiquidity. Amihud 2002 constructs a measure that captures theprice impact of trading volumes ( price move per unit volume). The measure is defined asthe one year moving average of the ratio of daily absolute price moves to traded volume.This measure is priced in the cross section of stock returns. Our expectation is that in theextreme deciles, limit to arbitrage arguments will result in differences in sensitivities tomodel factors. This may also allow us to endogenize cross sectional differences in loadings.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.) 7

Figure 2. Model fits for all the above models. All data and across the fourilliquidity groups. The first model is the random walk. Next the AR(1),followed by a model based on the BECM model and the fourth model isthe Corsi-CJ. The others explained in due course. The final model is ourproposed asymmetric and earnings announcement design.

How is news classified? How do you know that a particular story on forward guidanceshould be classified as a dividend story or a restructuring? Any number of keywordsmay be appropriate as it is not a unique mapping. We therefore examine each vendorsclassifications and break them into two groups, the first for very meaningful keywordslike acquisitions and defaults, and another for all others. The mapping is offered in anappendix. Dummy indicators are assigned to each.

Spillover effects. Preliminary examination of the news data suggest that it is rich for thelargest names, but decays rapidly with firm size. This paper constructs a industry levelaggregate news factor and asks if there are lead lag spillover effects with large industrynames leading smaller names. The method is based on Hou 2003. This all works so longas we stay within a region, so that asynchronous markets are not a problem.

VindS ,t = γ0 + γVindS ,t−1 + f(SentimentindL,t−1) + · · · + ε

Vi,t = β0 + βV i, t− 1 + βindSVindS ,t−1 + f(SentimentindL,t−1) + · · · + ε

Comparing models imperfect volatility proxies. If forecasting is difficult under thebest of circumstances, attempts to do so when the variable of interest is unobserved canonly complicate matters. Having an unbiased estimator helps. The daily squared returns,

8LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

(assuming zero drift), range estimators and realized volatility are well studied candidates.Patton 2010 examines the problem of latent volatility forecast comparisons under robustloss models. His work is general, volatility being just one type of latent variable forecasts.Further, the results hold for conditional forecasts as addressed by Giacomini and White2006 as well. The robust loss function families preserve the ranking of any two (possiblyimperfect) volatility forecasts by expected loss whether that ranking is done by the trueconditional variance or some conditionally unbiased proxy. The rankings will be invariantto the noise n the proxy, but the level of loss will be larger than if the true conditionalvariance. We exploit the results by comparing estimators under robust QLIKE loss usingthe Giacomini and White test for conditional predictive ability, as well as the standardDiebold Mariano and West test.

Conditional Predicitve Ability under a robust loss function.

Conditional Predictive Ability of the VIX.

Empirical Results

Our interest is in forecasting an unobservable variable in the hope that its proxy willnormalize returns. We should begin by examining the risk neutral proxy that is mostoften considered - the VIX. Alexander 2012 notes the tremendous growth in trade-ableinstruments based on the VIX and their behavior relative to the underlying CBOE futurescontract. For our purposes, we will examine the CBOE VIX at the open, and use this asour estimate for σS&P500 for the next day.

Better than a random walk? Can a VAR forecast of the VIX outperform a randomwalk? A VAR is constructed with the S&P500, the VIX and the Garman Klass rangeestimator for the VIX, our volatility of volatility. Table 2 lists the distributional prop-erties of our 10 years of data or 2,463 daily observations. Table 3 summarizes the VARresults which finds Granger causality for all three. Examining table 3, the first threecolumns summarize the in-sample VAR. The first column is for the S&P 500 insampleand the granger causality F-test marginal probabilities. Lagged S&P and lagged VIXgranger cause the S&P, vol2 does not for instance. Column two shows Granger causalityfor the VIX by both the lagged S&P 500 and the lagged VIX. V ol2 is granger caused bythe lagged VIX and the lagged V ol2. These are level σ regressions so the r-squareds andr-bars are rather large. The insample consists of the first 1,000 observations. We next testfor an integrated process. I(1) gives the augmented Dickey Fuller t-statistic. AR(1) para-meter is quite large for the VIX, .615 in-sample. The critical 5% value is based on a 21 lagautoregression. All three variables suggest an I(1) process. The cointegration tests followwith the (S&P, VIX) pair, next (VIX,V ol2) and finally (V ol2,S&P). The (S&P,VIX) pairare cointegrated for all three samples. Out of sample and for the full sample, the laggedS&P500 and VIX granger cause each other, and the vol2 is granger caused by all three.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.) 9

Obs mean stdev skew kurt min maxSP500 2,463 0.011 0.009 3.945 24.997 0.002 0.121VIX 2,462 0.011 0.005 2.280 6.593 0.005 0.042Vvol 2,463 0.063 0.037 2.722 13.955 0.008 0.472

Table 2. Full sample statistics for the SP500, VIX and Volvol

SP500in V IXin V olvolin SP500out V IXout V olvolout SP500 V IX V olvolSP500GR 0.001 0.001 0.229 0.000 0.000 0.008 0.000 0.000 0.002V IXGR 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.000V olvolGR 0.881 0.262 0.000 0.346 0.403 0.000 0.093 0.580 0.000r-sq 0.420 0.928 0.243 0.651 0.964 0.201 0.670 0.970 0.217r-bar 0.411 0.927 0.232 0.647 0.963 0.193 0.668 0.969 0.212nobs 995 1,458 2,453I(1) -21.769 -3.545 -22.929 -18.015 -3.559 -29.205 -22.817 -4.164 -37.096AR(1) 0.179 0.615 0.200 0.269 0.511 0.184 0.270 0.522 0.196Crit -2.871 -2.871 -2.871 -2.871 -2.871 -2.871 -2.871 -2.871 -2.871I(1)coi -6.782 -4.854 -1.928 -5.150 -4.541 -2.238 -6.677 -5.981 -2.679AR(1)coi -0.650 -0.382 -0.036 -0.297 -0.203 -0.016 -0.297 -0.210 -0.016Critcoi -3.359 -3.359 -3.359 -3.359 -3.359 -3.359 -3.359 -3.359 -3.359

Table 3. VAR fit, Granger causality, I(1) and cointegrated statistics forthe SP500, VIX and Volatility of VIX

To address these findings a Bayesian Vector Error Correction model is run on the firstdifferences of our independent variables, so the joint test of significance and r-squaredsare telling us that we have forecast-ability in and out of sample. The results in tables 5to 9 are stable in and out of sample. The smallest results are for the VIX with r-squaredsof 8 percent in sample and 6.5 percent out of sample. The error terms are generallymeaningful. These results, on the most followed of volatility instruments, suggests thatvolatility is forecastable.

Distributional Properties of Estimators. Out attention turns to the consensus volat-ility measure, the VIX. Are the normalized returns rSP500/σSP500 well behaved N(µ, 1)when the volatility is the VIX?

We compare ex-ante estimates σt−1 and ex-post σt. We also compare the list of rangeestimators for the S&P500 as well. Tables 14 and 15 are the in-sample results, followedby the out of sample and full sample results. The sample standard deviations are indeedclose to one for the VIX, the Parkinson, Garman Klass and Rogers Satchell estimators.The excess kurtosis and Kolomogorov-Smirnoff distances relative to N(0, 1) are also mod-est suggesting that the VIX is close to normal, as are the Garman-Klass and Parkinson.

10LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

coeffi tstatsi coeffo tstatso coeff tstatsSP500 L1 -0.235 -3.207 -0.295 -5.512 -0.309 -7.810SP500 L2 -0.203 -2.982 -0.260 -5.071 -0.271 -7.095SP500 L3 -0.237 -3.920 -0.196 -4.149 -0.213 -6.006SP500 L4 -0.157 -3.057 -0.105 -2.557 -0.120 -3.869SP500 L5 -0.104 -2.748 -0.021 -0.666 -0.035 -1.474VIX L1 0.661 3.586 1.666 9.465 1.538 11.697VIX L2 0.803 4.333 0.603 3.358 0.642 4.782VIX L3 0.671 3.675 0.583 3.255 0.610 4.540VIX L4 0.753 4.223 0.895 5.060 0.871 6.569VIX L5 0.343 1.978 0.527 3.076 0.492 3.815Volvol L1 -0.000 -0.035 0.014 1.603 0.014 2.311Volvol L2 0.003 0.407 0.006 0.676 0.008 1.338Volvol L3 0.005 0.844 0.008 1.017 0.009 1.545Volvol L4 0.002 0.369 -0.001 -0.181 0.001 0.162Volvol L5 0.002 0.411 -0.009 -1.658 -0.006 -1.534ec SP500 -0.001 -10.084 0.002 11.828 -0.002 -15.250ec VIX -0.000 -0.739 -0.000 -0.976 0.000 0.116ec Volvol -0.000 -1.138 0.000 0.904 0.000 1.136constant -0.000 -1.133 -0.001 -1.368 -0.001 -1.587SP500 200.898 29.796 256.468 29.796 408.930 29.796VIX 89.253 15.494 98.810 15.494 149.799 15.494Volvol 9.733 3.842 5.460 3.842 8.666 3.842R-sq 0.437 0.392 0.392Rbar 0.427 0.385 0.387nobs 994.000 1457.000 2457.000

Table 4. BEC model results for SP5

Table 5. BEC model equation for σSP500 in, out and total sample results.BECM with 5 lags.

Out of sample we see a pronounced difference in the estimators abilities to normalizeS&P500 returns. The VIX and all range estimators that use more than two daily datapoints have reasonably small K-S distances, but the standard deviation based on the VIXis very close to one. While the standard deviations of the Garman-Klass and Parkinsonare higher, the excess kurtosis is quite low, indicative of their ability to correctly move tocapture the fat tails driven by conditional volatility changes. The results are encouraging,we can normalize using the VIX, and the above mentioned estimators are also up to thetask, at the expense of standard deviations slightly higher than the more knowledgeableVIX. On the basis of these outcomes, our cross sectional distributional results and the

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)11

coeffi tstatsi coeffo tstatso coeff tstatsSP500 L1 -0.001 -0.095 -0.004 -0.478 -0.001 -0.131SP500 L2 0.009 0.667 -0.005 -0.581 -0.002 -0.250SP500 L3 0.023 2.032 -0.001 -0.145 0.003 0.488SP500 L4 0.006 0.595 0.008 1.162 0.009 1.656SP500 L5 -0.005 -0.659 0.005 1.026 0.005 1.135VIX L1 -0.161 -4.677 -0.104 -3.531 -0.120 -5.394VIX L2 -0.140 -4.043 -0.143 -4.773 -0.152 -6.672VIX L3 -0.061 -1.776 -0.113 -3.768 -0.109 -4.810VIX L4 -0.048 -1.441 -0.117 -3.968 -0.110 -4.916VIX L5 -0.025 -0.785 0.010 0.334 0.003 0.143Volvol L1 0.000 0.099 0.003 1.823 0.002 1.980Volvol L2 0.000 0.253 0.002 1.197 0.001 1.445Volvol L3 -0.001 -0.910 0.002 1.798 0.002 1.732Volvol L4 -0.000 -0.237 0.001 1.179 0.001 1.183Volvol L5 0.001 0.907 0.001 0.939 0.001 1.338ec SP500 0.000 1.682 -0.000 -1.954 0.000 2.232ec VIX 0.000 0.661 -0.000 -2.166 -0.000 -2.227ec Volvol -0.000 -3.080 0.000 2.253 0.000 2.868constant 0.000 2.870 0.000 3.207 0.000 3.482SP500 200.898 29.796 256.468 29.796 408.930 29.796VIX 89.253 15.494 98.810 15.494 149.799 15.494Volvol 9.733 3.842 5.460 3.842 8.666 3.842R-sq 0.080 0.065 0.059Rbar 0.063 0.053 0.052nobs 994.000 1457.000 2457.000

Table 6. BEC model results for VIX

Table 7. BEC model equation for VIX in, out and total sample results.

results of the Molnar paper, we choose the Garman Klass estimator with an overnightjump as our dependent variable.4

HAR-RNG-CJ: Jumps versus Intra-day Range. The Corsi model or HAR-RV canbe decomposed as in HAR-RV-CJ model of ABD2005. The Corsi assumption that theresponse to both jumps and the diffusion terms are equal is a nested model. Providesome evidence that the jumps are different

Results for benchmark models.

4The code allows us to examine any of our estimators, alone or with the overnight jump added. Thispaper can be redone with two line changes in the code and automatically updates all the tables. It is theenvy of sell side research - in other words, a single error means we are likely filled with errors.

12LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

coeffi tstatsi coeffo tstatso coeff tstatsSP500 L1 0.133 0.195 0.525 1.751 0.667 2.758SP500 L2 0.304 0.481 0.280 0.977 0.438 1.876SP500 L3 -0.532 -0.949 0.213 0.803 0.247 1.138SP500 L4 -0.784 -1.653 0.168 0.729 0.147 0.778SP500 L5 -1.101 -3.126 -0.053 -0.305 -0.119 -0.822VIX L1 4.480 2.622 4.346 4.410 4.076 5.073VIX L2 3.230 1.881 0.252 0.250 0.455 0.554VIX L3 6.150 3.633 0.472 0.470 1.124 1.368VIX L4 2.536 1.534 1.324 1.337 1.466 1.808VIX L5 2.647 1.646 0.890 0.927 1.090 1.383Volvol L1 -0.330 -5.099 -0.421 -8.446 -0.432 -11.684Volvol L2 -0.279 -4.565 -0.265 -5.390 -0.293 -7.992Volvol L3 -0.165 -2.915 -0.198 -4.333 -0.214 -6.209Volvol L4 -0.132 -2.722 -0.162 -3.926 -0.169 -5.475Volvol L5 -0.012 -0.319 -0.065 -2.060 -0.063 -2.639ec SP500 -0.003 -4.172 0.006 5.984 -0.006 -8.785ec VIX -0.006 -7.823 -0.008 -8.568 -0.007 -9.978ec Volvol -0.001 -1.053 0.000 0.357 0.000 0.527constant 0.011 2.842 0.025 5.932 0.020 7.362SP500 200.898 29.796 256.468 29.796 408.930 29.796VIX 89.253 15.494 98.810 15.494 149.799 15.494Volvol 9.733 3.842 5.460 3.842 8.666 3.842R-sq 0.398 0.378 0.376Rbar 0.387 0.370 0.371nobs 994.000 1457.000 2457.000

Table 8. BEC model results for V2

Table 9. BEC model equation for σV IX in, out and total sample results.

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.005 0.011 0.345 -1.016 299.225 0.382Range 0.005 0.002 0.061 -0.041 0.745 0.436Parkinson 0.160 0.039 1.237 0.282 1.112 0.074Garman Klass 0.155 0.038 1.207 0.210 0.968 0.079Rogers Satchell 0.153 0.039 1.234 0.097 1.080 0.080Meilijson 0.029 0.011 0.340 1.203 19.726 0.287VIX 0.081 0.029 0.912 -0.095 0.088 0.089

Table 10. Distribution results for ex-ante estimators: in-sample, rt/σt−1

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)13

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.009 0.004 0.117 -0.620 35.995 0.425Range 0.005 0.002 0.053 -0.182 -0.284 0.447Parkinson 0.107 0.028 0.886 -0.168 -0.965 0.070Garman Klass 0.112 0.031 0.972 -0.124 -0.434 0.070Rogers Satchell 0.107 0.035 1.106 -0.162 0.803 0.074Meilijson 0.036 0.015 0.485 2.030 21.496 0.273VIX 0.088 0.029 0.909 -0.050 0.084 0.088

Table 11. Distribution results for ex-post estimators: in-sample, rt/σt

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.018 0.007 0.251 1.029 17.349 0.336Range 0.005 0.002 0.082 -0.048 0.769 0.411Parkinson 0.148 0.033 1.270 0.297 1.321 0.065Garman Klass 0.142 0.032 1.221 0.308 1.168 0.065Rogers Satchell 0.142 0.032 1.228 0.329 1.423 0.063Meilijson 0.032 0.013 0.507 1.460 23.964 0.243VIX 0.062 0.025 0.938 -0.060 0.362 0.072

Table 12. Distribution results for ex-ante estimators: out-sample, rt/σt−1

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.011 0.004 0.152 0.514 12.595 0.383Range 0.004 0.002 0.072 -0.121 0.138 0.424Parkinson 0.090 0.023 0.869 -0.086 -0.982 0.054Garman Klass 0.091 0.025 0.940 -0.083 -0.414 0.052Rogers Satchell 0.083 0.027 1.033 -0.168 0.665 0.052Meilijson 0.011 0.016 0.622 -0.467 14.797 0.244VIX 0.059 0.024 0.931 -0.070 0.265 0.067

Table 13. Distribution results for ex-post estimators: out-sample, rt/σt

VAR factors inspired model. The design used the lagged dependent variable, together withthe lagged range (C) and jump (J) for the S&P and the VIX as well as the VIX itself.Only the lagged dependent variable varies cross sectionally. One could of course assignfactor loadings. That is the motive behind this design. The regressions are run againstthe entire set of assets, as well as against a split of the assets by Amihud illiquidity. Thepremise is that illiquid assets are harder to price, therefore their idiosyncratic volatilitywill be greater and a pooled model will have less explanatory power. The splits are forthe 10% most liquid assets in each cross section according to the lagged Amihud measure,

14LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.013 0.006 0.293 -0.313 240.451 0.354Range 0.005 0.001 0.074 -0.049 1.035 0.416Parkinson 0.153 0.025 1.256 0.291 1.247 0.067Garman Klass 0.147 0.025 1.215 0.269 1.091 0.067Rogers Satchell 0.146 0.025 1.230 0.234 1.284 0.069Meilijson 0.031 0.009 0.446 1.488 26.697 0.260VIX 0.070 0.019 0.927 -0.074 0.262 0.077

Table 14. Distribution results for ex-ante estimators: full sample, rt/σt−1

mean std error std dev skew kurtosis KS-statSimple Open to Close 0.010 0.003 0.139 0.255 18.296 0.391Range 0.005 0.001 0.065 -0.140 0.314 0.429Parkinson 0.097 0.018 0.876 -0.120 -0.974 0.058Garman Klass 0.100 0.019 0.953 -0.099 -0.419 0.057Rogers Satchell 0.093 0.021 1.063 -0.163 0.750 0.058Meilijson 0.021 0.012 0.570 0.136 17.230 0.253VIX 0.071 0.019 0.922 -0.063 0.200 0.074

Table 15. Distribution results for ex-post estimators: full sample, rt/σt

then the next 40%, the 50 to 90%, and finally the lowest 10%. All factors other than therange of the VIX are significant at the 1% level. The lowest illiquidity decile does fall infit out of sample.

HAR RNG CJ. Tables 19 to 21 represent our base design - the lagged range (C) andjump (J), the average of the next four lags t − 2 to t − 5, and the average of the nextseventeen lags t− 6 to t− 17. The Corsi and ABD papers use overlapping factors, so thelast factor would be over t− 1 to t− 22 for example. Each factor is significant at the 1%level and the results are quite well behaved in many respects. The r-squareds are largerout of sample and lowest for the most illiquid decile, but otherwise stable across illiquiditygroups. The range coefficients fall off in size as this is a one day forecasts. The J termsare significantly different than their nested restriction and lower than the correspondingC terms. In the low vol in sample period, the lagged jumps are mean reverting for themost liquid group, likely a consequence of these being the easiest to arbitrage by highfrequency traders. Out of sample the J5 and J22 rise with illiquidity, another indicationthat this is a sensible split.

HAR RNG CJplus. Tables 22 to 24 combine the last two models. All but a few instanceof the JV IX term are significant at the one percent level. Once we account for the model,the increase in overall fit is modest. This is going to be true of our introduction of seasonal

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)15

full top 10 upper 40 lower 40 bottom 10DepvarL1 0.463∗∗∗ 0.357∗∗∗ 0.457∗∗∗ 0.417∗∗∗ 0.426∗∗∗

CSP500 −0.030∗∗∗ 0.063∗∗ 0.013 0.003 0.009JSP500 −0.039∗∗∗ −0.039∗∗ −0.030∗∗∗ −0.035∗∗∗ -0.020V IX 0.830∗∗∗ 0.777∗∗∗ 0.731∗∗∗ 1.001∗∗∗ 0.961∗∗∗

CV IX -0.001 -0.004 −0.004∗∗∗ -0.001 −0.010∗∗∗

JV IX −0.003∗∗∗ 0.001 -0.001 -0.000 0.003R-sq 0.255∗∗∗ 0.214∗∗∗ 0.270∗∗∗ 0.243∗∗∗ 0.223∗∗∗

Rbar 0.255∗∗∗ 0.214∗∗∗ 0.270∗∗∗ 0.243∗∗∗ 0.223∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 16. VAR inspired model: in sample results. The lagged dependentvariable is augmented by the lagged range and jump for the S&P500, andthe VIX, as well as the VIX itself. In, out and full sample results across thefull cross section of assets, the 10% most liquid by Amihud measure, thenext 10 to 50%, 50 to 90% and the least liquid 10% of names. Significanceat the 1 (***), 5 (**) and 10 (*) percent levels

full top 10 upper 40 lower 40 bottom 10DepvarL1 0.538∗∗∗ 0.523∗∗∗ 0.569∗∗∗ 0.501∗∗∗ 0.440∗∗∗

CSP500 0.082∗∗∗ 0.076∗∗∗ 0.065∗∗∗ 0.105∗∗∗ 0.096∗∗∗

JSP500 −0.158∗∗∗ −0.138∗∗∗ −0.150∗∗∗ −0.167∗∗∗ −0.121∗∗∗

V IX 0.873∗∗∗ 0.830∗∗∗ 0.796∗∗∗ 0.954∗∗∗ 1.025∗∗∗

CV IX 0.010∗∗∗ 0.004∗ 0.003∗∗∗ 0.018∗∗∗ 0.028∗∗∗

JV IX −0.021∗∗∗ −0.014∗∗∗ −0.016∗∗∗ −0.024∗∗∗ −0.015∗∗∗

R-sq 0.468∗∗∗ 0.489∗∗∗ 0.508∗∗∗ 0.459∗∗∗ 0.345∗∗∗

Rbar 0.468∗∗∗ 0.489∗∗∗ 0.508∗∗∗ 0.459∗∗∗ 0.345∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 17. VAR inspired model: out of sample results

earnings announcements, jumps in volatility due to large negative returns. We anticiatethis will also be true of news.

HAR RNG CJplus and Seasonal Earnings Announcement Jumps. Tables 25 to 27 adda seasonal earnings announcement to the above model. Figure 3 shows the seasonalityand sparsity of the announcement data. The EAJ measure is the average of the lastfour announcement day dependent variable values. The coefficients are consistent acrossilliquidity groups and samples, about a quarter of the past EAJ.

Is there an asymmetry due to large negative returns? Yes, and it varies withliquidity. The additive slope designs are now introduced to address hypothesis about

16LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10DepvarL1 0.523∗∗∗ 0.501∗∗∗ 0.555∗∗∗ 0.489∗∗∗ 0.439∗∗∗

CSP500 0.108∗∗∗ 0.112∗∗∗ 0.094∗∗∗ 0.136∗∗∗ 0.093∗∗∗

JSP500 −0.134∗∗∗ −0.119∗∗∗ −0.131∗∗∗ −0.146∗∗∗ −0.090∗∗∗

V IX 0.804∗∗∗ 0.756∗∗∗ 0.723∗∗∗ 0.883∗∗∗ 1.106∗∗∗

CV IX 0.003∗∗∗ 0.000 -0.001 0.010∗∗∗ 0.003JV IX −0.019∗∗∗ −0.014∗∗∗ −0.015∗∗∗ −0.021∗∗∗ −0.009∗∗∗

R-sq 0.445∗∗∗ 0.454∗∗∗ 0.487∗∗∗ 0.449∗∗∗ 0.397∗∗∗

Rbar 0.445∗∗∗ 0.454∗∗∗ 0.487∗∗∗ 0.449∗∗∗ 0.397∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 18. VAR inspired model: full sample results.

full top 10 upper 40 lower 40 bottom 10C1 0.255∗∗∗ 0.287∗∗∗ 0.288∗∗∗ 0.254∗∗∗ 0.207∗∗∗

C5 0.259∗∗∗ 0.371∗∗∗ 0.289∗∗∗ 0.257∗∗∗ 0.211∗∗∗

C22 0.250∗∗∗ 0.329∗∗∗ 0.303∗∗∗ 0.244∗∗∗ 0.264∗∗∗

J1 0.186∗∗∗ 0.147∗∗∗ 0.207∗∗∗ 0.188∗∗∗ 0.161∗∗∗

J5 0.097∗∗∗ −0.016∗ 0.027∗∗∗ 0.104∗∗∗ 0.135∗∗∗

J22 0.152∗∗∗ -0.014 0.042∗∗∗ 0.159∗∗∗ 0.237∗∗∗

R-sq 0.348∗∗∗ 0.284∗∗∗ 0.342∗∗∗ 0.320∗∗∗ 0.329∗∗∗

Rbar 0.348∗∗∗ 0.284∗∗∗ 0.342∗∗∗ 0.320∗∗∗ 0.329∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 19. HAR-RNG-CJ: in sample results. The lagged dependent vari-able is augmented by the lagged range and jump for the S&P500, and theVIX, as well as the VIX itself. In, out and full sample results across thefull cross section of assets, the 10% most liquid by Amihud measure, thenext 10 to 50%, 50 to 90% and the least liquid 10% of names. Significanceat the 1 (***), 5 (**) and 10 (*) percent levels

the change in behavior under different states like large negative returns, announcementdates or news to name a few examples. The models nest the above HAR-RNG-CJ modelswhen the coefficients on the interactive terms are not significantly different from zero.What we find is that the behavior on days when the returns are −2σ days or worse is thatthere are meaningful changes to the slopes of most model parameters. The results are intables 28 to 30. In particular, there are differences across the liquidity groups, and theyare monotonic. IDownC1 and IDownC5 the coefficients on the range estimators for the dayand week, increase so we have momentum effects, but the size of these changes decreasesas we lose liquidity. IDownC22 mean reverts for largest half of liquidity, with the meanreversion falling as we lose liquidity.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)17

full top 10 upper 40 lower 40 bottom 10C1 0.392∗∗∗ 0.474∗∗∗ 0.383∗∗∗ 0.393∗∗∗ 0.330∗∗∗

C5 0.271∗∗∗ 0.265∗∗∗ 0.312∗∗∗ 0.267∗∗∗ 0.258∗∗∗

C22 0.146∗∗∗ 0.219∗∗∗ 0.200∗∗∗ 0.143∗∗∗ 0.164∗∗∗

J1 0.173∗∗∗ 0.160∗∗∗ 0.207∗∗∗ 0.149∗∗∗ 0.162∗∗∗

J5 0.129∗∗∗ 0.064∗∗∗ 0.088∗∗∗ 0.138∗∗∗ 0.135∗∗∗

J22 0.173∗∗∗ 0.053∗∗∗ 0.073∗∗∗ 0.185∗∗∗ 0.205∗∗∗

R-sq 0.534∗∗∗ 0.540∗∗∗ 0.563∗∗∗ 0.524∗∗∗ 0.419∗∗∗

Rbar 0.534∗∗∗ 0.540∗∗∗ 0.563∗∗∗ 0.524∗∗∗ 0.419∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 20. HAR-RNG-CJ: out of sample results.

full top 10 upper 40 lower 40 bottom 10C1 0.365∗∗∗ 0.445∗∗∗ 0.369∗∗∗ 0.371∗∗∗ 0.305∗∗∗

C5 0.273∗∗∗ 0.284∗∗∗ 0.311∗∗∗ 0.269∗∗∗ 0.252∗∗∗

C22 0.167∗∗∗ 0.234∗∗∗ 0.215∗∗∗ 0.160∗∗∗ 0.189∗∗∗

J1 0.178∗∗∗ 0.162∗∗∗ 0.209∗∗∗ 0.158∗∗∗ 0.164∗∗∗

J5 0.123∗∗∗ 0.052∗∗∗ 0.080∗∗∗ 0.133∗∗∗ 0.140∗∗∗

J22 0.172∗∗∗ 0.047∗∗∗ 0.072∗∗∗ 0.185∗∗∗ 0.227∗∗∗

R-sq 0.515∗∗∗ 0.510∗∗∗ 0.545∗∗∗ 0.515∗∗∗ 0.468∗∗∗

Rbar 0.515∗∗∗ 0.510∗∗∗ 0.544∗∗∗ 0.515∗∗∗ 0.468∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 21. HAR-RNG-CJ: full sample results.

Are announcement days different? Yes, it lasts for a week and does notdistinguish by illiquidity. The results are in tables 31 to 33.

Announcement and Large negative returns together. The results are in tables 34to 36.

Results for news sentiment augmented designs. Really a second paper, as weintroduce the models and apply the exact same design as Earnings or largedown days to news.

Are returns standardized by range estimates of volatility Gaussian?

18LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10CSP500 0.044∗∗∗ 0.074∗∗∗ 0.068∗∗∗ 0.075∗∗∗ 0.109∗∗∗

JSP500 −0.019∗∗∗ −0.034∗∗ −0.022∗∗∗ −0.016∗∗ -0.001V IX 0.277∗∗∗ 0.205∗∗∗ 0.230∗∗∗ 0.341∗∗∗ 0.321∗∗∗

CV IX 0.004∗∗∗ 0.001 0.003∗∗ 0.002 −0.008∗∗∗

JV IX 0.001 0.004∗ 0.000 0.004∗∗∗ 0.009∗∗∗

C1 0.242∗∗∗ 0.259∗∗∗ 0.268∗∗∗ 0.235∗∗∗ 0.197∗∗∗

C5 0.250∗∗∗ 0.347∗∗∗ 0.276∗∗∗ 0.243∗∗∗ 0.203∗∗∗

C22 0.250∗∗∗ 0.306∗∗∗ 0.299∗∗∗ 0.243∗∗∗ 0.267∗∗∗

J1 0.183∗∗∗ 0.144∗∗∗ 0.202∗∗∗ 0.181∗∗∗ 0.155∗∗∗

J5 0.092∗∗∗ −0.015∗ 0.021∗∗∗ 0.094∗∗∗ 0.128∗∗∗

J22 0.147∗∗∗ -0.009 0.032∗∗∗ 0.143∗∗∗ 0.228∗∗∗

R-sq 0.351∗∗∗ 0.288∗∗∗ 0.346∗∗∗ 0.325∗∗∗ 0.333∗∗∗

Rbar 0.351∗∗∗ 0.288∗∗∗ 0.346∗∗∗ 0.325∗∗∗ 0.333∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 22. HAR-RNG-CJp: in sample results. The HAR-RNG-CJ is aug-mented by the lagged range and jump for the S&P500, and the VIX, aswell as the VIX itself. In, out and full sample results across the full crosssection of assets, the 10% most liquid by Amihud measure, the next 10 to50%, 50 to 90% and the least liquid 10% of names. Significance at the 1(***), 5 (**) and 10 (*) percent levels

full top 10 upper 40 lower 40 bottom 10CSP500 0.050∗∗∗ -0.001 0.051∗∗∗ 0.067∗∗∗ 0.061∗∗∗

JSP500 −0.064∗∗∗ −0.060∗∗∗ −0.087∗∗∗ −0.061∗∗∗ −0.030∗∗

V IX 0.260∗∗∗ 0.333∗∗∗ 0.257∗∗∗ 0.275∗∗∗ 0.236∗∗∗

CV IX 0.017∗∗∗ 0.009∗∗∗ 0.012∗∗∗ 0.025∗∗∗ 0.029∗∗∗

JV IX −0.005∗∗∗ -0.002 −0.004∗∗∗ −0.006∗∗∗ 0.003C1 0.359∗∗∗ 0.446∗∗∗ 0.350∗∗∗ 0.349∗∗∗ 0.303∗∗∗

C5 0.255∗∗∗ 0.239∗∗∗ 0.292∗∗∗ 0.250∗∗∗ 0.243∗∗∗

C22 0.113∗∗∗ 0.148∗∗∗ 0.154∗∗∗ 0.111∗∗∗ 0.139∗∗∗

J1 0.181∗∗∗ 0.174∗∗∗ 0.225∗∗∗ 0.155∗∗∗ 0.158∗∗∗

J5 0.120∗∗∗ 0.063∗∗∗ 0.084∗∗∗ 0.126∗∗∗ 0.126∗∗∗

J22 0.164∗∗∗ 0.056∗∗∗ 0.074∗∗∗ 0.172∗∗∗ 0.192∗∗∗

R-sq 0.539∗∗∗ 0.545∗∗∗ 0.568∗∗∗ 0.531∗∗∗ 0.423∗∗∗

Rbar 0.539∗∗∗ 0.545∗∗∗ 0.568∗∗∗ 0.531∗∗∗ 0.423∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 23. HAR-RNG-CJp: out of sample results.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)19

full top 10 upper 40 lower 40 bottom 10CSP500 0.072∗∗∗ 0.027∗∗∗ 0.066∗∗∗ 0.089∗∗∗ 0.078∗∗∗

JSP500 −0.055∗∗∗ −0.053∗∗∗ −0.076∗∗∗ −0.056∗∗∗ −0.017∗

V IX 0.220∗∗∗ 0.289∗∗∗ 0.224∗∗∗ 0.243∗∗∗ 0.256∗∗∗

CV IX 0.012∗∗∗ 0.005∗∗∗ 0.009∗∗∗ 0.017∗∗∗ 0.011∗∗∗

JV IX −0.004∗∗∗ -0.001 −0.004∗∗∗ −0.004∗∗∗ 0.006∗∗∗

C1 0.334∗∗∗ 0.416∗∗∗ 0.337∗∗∗ 0.330∗∗∗ 0.282∗∗∗

C5 0.260∗∗∗ 0.262∗∗∗ 0.294∗∗∗ 0.253∗∗∗ 0.237∗∗∗

C22 0.141∗∗∗ 0.175∗∗∗ 0.177∗∗∗ 0.132∗∗∗ 0.160∗∗∗

J1 0.181∗∗∗ 0.170∗∗∗ 0.221∗∗∗ 0.159∗∗∗ 0.158∗∗∗

J5 0.114∗∗∗ 0.048∗∗∗ 0.074∗∗∗ 0.120∗∗∗ 0.130∗∗∗

J22 0.159∗∗∗ 0.045∗∗∗ 0.067∗∗∗ 0.166∗∗∗ 0.201∗∗∗

R-sq 0.520∗∗∗ 0.516∗∗∗ 0.550∗∗∗ 0.521∗∗∗ 0.472∗∗∗

Rbar 0.520∗∗∗ 0.515∗∗∗ 0.550∗∗∗ 0.521∗∗∗ 0.472∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 24. HAR-RNG-CJp: full sample results.

full top 10 upper 40 lower 40 bottom 10EAJ 0.253∗∗∗ 0.320∗∗∗ 0.284∗∗∗ 0.233∗∗∗ 0.226∗∗∗

CSP500 0.043∗∗∗ 0.077∗∗∗ 0.071∗∗∗ 0.071∗∗∗ 0.108∗∗∗

JSP500 −0.018∗∗∗ −0.030∗∗ −0.022∗∗∗ −0.016∗∗ 0.001V IX 0.283∗∗∗ 0.204∗∗∗ 0.233∗∗∗ 0.346∗∗∗ 0.324∗∗∗

CV IX 0.004∗∗∗ 0.001 0.003∗∗ 0.002 −0.008∗∗∗

JV IX 0.001∗ 0.005∗∗ 0.001 0.005∗∗∗ 0.009∗∗∗

C1 0.241∗∗∗ 0.258∗∗∗ 0.264∗∗∗ 0.233∗∗∗ 0.198∗∗∗

C5 0.250∗∗∗ 0.349∗∗∗ 0.277∗∗∗ 0.243∗∗∗ 0.202∗∗∗

C22 0.250∗∗∗ 0.309∗∗∗ 0.302∗∗∗ 0.241∗∗∗ 0.265∗∗∗

J1 0.178∗∗∗ 0.123∗∗∗ 0.189∗∗∗ 0.175∗∗∗ 0.151∗∗∗

J5 0.092∗∗∗ −0.018∗ 0.020∗∗∗ 0.094∗∗∗ 0.127∗∗∗

J22 0.149∗∗∗ -0.006 0.036∗∗∗ 0.146∗∗∗ 0.228∗∗∗

R-sq 0.354∗∗∗ 0.294∗∗∗ 0.352∗∗∗ 0.328∗∗∗ 0.336∗∗∗

Rbar 0.354∗∗∗ 0.294∗∗∗ 0.352∗∗∗ 0.328∗∗∗ 0.335∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 25. HAR-RNG-CJpE: in sample results. The HAR-RNG-CJ isaugmented by the lagged range and jump for the S&P500, and the VIX, aswell as the VIX itself. In, out and full sample results across the full crosssection of assets, the 10% most liquid by Amihud measure, the next 10 to50%, 50 to 90% and the least liquid 10% of names. Significance at the 1(***), 5 (**) and 10 (*) percent levels

20LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.261∗∗∗ 0.303∗∗∗ 0.265∗∗∗ 0.259∗∗∗ 0.240∗∗∗

CSP500 0.051∗∗∗ -0.001 0.052∗∗∗ 0.067∗∗∗ 0.065∗∗∗

JSP500 −0.061∗∗∗ −0.055∗∗∗ −0.081∗∗∗ −0.058∗∗∗ −0.029∗

V IX 0.261∗∗∗ 0.335∗∗∗ 0.256∗∗∗ 0.276∗∗∗ 0.234∗∗∗

CV IX 0.017∗∗∗ 0.009∗∗∗ 0.012∗∗∗ 0.025∗∗∗ 0.029∗∗∗

JV IX −0.005∗∗∗ -0.002 −0.004∗∗∗ −0.006∗∗∗ 0.004C1 0.358∗∗∗ 0.446∗∗∗ 0.349∗∗∗ 0.349∗∗∗ 0.302∗∗∗

C5 0.256∗∗∗ 0.243∗∗∗ 0.293∗∗∗ 0.251∗∗∗ 0.244∗∗∗

C22 0.111∗∗∗ 0.146∗∗∗ 0.152∗∗∗ 0.107∗∗∗ 0.138∗∗∗

J1 0.175∗∗∗ 0.164∗∗∗ 0.215∗∗∗ 0.150∗∗∗ 0.155∗∗∗

J5 0.120∗∗∗ 0.059∗∗∗ 0.084∗∗∗ 0.126∗∗∗ 0.127∗∗∗

J22 0.166∗∗∗ 0.058∗∗∗ 0.078∗∗∗ 0.174∗∗∗ 0.192∗∗∗

R-sq 0.543∗∗∗ 0.550∗∗∗ 0.572∗∗∗ 0.534∗∗∗ 0.426∗∗∗

Rbar 0.543∗∗∗ 0.550∗∗∗ 0.572∗∗∗ 0.534∗∗∗ 0.426∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 26. HAR-RNG-CJpE: out of sample results.

full top 10 upper 40 lower 40 bottom 10EAJ 0.258∗∗∗ 0.302∗∗∗ 0.266∗∗∗ 0.255∗∗∗ 0.239∗∗∗

CSP500 0.074∗∗∗ 0.027∗∗∗ 0.068∗∗∗ 0.089∗∗∗ 0.083∗∗∗

JSP500 −0.052∗∗∗ −0.047∗∗∗ −0.071∗∗∗ −0.053∗∗∗ -0.016V IX 0.217∗∗∗ 0.288∗∗∗ 0.221∗∗∗ 0.242∗∗∗ 0.251∗∗∗

CV IX 0.012∗∗∗ 0.005∗∗∗ 0.009∗∗∗ 0.017∗∗∗ 0.011∗∗∗

JV IX −0.004∗∗∗ -0.001 −0.003∗∗∗ −0.004∗∗∗ 0.006∗∗∗

C1 0.333∗∗∗ 0.416∗∗∗ 0.336∗∗∗ 0.329∗∗∗ 0.281∗∗∗

C5 0.261∗∗∗ 0.266∗∗∗ 0.296∗∗∗ 0.254∗∗∗ 0.237∗∗∗

C22 0.139∗∗∗ 0.173∗∗∗ 0.176∗∗∗ 0.129∗∗∗ 0.159∗∗∗

J1 0.176∗∗∗ 0.158∗∗∗ 0.210∗∗∗ 0.154∗∗∗ 0.155∗∗∗

J5 0.114∗∗∗ 0.044∗∗∗ 0.073∗∗∗ 0.120∗∗∗ 0.130∗∗∗

J22 0.161∗∗∗ 0.047∗∗∗ 0.071∗∗∗ 0.168∗∗∗ 0.201∗∗∗

R-sq 0.523∗∗∗ 0.520∗∗∗ 0.554∗∗∗ 0.524∗∗∗ 0.475∗∗∗

Rbar 0.523∗∗∗ 0.520∗∗∗ 0.554∗∗∗ 0.524∗∗∗ 0.475∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 27. HAR-RNG-CJpE: full sample results.

Concluding Remarks

To be completed.

Kiema Advisors 1

E-mail address: lscott@kiemaadvisors.com

Northfield Information Services 2

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)21

Figure 3. The seasonalilty of earnings announcements.

22LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.254∗∗∗ 0.320∗∗∗ 0.289∗∗∗ 0.233∗∗∗ 0.225∗∗∗

CSP500 0.045∗∗∗ 0.075∗∗∗ 0.078∗∗∗ 0.073∗∗∗ 0.110∗∗∗

JSP500 -0.001 -0.021 -0.003 0.003 0.018V IX 0.279∗∗∗ 0.206∗∗∗ 0.224∗∗∗ 0.338∗∗∗ 0.322∗∗∗

CV IX 0.004∗∗∗ 0.001 0.004∗∗ 0.002 −0.008∗∗∗

JV IX 0.001 0.005∗∗ 0.000 0.004∗∗∗ 0.008∗∗∗

C1 0.228∗∗∗ 0.257∗∗∗ 0.233∗∗∗ 0.221∗∗∗ 0.173∗∗∗

C5 0.243∗∗∗ 0.343∗∗∗ 0.276∗∗∗ 0.239∗∗∗ 0.202∗∗∗

C22 0.254∗∗∗ 0.304∗∗∗ 0.303∗∗∗ 0.239∗∗∗ 0.270∗∗∗

J1 0.172∗∗∗ 0.122∗∗∗ 0.179∗∗∗ 0.171∗∗∗ 0.149∗∗∗

J5 0.094∗∗∗ −0.021∗∗ 0.027∗∗∗ 0.093∗∗∗ 0.121∗∗∗

J22 0.145∗∗∗ -0.007 0.034∗∗∗ 0.139∗∗∗ 0.236∗∗∗

IDown 0.002∗∗∗ 0.002∗∗∗ 0.001∗∗∗ 0.002∗∗∗ 0.002∗∗∗

IDownC1 0.005 −0.100∗∗ 0.144∗∗∗ -0.018 0.137∗∗∗

IDownC5 0.105∗∗∗ 0.176∗∗ 0.119∗∗∗ 0.111∗∗∗ 0.011IDownC22 −0.035∗∗ 0.088 0.068∗ 0.057∗ -0.057IDownJ1 0.033∗∗∗ -0.003 0.065∗∗∗ 0.010 0.014IDownJ5 −0.063∗∗∗ 0.011 −0.120∗∗∗ −0.044∗∗∗ 0.013IDownJ22 0.036∗∗∗ 0.067 0.003 0.086∗∗∗ −0.119∗∗∗

R-sq 0.355∗∗∗ 0.295∗∗∗ 0.357∗∗∗ 0.330∗∗∗ 0.337∗∗∗

Rbar 0.355∗∗∗ 0.295∗∗∗ 0.356∗∗∗ 0.330∗∗∗ 0.337∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 28. HAR-RNG-CJpE: in sample results.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)23

full top 10 upper 40 lower 40 bottom 10EAJ 0.261∗∗∗ 0.306∗∗∗ 0.270∗∗∗ 0.258∗∗∗ 0.239∗∗∗

CSP500 0.068∗∗∗ 0.028∗∗ 0.073∗∗∗ 0.084∗∗∗ 0.074∗∗∗

JSP500 −0.040∗∗∗ −0.032∗∗∗ −0.051∗∗∗ −0.036∗∗∗ -0.015V IX 0.259∗∗∗ 0.315∗∗∗ 0.253∗∗∗ 0.272∗∗∗ 0.232∗∗∗

CV IX 0.017∗∗∗ 0.012∗∗∗ 0.013∗∗∗ 0.024∗∗∗ 0.028∗∗∗

JV IX −0.007∗∗∗ −0.003∗∗∗ −0.005∗∗∗ −0.008∗∗∗ 0.002C1 0.321∗∗∗ 0.372∗∗∗ 0.306∗∗∗ 0.314∗∗∗ 0.274∗∗∗

C5 0.250∗∗∗ 0.241∗∗∗ 0.285∗∗∗ 0.240∗∗∗ 0.257∗∗∗

C22 0.115∗∗∗ 0.170∗∗∗ 0.164∗∗∗ 0.108∗∗∗ 0.130∗∗∗

J1 0.166∗∗∗ 0.145∗∗∗ 0.180∗∗∗ 0.151∗∗∗ 0.160∗∗∗

J5 0.121∗∗∗ 0.062∗∗∗ 0.083∗∗∗ 0.126∗∗∗ 0.118∗∗∗

J22 0.166∗∗∗ 0.057∗∗∗ 0.074∗∗∗ 0.169∗∗∗ 0.211∗∗∗

IDown 0.002∗∗∗ 0.000 0.001∗∗∗ 0.003∗∗∗ 0.002∗∗∗

IDownC1 0.117∗∗∗ 0.240∗∗∗ 0.099∗∗∗ 0.088∗∗∗ 0.127∗∗∗

IDownC5 0.093∗∗∗ 0.270∗∗∗ 0.194∗∗∗ 0.134∗∗∗ −0.078∗∗

IDownC22 0.002 −0.096∗∗ −0.051∗∗∗ 0.012 0.052IDownJ1 0.036∗∗∗ 0.077∗∗∗ 0.189∗∗∗ −0.032∗∗∗ −0.050∗∗∗

IDownJ5 −0.047∗∗∗ −0.047∗∗∗ −0.093∗∗∗ −0.034∗∗∗ 0.041∗

IDownJ22 −0.019∗∗ -0.015 0.033∗∗ 0.013 −0.147∗∗∗

R-sq 0.546∗∗∗ 0.559∗∗∗ 0.580∗∗∗ 0.538∗∗∗ 0.428∗∗∗

Rbar 0.546∗∗∗ 0.559∗∗∗ 0.580∗∗∗ 0.538∗∗∗ 0.427∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 29. HAR-RNG-CJpE: out of sample results.

24LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.258∗∗∗ 0.305∗∗∗ 0.272∗∗∗ 0.254∗∗∗ 0.238∗∗∗

CSP500 0.087∗∗∗ 0.051∗∗∗ 0.088∗∗∗ 0.104∗∗∗ 0.092∗∗∗

JSP500 −0.033∗∗∗ −0.029∗∗∗ −0.043∗∗∗ −0.033∗∗∗ -0.002V IX 0.215∗∗∗ 0.270∗∗∗ 0.217∗∗∗ 0.239∗∗∗ 0.245∗∗∗

CV IX 0.012∗∗∗ 0.008∗∗∗ 0.010∗∗∗ 0.017∗∗∗ 0.010∗∗∗

JV IX −0.005∗∗∗ −0.002∗∗ −0.005∗∗∗ −0.006∗∗∗ 0.005∗∗

C1 0.300∗∗∗ 0.352∗∗∗ 0.294∗∗∗ 0.298∗∗∗ 0.252∗∗∗

C5 0.253∗∗∗ 0.262∗∗∗ 0.288∗∗∗ 0.244∗∗∗ 0.246∗∗∗

C22 0.144∗∗∗ 0.195∗∗∗ 0.188∗∗∗ 0.130∗∗∗ 0.156∗∗∗

J1 0.168∗∗∗ 0.142∗∗∗ 0.180∗∗∗ 0.154∗∗∗ 0.157∗∗∗

J5 0.115∗∗∗ 0.046∗∗∗ 0.074∗∗∗ 0.120∗∗∗ 0.122∗∗∗

J22 0.160∗∗∗ 0.045∗∗∗ 0.066∗∗∗ 0.162∗∗∗ 0.216∗∗∗

IDown 0.002∗∗∗ -0.000 0.001∗∗∗ 0.002∗∗∗ 0.002∗∗∗

IDownC1 0.106∗∗∗ 0.205∗∗∗ 0.111∗∗∗ 0.081∗∗∗ 0.137∗∗∗

IDownC5 0.100∗∗∗ 0.292∗∗∗ 0.193∗∗∗ 0.135∗∗∗ −0.062∗∗∗

IDownC22 -0.011 −0.090∗∗ −0.053∗∗∗ 0.005 0.033IDownJ1 0.038∗∗∗ 0.086∗∗∗ 0.174∗∗∗ −0.025∗∗∗ −0.038∗∗∗

IDownJ5 −0.047∗∗∗ −0.043∗∗∗ −0.093∗∗∗ −0.034∗∗∗ 0.039∗∗∗

IDownJ22 0.001 -0.003 0.037∗∗∗ 0.033∗∗∗ −0.133∗∗∗

R-sq 0.526∗∗∗ 0.528∗∗∗ 0.561∗∗∗ 0.527∗∗∗ 0.476∗∗∗

Rbar 0.526∗∗∗ 0.528∗∗∗ 0.561∗∗∗ 0.527∗∗∗ 0.476∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 30. HAR-RNG-CJpE: full sample results.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)25

full top 10 upper 40 lower 40 bottom 10EAJ 0.131∗∗∗ 0.296∗∗∗ 0.212∗∗∗ 0.108∗∗∗ 0.166∗∗∗

CSP500 0.041∗∗∗ 0.076∗∗∗ 0.067∗∗∗ 0.069∗∗∗ 0.109∗∗∗

JSP500 −0.010∗∗ −0.035∗∗ −0.018∗∗ -0.007 0.005V IX 0.278∗∗∗ 0.206∗∗∗ 0.231∗∗∗ 0.342∗∗∗ 0.319∗∗∗

CV IX 0.004∗∗∗ 0.001 0.003∗∗ 0.002 −0.008∗∗∗

JV IX 0.001 0.004∗ 0.001 0.004∗∗∗ 0.008∗∗∗

C1 0.239∗∗∗ 0.255∗∗∗ 0.267∗∗∗ 0.233∗∗∗ 0.196∗∗∗

C5 0.246∗∗∗ 0.346∗∗∗ 0.268∗∗∗ 0.239∗∗∗ 0.203∗∗∗

C22 0.250∗∗∗ 0.298∗∗∗ 0.303∗∗∗ 0.242∗∗∗ 0.262∗∗∗

J1 0.179∗∗∗ 0.155∗∗∗ 0.196∗∗∗ 0.175∗∗∗ 0.154∗∗∗

J5 0.092∗∗∗ −0.018∗∗ 0.020∗∗∗ 0.094∗∗∗ 0.123∗∗∗

J22 0.150∗∗∗ -0.007 0.035∗∗∗ 0.147∗∗∗ 0.233∗∗∗

IEarnAC 0.003∗∗∗ 0.003∗∗ 0.002∗∗∗ 0.003∗∗∗ 0.002∗∗

IEarnACC1 0.063∗∗∗ -0.094 −0.138∗∗∗ 0.018 0.159∗

IEarnACC5 0.188∗∗∗ -0.045 0.538∗∗∗ 0.292∗∗∗ -0.055IEarnACC22 0.191∗∗∗ 0.787∗∗∗ -0.022 0.061 0.218∗∗

IEarnACJ1 −0.046∗∗∗ −0.163∗∗∗ −0.073∗∗∗ -0.006 −0.081∗∗∗

IEarnACJ5 −0.036∗∗ -0.024 -0.030 -0.024 0.282∗∗∗

IEarnACJ22 −0.049∗∗ -0.000 0.045 -0.006 −0.287∗∗∗

R-sq 0.355∗∗∗ 0.298∗∗∗ 0.354∗∗∗ 0.329∗∗∗ 0.336∗∗∗

Rbar 0.355∗∗∗ 0.297∗∗∗ 0.354∗∗∗ 0.329∗∗∗ 0.336∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 31. HAR-RNG-CJpEAS: in sample results.

26LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.120∗∗∗ 0.158∗∗∗ 0.121∗∗∗ 0.100∗∗∗ 0.118∗∗∗

CSP500 0.047∗∗∗ -0.004 0.051∗∗∗ 0.060∗∗∗ 0.064∗∗∗

JSP500 −0.057∗∗∗ −0.050∗∗∗ −0.080∗∗∗ −0.052∗∗∗ -0.025V IX 0.258∗∗∗ 0.332∗∗∗ 0.252∗∗∗ 0.275∗∗∗ 0.224∗∗∗

CV IX 0.017∗∗∗ 0.009∗∗∗ 0.012∗∗∗ 0.024∗∗∗ 0.028∗∗∗

JV IX −0.006∗∗∗ −0.002∗ −0.005∗∗∗ −0.007∗∗∗ 0.003C1 0.354∗∗∗ 0.444∗∗∗ 0.341∗∗∗ 0.345∗∗∗ 0.300∗∗∗

C5 0.255∗∗∗ 0.238∗∗∗ 0.291∗∗∗ 0.250∗∗∗ 0.238∗∗∗

C22 0.112∗∗∗ 0.149∗∗∗ 0.156∗∗∗ 0.107∗∗∗ 0.139∗∗∗

J1 0.177∗∗∗ 0.164∗∗∗ 0.222∗∗∗ 0.150∗∗∗ 0.156∗∗∗

J5 0.121∗∗∗ 0.058∗∗∗ 0.083∗∗∗ 0.127∗∗∗ 0.131∗∗∗

J22 0.167∗∗∗ 0.060∗∗∗ 0.079∗∗∗ 0.173∗∗∗ 0.192∗∗∗

IEarnAC 0.005∗∗∗ 0.005∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗

IEarnACC1 0.316∗∗∗ 0.110∗∗ 0.387∗∗∗ 0.314∗∗∗ 0.193∗∗

IEarnACC5 0.194∗∗∗ 0.478∗∗∗ 0.173∗∗∗ 0.258∗∗∗ 0.514∗∗∗

IEarnACC22 -0.013 −0.181∗∗ -0.054 -0.018 0.023IEarnACJ1 −0.058∗∗∗ -0.022 −0.105∗∗∗ −0.052∗∗∗ -0.033IEarnACJ5 −0.095∗∗∗ -0.042 −0.062∗∗∗ −0.082∗∗∗ −0.296∗∗∗

IEarnACJ22 0.022 -0.086 −0.122∗∗∗ 0.111∗∗∗ 0.050R-sq 0.544∗∗∗ 0.551∗∗∗ 0.574∗∗∗ 0.536∗∗∗ 0.428∗∗∗

Rbar 0.544∗∗∗ 0.551∗∗∗ 0.574∗∗∗ 0.536∗∗∗ 0.427∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 32. HAR-RNG-CJpEAS: out of sample results.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)27

full top 10 upper 40 lower 40 bottom 10EAJ 0.130∗∗∗ 0.176∗∗∗ 0.142∗∗∗ 0.099∗∗∗ 0.127∗∗∗

CSP500 0.070∗∗∗ 0.024∗∗ 0.066∗∗∗ 0.082∗∗∗ 0.080∗∗∗

JSP500 −0.048∗∗∗ −0.046∗∗∗ −0.069∗∗∗ −0.047∗∗∗ -0.012V IX 0.217∗∗∗ 0.287∗∗∗ 0.220∗∗∗ 0.242∗∗∗ 0.246∗∗∗

CV IX 0.012∗∗∗ 0.005∗∗∗ 0.009∗∗∗ 0.017∗∗∗ 0.011∗∗∗

JV IX −0.004∗∗∗ -0.002 −0.004∗∗∗ −0.005∗∗∗ 0.005∗∗∗

C1 0.329∗∗∗ 0.412∗∗∗ 0.329∗∗∗ 0.326∗∗∗ 0.279∗∗∗

C5 0.258∗∗∗ 0.261∗∗∗ 0.292∗∗∗ 0.252∗∗∗ 0.233∗∗∗

C22 0.139∗∗∗ 0.175∗∗∗ 0.179∗∗∗ 0.128∗∗∗ 0.159∗∗∗

J1 0.177∗∗∗ 0.165∗∗∗ 0.218∗∗∗ 0.154∗∗∗ 0.156∗∗∗

J5 0.115∗∗∗ 0.043∗∗∗ 0.073∗∗∗ 0.121∗∗∗ 0.132∗∗∗

J22 0.162∗∗∗ 0.047∗∗∗ 0.071∗∗∗ 0.167∗∗∗ 0.202∗∗∗

IEarnAC 0.004∗∗∗ 0.004∗∗∗ 0.004∗∗∗ 0.005∗∗∗ 0.004∗∗∗

IEarnACC1 0.259∗∗∗ 0.121∗∗∗ 0.301∗∗∗ 0.257∗∗∗ 0.194∗∗∗

IEarnACC5 0.194∗∗∗ 0.404∗∗∗ 0.251∗∗∗ 0.274∗∗∗ 0.349∗∗∗

IEarnACC22 0.030 -0.113 −0.087∗∗∗ 0.004 0.108IEarnACJ1 −0.057∗∗∗ −0.076∗∗∗ −0.103∗∗∗ −0.038∗∗∗ −0.043∗∗

IEarnACJ5 −0.079∗∗∗ -0.033 −0.053∗∗∗ −0.076∗∗∗ −0.141∗∗∗

IEarnACJ22 0.015 -0.040 −0.073∗∗∗ 0.098∗∗∗ -0.029R-sq 0.524∗∗∗ 0.522∗∗∗ 0.555∗∗∗ 0.526∗∗∗ 0.476∗∗∗

Rbar 0.524∗∗∗ 0.521∗∗∗ 0.555∗∗∗ 0.526∗∗∗ 0.476∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 33. HAR-RNG-CJpEAS: full sample results.

28LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.136∗∗∗ 0.300∗∗∗ 0.222∗∗∗ 0.122∗∗∗ 0.172∗∗∗

CSP500 0.043∗∗∗ 0.074∗∗∗ 0.074∗∗∗ 0.072∗∗∗ 0.111∗∗∗

JSP500 0.008∗ −0.027∗ 0.001 0.011 0.021V IX 0.274∗∗∗ 0.208∗∗∗ 0.222∗∗∗ 0.333∗∗∗ 0.317∗∗∗

CV IX 0.004∗∗∗ 0.001 0.004∗∗ 0.002 −0.008∗∗∗

JV IX 0.000 0.004 -0.000 0.003∗∗∗ 0.008∗∗∗

C1 0.227∗∗∗ 0.255∗∗∗ 0.236∗∗∗ 0.221∗∗∗ 0.172∗∗∗

C5 0.240∗∗∗ 0.339∗∗∗ 0.268∗∗∗ 0.234∗∗∗ 0.203∗∗∗

C22 0.253∗∗∗ 0.293∗∗∗ 0.304∗∗∗ 0.240∗∗∗ 0.267∗∗∗

J1 0.174∗∗∗ 0.154∗∗∗ 0.186∗∗∗ 0.172∗∗∗ 0.152∗∗∗

J5 0.095∗∗∗ −0.022∗∗ 0.028∗∗∗ 0.094∗∗∗ 0.117∗∗∗

J22 0.145∗∗∗ -0.007 0.032∗∗∗ 0.139∗∗∗ 0.241∗∗∗

IEarnAC 0.292∗∗∗ 0.264∗∗ 0.204∗∗∗ 0.242∗∗∗ 0.139∗

IEarnACC1 0.058∗∗∗ -0.109 −0.142∗∗∗ 0.020 0.145∗

IEarnACC5 0.178∗∗∗ -0.051 0.531∗∗∗ 0.297∗∗∗ -0.069IEarnACC22 0.200∗∗∗ 0.791∗∗∗ -0.033 0.036 0.227∗∗

IEarnACJ1 −0.044∗∗∗ −0.163∗∗∗ −0.060∗∗∗ -0.008 −0.080∗∗∗

IEarnACJ5 −0.037∗∗ -0.019 -0.033 -0.022 0.279∗∗∗

IEarnACJ22 −0.054∗∗ 0.005 0.048 -0.018 −0.285∗∗∗

IDown 0.207∗∗∗ 0.180∗∗∗ 0.081∗∗∗ 0.172∗∗∗ 0.155∗∗∗

IDownC1 0.003 −0.107∗∗∗ 0.143∗∗∗ -0.018 0.139∗∗∗

IDownC5 0.099∗∗∗ 0.175∗∗ 0.117∗∗∗ 0.113∗∗∗ 0.010IDownC22 −0.029∗ 0.100 0.067∗ 0.053 -0.062IDownJ1 0.032∗∗∗ 0.002 0.059∗∗∗ 0.008 0.011IDownJ5 −0.062∗∗∗ 0.012 −0.118∗∗∗ −0.044∗∗∗ 0.008IDownJ22 0.036∗∗∗ 0.065 0.005 0.085∗∗∗ −0.114∗∗∗

R-sq 0.356∗∗∗ 0.299∗∗∗ 0.358∗∗∗ 0.331∗∗∗ 0.338∗∗∗

Rbar 0.356∗∗∗ 0.298∗∗∗ 0.357∗∗∗ 0.331∗∗∗ 0.338∗∗∗

nobs 349317.000∗∗∗ 20734.000∗∗∗ 90430.000∗∗∗ 107769.000∗∗∗ 45023.000∗∗∗

Table 34. HAR-RNG-CJpEADS: in sample results.

VOLATILITY FORECASTS WITH THE HAR RNG CJ. (PRELIMINARY, COMMENTS WELCOMED.)29

full top 10 upper 40 lower 40 bottom 10EAJ 0.131∗∗∗ 0.168∗∗∗ 0.138∗∗∗ 0.110∗∗∗ 0.127∗∗∗

CSP500 0.064∗∗∗ 0.024∗∗ 0.071∗∗∗ 0.078∗∗∗ 0.072∗∗∗

JSP500 −0.037∗∗∗ −0.027∗∗∗ −0.050∗∗∗ −0.032∗∗∗ -0.012V IX 0.257∗∗∗ 0.312∗∗∗ 0.249∗∗∗ 0.271∗∗∗ 0.222∗∗∗

CV IX 0.017∗∗∗ 0.012∗∗∗ 0.012∗∗∗ 0.023∗∗∗ 0.027∗∗∗

JV IX −0.007∗∗∗ −0.004∗∗∗ −0.006∗∗∗ −0.009∗∗∗ 0.002C1 0.317∗∗∗ 0.372∗∗∗ 0.301∗∗∗ 0.309∗∗∗ 0.271∗∗∗

C5 0.248∗∗∗ 0.235∗∗∗ 0.282∗∗∗ 0.239∗∗∗ 0.251∗∗∗

C22 0.115∗∗∗ 0.173∗∗∗ 0.167∗∗∗ 0.109∗∗∗ 0.131∗∗∗

J1 0.168∗∗∗ 0.144∗∗∗ 0.186∗∗∗ 0.151∗∗∗ 0.161∗∗∗

J5 0.123∗∗∗ 0.062∗∗∗ 0.084∗∗∗ 0.127∗∗∗ 0.123∗∗∗

J22 0.167∗∗∗ 0.059∗∗∗ 0.075∗∗∗ 0.169∗∗∗ 0.211∗∗∗

IEarnAC 0.499∗∗∗ 0.432∗∗∗ 0.493∗∗∗ 0.553∗∗∗ 0.572∗∗∗

IEarnACC1 0.309∗∗∗ 0.008 0.353∗∗∗ 0.336∗∗∗ 0.209∗∗

IEarnACC5 0.193∗∗∗ 0.546∗∗∗ 0.189∗∗∗ 0.243∗∗∗ 0.516∗∗∗

IEarnACC22 -0.013 −0.160∗ -0.063 -0.032 -0.010IEarnACJ1 −0.055∗∗∗ -0.000 −0.081∗∗∗ −0.053∗∗∗ -0.037IEarnACJ5 −0.101∗∗∗ -0.064 −0.070∗∗∗ −0.080∗∗∗ −0.305∗∗∗

IEarnACJ22 0.012 −0.098∗ −0.150∗∗∗ 0.098∗∗∗ 0.075IDown 0.223∗∗∗ 0.023 0.128∗∗∗ 0.246∗∗∗ 0.240∗∗∗

IDownC1 0.115∗∗∗ 0.235∗∗∗ 0.084∗∗∗ 0.093∗∗∗ 0.126∗∗∗

IDownC5 0.094∗∗∗ 0.275∗∗∗ 0.200∗∗∗ 0.134∗∗∗ −0.077∗∗

IDownC22 0.002 −0.096∗∗ −0.042∗∗ 0.009 0.051IDownJ1 0.035∗∗∗ 0.077∗∗∗ 0.189∗∗∗ −0.035∗∗∗ −0.048∗∗∗

IDownJ5 −0.049∗∗∗ −0.052∗∗∗ −0.097∗∗∗ −0.033∗∗∗ 0.037∗

IDownJ22 −0.020∗∗ -0.016 0.031∗∗ 0.010 −0.149∗∗∗

R-sq 0.547∗∗∗ 0.560∗∗∗ 0.581∗∗∗ 0.539∗∗∗ 0.429∗∗∗

Rbar 0.547∗∗∗ 0.560∗∗∗ 0.581∗∗∗ 0.539∗∗∗ 0.429∗∗∗

nobs 573429.000∗∗∗ 58993.000∗∗∗ 229313.000∗∗∗ 212131.000∗∗∗ 34605.000∗∗∗

Table 35. HAR-RNG-CJpEADS: out of sample results.

30LOUIS SCOTT, KIEMAADVISORS 1 ANDDANDIBARTOLOMEO, NORTHFIELD INFORMATION SERVICES 2

full top 10 upper 40 lower 40 bottom 10EAJ 0.138∗∗∗ 0.185∗∗∗ 0.157∗∗∗ 0.108∗∗∗ 0.132∗∗∗

CSP500 0.083∗∗∗ 0.047∗∗∗ 0.085∗∗∗ 0.097∗∗∗ 0.090∗∗∗

JSP500 −0.029∗∗∗ −0.027∗∗∗ −0.042∗∗∗ −0.028∗∗∗ 0.001V IX 0.215∗∗∗ 0.269∗∗∗ 0.216∗∗∗ 0.239∗∗∗ 0.241∗∗∗

CV IX 0.011∗∗∗ 0.008∗∗∗ 0.009∗∗∗ 0.017∗∗∗ 0.010∗∗∗

JV IX −0.006∗∗∗ −0.003∗∗∗ −0.005∗∗∗ −0.006∗∗∗ 0.004∗∗

C1 0.297∗∗∗ 0.351∗∗∗ 0.290∗∗∗ 0.294∗∗∗ 0.249∗∗∗

C5 0.251∗∗∗ 0.257∗∗∗ 0.284∗∗∗ 0.242∗∗∗ 0.242∗∗∗

C22 0.144∗∗∗ 0.196∗∗∗ 0.191∗∗∗ 0.130∗∗∗ 0.156∗∗∗

J1 0.169∗∗∗ 0.147∗∗∗ 0.185∗∗∗ 0.155∗∗∗ 0.159∗∗∗

J5 0.116∗∗∗ 0.046∗∗∗ 0.074∗∗∗ 0.121∗∗∗ 0.124∗∗∗

J22 0.160∗∗∗ 0.046∗∗∗ 0.067∗∗∗ 0.162∗∗∗ 0.217∗∗∗

IEarnAC 0.397∗∗∗ 0.387∗∗∗ 0.388∗∗∗ 0.459∗∗∗ 0.364∗∗∗

IEarnACC1 0.252∗∗∗ 0.036 0.270∗∗∗ 0.276∗∗∗ 0.205∗∗∗

IEarnACC5 0.188∗∗∗ 0.452∗∗∗ 0.263∗∗∗ 0.265∗∗∗ 0.347∗∗∗

IEarnACC22 0.035∗ -0.090 −0.094∗∗∗ -0.011 0.087IEarnACJ1 −0.053∗∗∗ −0.059∗∗∗ −0.078∗∗∗ −0.038∗∗∗ −0.045∗∗

IEarnACJ5 −0.083∗∗∗ -0.050 −0.062∗∗∗ −0.075∗∗∗ −0.148∗∗∗

IEarnACJ22 0.004 -0.051 −0.100∗∗∗ 0.084∗∗∗ -0.015IDown 0.163∗∗∗ -0.013 0.098∗∗∗ 0.188∗∗∗ 0.199∗∗∗

IDownC1 0.104∗∗∗ 0.202∗∗∗ 0.100∗∗∗ 0.086∗∗∗ 0.138∗∗∗

IDownC5 0.099∗∗∗ 0.296∗∗∗ 0.197∗∗∗ 0.136∗∗∗ −0.063∗∗∗

IDownC22 -0.010 −0.089∗∗ −0.047∗∗∗ 0.001 0.033IDownJ1 0.038∗∗∗ 0.083∗∗∗ 0.173∗∗∗ −0.028∗∗∗ −0.037∗∗∗

IDownJ5 −0.048∗∗∗ −0.046∗∗∗ −0.095∗∗∗ −0.034∗∗∗ 0.036∗∗

IDownJ22 0.001 -0.004 0.035∗∗∗ 0.031∗∗∗ −0.132∗∗∗

R-sq 0.527∗∗∗ 0.529∗∗∗ 0.562∗∗∗ 0.529∗∗∗ 0.477∗∗∗

Rbar 0.527∗∗∗ 0.529∗∗∗ 0.562∗∗∗ 0.529∗∗∗ 0.477∗∗∗

nobs 922746.000∗∗∗ 79727.000∗∗∗ 319743.000∗∗∗ 319900.000∗∗∗ 79628.000∗∗∗

Table 36. HAR-RNG-CJpEADS: full sample results.