Tail Behavior and Dependence Structure in the APARCH Model

Farrukh Javed; Krzysztof Podgórski

doi:10.1515/jtse-2016-0002

Article

Tail Behavior and Dependence Structure in the APARCH Model

Published/Copyright: December 3, 2016

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From the journal Journal of Time Series Econometrics Volume 9 Issue 2

Abstract

The APARCH model attempts to capture asymmetric responses of volatility to positive and negative ‘news shocks’ – the phenomenon known as the leverage effect. Despite its potential, the model’s properties have not yet been fully investigated. While the capacity to account for the leverage is clear from the defining structure, little is known how the effect is quantified in terms of the model’s parameters. The same applies to the quantification of heavy-tailedness and dependence. To fill this void, we study the model in further detail. We study conditions of its existence in different metrics and obtain explicit characteristics: skewness, kurtosis, correlations and leverage. Utilizing these results, we analyze the roles of the parameters and discuss statistical inference. We also propose an extension of the model. Through theoretical results we demonstrate that the model can produce heavy-tailed data. We illustrate these properties using S&P500 data and country indices for dominant European economies.

Keywords: time series models; heavy tails; leverage effect; estimation

MSC 2010: C13; C32; C58

Funding statement: The authors were supported by the Riksbankens Jubileumsfond Grant Dnr: P13-1024:1 and the Swedish Research Council Grant Dnr: 2013–5180.

Appendix

Recurrent Formula for Moments of λt

To evaluate moments of the model, we use the integer moments of λt defined by

[35]λt=α(1−θ)δet+δ+(1+θ)δet−δ+β.

Our formula involves e(p)=Eet+p=Eet−p given in the following

[36]e(p)=2(p−1)/22π∫0∞x(p−1)/2e−xdx=2p/2−1πΓp+12.

We formulate this as a lemma and skip the proof which is straightforward.

Lemma 1:

The integer valued moments mk=E(λtk) of λt can obtained from the following recurrent relation

mk=∑r=0kkrαr(1−θ)rδ+(1+θ)rδe(rδ)βk−r.

Corollary 1:

Letm1andm2be the first and second moment ofλt. Then

m1=Γδ+12α(1−θ)δ+(1+θ)δ2δ/2−1π+β,

m2=2δ−1α2π(1−θ)2δ+(1+θ)2δΓδ+12+

+2δ/2αβπ(1−θ)δ+(1+θ)δΓδ+12+β2.

Moreover, varianceσ2ofλtis given as

σ2=2δ−1α2πΓ2δ+12πΓδ+12Γ2δ+12−12(1−θ)2δ+(1+θ)2δ−1−θ2δ

Stationarity Condition for the Volatility Model

We provide with the conditions for the existence of a stationary solution to the following series, which also satisfies the recurrence relation eq. [3]:

[37]ρtδ=α0λ+∑k=1∞λt−1⋯λt−k.

We consider convergence of the above series in the mean-square sense and, more generally, in Lq-sense, where q>0 and Lq is the space of random variables with the finite q moment. Since the Lq-spaces are Banach spaces, the absolute convergence of the series implies its convergence in the Lq-norm. We note that by proving the Lq convergence for ρtδ we show that ρt belongs to Lp space with p=qδ. The absolute convergence means that

∑k=1∞E(λt−1⋯λt−k)q1/q<∞

and since E(λt−1⋯λt−k)q=mqk, where mq=Eλtq, the condition for convergence reduces to mq<1. By Hölder’s inequality, we note that if 0<p<q, then mp<1 is less restrictive than mq<1.

Note that by the triangle inequality for the q-norm

E(λtq)1q≤α(1−θ)δ+(1+θ)δe1q(p)+2β.

From this observation we obtain immediately the following result.

Proposition 2:

A sufficient condition for existence of the strictly stationary solution ρt in eq. [1] such that it belongs to Lp is given by the inequality

[38]α(1−θ)δ+(1+θ)δe1q(p)+2β<1,

where q=p/δ and α, β, δ and θ are parameters in the model given in eq. [1], while m(p) is given in eq. [36]. We note that for a symmetric case (θ=0) we obtain simply

[39]αe1q(p)+β>12.

For the important special case of q=1, we have the explicit value for E(λt)q given in Corollary 1. This allows to obtain the sufficient and necessary condition for the absolute convergence of the series defining ρtδ in the Lp-norm. Namely, we have the followng result.

Proposition 3:

A sufficient and necessary condition for the existence of a strictly stationary solution ρt to eq. [1] that belongs to Lδ (in the absolute convergence of the series) is given by the inequality for the parameters α>0, β∈[0,1], and δ>0:

[40](1−θ)δ+(1+θ)δ<π1−βα 21−δ/2Γδ+12.

We also note two important special cases. Firstly, δ=1 yields the condition that does not depend on θ:

[41]1<π21−βα.

Secondly, the case ofδ=2yields

[42]1+θ2<1−βα.

Similarly, we can obtain a stronger sufficient and necessary condition for the case q=2, for which we have also exact value for the moment in Corollary 1.

Proposition 4:

A sufficient and necessary condition for the existence of a strictly stationary solution ρt to eq. [1] that belongs to L2δ (in the absolute convergence of the series) is given by the inequality

2δ/2−1Γδ+12α(1−θ)2δ+(1+θ)2δ+Γδ+12β(1−θ)δ+(1+θ)δ<

[43]<π2δ1−β2α.

We note two important special cases. Firstly, δ=1 yields the condition

[44]α1+θ2+23/2πβ<1−β2α

Secondly, the case of δ=2 yields

[45]3α(1+6θ2+θ4)+2β(1+θ2)<1−β2α.

In our estimation strategies, we have emphasized the importance to limit the range of parameters so appropriate moments (the kurtosis, for example) exist. It is clear that the n the moment of yt exists whenever the n the moment of ρt is finite. The fourth moment is needed to capture tails through kurtosis. For δ=1, we can use Proposition 2 with p=4 that leads to

[46]2α324+2β<1.

However, in order to have flexibility in modeling kurtosis and thus tails, we provide a stronger result for this case that is based on exact formula for the fourth moment given in eq. [18] to obtain a lengthy but elementary sufficient condition.

Proposition 5:

For the caseδ=1, volatilityρtexists in theL4sense if

β4+4αβ32π+6α2β2(1+θ2)+8α3β(1+3θ2)2π+3α4(1+6θ2+θ4)≤1.

On the other hand, for δ=2 one can use the sufficient condition for finite kurtosis given in eq. [45] listed in Proposition 4.

For the leverage effect, restrictions on the parameters depend on what measure of the leverage is considered. This was discussed in Subsection 4.2. There for δ=1, two convenient characteristics were explicitly evaluated: r(ρt,εt−1) and r(ρt2,εt−1). They require only the second and third moment of ρt, respectively. Thus the condition following from the existence of kurtosis implies existence of both leverage characteristics. For δ=2, the convenient characteristics are r(ρt2,rt−12) and r(ρt4,rt−12). For the first one the existence of the kurtosis suffices, while for the second the sixth moment is needed. For the latter one can write the parameter restriction using Proposition 2.

Moments ofρt

Here we collect some results on the moments of ρt that are used throughout the paper. First, by the power of a sum algebraic formula we have a relation between these moments and moments of the series L=∑k=1∞λ−1…λt.

Proposition 6:

We have

E(ρtkδ)=α0k∑r=0kkrλrMk−r,

where Mk is the k-th moment of the series L.

For computing moments explicitely in terms of the actual parameters of the model one can utilize the following lemma together with Lemma 1.

Lemma 2:

Letmkbe thek-th moment ofλt. Then

M1=p1,

M2=p2(1+2p1),

M3=p3(1+3(p1+p2)+6p1p2),

M4=p4(1+4(p1+p3)+6p2+12(p1p2+p1p3+p2p3)+24p1p2p3),

wherepk=mk/(1−mk).

Proof:

The proof of the first and second moments can be seen from the previous results. For the sake of brevity, we just present the fourth moment argument – the third moment can be obtained in a similar fashion. We note that after some combinatorics and algebra, the fourth moment can be simplified as

L4=∑kjlm=1∞λ−1…λ−kλ−1…λ−jλ−1…λ−lλ−1…λ−m=∑i=1∞λ−1…λ−i4Li,

where

Li=1+4∑m=1∞λ−i−13…λ−i−m3+4∑m=1∞λ−i−1…λ−i−m+6∑m=1∞λ−i−12…λ−i−m2

+12(∑m=1∞λ−i−12…λ−i−m2∑n=1∞λ−i−m−1…λ−i−m−n

+∑m=1∞λ−i−13…λ−i−m3∑n=1∞λ−i−m−1…λ−i−m−n

+∑m=1∞λ−i−13…λ−i−m3∑n=1∞λ−i−m−12…λ−i−m−n2)

+24∑m=1∞λ−i−13…λ−i−m3∑n=1∞λ−i−m−12…λ−i−m−n2∑l=1∞λ−i−m−n−1…λ−i−m−n−l

Applying the expectations we get,

L4=∑i=1∞m4i(1+4∑m=1∞m3m+4∑m=1∞m1m+6∑m=1∞m2m+12(∑m=1∞m2m∑n=1∞m1n

+∑m=1∞m3m∑n=1∞m1+∑m=1∞m3m∑n=1∞m2n)+24(∑m=1∞m3m∑n=1∞m2n∑l=1∞m1l))

=m41−m4[1+4(m31−m3+m11−m1)+6m21−m2+12(m11−m1m21−m2

+m11−m1m31−m3+m21−m2m31−m3)+24m11−m1m21−m2m31−m3].

Autocorrelation of Volatility and Heteroskedastic Innovations

It is important for both theoretical and practical considerations to have insight into time dependence in our volatility model ρt as well as in the conditionally heteroskedastic (CH) innovations εt=ρtet. The explicit formulas for the correlations of these two processes are not trackable for arbitrary δ. However one can get relatively simple formulas for autocorrelations of ρtδ and εtδ. While some results on the autocorrelation of εt for the Gaussian innovation can be found in He (1997), our derivations are more general since they cover our extended scale-location model [2], including non-Gaussian innovations. Moreover, we give the autocorrelation of the powers of volatility ρt.

Proposition 7:

Let random variablesλi’s in eq. [2] be iid such thatσ2+m2<1, wheremandσ2are their mean and variance, respectively. Then fort≥0the correlation ofρtδis given byr(ρtδ,ρ0δ)=mt.

Proof:

It is enough to consider the case α0=1 for which we have

Cov(ρtδ,ρ0δ)=Cov∑k=1∞λt−1⋯λt−k,∑k=1∞λ−1⋯λ−k

=Cov∑k=1tλt−1⋯λt−k+∑k=t+1∞λt−1⋯λt−k,∑k=1∞λ−1⋯λ−k

=Cov∑k=1tλt−1⋯λt−k,∑k=1∞λ−1⋯λ−k+

+Cov∑k=t+1∞λt−1⋯λt−k,∑k=1∞λ−1⋯λ−k

=Eλt−1⋯λ0Var∑k=1∞λ−1⋯λ−k

=mtVar(ρtδ),

where the last two equations are due to independence between λi’s.

We can also give the correlation structure for the δ power of the absolute value of the innovations εt=ρtet in APARCH models with unspecified noise et.

Proposition 8:

Let random variablesλi’s be as in eq. [2]. Bymandσ2we denote their mean and variance, respectively. Additionally we assume thatλtis a non-random function ofet, whereetare iid random variables. Denoteγδ=Cov(λ0,|e0|δ)andνδ=E(|e0|δ). Then the autocorrelation ofεtδ=ρtδetδis given

r(εtδ,ε0δ)=(1−m2)γδ+σ2mνδσ2ν2δ+(λ+m(1−λ))2(1−σ2−m2)(ν2δ−νδ2)νδmt−1

and

Var|εt|δ=α01−m2σ2ν2δ+λ+m(1−λ)21−σ2−m2(ν2δ−νδ2)1−σ2−m2.

Proof:

Let us start with computing variance of |εt|δ. By independence of ρ0 and e0 we have

Var(ρ0δ|e0|δ)=Eρ02δE|e0|2δ−Eρ0δ2E|e0|δ2

=Varρ0δν2δ+(ν2δ−νδ2)Eρ0δ2

=α01−m2σ2ν2δ+λ+m(1−λ)21−σ2−m2(ν2δ−νδ2)1−σ2−m2.

To compute autocovariance note that for t≥1

Cov(εtδ,ε0δ)=Cov(ρtδ|et|δ,ρ0δ|e0|δ)

=EetδCovρtδ,ρ0δ|e0|δ

=νδCov(ρtδ,ρ0δ\tildeδ)+νδCov(ρtδ,ρ0δ),

where e˜δ=|e0|δ−νδ. From Proposition 7:

Cov(ρtδ,ρ0δ)=Var(ρ0δ)mt

=α01−m2σ21−σ2−m2mt

Note that for t≥1 we can write

∑k=1∞λt−1⋯λt−k=∑k=1tλt−1⋯λt−k+λt−1…λ01+∑j=1∞λ−1…λ−j,

with the first term be independent of both e˜δ and ρ0. Thus we have for t≥1:

Cov(ρtδ,ρ0δe˜δ)=α02Cov∑k=1∞λt−1⋯λt−k,1+∑k=1∞λ−1⋯λ−ke˜δ

=α02Covλt−1…λ01+∑k=1∞λ−1⋯λ−k,∑k=1∞λ−1⋯λ−ke˜δ

+α02Covλt−1…λ01+∑k=1∞λ−1⋯λ−k,e˜δ

=α02mt−1Covλ01+∑k=1∞λ−1⋯λ−k,∑k=1∞λ−1⋯λ−ke˜δ

+α02mt−1Covλ01+∑k=1∞λ−1⋯λ−k,e˜δ

=α02mt−1γδE∑k=1∞λ−1⋯λ−k2+E∑k=1∞λ−1⋯λ−k

+α02mt−1Covλ0,e˜δ+Covλ0∑k=1∞λ−1⋯λ−k,e˜δ

=α02mt−1γδVar∑k=1∞λ−1⋯λ−k+m1−m1+m1−m+

+α02mt−1γδ1−m

=α01−m2mt−1γδσ21−σ2−m2+m

=α02(1−m)2(1−m2)1−σ2−m2γδmt−1.

Thus combining the two results we obtain the formula for the autocovariance function

Cov(εtδ,ε0δ)=α01−m21−m2γδ+σ2mνδ1−σ2−m2νδmt−1.

Corollary 2:

Let us assume that e0 has a symmetric distribution around zero. Then

γδ=α2(1+θ)δ+(1−θ)δν2δ−νδ2

and

m=α2(1+θ)δ+(1−θ)δνδ+β,

σ2=α242ν2δ(1+θ)2δ+(1−θ)2δ−νδ2(1+θ)δ+(1−θ)δ2.

Proof:

Be the assumed symmetry of e0, we have

Cov(λ0,|e0|δ)=α((1+θ)δ+(1−θ)δ)Cov(e0+δ, | e0|δ)

=α2(1+θ)δ+(1−θ)δVar|e0|δ

=α2(1+θ)δ+(1−θ)δν2δ−νδ2.

Let ρt be the time varying volatility process, yt be the returns in the APARCH model [1] with a constant function f and the δ-powers returns be defined as rtδ=yt+δ−yt−δ. Then the correlation between the δ-powers of volatility, ρtδ, and lagged centered returns, εt−1(δ), can be regarded as a measure of the leverage effect. Alternatively, one can view the square of standard deviation (variance) as the measure of volatility, so that the leverage effect can be described through the correlation between of ρt2δ and the lagged returns. Denote as before γδ=Cov(λ0,|e0|δ). The explicit relations for these correlation are given in the following result.

Proposition 9:

Let us assume that the random variables et have a distribution that is symmetric around zero. With the notation introduced above we have

r(ρtδ,εt−1(δ))=αν2δ(1+cv1−2)(1−θ)δ−(1+θ)δ2,

rρt2δ,εt−1(δ)=2rρtδ,εt−1(δ)cv1cv2α0+α1−θδ+1+θδ2ν3δν2δ+βEρ03δEρ02δEρ0δEρ02δ,

where cv1 is the coefficient of variation for ρ0δ while cv2 is the coefficient of variation of ρ02δ.

Proof:

We note that because the expected value of εt−1 is zero (the symmetry of e0):

Cov(ρtkδ,εt−1(δ))=Covα0+ρ0δλ0k,ρ0δe0+δ−e0−δ

=Eρ0δCov(α0+ρ0δλ0)k,e0+δ−e0−δ

For k=1 and through conditioning on ρ0 that is independent of e0 and of λ0, we have

Eρ0δCovα0+ρ0δλ0,e0+δ−e0−δ

=αEρ02δCov1−θδe0+δ+1+θδe0−δ,e0+δ−e0−δ

=αEρ02δE1−θδe0+2δ−1+θδe0−2δ

=α2Eρ02δ1−θδ−1+θδν2δ.

The first part of the result follows if we note that

Var(ρ0δ)=E(ρ02δ)cv121+cv12,

Var(r0δ)=E(ρ02δ)ν2δ.

For k=2, let us note that

E(ρ0δ)Covα0+ρ0δλ02,e0+δ−e0−δ=α0αEρ02δ1−θδ−1+θδν2δ+

+Eρ03δEα1−θδe0+δ+1+θδe0−δ+β2e0+δ−e0−δ

=αν2δ1−θδ−1+θδα0Eρ02δ+βEρ03δ+

+α22ν3δEρ03δ1−θ2δ−1+θ2δ

=α2Eρ02δν2δ1−θδ−1+θδ×

×2α0+2β+α1−θδ+1+θδν3δν2δEρ03δEρ02δ.

We also observe that

Varρ2δ=Varρδcv2cv1Eρ2δEρδ.

This combined with the first part completes the proof.

Remark 6:

The coefficients of covariation that are presented in the above result and other moments of ρ0 can be computed explicitly in terms of the model parameters using Proposition 6 and, in particular, Lemma 1, Lemma 2, Remark 2 and Corollary 2.

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Published Online: 2016-12-3

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https://doi.org/10.1515/jtse-2016-0002

Keywords for this article

time series models; heavy tails; leverage effect; estimation