Leverage, asymmetry and heavy tails in the high-dimensional factor stochastic volatility model Mengheng Li, [email protected], +316 141 519 03 Department of Econometrics, VU University Amsterdam, NL Marcel Scharth Discipline of Business Analytics, The University of Sydney Business School, AU Abstract We propose a flexible high-dimensional factor stochastic volatility (SV) model with leverage effect based on the generalised hyperbolic skew Stu- dent’s t -error to model asymmetry and heavy tails. With shrinkage, the model leads to different parsimonious forms, and thus is able to disengage system- atic leverage effect and skewness from asset-specific ones. We also develop a highly efficient Markov chain Monte Carlo estimation procedure to analyse the univariate version of the model based on efficient importance sampling. With marginalisation of factors, extension to high-dimensional factor model is achieved with computational complexity shown to be linear in the number of factors and assets. Introduction In the literature of univariate SV models, parameter-driven SV models and observation-driven (G)ARCH type of models (Kim et al., 1998) are mainly used to model the following “stylised facts” of asset re- turns: • Volatility clustering (consecutive volatile periods); • Heavy tails of asset returns (extreme events); • Asymmetry or skewness of asset returns (loss bigger than gains); • Leverage effect (loss correlates with higher volatility). When modelling a portfolio of asset returns, “stylised facts” of in- dividual asset returns are linked via the hypothetical market portfo- lio (Fama and French, 1993), leading to non-diagonal higher-moment matrices. This makes modelling high-dimensional portfolios expo- nentially complex due to the “curse of dimensionality”. Factor SV model A straightforward remedy is through a factor model. A factor model naturally delivers systematic interpretation for the multivariate dy- namics of a vector of time series, which can help track the sources of observed “stylised facts”. In the spirit of Chib et al. (2006), we propose the following factor SV model y t =Λf t + u t , t =1, ..., T , {f j,t } T t=1 ∼ Model (1), ∀j ∈{1, ..., p}, {u i,t } T t=1 ∼ Model (1), ∀i ∈{1, ..., n}. Model (1) is sufficiently flexible for capturing “stylised facts”. A good candidate of such a model is the following SV model of Nakajima and Omori (2012), y t = ν t exp(h t /2), t =1, ..., T , ν t = α + β W t + p W t t , t =1, ..., T , h t+1 = μ(1 - φ)+ φh t + η t ,t =1, ..., T - 1, t η t ∼ N ( 0 0 , 1 ρσ ρσσ 2 ) ,t =1, ..., T , W t ∼ IG( ζ 2 , ζ 2 ), t =1, ..., T , (1) 2 1 0 - f(8 ; -2<- <2, 1 =10) -1 -2 -6 -4 -2 0 8 2 4 6 0 0.25 0.15 0.3 0.1 0.35 0.05 0.2 20 15 1 10 f(8 ; - =-2, 5<1 <20) 5 -6 -4 -2 0 8 2 4 6 0.05 0 0.3 0.25 0.2 0.15 0.1 Figure 1: Different density shapes of generalised hyperbolic skew Student’s t - distribution. Left: varying β with ζ = 10; Right: varying ζ with β = -2. MCMC sampler for the factor SV model The sampling scheme is straightforward, and with marginalisation of factors f t , the loadings Λ can be sampled efficiently. Notice the factor SV model gives y i,t = p X j =1 Λ ij (α f j + β f j W j,t + q W j,t ξ j,t )e h j,t /2 +(α u i + β u i Q i,t + q Q i,t i,t )e l i,t /2 . The MCMC sampler iterates over • Sample f t |y 1:T , Λ,θ ; • Marginalise f j,t = ˜ f j,t + ξ j,t ; Sample Λ|y 1:T , { ˜ f j,1:T } p j =1 ,θ ; • Obtain u t = y t - Λf t ; • Sample SV h t and l t , mixture series W t and Q t , as well as other hyperparameters θ using sampling scheme for Model (1). MCMC sampler for the univariate SV model The last step of the sampler for the factor SV model is to apply an MCMC sampler for the p + n individual univariate SV model (1). Thus our model is linearly scalable in number of assets n and number of factors p. The MCMC sampler has two parts. 1. Sampling (h 1:T ,W 1:T )|y 1:T ,θ ; 2. Sampling θ |y 1:T ,h 1:T ,W 1:T . PGAS-EIS sampler We term our sampler particle Gibbs with ancestor sampling using effi- cient importance sampling (EIS-PGAS). EIS stems from particle effi- cient importance sampling (PEIS) of Scharth and Kohn (2016) which builds a globally optimal importance density. Based on the fact that exponential family kernels are closed under multiplication, a sequen- tial Monte Carlo sampler with EIS is given by h t ,W t |h t-1 ,W t-1 ,y 1:T ∼ N (μ t ,v t ) · IG( ζ 2 + s t , ζ 2 + r t ), where ν t = (1 - ρ 2 )σ 2 1 + (1 - ρ 2 )σ 2 c t ,μ t = ν t b t + μ(1 - φ)+ φh t-1 + ρσ ¯ t-1 (1 - ρ 2 )σ 2 , and ¯ t-1 =(y t-1 e -h t-1 /2 - α - βW t-1 )/ p W t-1 . b t , c t , s t and r t are importance parameters determined by a sequence of simple OLS, minimising the χ 2 -divergence between the importance density and the conditional posterior. Figure 2: Posterior estimate of the SV series h 1:T . Left: Particle Gibbs with a bootstrap filter; Right: Particle Gibbs with EIS importance density. As is seen, with the constructed globally optimal importance density particles receive higher weights and the degeneration of particle system is mitigated. The sampler is augmented with the ancestor sampling devise of Lind- sten et al. (2014), leading to improved mixing. 5 10 15 20 25 30 35 40 45 50 -2 -1 0 1 2 Time (t) State (x t ) 5 10 15 20 25 30 35 40 45 50 -2 -1 0 1 2 Time (t) State (x t ) Figure 3: Particle degeneration or impoverishment is inevitable. Left: Without an- cestor sampling, particle Gibbs repeats the previous draw with higher probability as t increases; Right: ancestor sampling breaks the reference trajectory into pieces, which means new draw differs from previous draw with high probability. Theorem 1 (Invariance). The EIS-PGAS kernel K M θ parametrised by θ ∈ Θ with any M ≥ 0 leaves the posterior probability density func- tion p(x 1:T |y 1:T ) invariant: Z B p(x 1:T |y 1:T )dx 1:T = Z K M θ (x ? 1:T ,B )p(x ? 1:T |y 1:T )dx ? 1:T , ∀B ∈F 1:T . Theorem 2 (Ergodicity). Suppose for any t =1, ..., T and θ ∈ Θ, given {x i 1:t-1 } M +1 i=1 , B ∈F 1:T , and sup x t ( max i ω i θ,t ) ≤ ¯ ω θ < ∞. Then for any M ≥ 1 and θ ∈ Θ, there exists some ϕ ∈ [0, 1) such that (K M θ ) n (x ? 1:T ,B ) - Z B p(x 1:T |y 1:T )dx 1:T TV ≤ p(y 1:T ) N - 1 N ¯ ω θ T ϕ n . Shrinkage on leverage effect and asymmetry To investigate the systematic content of factors, we modify the conju- gate normal prior for p + n skewness parameters β k , k =1, ..., p + n using a Bayesian shrinkage prior akin to Bayesian variable selection. This results in a shrinkage posterior β k |· ∼ Δ β k D 0 (β k ) + (1 - Δ β k )N (μ β k ,σ 2 β k ), where Δ β k = 1 - Δ β Δ β ˜ σ 2 β k +1 - Δ β , with ˜ σ 2 β k = σ β k v β exp( μ 2 β k 2σ 2 β k ). The shrinkage probability Δ β has a conjugate beta prior. The leverage effect parameter ρ k can be shrunken similarly via a conjugate normal- inverse-gamma prior. Empirical applications Some results -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 leverage effect ; Pos. mean 1st factor 2nd factor 3rd factor 4th factor -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 skewness - Pos. mean 1st factor 2nd factor 3rd factor 4th factor 0 0.2 0.4 0.6 0.8 1 P(;|.)=0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P(-|.)=0 individual asset 1st factor 2nd factor 3rd factor 4th factor Figure 4: Sorted posterior estimate of ρ and β . Left / Middle: posterior mean esti- mate of leverage effect ρ / skewness parameter β with 95% credible interval; Right: posterior zero probability of β against that of ρ. Coloured dots indicate the parame- ters corresponding to four factors. We can calculate the implied time-varying correlation as d Corr ij,t = ∑ 4 k =1 Λ ik Λ jk exp( ˆ h k,t ) ˆ σ i,t ˆ σ j,t . 01/95 07/97 01/00 07/02 01/05 07/07 01/10 07/12 01/15 07/17 0 0.05 0.1 0.15 0.2 0.25 1st factor 4st factor 01/95 07/97 01/00 07/02 01/05 07/07 01/10 07/12 01/15 07/17 0 0.05 0.1 0.15 The Dow Chemical Company Walgreens Boots Alliance, Inc. PayPal Holdings 01/95 07/97 01/00 07/02 01/05 07/07 01/10 07/12 01/15 07/17 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 The Dow Chemical vs. Walgreens Boots The Dow Chemical vs. PayPal Walgreens Boots vs. PayPal Figure 5: Posterior mean of factor and stochastic volatility process. Left: SV exp( ˆ h j,t /2) from the j =1, 4-th factors; Middle: Volatility ˆ σ i,t of three chosen as- set returns; Right: Implied time-varying correlations d Corr ij,t among the three asset returns. In the year of financial crisis the three correlation series start climbing up, one of which even shoots up to over 0.4. Yet outside the crisis pe- riod the correlation can be low in absolute value. So equicorrelation models are suboptimal in diversification. Dynamic portfolio management The solution of this MVP problem is given by ω t+h|t = Ω -1 t+h|t /( 0 Ω -1 t+h|t ). So, the value-at-risk (VaR) is V aR p,t+1|t (α)= q ω 0 t+1|t Ω t+1|t ω t+1|t F -1 y p,t+1|t (α). Sharpe ratio and information ratio are defined similarly using filtered estimate of covariance matrix. Figure 6: The table shows p-values of coverage ratio tests for the US portfolio. Port- folio weights are updated weekly based on one-step ahead forecast of covariance matrix. α is the nominal level of VaR. Shaded cells indicate rejection of coverage ratio test at 10% level. Figure 7: The table shows the average weekly MVP returns and variances for the U.S. portfolio under different models. EquWgt denotes a equal weighted portfolio. SR and IR are also reported where the latter is relative to S&P 100 index return. One- and two-week rebalancing policies are considered. Shaded cells indicate the best performer with lowest variance, highest mean, highest SR, or highest IR. Conclusion We propose a high-dimensional factor SV model with leverage ef- fect using the generalised hyperbolic skew Student’s t-error to address asymmetry and heavy tails of equity returns, as well as a highly effi- cient MCMC algorithm for Bayesian inference. The model is shown to be flexible enough to distinguish asset-specific mean and volatility dynamics from common factors. With shrinkage, the model helps an- swer whether leverage effect and return asymmetry are systematic or idiosyncratic. References Chib, S., Nardari, F., and Shephard, N. (2006). Analysis of high dimensional multi- variate stochastic volatility models. Journal of Econometrics, 134(2):341–371. Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33(1):3–56. 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