modeling the interactions between volatility and …...modeling the interactions between volatility...

Modeling the Interactions between Volatility andReturns

PRELIMINARY DRAFT**

Andrew Harvey and Rutger-Jan LangeFaculty of Economics, Cambridge University

July 21, 2014

Abstract

Volatility of a stock may incur a risk premium, leading to a positivecorrelation between volatility and returns. On the other hand the leverageeffect, whereby negative returns increase volatility, acts in the opposite di-rection. We propose a two component ARCH in Mean model to separatethe two effects; such a model also picks up the long memory features ofthe data. An exponential formulation, with the dynamics driven by thescore of the conditional distribution, is shown to be theoretically tractableas well as practically useful. In particular it enables us to write down theasymptotic distribution of the maximum likelihood estimator, somethingthat has not proved possible for standard formulations of ARCH in Mean.Our EGARCH-M model in which the returns have a conditional skewedgeneralized-t distribution is shown to give a good fit to daily S&P500 excessreturns over a 60-year period (4 Jan 1954 - 30 Dec 2013).

Keywords: Dynamic conditional score (DCS) model; generalized t-distribution;leverage; returns; risk premium; two component model; volatility.

1 Introduction

Volatility of a stock may incur a risk premium. A popular textbook model forintroducing a time-varying risk premium into a return is Autoregressive Condi-tional Heteroscedasticity in Mean, or simply ARCH-M; see, for example, Taylor(p 205, p 252-4), Mills and Markellos (2008, 3rd ed, p. 287-93) and Martin et al(2013). Recent papers on the topic include Baillie and Morana (2009), Lundblad

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(2007), Guo and Neely (2008) and Christiansen et al (2010, 2012); Koopman andUspensky (2002) approach the problem in an unobserved components framework.

Current ARCH-M models lack a comprehensive asymptotic theory for the max-imum likelihood (ML) estimator. Here we introduce an exponential formulation,with the dynamics driven by the score of the conditional distribution, and showthat it is theoretically tractable as well as practically useful. Models constructedusing the conditional score were introduced into the literature by Creal, Koop-man and Lucas (2011, 2013), where they are called Generalized AutoregressiveScore (GAS) models and Harvey (2013), where they are called Dynamic Condi-tional Score (DCS) models. The asymptotic theory for the proposed class of DCSEGARCH-M models can be obtained by extending the results in Harvey (2013)and Blasques et al (2014). Other theoretical results, such as the existence of mo-ments for a conditional Student t distribution, may also be derived. As regardsthe practical value of DCS models, there is already a good deal of evidence (inthe references cited) to show that they tend to outperform standard models, oneof the principal reasons being the way in which the dynamic equations deal withobservations which, in a Gaussian world, would be regarded as outliers. Whencoupled with their theoretical advantages, this makes them extremely attractive.

A model with two components of volatility has a number of attractions, one ofwhich is to account for the long memory behavior often seen in the autocorrelationsof absolute values of returns or their squares. We develop this model further byinvestigating the viability of a long-run, or persistent, component that derives froma specification which, in a linear state space model, yields smoothed estimates thatare essentially a cubic spline. Hence it plays a similar role to the splines used byEngle and Rangel (2008) to capture long-term movements in volatility. It alsoprovides an alternative to the trigonometric terms used by Baillie and Morana(2009), Christiansen et al (2010) and others. Our stochastic specification can beincorporated into the filter, and hence updated with each new observation, and itcan be modified so as to make it stationary.

A two-component model enables the researcher to distinguish between the ef-fects of short and long-run volatility. This distinction is similar to that impliedby the ‘news effect’ described by Chou (1988) and Schwert (1989), which predictsa negative correlation between unexpected (i.e. short-run) volatility and returnbecause it makes investors nervous of risk, as opposed to that emanating fromMerton’s (1973) intertemporal capital asset pricing model (ICAPM), which pre-dicts a positive correlation between expected (i.e. long-run) volatility and return;see French et al (1987). Failure to model both types of volatility has led to in-conclusive results regarding the sign of the risk premium. For example, the riskpremium is negative and significant according to Campbell (1987) and Nelson(1991), positive but insignificant according to French et al. (1987) and Campbell

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and Hentschel (1992), and positive or negative (depending on the method) accord-ing Glosten et al. (1993) and Turner et al (1989). Indeed Lundblad (2007) arguesthat, when existing GARCH-M models are used, even a hundred years of data maynot be sufficient to draw clear inferences. Here we show that a carefully specifiedtwo-component model can resolve these apparent contradictions because it enablesthe researcher to investigate the possibility that when long-run volatility goes upit tends to be followed by an increasing level of returns, whereas an increase inshort-run volatility leads to a fall.

Leverage plays an important role in volatility models. It acts in the oppositedirection to the risk premium effect, in that negative returns are associated withincreased volatility; see Bekeart and Wu (2000) for an excellent discussion of thefinance issues involved. Figure 1 highlights some of the relationships betweenvolatility and returns. Again a two component model plays an important role inthat it allows us to whether the effect of leverage is different in the short-run andthe long-run. For example it may be that leverage mainly affects the volatilityshort-run component, as found by Engle and Lee (1999) and others. Indeed short-run volatility may even decrease after a good day, because of the calming effect thishas on the market. Standard GARCH models are unable to identify this possibilitybecause they are constrained to model leverage with an indicator variable ratherthan the sign variable which can be adopted in EGARCH. It is interesting to notethat Chen and Ghysels (2008) have recently identified the presence of such effectsin diurnal data.

The next section sets out the DCS formulation of the EGARCH-M model andsection 3 gives the large sample distribution. Section 4 establishes the conditionsfor the existence of moments, explores the patterns of autocorrelation functions

Figure 1: Some interactions between volatility and returns.

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and discusses predictions. Section 5 introduces the skew generalized Student’s t-distribution for returns. Section 6 fits the new DCS EGARCH models to S&P500excess returns over a 60-year period (4 Jan 1954 - 30 Dec 2013) and compares themwith standard GARCH-M models. Section 7 indicates how the DCS EGARCHmodeling framework can be generalized to multivariate time series. Section 8 con-cludes by summarizing the extent to which our econometric modeling has suceededin disentangling the various interactions between returns and volatility.

2 Model Formulation

In the ARCH-M model, as introduced by Engle et al (1987), the model for a seriesof returns subject to a time-varying risk premium is

yt = µ+ ασmtpt−1 + εtσtpt−1, t = 1, ..., T, (1)

where σtpt−1 is the conditional standard deviation, εt is a serially independentstandard normal variable, that is εt ∼ NID(0, 1), µ and α are parameters and mis typically set to one or two (or it can be estimated). The conditional variance,σ2tpt−1, is assumed to be generated by a dynamic equation that is dependent on

squared observations, so in the GARCH(1,1) case

σ2t+1|t = ω(1− φ) + φσ2

tpt−1 + κ(y2t − σ2tpt−1), (2)

where ω, φ and κ are parameters.Engle et al (1987), and most subsequent studies, find the standard deviation,

that ism = 1, gives a better fit than variance (although the ICAPM theory suggeststhe latter). This is fortunate because it turns out that the asymptotic theory forML estimators of DCS EGARCH-M models is most easily developed with thestandard deviation or, more generally, the scale. Thus the DCS EGARCH-M willbe set up as

yt = µ+ α exp(λtpt−1) + εt exp(λtpt−1), t = 1, ..., T, (3)

where exp(λtpt−1) is the scale, with the dynamic equation for λtpt−1 driven by thescore of the conditional distribution of yt at time t, that is the first derivative ofthe logarithm of the probability density function at time t. When εt is NID(0, 1),in which case the scale is the standard deviation, the score with respect to λtpt−1 is

ut = (yt − µ)e−λtpt−1((yt − µ)e−λtpt−1 − α)− 1, t = 1, ..., T, (4)

which, on substituting (yt − µ)e−λtpt−1 = α + εt, becomes

ut = ε2t − 1 + αεt, t = 1, ..., T. (5)

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At the true parameter values, the scores are IID(0, σ2u), where σ2

u = E[(ε2t − 1 +αεt)

2] = 2 + α2 because ε2t ∼ χ21 and so its variance is two.

In a pure DCS-EGARCH volatility model, that is α = 0, the dynamics aredriven by the squares of the ε′ts when the conditional distribution is Gaussian. Inthe DCS-EGARCH-M model, the value of εt itself appears, reflecting the fact thatit, too, is informative about the movements in λt+1|t. This term does not appear inthe GARCH-M model, (2), where the variable corresponding to ut is y2t − σ2

tpt−1 =σ2tpt−1(ε

2t − 1). Having said that, it is important to note that the inclusion in ut

of αεt, which we will hereafter call the ARCH-M score term, is not crucial to themodel because α is typically very small: dropping it makes very little practicaldifference and the theoretical properties of the model as a whole become muchsimpler.

Remark 1 Note that αεt is not a leverage term, despite the superficial resem-blance to the leverage term in the classic EGARCH model of Nelson (1991). Indeedit will normally act in the other direction because, as a rule, α will be positive. Ina Gaussian DCS EGARCH model leverage is captured by the variable ε2t sgn(−εt);see the discussion in sub-section 2.2.

Using the variance as the risk premium variable for the Gaussian model, thatis m = 2 in (1), gives an ARCH-M score term of αεt exp(λtpt−1). More generally,for any m > 0, ut = ε2t − 1 +αmεt exp(λtpt−1(m− 1)). The presence of λtpt−1 in thescore for m 6= 1 leads to a less tractable asymptotic theory. The use of ln σtpt−1,as suggested by Engle et al (1987) for some series, means that αεt exp(λtpt−1) isreplaced by αλtpt−1 and the ARCH-M score term again depends on λtpt−1.

2.1 Student-t distribution

The Student-t distribution usually improves the fit to returns. The score for aconditional t-distribution with ν degrees of freedom and scale exp(λtpt−1) is

ut = (ν + 1)bt − 1 + α(1− bt)[(ν + 1)/ν]εt, (6)

where

bt =ε2t/ν

1 + ε2t/ν, 0 ≤ bt ≤ 1, 0 < ν <∞, (7)

is distributed as beta(1/2, ν/2) at the true parameter values. In the absence of theARCH-M effect, this model is known1 as Beta-t-EGARCH.

1The (potential) instability in conventional EGARCH-M models noted by St. Pierre (1998)seems not to be an issue with Beta-t-EGARCH-M, presumably because the score downweightsoutliers.

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A Student-t distribution, or a generalized Student-t distribution as in section5, is assumed when models are estimated. However, the formulae associated withthe Gaussian distribution (obtained by ν → ∞) are often much simpler and aresometimes employed to illustrate theoretical points.

2.2 Dynamics and leverage

The basic dynamic model for λt|t−1 is the stationary first-order process2

λt+1|t = ω(1− φ) + φλt|t−1 + κut, |φ| < 1, (8)

where ω, φ and κ are parameters, with ω denoting the unconditional mean, andλ1|0 = ω.

The leverage effect in finance suggests that volatility rises when the asset pricefalls. This asymmetry can be captured by a DCS EGARCH model by modifyingthe dynamic equation in (8) to

λt+1|t = ω (1− φ) + φλt|t−1 + κut + κ∗sgn(−εt)(ut + 1), (9)

where κ∗ is a new parameter which, because the negative of the sign of the returnis taken, is usually expected to be positive; see Harvey (2013, p109). The rise involatility following a fall in the asset price need not necessarily be due to leverageas such3. For example a sharp fall in asset price may induce more uncertainty andhence higher variability.

The standard way of incorporating leverage effects into GARCH models isby including a variable in which the squared observations are multiplied by anindicator, I(yt < 0), taking the value one for yt < 0 and zero otherwise; see Taylor(2005, p. 220-1). The sign cannot be used because negative values could give anegative conditional variance.

2.3 Two components

Instead of capturing long memory by a fractionally integrated process, as in therecent paper by Christensen et al (2010), two components may be used. Thus

λt|t−1 = ω + λ1,t|t−1 + λ2,t|t−1,

λi,t+1|t = φi λi,t|t−1 + κi ui,t + κ∗i sgn(−εt) (ui,t + 1), i = 1, 2,(10)

2More general dynamic models can be constructed and explanatory variables may be added;see Harvey (2013).

3Bekaert and Wu (2002) write ‘To many, leverage effects have become synonymous withasymmetric volatility, the asymmetric nature of the volatility response to return shocks.’

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where φ1 > φ2 if λ1,tpt−1 denotes the long-run component. The scores, ui,t, willusually be the same but may be different when the ARCH-M effect is not drivenby total volatility.

An attraction of a two component model is that it allows the leverage effectto impact differently in the long run and short run. For example, Engle and Lee(1999) and Harvey (2013, p 134-5) find that leverage is confined to the short-termcomponent. If this is the case, the evolution of the long-run component is evenless susceptible to the influence of strongly negative returns and so may be moresuitable for capturing the ARCH-M effect. If φ1 is set to one (and ω = 0), λ1,t+1|t isthe long-run forecast of volatility and α exp(λ1,t+1|t) is the corresponding forecastof the risk premium.

The possibility of including a short-term ARCH-M effect may also be investi-gated. Equation (3) may be amended to become

yt = µ+α1 exp(ω+λ1,tpt−1)+α2[exp(λ2,tpt−1)−1]+εt exp(λtpt−1), t = 1, ..., T, (11)

so that when volatility is at its equilibrium level, the risk premium is µ+α1 exp(ω).

2.4 Damped spline filter

Engle and Rangel (2008) propose the use of cubic splines for capturing slowlychanging movements in volatility. Their approach is to fit the splines nonpara-metrically with knots at selected points throughout the sample. However, thesmoothed estimates of an integrated random walk trend in a linear unobservedcomponents model are known to be closely related to a cubic spline; see Har-vey (2013, pp 77-9). The same idea may be adapted for modeling the long-runcomponent of volatility in a DCS EGARCH model. The filter is

λ1,t+1|t = λ1,t|t−1 + λβ1,t|t−1 + κ1ut

λβ1,t+1|t = λβ1,t|t−1 +κ21

2− κ1ut,

where the starting values, λ1,1|0 and λβ1,1|0, are treated as fixed and estimated alongwith the other parameters. A persistent stationary component may be constructedby introducing a damping factor, φ, which is close to one, into the equations. Thus

λ1,t+1|t = φ1 λ1,t|t−1 + λβ1,t|t−1 + κ1 ut, |φ1| < 1, (12)

λβ1,t+1|t = φ1 λβ1,t|t−1 +

κ212− κ1

ut.

We will refer to this (constrained) second-order model as the damped spline filter.

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Remark 2 The use of the score to drive the dynamics poses one further issue.As noted in Remark 1, the ARCH-M score term, αεt in the Gaussian model, willnormally act in the opposite direction from that of the leverage term. One solutionis to drop the ARCH-M score term from the dynamic equation. However, if theARCH-M effect depends only on λ1,tpt−1, the DCS model has a short-term dynamicequation4 that does not contain an ARCH-M term in the score.

Remark 3 Identifiability requires φ1 6= φ2, which we have implicitly assumed bysetting φ1 > φ2, together with κ1 6= 0 and κ2 6= 0.

3 Large sample distribution of the maximum like-

lihood estimator

When the conditional distribution of returns is Student’s t and µ is known tobe zero, an analytic expression for the information matrix can be derived. Thereason this is possible is that the information matrix for the static model, that isλt|t−1 = λ, is independent of λ, as are E (∂ut/∂λ) and E(ut∂ut/∂λ).

Proposition 4 Suppose that the distribution of yt conditional on past observationsin (3) is Student’s t with scale expλt|t−1 and mean α expλt|t−1, where λt|t−1 isgenerated by the stationary first-order dynamic process (8). The degrees of freedom,ν, is assumed to be known. Let ψ = (κ, φ, ω)′ and define a, b and c as

a = φ+ κE

(∂ut∂λ

)(13)

b = φ2 + 2φκE

(∂ut∂λ

)+ κ2E

(∂ut∂λ

)2

≥ 0

c = κE

(ut∂ut∂λ

),

where the model is formulated in such a way that the unconditional and conditional

expectations in (13) are the same. Assuming that b < 1 and κ 6= 0, then (ψ̃′, α)′,

the ML estimator of (ψ′, α)′, is consistent and the limiting distribution of√T (ψ̃

′−

ψ′, α̃ − α)′ is multivariate normal with mean vector zero and covariance matrix

given by V ar(ψ̃, α̃) = I−1(ψ, α), where the information matrix is

I

[ψα

]=ν + 1

ν + 3

[(2(ν/(ν + 1)) + α2)D(ψ) αd(ψ)

αd′(ψ) 1

](14)

with d(ψ) = (0, 0, (1− φ)/(1− a)) and D(ψ) as in Appendix A.

4The score for the long-term component is then ut = ε2t +αεt exp(−λ2,tpt−1)−1 in the Gaussianmodel. More generally the ARCH-M score term is multiplied by exp(−λ2,tpt−1).

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Remark 5 When ν is estimated, (14) is extended accordingly; see Harvey (2013,p 116). Note that E(∂ ln ft/∂α.∂ ln ft/∂ν) = 0.

The form of (14) follows from Corollary 9 in Harvey (2013, p 47). The expec-tations of terms involving ut and its derivatives can be found from properties ofthe beta distribution. For example,

σ2u = E[((ν + 1)bt − 1)2] + 2α[(ν + 1)/ν]E[(1− bt)((ν + 1)bt − 1))εt]

+α2 [(ν + 1)/ν]2E[((1− bt)εt)2]

=2ν

ν + 3+ 0 + α2 (ν + 1)2

ν2ν2

(ν + 3)(ν + 1)=

2ν

ν + 3+ α2ν + 1

ν + 3

The expression for σ2u gives a in (13) because it is the same as E (∂ut/∂λ) . It is

straightforward but tedious to derive b and c. However, for a Gaussian conditionaldistribution, it is easy to see that

E(u′2t ) = E(2ε2t + 3αεt + α2)2 = 12 + 13α2 + α4 and E(utu′t) = −4− 3α2,

and so

a = φ− κ(2 + α2), c = −κ(4 + 3α2), (15)

b = φ2 − 2φκ(2 + α2) + κ2(12 + 13α2 + α4) ≥ 0.

Because α is typically very small, the inclusion of terms involving α2 and α4 inthe above formulae makes very little practical difference. If the ARCH-M scoreterm is dropped, or simply ignored in the evaluation of the information matrix,the expressions for the elements of the D(ψ) matrix for the t−distribution revertto those in the Beta-t-EGARCH model. Leverage effects can then be included asset out in Harvey (2013, pp 121-4).

A small Monte Carlo experiment was run in order to investigate the finitesample properties of the ML estimator. A Gaussian model was simulated and MLestimation carried out for 10,000 iterations with typical values of the parameters;but note that, according to (14) the asymptotic distribution does not depend on ω.As can be seen from Table 1, the simulated RMSEs are very close to the asymptoticstandard errors (ASEs) for T = 10, 000. They are similarly close for T = 1000,except for φ. However, this may not be too surprising because the value of φ = 0.98is quite close to unity, and the estimates of φ were constrained to be less than one

True T = 10, 000 T = 1000parameter Mean RMSE ASE Mean RMSE ASE

κ 0.04 0.0399 0.0024 0.0023 0.0397 0.0077 0.0074φ 0.98 0.9794 0.0030 0.0028 0.9734 0.0155 0.0090ω 0.10 0.0984 0.0350 0.0354 0.0882 0.1123 0.1119α 0.05 0.0499 0.0099 0.0100 0.0498 0.0314 0.0316

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Table 1: Monte Carlo results for Gaussian EGARCH-M model

When there is leverage in Beta-t-EGARCH, the information matrix is as inHarvey (2013, pp. 121-4). The main point to note is that identifiability requiresonly that either κ or κ∗ be non-zero. (When there is no leverage term, D(ψ) issingular if κ = 0 and the model is underidentified.) Thus Wald and LR tests of thenull hypothesis that either κ or κ∗ is zero can be carried out and this continuesto be the case when there is an ARCH-M term.

Remark 6 When µ is included in the model, the score with respect to µ is εt exp(λtpt−1).The static information matrix for the Gaussian model is then

I

λαµ

=

2 + α2 α αe−λ

α 1 0αe−λ 0 e−2λ

, (16)

and a decomposition like the one in (14) is not possible when λ is stochastic becauseof the presence of e−λtpt−1 in the off-diagonal elements of the information matrixof the t-th observation conditional on the observations available at time t− 1.

Remark 7 Because the above information matrix in (14) does not depend on αwhen α = 0, a Lagrange multiplier test of the null hypothesis that α = 0 against thealternative α 6= 0 is very easy. To be specific, the statistic, which is asymptoticallydistributed as χ2

1 under the null hypothesis of a Beta-t-EGARCH model, is

LM =(ν + 1)(ν + 3)

Tν2

T∑t=1

(yt − µ̃

exp(λtpt−1)

)2

.

4 Moments, autocorrelations and predictions

One of the important features of the Beta-t-EGARCH model, is that when λtpt−1is stationary, the unconditional moment of the observations exists whenever thecorresponding conditional moment exists5. This is not generally true for GARCHmodels where issues surrounding the existence of unconditional moments can bequite complex. Furthermore exact analytic expressions for moments and autocor-relations of absolute values of the observations raised to any non-negative powermay be derived; see Harvey (2013, ch 4). This section investigates the extent towhich it is possible to generalize the Beta-t-EGARCH results to a model whichincludes an ARCH-M term.

5In Nelson’s classic EGARCH model, the dynamics are driven by absolute values of returns;see, for example, Christensen et al (2010). If the conditional distribution is Student’s t, thereturns have no moments of any order.

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4.1 ACFs of powers of absolute values

For the DCS EGARCH-M model the addition of the ARCH-M score term, α(1−bt)[(ν + 1)/ν]εt, makes it difficult to derive corresponding analytic expressions,but the boundedness of the score, and hence exp(mλtpt−1), for the t-distributionmeans that the unconditional m − th moment exists whenever the correspondingconditional m− th moment does.

Because α is typically small, the moments and autocorrelations of absolutevalues are unlikely to be very different from those of the Beta-t-EGARCH model.The (unconditional) expectation of volatility in the stationary Beta-t-EGARCHmodel is

E (expλtpt−1) = eω∞∏j=1

e−ψjβν(ψj), (17)

where ψj, j = 1, 2, .. is the coefficient of ut−j in the moving average representationof λtpt−1 and βν(a) is Kummer’s (confluent hypergeometric) function.The correction

factor∞∏j=1

e−ψjβν(ψj) is greater than or equal to one because E (expλtpt−1) ≥ eω

by Jensen’s inequality.

4.2 ACF of returns

The unconditional moments about µ are given by E(α+ εt)mE(expmλtpt−1), m =

1, 2, .... Thus the mean is µ+ αE(expλtpt−1) = µ+ µλ, whereas the variance is

V ar(yt) = (σ2ε+α

2)E(exp 2λtpt−1)−µ2λ = (σ2

ε+α2)E(exp 2λtpt−1)−α2[E(expλtpt−1)]

2

The autocovariance of yt at lag τ , is

γ(τ) = E(yt − µ− µλ)(yt−τ − µ− µλ)]= α2E(exp(λtpt−1 + λt−τ pt−τ−1))

+αE(εt−τ exp(λtpt−1 + λt−τ pt−τ−1))− µ2λ, (18)

because the two terms containing εt have zero expectation (by the law of iter-ated expectations). The term E(εt−τ exp(λtpt−1 + λt−τ pt−τ−1)) is also zero because,although λtpt−1 depends on εt−τ , we can write

E(εt−τ exp(λtpt−1 + λt−τ pt−τ−1)) = Pr(εt−τ > 0)E(|εt−τ | exp(λtpt−1 + λt−τ pt−τ−1))

−Pr(εt−τ < 0)E(|εt−τ | exp(λtpt−1 + λt−τ pt−τ−1)) (19)

and this is zero when there are no leverage effects and εt is symmetric. Thus

γ(τ) = α2(E(exp(λtpt−1 + λt−τ pt−τ−1))− [E(expλtpt−1)]2),

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and

ρ(τ) =E(exp(λtpt−1 + λt−τ pt−τ−1))− [E(expλtpt−1)]

2

(σ2ε/α

2)E(exp 2λtpt−1) + E(exp 2λtpt−1)− [E(expλtpt−1)]2, τ ≥ 1.

These autocorrelations do not depend on ω. Clearly the autocorrelations are zerofor α = 0. For non-zero α, ρ(τ) has the same sign as the corresponding autocorrela-tion of the volatility, expλtpt−1, with a pattern derived from that of the λ′tpt−1s. Notethat in the GARCH-M model, the autocorrelations of the y′ts must all be positivebecause of the positivity constraints on the GARCH parameters; see Hong(1991).(Of course in practice the autocorrelations in the first-order model, (8), are almostinvariably positive.)

When the dynamic equation contains a leverage term, the second term in (19)will be greater than the first6 for κ∗ > 0 and so the second term in (18) will benegative for positive α, so weakening the autocorrelations. In fact they may nolonger be positive.

4.3 Predictions

Expressions for conditional moments of future observations can be worked outfor the pure EGARCH models. To do so here is more difficult because of theadditional term in ut. But the full predictive distribution is often what is neededand simulating it is not difficult because it depends only on independent Student-tvariates.

Of course if the ARCH-M score term is dropped from the dynamic equation,predictions of moments can be made as in a Beta-t-EGARCH model, but with theadditional feature that the conditional mean is no longer constant. The MMSEpredictor of the observation ` steps ahead is

ET (yT+`) = αET(eλT+`pT+`−1

)= αeλT+`pT

`−1∏j=1

e−ψjβν(ψj), ` = 2, 3, .. (20)

where λT+`pT is the MMSE predictor of λT+`pT+`−1; see expression (4.30) in Harvey(2013, p 111).

5 Skew generalized Student-t distribution

In the Gamma-GED-EGARCH model, the conditional distribution of yt is a gen-eral error distribution (GED), with time-varying scale parameter exp(λtpt−1) and

6A similar term appears in the ACF of the Beta–t-EGARCH model and can be evaluated asin Harvey (2013, p 239-41).

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λtpt−1 evolving as a linear function of the conditional score variable

ut =υ

2

(|yt|

exp(λtpt−1)

)υ− 1, t = 1, ..., T, (21)

where υ is a shape parameter that is two for the normal and one for the Laplace,or double exponential, distribution. When υ < 2, the response is less sensitive tooutliers than it is for a normal distribution, but it is far less robust than for a Beta-t-EGARCH model with small degrees of freedom; it is clear from (21) that ut isnot bounded. At the true parameter values, the u′ts are independently distributedas gamma variates with shape parameters 2 and 1/υ; see Harvey (2013, section4.4).

The generalized Student-t distribution contains the GED and Student-t distri-butions as limiting cases. It was introduced by McDonald and Newey (1987), whoproposed its use for static regression models, and it was subsequently employedby Theodossiou (1998) for financial data. The generalized Student-t distributioncan be used to construct a general DCS EGARCH model, which we will call Beta-Gen-t-EGARCH. A generalized t variable, y, is such that |y| is distributed asa generalized beta of the second kind (GB2) with two of the shape parametersconstrained so that the mode is at the origin. The PDF is

f(y) =υ

2ϕν1/υ1

B(ν2, 1υ)

(1 +

1

ν

|y − µ|υ

ϕυ

)−( ν2+ 1υ)

, ∞ < y <∞, (22)

where µ is a location parameter, ϕ is a scale parameter, wheras υ and ν arepositive shape parameters7. Setting υ = 2 gives tν , whereas GED(υ) is obtainedwhen ν → ∞. The tail index is υν/2; hence it is ν for Student’s t, but ν/2for generalized Pareto, a distribution given by setting υ = 1. The generalizedStudent-t introduces more flexibility into the shape of the distribution: for a giventail index and standard deviation, the distribution is more peaked for lower υwhereas, when υ is fixed, the peak becomes higher as the tail index decreases. Themoments about the mean for m = 2, 4, .... are8

E[(y − µ)m] =Γ( 1

υ+ m

υ)Γ(ν

2− m

υ)

Γ( 1υ)Γ(ν

2)

νmυ ϕm, −1 < m < υν/2.

The Beta-Gen-t-EGARCH model has ϕ = exp(λtpt−1) and the score is

ut =∂ ln ft∂λtpt−1

=(υν

2+ 1)bt − 1, (23)

7Note that ∂f(y)/∂y = 0 at y = µ for υ > 1. (For υ < 1, ∂f(y)/∂y = ∞.)8For example m = 2 with t gives V ar(y) = ϕ2ν/(ν − 2).

13

where

bt =(|yt − µ| e−λtpt−1)υ/ν

(|yt − µ| e−λtpt−1)υ/ν + 1

is distributed as beta(1/υ, ν/2) at the true parameter values. For υ = 2 we obtainthe score as in (6). Letting ν →∞ gives the score as in (21)9. The score is boundedfor ν <∞. A bounded score means that the existence of moments is not affectedby volatility, although they may become hugely inflated. Exact expressions for theunconditional moments can be derived in the same way as for Beta-t-EGARCH;see subsection 4.1. When ν =∞, the existence of unconditional moments requiresconstraints on the parameters. Thus for the first-order model, (8), the m−th ordermoment only exists for m < 1/υκ. When the model is estimated by ML, an exactexpression for the information matrix for a single component first-order model canbe constructed in a similar way as was done for the GB2 model in Harvey (2013,sub-section 5.3).

Skewness can be introduced into the distribution by means of the Fernandezand Steel (1998) method, as used by Harvey and Sucarrat (2014) for the Student-tdistribution. The PDF in (22) then becomes

f(y) =2

γ + 1γ

υ

2ϕν1/υ1

B(ν2, 1υ)

(1 +

1

ν

|y − µ|υ

γυ sgn(y−µ)ϕυ

)−( ν2+ 1υ)

, ∞ < y <∞, (24)

where 0 < γ < ∞ and γ = 1 gives symmetry. This distribution is similar to,but not quite the same as, the skew generalized t distribution of McDonald et al(2009), which is based on the skew-t distribution of Hansen (1994). The score inthe EGARCH model now becomes

ut =(υ ν

2+ 1) (|yt − µ| e−λtpt−1)υ/ν

(|yt − µ| e−λtpt−1)υ/ν + γυ sgn(yt−µ)− 1. (25)

The introduction of skewness implies that µ is still the mode but no longerthe mean. This raises the new issue that the conditional expectation of εt is nolonger zero10. As a result, the ARCH-M effect is confounded with the conditionalexpectation εt exp(λtpt−1). The mean of a skewed distribution is (γ + γ−1)E(|ε|),where E(|ε|) refers to the original (non-skewed) variable; see e.g. Harvey andSucarrat (2014). For the skewed generalized Student-t distribution, we have

µε =

(γ − 1

γ

)E(|ε|) =

(γ − 1

γ

)Γ(2/υ)Γ(ν/2− 1/υ)

Γ(1/υ)Γ(ν/2)ν1/υ

9There may therefore be a case for approximating Gamma-GED-EGARCH by a Beta-Gen-t-EGARCH in which ν is large but bounded.

10A similar problem was noted in Harvey and Sucarrat (2014), where a correction for skew-ness was made in order for the returns to be martingale differences. There the model wasre-formulated. Here it is re-arranged.

14

and so the conditional expectation of yt in (3) is µ + (α + µε) exp(λtpt−1). Thusthe ARCH-M effect11 is not given by the estimate of α in (3) but by α† = α+ µε.When γ is less than one, which is typically the case, µε is negative and hence theARCH-M effect is reduced. A more convenient model formulation is obtained bysubtracting µε from the disturbance term so that

yt = µ+ α† expλtpt−1 + (εt − µε) expλtpt−1,

where α† = α + µε. For the two-component model, a modification of (11) yields

yt = µ+α1 exp(ω+λ1,tpt−1)+α2[exp(λ2,tpt−1)−1]+(εt−µε) exp(λtpt−1), t = 1, ..., T,(26)

where α1 plays the same role as α† in the previous expression. If the mean, µ,can be dropped, the ARCH-M effects are estimated more precisely; see Lanne andSaikkonen (2006).

6 A location-scale model for the S&P500

This section investigates the relationship between the conditional mean and con-ditional variance of S&P500 excess returns, but without imposing a particularrelationship. Instead, we allow the filtered location and the filtered scale to varyindependently. In particular, we assume the daily excess return yt is given by

yt = ρ yt−1 + µt|t−1 + (εt − µε) exp(λt|t−1), (27)

where ρ captures possible auto-correlation in the returns, µt|t−1 is the filteredlocation, and exp(λt|t−1) is the filtered scale. The residuals εt are i.i.d., but mayhave non-zero mean due to skewness. The mean of the residuals is denoted by µε,such that the scale is multiplied by an i.i.d. random variable with zero mean. Thissection makes no a priori assumption as to how location and scale should movetogether, but examines a possible intertemporal relationship a posteriori.

Typically, in (27), the filtered location is two orders of magnitudes smallerthan the filtered scale. Indeed, the ambiguous results in the ARCH-M literatureare caused mainly by the fact that it is hard to filter location accurately. During1994-2014, daily excess returns had an unconditional mean (standard deviation)of 0.016% (1.22%). The signal to noise ratio regarding the location, therefore, isvery low. Further, the location filter must be robust to large values of |yt|, whichoccur frequently as financial returns have fat tails.

11This assumes that the total volatility, rather than just the long-run component, enters intothe ARCH-M effect.

15

This paper assumes that the residuals εt are distributed as a (possibly skewed)generalized Student-t. The generalized Student-t distribution was introduced byMcDonald and Newey (1987) and employed by Theodossiou (1998) for financialdata. The p.d.f. for εt with unit scale is

P (εt ∈ dx) = p(x) dx =υ

2 ν1/υ1

B(ν2, 1υ)

(1 +|x|υ

ν

)−( ν2+ 1υ)

dx,

for x ∈ (−∞,∞), and where υ and ν are positive shape parameters. For υ = 2,the Student-t distribution with ν degrees of freedom is obtained. For ν →∞, theGED with shape parameter υ is obtained. The tail index is υν/2. The mean andmode are both at zero. Skewness can be introduced by method in Fernandez andSteel (1998), such that the p.d.f. becomes

p(x) =2

γ + 1γ

υ

2 ν1/υ1

B(ν2, 1υ)

(1 +

1

ν

|x|υ

γυ sgn(x)

)−( ν2+ 1υ)

, (28)

for positive γ. The choice γ = 1 gives symmetry, whereas γ < 1 implies negativeskew. If γ 6= 1, then E(εt) 6= 0. In Harvey and Sucarrat (2014) it is shown thatE(εt) is

µε := E (εt) =

(γ − 1

γ

)Γ(2/υ)Γ(ν/2− 1/υ)

Γ(1/υ)Γ(ν/2)ν1/υ,

such that E(εt) depends on three shape parameters. If γ = 1, then µε = 0. In(27), the scale is multiplied by the zero-mean i.i.d. random variable εt− µε. Fromthe p.d.f. of εt, it follows that the p.d.f. of yt equals

f(yt) = exp(−λt|t−1) p(yt − µt|t−1 − ρ yt−1

exp(λt|t−1)+ µε

)= exp(−λt|t−1) p (εt) , (29)

where p is given by (28) and εt = (yt − ρ yt − µt|t−1) exp(−λt|t−1) + µε is simplythe measurement equation (27) re-written. In dynamic conditional score (DCS)models, the dynamics of a time-varying parameter is driven by an innovation thatis proportional to the conditional score. The score u·,t is defined as the derivativeof the log-likelihood with respect to the time-varying parameter, i.e.:

uλ,t :=∂ ln f(yt)

∂λt|t−1=(ν υ

2+ 1) (

1− µεεt

)|εt|υ

|εt|υ + ν γυ sgn(εt)− 1,

uµ,t :=∂ ln f(yt)

∂µt|t−1=(ν υ

2+ 1)(

1− µεεt

)|εt|υ−2

|εt|υ + ν γυ sgn(εt)

εtexp(λt|t−1)

,(30)

where f is the predictive density (29). The scores can be written as functions ofyt or of εt; the latter leads to more compact expressions. For Gaussian residuals

16

(γ = 1, µε = 0, υ = 2 and ν →∞), the scores are

uλ,t = limν→∞

ν + 1

ε2t + νε2t − 1 = ε2t − 1,

uµ,t = limν→∞

ν + 1

ε2t + ν

εtexp(λt|t−1)

=εt

exp(λt|t−1).

The dynamic equation for the location and the scale involve the scores and are asfollows:

λt+1|t = ωλ (1− ϕλ) + ϕλ λt|t−1 + κλ uλ,t + κ∗ sgn(−εt) (uλ,t + 1),

µt+1|t = ωµ (1− ϕµ) + ϕµ µt|t−1 + κµ uµ,t,

where, in both equations, ω denotes the long-term average, ϕ < 1 denotes thepersistence, and κ and κ∗ govern the sensitivity with respect to the score. Involatility modeling, it is well-known that positive returns and negative returns mayhave different impact on the volatility. The asymmetric response of volatility toreturns has become known as the ‘leverage effect’. The leverage parameter κ∗ > 0implies that a fall in the asset price increases uncertainty by a larger amount thana corresponding rise in the asset price would have done. The leverage parameter κ∗

is multiplied by (uλ,t + 1), such that λt+1|t is a continuous function of yt (comparewith (30)).

7 ARCH-M in SP500

In this section we fit a variety of EGARCH-M models to the S&P500 stock marketindex. The return on day t is defined as the continuously compounded percentage100 ∗ [log(It)− log(It−1)], where It is the adjusted closing price of the index.

The data are drawn from Yahoo Finance (yahoo.finance.com), spanning a 60-year period from 4 January 1954 to 31 December 2013; a total of 15, 104 obser-vations. No adjustment was made for dividends, as the consensus (e.g. French etal. 1987, Poon & Taylor 1992 and Koopman & Uspensky 2002) seems to be thatthey have little or no effect on the estimates.

The risk-free return is proxied by the secondary market for 3-month Treasurybills12. The risk-free return, rf,t, is defined as the continuously compounded inter-est rate per business day.

12This data is available from Table H.15 of the Federal Reserve(http://www.federalreserve.gov/releases/h15/data.htm), from 4 January 1954 onwards.The table cites yearly interest rates. To calculate the (continuously compounded) interestrate per business day, we assume 250 business days per year. Thus the daily interest rate inpercentages is rf,t = 100/250 ∗ log(1+yearly interest rate in percentages/100).

17

The daily excess return, yt, is defined as the daily S&P500 return minus thedaily risk-free return, i.e. yt = 100∗ [log(It)− log(It−1)]−rf,t. Our analysis focuseson the time series yt.

Four basic models were estimated (see Appendix 2 for details). In all cases theconditional distribution was assumed to be skew generalized Student-t. In somecases we set γ = 1 and υ = 2, such that the conditional distribution is simplyStudent-t.

Model 1 has a single component and is specified as in (3) with dynamicequation (9) driven by a score as in (25). All other models have two components,as in (10). They differ in the way in which volatility affects returns. In Model2, the EGARCH-M effect is driven by the overall volatility, that is ω and the twocomponents, with identifiability enforced by the restriction φ1 > φ2. In Model 3,it is driven instead by the sum of ω and the long-run component, λ1,t|t−1. The factthat only one component is associated with the EGARCH-M effect is enough toensure identifiability; in fact unrestricted estimation always yields a model in whichthe EGARCH-M effect is associated with the more persistent component. Model4 permits two separate EGARCH-M effects; one associated with the long-run andone with the short-run component of volatility.

In Models 2b, 3b and 4b, the dynamic equation for the first component isadjusted to include a damped cubic spline as in equation (12). In Models 1c, 2c,3c and 4c, the risk premium parameter α is set to zero in the score and appearsonly in the predictive density. Models 2bc, 3bc and 4bc both include a splineand have the risk premium parameter set to zero in the score.

Tables 2 and 3 in Appendix 3 show the results. For all models reported in Table2, εt is Student-t distributed. For all models in Table 3, εt is skew generalized tdistributed.

Model 1 in Table 2 shows that leverage is necessary to find a positive ARCH-Meffect. When leverage is not explicitly included in the model, α takes the role ofleverage and is estimated to be negative. This can be explained by the appearanceof α in the score, where it acts like a leverage term if α is negative. To separatethe effect of leverage from the effect of the risk premium, we may set α to zero inthe score (Model 1c). Then the estimate of α is positive even when the leverageterm as such is excluded from the model. Leverage is crucial for a good fit, andModel 1c outperforms the GARCH model when it is included.

CAPM suggests that excess returns are possible only at the cost of extra risk.This suggests that µ must be set to 0, thus eliminating a ‘free lunch’. We explicitlytest for this by incorporating both µ and α in Model 1. Model 1 with α (and no µ)and Model 1 with µ (and no α) perform similarly in terms of fit. This suggests thatthe ARCH-M effect is fairly constant or moving slowly over time, thus motivatinga two-component model.

18

Model 2 is a marked improvement in log-likelihood over Model 1; althoughthe EGARCH-M effect is still associated with the overall volatility, the volatilityitself is modelled using two components. However, the estimate of α remainsinsignificant unless µ is set to 0.

In Models 2 and 3 we find κ∗2 > κ2, implying that short-run volatility decreasesafter a good day (see also Figure 3 in Appendix 2). Herein lies the added value ofour EGARCH formulation, which allows us to expressly identify the calming effectof a good day13. In contrast, the very setup of the GARCH model precludes sucha finding; its dynamic equation uses the indicator rather than the sign. At most,a GARCH model can find that volatility does not go up because the coefficientof the indicator, κ2, is restricted to be non-negative to prevent variance becomingnegative. If the constraint on κ2 is not imposed, it is estimated to be negative,mimicking the calming effect we find with our EGARCH model.

Model 3 clearly outperforms the two-component GARCH model in terms of fit,but the estimates for α are insignificant unless µ is dropped.

In Table 3, the estimates for α1 are positive and significant, irrespective ofwhich model is used: Models 3 and 4 both find a slowly varying risk premium.Specifically, Model 3 rejects a constant risk premium (µ is insignificant), whileModel 4 rejects a quickly varying risk premium (α2 is insignificant). In fact, thebest BIC is obtained by setting both µ and α2 to zero, as in Model 3c (withoutµ).

To corroborate this finding, we also consider a model that simultaneously in-corporates all three parameters µ, α1 and α2. All three are now significant: specifi-cally, short-term volatility drives prices down (i.e. α2 < 0) and long-term volatilitydrives prices up (α1 > 0). The former can be tentatively ascribed to the ‘newseffect’ as described in Chou (1988) and Schwert (1989), where unanticipated (i.e.short-term) volatility drives prices down as risk-averse investors flee. The lattercomes into play once the higher volatility has been incorporated into the (lower)price, after which the now anticipated (i.e. long-term) volatility is expected todrive prices up. Investors willing to bear the risk can then reap the reward of arisk premium. This positive correlation between risk and return may partly hingeon the preceding negative one, in that prices must come down before they areexpected to go up again. Given the subtle interplay between the three parame-ters, Model 4 should be interpreted with caution. Still, it makes intuitive sensein that both the news effect and the risk premium point in the expected directionand, moreover, Model 4 with all three parameters has the best AIC of all models.Both the news effect and the risk premium have been discussed at length in theliterature; to date, however, it has not yet been found that both could be true,

13Of course, this advantage of EGARCH applies only to the short-run component; the long-runcomponent of volatility is affected by big swings in either direction.

19

separated only in time.A comparison of Table 3 (with the skew generalized t distribution) and Table

2 (without it) suggests that the skew generalized t distribution is crucial to estab-lishing the existence of a risk premium. In Table 3, the parameter estimates for γand υ always differ significantly from 1 and 2, respectively. The need for negativeskewness in models for stock market returns is well known; the need for a gener-alized Student-t is less so. The latter is especially valuable in combination withskewness, however, because skewness pushes probability into the tail. In doing soit lowers the central peak, but the flexibility of the generalized Student-t is ableto counteract this undesirable tendency.

The introduction of negative skew implies a negative mean for the residualsεt. Without µ or α, therefore, the stock market would go down on average. Thenegative mean of εt allows both µ and α to take higher absolute values. Thismakes it easier to distinguish between them, and α1 is always favoured over µ inthis setting.

What do our findings imply about annual excess returns? In Model 3c withoutµ and a generalized t distribution, the expected daily excess return is

E

[εt exp(λt|t−1) + α1 exp(ω + λ1,t|t−1)

]= 0.0104%.

The first term contributes to the expectation, since εt is negatively skewed and hasnegative expectation. As daily returns are continuously compounded, the expectedannual excess return is

100

[exp

(250 ∗ 0.0104

100

)− 1

]= 2.63%

where we have assumed 250 business days. Historically, excess returns on the S&Phave been ∼ 2.8%14.

Skewness resolves difficulties noted in Guo and Neely (2008).??This section has shown that the risk premium is positive, significant and slowly

varying when estimated with a two-component DCS EGARCH model, (26) and(10), in which leverage enters differently into long and short run components. Wealso find that skewness must be complemented by the generalized t distribution forthe best results, and that with the EGARCH model we can, and do, find a calmingeffect of positive returns on short-run volatility. Finally there is evidence that high

14One dollar, invested in the S&P500 on 1 Jan 1954, would have grown to $74.5 over thecourse of the next 60 years, representing an annual return of ∼ 7.5%. The same dollar investedin 3-month U.S. Treasury bills would have accumulated only to a fifth of that, i.e. to $15.4,equivalent to an annual return of ∼ 4.7%. The average annual excess return over the period1954-2013 has thus been ∼ 2.8%

20

short-term volatility may cause returns to go down. These results demonstratethe versatility and viability of our EGARCH-M formulation. However, they areobtained from a very long sample for just one stock index, albeit a very importantone. From the practitioners point of view the next question to ask is whether thesubstantive findings carry over to shorter time periods and other indices.

8 Conclusions and extensions

The fact that the asymptotic theory can be established for a basic version ofthe DCS EGARCH-M model provides re-assurance on the viability of the formadopted, that is modeling the logarithm of the scale by a linear function of theconditional scores of past observations and letting the risk premium effect dependon the scale (or standard deviation). More general models constructed along theselines are likely to be equally effective.

The simulation evidence shows that the asymptotic theory gives a good ap-proximation in moderate size samples and the applications indicate that our DCSmodel outperforms other specifications in terms of goodness of fit.

Leverage effects play an important role in volatility models and the use of twocomponents, coupled with the EGARCH formulation, reveals some interesting, andperhaps surprising, aspects of leverage: positive returns seem to actively reduceshort-term volatility. This suggests that short-term volatility can be thought ofas ‘bad sentiment’, which drives prices down, increases after negative shocks andis tempered by positive ones. Long-term volatility, on the other hand, increaseswith large swings in either direction. In other words, negative shocks have anupward effect on both short- and long-term volatility, whereas positive shocksreduce short-term but not long-term volatility.

Our results allow us to reject both a constant and a rapidly varying risk pre-mium; instead, we have demonstrated that the risk premium is associated withthe slowly varying component of volatility. The inclusion of skewness seems tobe key to this result, and is ideally complemented with the generalized Student-tdistribution.

21

9 Appendix 1: Asymptotics

When all other parameters are known, the information matrix for ψ = (φ, κ, ω)′

for a single observation in a DCS model is given by I(ψ) = I.D(ψ), where I is theinformation quantity for a single observation and

D(ψ) = D

κφω

=1

1− b

A D ED B FE F C

, b < 1, (31)

with

A = σ2u, B =

κ2σ2u(1 + aφ)

(1− φ2)(1− aφ), C =

(1− φ)2(1 + a)

1− a,

D =aκσ2

u

1− aφ, E = c(1− φ)/(1− a) and F =

acκ(1− φ)

(1− a)(1− aφ).

The extension to situations in which there are fixed parameters in addition tothose in the dynamic equation for λt|t−1 is discussed in Harvey (2013, p 47).

22

11 Appendix 3: Parameter estimates

1954-2013 Distr. Vol.

15,104 obs for e k_2 lev_2 phi_2 k_1 lev_1 phi_1 om. mu a_1 a_2 nu upsil gam. LogL AIC BIC e e^2 u_1 u_2

GARCH t 0.023 0.093 0.921 0.008 0.022 0.009 7.55 -17866.2 2.3667 2.3702 137.0 15.4

0.004 0.007 0.003 0.001 0.016 0.022 0.42 0.00 0.75

Model 1 t 0.043 0.990 -0.388 0.111 -0.128 7.28 -17952.7 2.3780 2.3810 132.5 80.4 95.7

without leverage 0.002 0.001 0.049 0.015 0.025 0.41 0.000 0.000 0.000

Model 1c t 0.042 0.991 -0.378 0.015 0.041 7.38 -17964.3 2.3795 2.3826 133.1 87.9 102.6

without leverage 0.002 0.001 0.050 0.016 0.026 0.42 0.000 0.000 0.000

Model 1c t 0.036 0.023 0.989 -0.456 0.025 0.001 7.64 -17835.2 2.3626 2.3661 120.3 76.9 65.8

0.002 0.002 0.001 0.042 0.016 0.026 0.43 0.000 0.000 0.000

Model 1c t 0.036 0.023 0.989 -0.456 0.026 7.64 -17835.2 2.3624 2.3655 120.2 77.0 65.9

without alpha 0.002 0.001 0.001 0.042 0.005 0.43 0.000 0.000 0.000

Model 1c t 0.036 0.023 0.988 -0.447 0.040 7.69 -17836.4 2.3626 2.3656 123.4 76.6 66.2

without mu 0.002 0.002 0.001 0.040 0.009 0.43 0.000 0.000 0.000

Model 2c t 0.007 0.035 0.903 0.024 0.007 0.997 -0.498 0.040 -0.024 7.79 -17756.7 2.3526 2.3576 133.1 75.3 23.3

0.004 0.003 0.017 0.002 0.002 0.001 0.075 0.015 0.025 0.44 0.00 0.00 0.27

GARCH t 0.000 0.146 0.849 0.011 0.000 0.975 0.002 0.015 0.023 7.80 -17812.0 2.3599 2.3650 150.2 16.8

0.007 0.012 0.015 0.002 0.005 0.002 0.000 0.019 0.035 0.46 0.00 0.67

GARCH t -0.019 0.167 0.877 0.022 -0.022 0.974 0.002 0.010 0.033 7.79 17803.0

unconstrained 0.007 0.011 0.010 0.003 0.005 0.002 0.000 0.019 0.037 0.46

Model 3 t 0.007 0.034 0.903 0.023 0.006 0.997 -0.515 0.036 -0.016 7.81 -17757.0 2.3526 2.3577 135.4 75.1 23.3 23.4

0.004 0.003 0.016 0.002 0.002 0.001 0.080 0.016 0.027 0.44 0.00 0.00 0.27 0.27

Model 3b t 0.013 0.033 0.910 0.019 0.007 0.992 -0.463 0.034 -0.013 7.82 -17763.5 2.3535 2.3585 134.1 68.6 27.8 28.0

0.004 0.003 0.016 0.002 0.002 0.001 0.062 0.017 0.028 0.44 0.00 0.00 0.11 0.11

Model 3c t 0.007 0.034 0.904 0.023 0.006 0.996 -0.497 0.028 -0.002 7.81 -17757.2 2.3526 2.3577 135.4 74.9 23.4 23.4

0.004 0.003 0.016 0.002 0.002 0.001 0.073 0.016 0.027 0.44 0.00 0.00 0.27 0.27

Model 3bc t 0.013 0.033 0.911 0.019 0.007 0.992 -0.456 0.027 0.000 7.83 -17763.7 2.3535 2.3586 134.1 68.3 27.9 27.9

0.004 0.003 0.016 0.002 0.002 0.001 0.059 0.017 0.029 0.44 0.00 0.00 0.11 0.11

ARCH at 20 lagsShort-run Long-run ARCH-M Generalised-T Fit

Table 2: Parameter estimates for one and two component GARCH-M models, and corresponding DCS EGARCH-M models. All models haveεt distributed as Student-t.

26

1954-2013 Distr. Vol.

15,104 obs for e_t k_2 lev_2 phi_2 k_1 lev_1 phi_1 om. mu a_1 a_2 nu upsil gam. LogL AIC BIC e e^2 u_1 u_2

Model 3 Gen-t 0.010 0.034 0.917 0.023 0.007 0.997 -0.491 0.032 0.088 10.01 1.869 0.939 -17737.7 2.3503 2.3564 129.1 58.2 22.5 21.8

0.004 0.003 0.013 0.002 0.002 0.000 0.088 0.017 0.035 1.42 0.065 0.010 0.00 0.00 0.31 0.35

Model 3b Gen-t 0.017 0.032 0.925 0.018 0.007 0.993 -0.485 0.027 0.099 10.75 1.840 0.939 -17743.4 2.3511 2.3571 129.1 50.0 26.4 25.5

0.003 0.003 0.012 0.002 0.002 0.001 0.069 0.018 0.037 1.64 0.065 0.010 0.00 0.00 0.15 0.19

Model 3c Gen-t 0.011 0.033 0.915 0.023 0.006 0.997 -0.610 0.025 0.107 10.62 1.849 0.937 -17736.1 2.3501 2.3562 129.0 56.6 22.2 22.2

0.004 0.003 0.013 0.002 0.002 0.000 0.086 0.018 0.037 1.61 0.065 0.010 0.00 0.00 0.33 0.33

Model 3bc Gen-t 0.017 0.031 0.924 0.018 0.007 0.993 -0.548 0.019 0.118 10.85 1.840 0.936 -17741.9 2.3509 2.3569 128.4 50.8 26.3 26.3

0.003 0.003 0.012 0.002 0.002 0.001 0.068 0.019 0.038 1.66 0.065 0.010 0.00 0.00 0.16 0.16

Model 3 Gen-t 0.012 0.034 0.922 0.022 0.007 0.996 -0.420 0.145 10.57 1.848 0.939 -17739.2 2.3504 2.3560 129.2 55.3 22.7 21.2

without mu 0.003 0.003 0.012 0.002 0.002 0.000 0.073 0.021 1.59 0.065 0.010 0.00 0.00 0.30 0.39

Model 3b Gen-t 0.018 0.031 0.930 0.017 0.007 0.993 -0.456 0.148 11.00 1.832 0.938 -17744.5 2.3511 2.3566 129.3 48.2 26.4 24.8

without mu 0.003 0.003 0.011 0.001 0.002 0.001 0.065 0.021 1.71 0.065 0.010 0.00 0.00 0.15 0.21

Model 3c Gen-t 0.011 0.033 0.918 0.022 0.006 0.996 -0.590 0.150 10.72 1.847 0.936 -17737.0 2.3501 2.3557 128.9 56.1 22.5 22.5

without mu 0.003 0.003 0.013 0.002 0.002 0.000 0.078 0.021 1.62 0.065 0.010 0.00 0.00 0.32 0.32

Model 3bc Gen-t 0.017 0.031 0.927 0.018 0.006 0.993 -0.543 0.151 10.93 1.838 0.936 -17742.4 2.3508 2.3564 128.4 50.3 26.2 26.2

without mu 0.003 0.003 0.011 0.001 0.002 0.001 0.065 0.021 1.67 0.065 0.010 0.00 0.00 0.16 0.16

Model 4 Gen-t 0.011 0.034 0.920 0.022 0.007 0.997 -0.439 0.126 0.011 10.54 1.849 0.938 -17739.0 2.3505 2.3566 130.4 55.7 22.3 21.3

without mu 0.004 0.003 0.012 0.002 0.002 0.000 0.081 0.037 0.019 1.59 0.065 0.010 0.00 0.00 0.32 0.38

Model 4b Gen-t 0.018 0.032 0.929 0.018 0.007 0.993 -0.458 0.142 0.003 11.00 1.832 0.938 -17744.5 2.3512 2.3573 129.6 48.4 26.3 24.9

without mu 0.003 0.003 0.011 0.001 0.002 0.001 0.065 0.031 0.013 1.71 0.065 0.010 0.00 0.00 0.16 0.21

Model 4c Gen-t 0.011 0.033 0.918 0.022 0.006 0.997 -0.593 0.143 0.004 10.71 1.847 0.936 -17737.0 2.3502 2.3563 129.4 56.1 22.4 22.4

without mu 0.004 0.003 0.013 0.002 0.002 0.000 0.079 0.037 0.019 1.62 0.065 0.010 0.00 0.00 0.32 0.32

Model 4bc Gen-t 0.017 0.031 0.927 0.018 0.006 0.993 -0.543 0.154 -0.001 10.93 1.838 0.936 -17742.4 2.3510 2.3570 128.3 50.3 26.2 26.2

without mu 0.003 0.003 0.011 0.001 0.002 0.001 0.065 0.039 0.020 1.67 0.065 0.010 0.00 0.00 0.16 0.16

Model 4 Gen-t 0.006 0.032 0.899 0.026 0.011 0.997 -0.589 0.208 0.064 -0.155 10.70 1.841 0.947 -17733.3 2.3499 2.3564 121.0 61.3 25.0 22.6

0.004 0.003 0.023 0.003 0.003 0.000 0.103 0.090 0.031 0.074 1.65 0.066 0.010 0.00 0.00 0.20 0.31

Model 4b Gen-t 0.015 0.027 0.900 0.022 0.013 0.991 -0.532 0.211 0.098 -0.173 11.06 1.826 0.943 -17739.4 2.3507 2.3573 120.6 48.5 30.7 29.0

0.004 0.004 0.024 0.002 0.003 0.001 0.078 0.093 0.043 0.076 1.75 0.066 0.011 0.00 0.00 0.06 0.09

ARCH at 20 lagsShort-run Long-run ARCH-M Generalised-T Fit

Table 3: Parameter estimates for DCS EGARCH-M models with εt skew generalized t-distributed.

27

Table 1: Parameter estimates for DCS EGARCH models1954-2013 Short-run Long-run Vol. ARCH-M Generalised-T Fitn=15,104 κ2 κ∗2 φ2 κ1 κ∗1 φ1 ω µ α1 α2 ν υ γ LogL AIC BICModel 3 0.010 0.034 0.917 0.023 0.007 0.997 -0.491 0.032 0.088 10.01 1.869 0.939 -17737.7 2.3503 2.3564

0.004 0.003 0.013 0.002 0.002 0.000 0.088 0.017 0.035 1.423 0.065 0.01

Model 3b 0.017 0.032 0.925 0.018 0.007 0.993 -0.485 0.027 0.099 10.75 1.840 0.939 -17743.4 2.3511 2.35710.003 0.003 0.012 0.002 0.002 0.001 0.069 0.018 0.037 1.635 0.065 0.01

Model 3c 0.011 0.033 0.915 0.023 0.006 0.997 -0.610 0.025 0.107 10.62 1.849 0.937 -17736.1 2.3501 2.35620.004 0.003 0.013 0.002 0.002 0.000 0.086 0.018 0.037 1.606 0.065 0.01

Model 3bc 0.017 0.031 0.924 0.018 0.007 0.993 -0.548 0.019 0.118 10.85 1.840 0.936 -17741.9 2.3509 2.35690.003 0.003 0.012 0.002 0.002 0.001 0.068 0.019 0.038 1.658 0.065 0.01

Model 3 0.012 0.034 0.922 0.022 0.007 0.996 -0.420 0.145 10.57 1.848 0.939 -17739.2 2.3504 2.35600.003 0.003 0.012 0.002 0.002 0.000 0.073 0.021 1.591 0.065 0.01

Model 3b 0.018 0.031 0.930 0.017 0.007 0.993 -0.456 0.148 11.00 1.832 0.938 -17744.5 2.3511 2.35660.003 0.003 0.011 0.001 0.002 0.001 0.065 0.021 1.71 0.065 0.01

Model 3c 0.011 0.033 0.918 0.022 0.006 0.996 -0.590 0.150 10.72 1.847 0.936 -17737.0 2.3501 2.35570.003 0.003 0.013 0.002 0.002 0.000 0.078 0.021 1.622 0.065 0.01

Model 3bc 0.017 0.031 0.927 0.018 0.006 0.993 -0.543 0.151 10.93 1.838 0.936 -17742.4 2.3508 2.35640.003 0.003 0.011 0.001 0.002 0.001 0.065 0.021 1.673 0.065 0.01

Model 4 0.011 0.034 0.920 0.022 0.007 0.997 -0.439 0.126 0.011 10.54 1.849 0.938 -17739.0 2.3505 2.35660.004 0.003 0.012 0.002 0.002 0.000 0.081 0.037 0.019 1.586 0.065 0.01

Model 4b 0.018 0.032 0.929 0.018 0.007 0.993 -0.458 0.142 0.003 11.00 1.832 0.938 -17744.5 2.3512 2.35730.003 0.003 0.011 0.001 0.002 0.001 0.065 0.031 0.013 1.713 0.065 0.01

Model 4c 0.011 0.033 0.918 0.022 0.006 0.997 -0.593 0.143 0.004 10.71 1.847 0.936 -17737.0 2.3502 2.35630.004 0.003 0.013 0.002 0.002 0.000 0.079 0.037 0.019 1.622 0.065 0.01

Model 4bc 0.017 0.031 0.927 0.018 0.006 0.993 -0.543 0.154 -0.001 10.93 1.838 0.936 -17742.4 2.3510 2.35700.003 0.003 0.011 0.001 0.002 0.001 0.065 0.039 0.02 1.674 0.065 0.01

Model 4 0.006 0.032 0.899 0.026 0.011 0.997 -0.589 0.208 0.064 -0.155 10.70 1.841 0.947 -17733.3 2.3499 2.35640.004 0.003 0.023 0.003 0.003 0.000 0.103 0.09 0.031 0.074 1.651 0.066 0.01

Model 4b 0.015 0.027 0.900 0.022 0.013 0.991 -0.532 0.211 0.098 -0.173 11.06 1.826 0.943 -17739.4 2.3507 2.35730.004 0.004 0.024 0.002 0.003 0.001 0.078 0.093 0.043 0.076 1.754 0.066 0.011

28

Table 2: Estimates for Model 1A

Volatility Mean Shape Fit ARCH at 100 lagsκ κ∗ ϕ ω ρ µ ν υ γ LogL AIC BIC εt uµ,t ε2t uλ,t

Full sample 0.038 0.027 0.988 -0.392 0.088 0.012 10.38 1.851 0.948 -17757.9 2.3526 2.3572 108.4 145.3 200.1 117.2[1954, 2014) 0.002 0.002 0.001 0.054 0.008 0.006 1.54 0.065 0.011 0.27 0.00 0.00 0.12

Subsample I 0.050 0.039 0.969 -0.626 0.217 0.011 6.34 2.083 0.948 -4499.3 1.7993 1.8111 181.9 186.4 231.7 123.4[1954, 1974) 0.005 0.004 0.005 0.061 0.014 0.008 1.49 0.154 0.019 0.00 0.00 0.00 0.06

Subsample II 0.026 0.011 0.991 -0.301 0.091 -0.004 8.87 1.930 0.996 -6253.9 2.4769 2.4885 86.0 104.9 106.4 149.4[1974, 1994) 0.003 0.002 0.002 0.078 0.014 0.011 2.32 0.131 0.020 0.84 0.35 0.31 0.00

Subsample III 0.036 0.040 0.985 -0.273 -0.038 0.025 23.81 1.613 0.893 -6857.2 2.7268 2.7385 137.0 136.9 99.3 126.9[1994, 2014) 0.003 0.003 0.002 0.068 0.007 0.005 9.02 0.079 0.017 0.01 0.01 0.50 0.04

Table 3: Estimates for Model 1B

Volatility Mean Shape Fit ARCH at 100 lagsκ κ∗ ϕ ω ρ κµ ϕµ ωµ ν υ γ LogL AIC BIC εt uµ,t ε2t uλ,t

Full sample 0.036 0.029 0.989 -0.380 0.081 0.006 0.984 0.007 9.82 1.866 0.947 -17746.9 2.3514 2.3570 115.6 143.6 197.4 110.8[1954, 2014) 0.002 0.002 0.001 0.060 0.008 0.001 0.001 0.008 1.58 0.073 0.011 0.14 0.00 0.00 0.22

Subsample I 0.045 0.040 0.976 -0.610 0.204 0.007 0.964 0.005 6.07 2.100 0.945 -4495.4 1.7986 1.8129 185.3 183.4 227.7 119.5[1954, 1974) 0.005 0.004 0.003 0.069 0.015 0.004 0.014 0.010 1.58 0.175 0.020 0.00 0.00 0.00 0.09

Subsample II 0.026 0.010 0.991 -0.313 0.119 -0.026 0.787 -0.002 9.11 1.905 0.995 -6250.3 2.4763 2.4905 83.7 97.3 112.7 150.1[1974, 1994) 0.003 0.002 0.001 0.080 0.020 0.013 0.085 0.010 2.72 0.144 0.020 0.88 0.56 0.18 0.00

Subsample III 0.033 0.041 0.987 -0.276 -0.043 0.004 0.994 0.037 21.67 1.628 0.890 -6851.8 2.7255 2.7397 142.8 141.6 96.1 117.1[1994, 2014) 0.003 0.003 0.001 0.089 0.022 0.002 0.001 0.015 10.43 0.114 0.017 0.00 0.00 0.59 0.12

29

Table 4: Estimates for Model 1C

Volatility Mean Shape Fit ARCH at 100 lagsκ κ∗ ϕ ω ρ µ α120 ν υ γ LogL AIC BIC εt uµ,t ε2t uλ,t

Full sample 0.038 0.028 0.987 -0.398 0.088 -0.026 0.060 10.34 1.855 0.946 -17752.4 2.3520 2.3571 108.4 145.7 204.6 116.6[1954, 2014) 0.002 0.002 0.001 0.050 0.008 0.013 0.018 1.46 0.063 0.011 0.27 0.00 0.00 0.12

Subsample I 0.049 0.040 0.969 -0.624 0.216 -0.035 0.086 6.44 2.073 0.944 -4496.1 1.7985 1.8115 182.1 185.4 228.1 117.9[1954, 1974) 0.005 0.004 0.005 0.060 0.014 0.020 0.035 1.42 0.142 0.019 0.00 0.00 0.00 0.11

Subsample II 0.026 0.011 0.991 -0.306 0.091 -0.028 0.034 8.91 1.928 0.996 -6253.7 2.4772 2.4901 86.0 104.9 106.6 150.1[1974, 1994) 0.003 0.002 0.002 0.077 0.014 0.038 0.052 2.23 0.125 0.020 0.84 0.35 0.31 0.00

Subsample III 0.035 0.040 0.984 -0.296 -0.038 0.000 0.039 23.80 1.614 0.895 -6856.0 2.7268 2.7397 137.4 137.4 100.0 125.4[1994, 2014) 0.004 0.003 0.002 0.091 0.005 0.015 0.013 12.57 0.109 0.015 0.01 0.01 0.48 0.04

Table 5: Estimates for Model 1C with µ dropped

Volatility Mean Shape Fit ARCH at 100 lagsκ κ∗ ϕ ω ρ α120 ν υ γ LogL AIC BIC εt uµ,t ε2t uλ,t

Full sample 0.038 0.027 0.988 -0.420 0.088 0.027 10.34 1.853 0.950 -17754.5 2.3522 2.3567 108.4 117.1 202.9 116.6[1954, 2014) 0.002 0.002 0.001 0.051 0.008 0.008 1.463 0.063 0.010 0.267 0.117 0.00 0.123

Subsample I 0.050 0.039 0.969 -0.645 0.215 0.031 6.36 2.080 0.950 -4497.6 1.7987 1.8104 181.6 100.0 230.4 121.2[1954, 1974) 0.005 0.004 0.005 0.059 0.014 0.013 1.398 0.144 0.019 0.00 0.482 0.00 0.073

Subsample II 0.026 0.010 0.991 -0.306 0.091 -0.002 8.86 1.931 0.997 -6253.9 2.4769 2.4886 86.0 125.3 106.6 149.6[1974, 1994) 0.003 0.002 0.002 0.078 0.014 0.015 2.215 0.125 0.020 0.839 0.044 0.308 0.001

Subsample III 0.035 0.040 0.984 -0.294 -0.038 0.039 23.83 1.615 0.894 -6856.0 2.7264 2.7380 137.5 46.8 100.1 125.4[1994, 2014) 0.003 0.003 0.002 0.081 0.014 0.014 10.356 0.099 0.014 0.008 1.00 0.479 0.044

30

Table 6: Estimates for Model 2A

Volatility Mean Shape Fit ARCH at 100 lagsκ2 κ∗2 ϕ2 κ1 κ∗1 ϕ1 ω ρ µ ν υ γ LogL AIC BIC εt uµ,t ε2t uλ,t

Full sample 0.013 0.037 0.909 0.022 0.007 0.997 -0.417 0.090 0.012 9.53 1.908 0.948 -17681.8 2.3429 2.3490 112.9 140.0 214.5 119.2[1954, 2014) 0.004 0.003 0.014 0.002 0.002 0.001 0.079 0.008 0.006 1.33 0.066 0.011 0.18 0.01 0.00 0.09

Subsample I 0.032 0.047 0.891 0.017 0.009 0.998 -0.565 0.217 0.011 5.61 2.200 0.940 -4468.9 1.7884 1.8040 184.5 184.2 226.3 100.3[1954, 1974) 0.007 0.005 0.024 0.003 0.003 0.001 0.152 0.014 0.007 1.13 0.151 0.019 0.00 0.00 0.00 0.47

Subsample II -0.009 0.020 0.825 0.028 0.005 0.992 -0.298 0.088 -0.004 8.52 1.955 0.999 -6245.1 2.4746 2.4901 85.8 104.3 120.1 148.0[1974, 1994) 0.009 0.005 0.095 0.004 0.003 0.002 0.079 0.014 0.011 2.33 0.138 0.020 0.84 0.37 0.08 0.00

Subsample III -0.049 0.049 0.578 0.040 0.032 0.986 -0.261 -0.032 0.025 15.81 1.704 0.895 -6813.5 2.7107 2.7262 138.4 138.3 95.3 83.6[1994, 2014) 0.011 0.007 0.089 0.004 0.004 0.003 0.100 0.013 0.014 5.39 0.102 0.017 0.01 0.01 0.62 0.88

31

12 Appendix 4: Graphs

This section contains some graphs for different models.

1954 1964 1974 1984 1994 2004 20140

20406080

Value of 1 dollar invested 1 Jan 1954

1954 1964 1974 1984 1994 2004 2014

123456

Ratio S&P500 over T-bills

exp(∑t

i=1 yi)

S&P5003 month US T-bills

Figure 1: The bottom graphs shows the value of one dollar invested in the S&P500 versus the value of a dollar invested in 3-month Treasurybills. By continuously compounding the excess returns yt, the top graph is obtained. Its value is equal to the ratio of the S&P500 over theT-bills from the bottom graph.

32

−15 −10 −5 0 5 10 15

−0.5

−0.25

0

0.25

0.5

Realisation of ǫt

Impact

Estimated impact curves κiut + κ∗

i sgn(−ǫt)(ut + 1)

Impact on λt+1|t

Figure 2: Model 1A, full sample. Estimated news impact curve.

33

−15 −10 −5 0 5 10 15

−0.5

−0.25

0

0.25

0.5

Realisation of ǫt

Impact



Impact on λt+1|t

Figure 3: Model 1A, Subsample I. Estimated news impact curve, showing a high degree of asymmetry.

34

−15 −10 −5 0 5 10 15

−0.5

−0.25

0

0.25

0.5

Realisation of ǫt

Impact



Impact on λt+1|t

Figure 4: Model 1A, Subsample II. The estimated news impact curve is quite symmetric.

35

−15 −10 −5 0 5 10 15

−0.5

−0.25

0

0.25

0.5

Realisation of ǫt

Impact



Impact on λt+1|t

Figure 5: Model 1A, Subsample III. Estimated news impact curve, showing that positive returns have a calming effect on the market.

36

−15 −10 −5 0 5 10 15−0.02

−0.01

0

0.01

0.02

Realisation of ǫt

Impact

Estimated location impact curves

Impact on µt+1|t

Figure 6: Model 1B, full sample. The filtered location increases when estimated εt is positive, and decreases when it is negative. It does notdo so linearly, however. This graph shows that unexpected excess returns of roughly 3% or more affect the location less than linearly. In effect,very large returns attributed to the heavy tail of εt and are thus down-weighted as far as the updating of the location is concerned.

37

1954 1964 1974 1984 1994 2004

−60

−40

−20

0

20

40

60

Time

Return

S&P500 return vs. long-term location

Annual excess S&P500 returnµ1,t|t−1

Figure 7: Model 1B, full sample. Estimated location µt|t−1, multiplied by 250, tracks roughly the historic 250-day return.

38

1954 1964 1974 1984 1994 2004 2014−20

−15

−10

−5

0

5

10

15

20

Time

Return

S&P500 return vs. long-term location

Daily excess S&P500 returnµ1,t|t−1

Figure 8: Model 1B, full sample. Estimated location µt|t−1 and daily excess returns. Clearly, the filtered location is very low in comparisonwith the noise. Typically, the filtered location is two orders of magnitude lower than the filtered scale.

39

1 2 3 4 5−0.2

−0.1

0

0.1

0.2

y = (0.044)+(−0.064)*x+(0.014)*x2

R2 = 20%

exp(λt|t−1)

µt|t−1

Contemporaneous relation between location and scale

Figure 8: Model 1B, full sample. Contemporaneous relation between filtered location and filtered scale. Due to the leverage effect, the filteredscale is likely to be high after a series of negative returns, i.e. when the location is low. This leads to a negative contemporaneous correlationbetween location and scale.

40

1 2 3 4 5

−0.2

−0.1

0

0.1

0.2

y = (−0.034)+(0.047)*x+(−0.008)*x2

R2 = 11%

Daily standard deviation (%) at time t

µ(t+101)−µ(t)

Shape of the risk-return relationship

Figure 8: Model 1B, full sample. To investigate the intertemporal relationship between location and scale, we plot the difference in thelocation over a 100-day horizon. Clearly, the relationship is now positive and R2 = 11% is reasonably large.

41

0 1 2 3 4 5 6

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Daily standard deviation at time t in %

Changein

locationµt+k−µt

Estimated risk-return relation over different horizons

10 days20 days40 days80 days120 days180 days260 days

Figure 8: Model 1B, full sample. Estimated risk-return relationship for different horizons. For 10-40 days, the risk-return relationship isrelatively flat (and explanatory power is low, see next graph). For 80-180 days, the relationship is fairly strong and explanatory power isrelatively high (see next graph).

42

100 200 300

2

4

6

8

10

12

14

Days since volatility event

R2ofrisk-return

relation(%

)

Risk premium needs about 120 days to materialize fully

Figure 8: Model 1B, full sample. Explanatory power for the risk-return relationship over different horizons. The R2 increases over time,showing that the risk premium needs about 120 days to materialize, on average.

43

0 50 100 150 200 250 300−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Lag k

Estim

ate

αk

Lagged EGARCH-M effect yt = µ+ αk exp(λt−k|t−k−1 ) + (εt − µǫ) exp(λt|t−1)

αkαk + 2 std. err.αk − 2 std. err.

Figure 8: Model 1C, full sample. Estimates of αk as a function of k. Clearly, the risk premium is significantly positive for k roughly between60 and 160.

44

−300 −200 −100 0 100 200 300

0

k

Correlation

Correlation between λ1,t|t−1 and ǫt+k

Figure 8: Model 2A, full sample. The cross correlation between exp(λ1,t|t−1) and εt+k shows that high long-term volatility is predictive ofpositive residuals up to 20 days.

45

−15 −10 −5 0 5 10 15−0.4

−0.2

0

0.2

0.4

Realisation of ǫt

Impact



Impact on λ1,t+1|t

Impact on λ2,t+1|t

Figure 3: This graph shows the impact of εt on both components of volatility. The dots correspond to actual realizations of εt. It is seen thatpositive returns reduce short-term volatility.

46

1954 1964 1974 1984 1994 2004 2014

1

2

3

4

5Level

/scale

S&P500 and estimated scale

exp(∑t

i=1 yi)exp(ω + λ1,t|t−1 + λ2,t|t−1)

Figure 4: This graph shows the filtered estimate of the scale. The scale multiplies εt, which is distributed as a skew generalized Student-t. Forparameter estimates in Model 3c, a scale of around 4 implies a 95% confidence interval of around [−9%, 8%], which is a huge range for dailyreturns.

47

1954 1964 1974 1984 1994 2004 2014

1

2

3

4

5Level

/scale

S&P500 with short-run and long-run volatility

exp(∑t

i=1 yi)exp(λ2,t|t−1)

exp(ω + λ1,t|t−1)

Figure 5: The estimated scale is separated as a product of two terms: exp(ω + λ1,t|t−1) and exp(λ2,t|t−1). The first is highly persistent, whilesecond mean-reverts relatively quickly to its equilibrium level of 1.

48

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24200

250

300

350

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24−20

0

20

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 240

2

4

S&P500 around Black Monday (19 Oct 1987)

Estimated scale

Excess return

Adjusted close

Figure 6: Behavior of the stock market and estimated scale around Black Monday (19 Oct 1987).

49

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24−20

0

20

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24

1

2

3

S&P500 around Black Monday (19 Oct 1987)

exp(λ2,t|t−1)exp(ω + λ1,t|t−1)

Excess return

Figure 7: Behavior of estimated long-run and short-run volatility around Black Monday (19 Oct 1987).

50

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24600

1000

1400

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24−10

0

10

1 Sep 9 16 23 30 7 Oct 14 21 28 4 Nov 11 18 25 3 Dec 10 17 24

1

2

3

S&P500 at Lehman collapse (15 Sep 2008)

exp(λ2,t|t−1)


Excess return

Adjusted close

Figure 8: Behavior of long-run and short-run volatility around the collapse of Lehman Brothers (15 Sep 2008). Due to a combination of largenegative and large positive shocks, most uncertainty is incorporated into the long-run component of volatility.

51

1 June 8 15 22 29 7 July 14 21 28 4 Aug 11 18 25 1 Sep 9 16 23440

460

480

1 June 8 15 22 29 7 July 14 21 28 4 Aug 11 18 25 1 Sep 9 16 23−2−1

012

1 June 8 15 22 29 7 July 14 21 28 4 Aug 11 18 25 1 Sep 9 16 23 300

1

2

S&P500 in Booming Nineties (Summer 1994)

exp(λ2,t|t−1)


Excess return

Adjusted close

Figure 9: During the Booming Nineties, the longest period on record of continuous economic expansion in the US, long-term volatility is lowand flat.

52

1955 1965 1975 1985 1995 2005 2014−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

50S&P500 rolling-window excess return

Realized excess return (%), p.a.

Expected return (%), based on risk premium

Figure 10: Rolling annual realized return, and predicted return based on risk premium.

53

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

Residuals ǫt

Probability

Analysis of residuals ǫt

Residuals ǫt of Model 3cEstimated Gen-skew-t of Model 3cStudent-t with ν = 7.8

Figure 11:Implied residuals and estimated distribution of the εt in Model 3c. This distribution is compared with a (non-skew, non-generalized)Student-t with ν = 7.8, as typically implied by other models.

54

13 Acknowledgements

Rutger-Jan Lange is a Post-Doctoral Research Associate on the project DynamicModels for Volatility and Heavy Tails. We are grateful to the Keynes Fund forfinancial support.

Earlier version sof this paper were presented at the conference Recent Develop-ments in Financial Econometrics and Empirical Finance at the University of Essexin June 2014 and at the Cambridge Finance-Tinbergen Institute meeting in Am-sterdam in May 2014. We are grateful to Peter Boswijk, Andre Lucas, Siem-JanKoopman, Mark Salmon and Robert Taylor for helpful comments

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