Time-varying asymmetry and tail thickness in long series of daily financial returns

Błażej Mazur; Mateusz Pipień

doi:10.1515/snde-2017-0071

Abstract

We demonstrate that analysis of long series of daily returns should take into account potential long-term variation not only in volatility, but also in parameters that describe asymmetry or tail behaviour. However, it is necessary to use a conditional distribution that is flexible enough, allowing for separate modelling of tail asymmetry and skewness, which requires going beyond the skew-t form. Empirical analysis of 60 years of S&P500 daily returns suggests evidence for tail asymmetry (but not for skewness). Moreover, tail thickness and tail asymmetry is not time-invariant. Tail asymmetry became much stronger at the beginning of the Great Moderation period and weakened after 2005, indicating important differences between the 1987 and the 2008 crashes. This is confirmed by our analysis of out-of-sample density forecasting performance (using LPS and CRPS measures) within two recursive expanding-window experiments covering the events. We also demonstrate consequences of accounting for long-term changes in shape features for risk assessment.

Keywords: density forecasting; Flexible Fourier Form; GARCH models; generalized asymmetric Student t distribution; tail asymmetry

JEL Classification: C11; C58; G10

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2013/09/B/HS4/01945 and 2017/25/B/HS4/02529

Funding statement: The research was supported by the National Science Centre, Poland (NCN) research grants (2013/09/B/HS4/01945 and 2017/25/B/HS4/02529).

Acknowledgment

We would like to thank the anonymous referee for very useful comments and suggestions. We are grateful for helpful remarks by Timo Teräsvirta as well as participants of the 3rd International Workshop on Financial Markets and Nonlinear Dynamics (FMND) and the 25th Annual SNDE Symposium.

Appendix A

The AST distribution as given by Zhu and Galbraith (2010), Eq. (7), has the following form:

(10)fAST(z|m,s,α,ν1,ν2)={1s[1+1ν1(z−m2αsK(ν1))2]−ν1+12,z≤m1s[1+1ν2(z−m2αsK(ν2))2]−ν2+12,z>m,

where s > 0, 0 < α < 1, ν₁ > 0, ν₂ > 0 , m denotes mode of the distribution, s controls its scale and K(ν) is given by:

K(ν)=Γ(ν+1)/2πνΓ(ν/2).

Moreover,

α∗=αK(ν1)B,B=αK(ν1)+(1−α)K(ν2).

However, setting:

m=−c(α,ν1,ν2)d(α,ν1,ν2)

and

s=1Bd(α,ν1,ν2)

with

c(α,ν1,ν2)=4B[−α∗2ν1ν1−1+(1−α∗)2ν2ν2−1]

and

d(α,ν1,ν2)=4[αα∗2ν1ν1−2+(1−α)(1−α∗)2ν2ν2−2]−[c(α,ν1,ν2)]2

which requires ν₁ > 2, ν₂ > 2, implies that the random variable under consideration has zero mean and unit variance. This standardization is not considered by Zhu and Galbraith. In the paper we make explicit use of its re-scaled version, having zero mean and variance σ²:

(11)fAST−ST(x|σ,α,ν1,ν2)={Bd(α,ν1,ν2)σ[1+1ν1(2α∗)2(xd(α,ν1,ν2)σ+c(α,ν1,ν2))2]−ν1+12,x≤mSBd(α,ν1,ν2)σ[1+1ν2(2(1−α∗))2(xd(α,ν1,ν2)σ+c(α,ν1,ν2))2]−ν2+12,x>mS,

with

mS=−σc(α,ν1,ν2)d(α,ν1,ν2).

The AST-ST distribution has mean equal to zero, variance equal to σ², is unimodal with mode at m_S with P{X < m_s}= α. The difference between α^* and α stems from the fact that ν₁ ≠ ν₂ in general. There are two sources of asymmetry: α ≠ 0.5 induces skewness while ν₁ ≠ ν₂ generates tail asymmetry (with ν₁ and ν₂ controlling thickness of the lower and the upper tail, respectively). Imposing ν = ν₁ = ν₂ and α = 0.5 results in reduction to t distribution with ν degrees of freedom.

Appendix B

Baillie and Morana (2009) developed a long-memory FIGARCH-type model with Flexible Fourier component in the intercept of the volatility equation. We analyze empirical properties of two additional specifications similar to their A-FIGARCH(1,d,1,k) model in order to investigate adequacy of long memory volatility effects in our data. In order to do so we replace the GJR-type short-term volatility equation given above with formulation analogous to Eq. (9) of Baillie and Morana (adjusted to our notation):

(12)ht=ωt+η(L)ξt2

with η corresponding to λ of Baillie and Morana (2009). The first model, labeled AF-FIGARCH (Additive Fourier FIGARCH), assumes additive decomposition of conditional volatility [analogous to Eq. (7) of Baillie and Morana (2009)] by setting:

(13)ωt=ω0+exp⁡[fω(t)]

where f_ω(t) is given by the form (4) and f_v(t) = 0 (or F_v = 0). The second model, labeled MF-FIGARCH (Multiplicative Fourier FIGARCH), is based on multiplicative decomposition of volatility used in our paper. It assumes that ω_t = ω₀ (i.e. F_ω = 0) while allowing for (F_v > 0). Consequently, in the two formulations long-run volatility behavior is driven either by f_ω(t) or by f_v(t). At this stage we do not consider time-variation in asymmetry/tail parameters, therefore we set F_α = F_lt = F_rt = 0, we also assume p = q = 1. Note that (unlike Baillie and Morana) we make use of unconstrained and estimated frequency parameters. The conditional distribution is assumed to be the standardized AST (the AST-ST form in Appendix A). By setting F_ω = 0 in AF-FIGARCH or F_v = 0 in MF-FIGARCH we obtain the same model: AR-AST-FIGARCH(1,d,1). We assume that the conditional location parameter is given by (5) to allow for direct comparison with the other results in this paper. Estimation results comparing model fit are given in Table 4 (the entries are analogous to those of Table 1, the reference model is M_0,0,0,0,0 i.e. the AST-GJR GARCH). The results of Table 4 suggest that accounting for GJR-type asymmetric news impact curve is (empirically) more relevant than modelling of long memory in volatility equation. This conclusion reinforces our choice of the AST-GJR specification as the baseline case. Secondly, the differences in empirical adequacy between the additive (AF-FIGARCH) and the multiplicative (MF-FIGARCH) long-term volatility modelling are not large. Bayesian estimates of parameters (posterior means and standard deviations) are included in Table 5. Parameters β, ϕ, and d are defined as in Baillie and Morana (2009). Introduction of Flexible Fourier long-term volatility component is supported by Bayes factors but not by BIC (the latter result indicates unparsimonious specification).

Table 4:

Goodness of fit comparison (full sample): differences in marginal data density and BIC score values against the M(0,0,0,0,0).

F_v/F_ω	No. par.	log₁₀ of Bayes factor vs. M_0,0,0,0,0		BIC differences vs. M_0,0,0,0,0
F_v/F_ω	No. par.	MF − FIGARCH	AF − FIGARCH	MF − FIGARCH	AF − FIGARCH
0	10	−29.71		133.43
1	13	−27.38	−27.10	140.61	135.25
2	16	−26.58	−26.38	157.30	155.12
3	19	−25.02	−25.17	177.39	173.70
4	21	−24.09	−24.69	187.23	191.24

Table 5:

Posterior characteristics: means and standard deviations (in italics) of AF/MF-FIGARCH parameters.

F_v/F_ω	δ	ρ	α₀	ϕ	β	d	λ₁	ν₁	ν₂	α₀₀
0	0.022	0.156	0.037	0.651	0.751	0.533	0.034	5.49	7.18	0.500
	0.017	0.0078	0.0058	0.0519	0.0200	0.0337	0.0257	0.3714	0.6315	0.0077
MF − FIGARCH
1	0.022	0.157	0.043	0.677	0.737	0.505	0.034	5.43	7.29	0.499
	0.0178	0.0082	0.0063	0.0598	0.0276	0.0432	0.0276	0.3555	0.6903	0.0079
2	0.024	0.157	0.045	0.686	0.732	0.496	0.031	5.43	7.18	0.499
	0.0186	0.0079	0.0069	0.0574	0.0265	0.0404	0.0284	0.3829	0.6444	0.0081
3	0.021	0.159	0.048	0.702	0.723	0.479	0.036	5.36	7.26	0.497
	0.0173	0.0075	0.0077	0.0598	0.0294	0.0448	0.0272	0.3779	0.6448	0.0084
4	0.024	0.157	0.050	0.711	0.720	0.471	0.030	5.39	7.20	0.498
	0.0187	0.0082	0.0080	0.0607	0.0279	0.0422	0.0290	0.3585	0.6818	0.0078
AF − FIGARCH
1	0.022	0.157	0.037	0.685	0.729	0.493	0.033	5.43	7.25	0.499
	0.0180	0.0081	0.0067	0.0586	0.0277	0.0414	0.0275	0.3565	0.6839	0.0082
2	0.023	0.158	0.036	0.696	0.725	0.483	0.032	5.44	7.17	0.499
	0.0186	0.0077	0.0072	0.0571	0.0275	0.0386	0.0291	0.3747	0.6603	0.0079
3	0.021	0.158	0.036	0.700	0.722	0.479	0.035	5.38	7.23	0.497
	0.0177	0.0088	0.0082	0.0593	0.0284	0.0409	0.0273	0.3868	0.6403	0.0087
4	0.024	0.158	0.038	0.715	0.715	0.466	0.031	5.48	7.11	0.501
	0.0193	0.0080	0.0081	0.0566	0.0293	0.0400	0.0299	0.3686	0.6310	0.0077

In line with estimates reported by Baillie and Morana we find that increased flexibility of the long term volatility component (indicated by larger values of F) is accompanied by a decrease in estimates of d. In our case the decrease is rather moderate but this might be due to the fact that our sample does not cover the pre-war period (with the Great Depression). Again, we find support for tail asymmetry (ν₁ ≠ ν₂) but not for skewness (α ≠ 0.5, the reported estimates of α_0,0 correspond to values of α), as in the other models we consider. Estimates of ν₁ and ν₂ imply rejection of conditional normality (the assumption was used by Baillie and Morana who analyze S&P500 data at weekly frequency).

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Supplementary Material

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