Bayesian time-varying quantile forecasting for. Value-at-Risk in financial markets

Transcription

1 Bayesian time-varying quantile forecasting for Value-at-Risk in financial markets Richard H. Gerlach a, Cathy W. S. Chen b, and Nancy Y. C. Chan b a Econometrics and Business Statistics, University of Sydney, Australia. Tel: Fax: R.Gerlach@econ.usyd.edu.au. b Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taiwan. Tel: ext Fax: chenws@fcu.edu.tw. Corresponding author is Cathy W. S. Chen. chenws@fcu.edu.tw. 1

2 Abstract Recently, Bayesian solutions to the quantile regression problem, via the likelihood of a Skewed-Laplace distribution, have been proposed. These approaches are extended and applied to a family of dynamic conditional autoregressive quantile models. Popular Value at Risk models, used for risk management in finance, are extended to this fully nonlinear family. An adaptive Markov chain Monte Carlo scheme is designed for estimation and inference. Simulation studies illustrate favorable performance, compared to the standard numerical optimization of the usual non-parametric quantile criterion function. An empirical study employing ten major financial stock indices and Value at Risk forecasting finds significant nonlinearity in dynamic quantiles and evidence favoring the proposed model family, for lower level quantiles, compared to standard parametric volatility and risk models in the literature. Keywords: CAViaR model; Asymmetric; Skew-Laplace distribution; Value-at-Risk; GARCH; Regression quantile.

3 1 Introduction The quantile regression problem was first proposed and solved by Koenker and Bassett (1978). The problem involved estimating the parameters of a standard linear regression model, whose aim was to specify the regression line for the response variable measured at a specific quantile of the distribution, rather than at the mean of the response. Applications of this method have been wide ranging and include: labour and wage economics (e.g. Machado and Mata, 2005); medicine (e.g. Wei et al. 2006); ecology (e.g. Cade and Noon, 1996), among many others. The quantile regression solution involved the criterion function that, when minimised, returned the optimal quantile regression estimator. This criterion required no distributional assumption, as such quantile regression is often regarded as a non-parametric method, despite mostly parametric forms being assumed for the regression relationship between the response and the regressors; most examples of its use thus make it semi-parametric in nature. Recently, various authors have noted that the quantile regression criterion function is related to the likelihood for the Skewed-Laplace (SL) distribution; see e.g. Yu and Moyeed (2001) and Tsionas (2003). This discovery allows likelihood estimation, which has motivated Bayesian solutions to this problem. These proposals, see also Yu and Zhang (2005) and Geraci and Bottai (2007), all involve Markov chain Monte Carlo (MCMC) computational methods due to the non-standard form of the SL likelihood. We extend these MCMC methods to be adaptive and to cover a family of nonlinear dynamic autoregressive conditional quantile models. Value-at-Risk (VaR) forecasting is required by financial institutions worldwide (see Basel II: for capital allocation and risk management. Thus, in recent years, volatility forecasting and measures of market risk have been very important for financial institutions. VaR forecasts the minimum loss over a given time interval, under normal market conditions, at a given confidence level, for an investment portfolio (Jorion, 1996); VaR is thus a function of the quantiles of an asset return distribution. Many competing econometric and time series methods have been used in the literature to forecast quantiles, and hence VaR (see Kuester et al, 2006 for a review). 1

4 Engle and Manganelli (2004) proposed to use dynamic quantile regression to model VaR directly, introducing some conditional autoregressive VaR (CAViaR) dynamic quantile models. GARCH-type models, see Engle (1982) and Bollerslev (1986), with parametric specified error distributions, are also popular in this area, see e.g. Chen and So (2006). Black (1976) first discovered the asymmetric volatility phenomenon in financial markets. Many nonlinear models have been proposed to capture this trait, including some simple nonlinear CAViaR specifications. In this paper we extend the existing CAViaR model forms into a fully nonlinear family of dynamic models, in the spirit of threshold GARCH modelling (see Zakoian, 1994 and Brooks, 2001). We design an adaptive MCMC sampling scheme, extending work by Yu and Moyeed (2001), Tsionas (2003), Geraci and Bottai (2007) and Chen and So (2006), to facilitate efficient Bayesian estimation, inference and forecasting for the proposed dynamic quantile model family. The proposed model and MCMC methods are examined firstly through a simulation study, and secondly through application to various financial market stock indices in a study of VaR forecasting. The simulation study illustrates favourable estimation performance compared with numerically optimising the quantile criterion function. The empirical study puts forward evidence of strong nonlinearity in dynamic quantiles in financial markets, supporting the extended fully nonlinear CAViaR model. Secondly, the VaR forecasting study illustrates that CAViaR models performed favourably compared to RiskMetrics and general GARCH estimators at VaR forecasting during our forecast period of 2005 to 2007, especially at low quantiles. Section 2 discusses dynamic quantiles and the link with the SL distribution. Section 3 proposes the extended family of dynamic autoregressive quantile models. Section 4 presents the MCMC methods employed here. Sections 5 and 6 respectively present the simulation and empirical studies. Section 7 concludes. 2 Dynamic quantiles and Value at Risk This section discusses the general dynamic quantile problem and the relation to the SL distribution. 2

5 2.1 General problem The general dynamic quantile regression problem may be written: y t = f t (β, x t 1 ) + u t, (1) where y t is a dynamic observation at time t; x t 1 is a set of explanatory variables, which could include lagged values of the response y t k ; k > 0; β are the unknown parameters and u t is an unknown error term, with a generally unspecified distribution. The exogenous variables are lagged so as to accommodate forecasting, which is highly relevant in the financial examples we consider. The conditional quantile at probability level α, is then: q α (y t β, x t 1 ) = f t (β α, x t 1 ) where β α is the solution to: min β ρ α (y t f t (β, x t 1 )). (2) t The function ρ(.) is a loss function, usually specified as ρ α (u) = u(α I(u < 0)). The function f t (.) defines the dynamic link between the response y t and the explanatory information x t 1. It is usually linear in the parameters and x, which will be extended in this paper. 2.2 The Skewed-Laplace connection Yu and Moyeed (2001) and Tsionas (2003) illustrated the link between the solution to the quantile estimation problem and the likelihood for the SL distribution, as follows. The SL location-scale family (SL(µ, τ, α)) of distributions has density function: p α (u) = α(1 α) τ [ exp ρ α ( u µ τ )], (3) where µ is the mode and τ > 0 is a scale parameter. If it is assumed, in model (1), that u t SL(0, τ, α) and is i.i.d., then the likelihood function becomes: { L α (β, τ; y, X) τ n exp τ 1 n } (y t f t (β))[α I (,0) (y t f t (β))]. (4) t=1 3

6 Since (2) is contained in the exponent of the likelihood, the maximum likelihood estimate for β is equivalent to the quantile estimator in (2). It is important to emphasize that, though we treat (4) exactly as a likelihood function, the methods here do not actually assume that the observations y follow a Skewed-Laplace distribution. The likelihood form here is only employed because it leads to a mathematically equivalent estimator to (2). This allows us to consider powerful computational methods, such as adaptive MCMC algorithms, that employ a likelihood function. Authors such as Yu and Moyeed (2001) and others, have illustrated that accurate inference can be achieved by adopting such an approach. We further investigate and add to these findings for dynamic nonlinear quantile models in this paper. 2.3 Value at Risk The Basel Capital Accord, originally signed by the Group of Ten countries in 1988, requires Authorized Deposit-taking Institutions (ADIs) to hold sufficient capital to provide a cushion against unexpected losses. Value-at-Risk (VaR) is a procedure designed to forecast the worst expected loss over a given time interval under normal market conditions, at a given confidence level α (Jorion, 1996). That is, α = Pr(y t < V ar t y 1,t 1 ) The VaR is thus proportional to a quantile in the conditional one-step-ahead forecast distribution for the observations. Kuester et al (2006) review many methods for estimating VaR, which can be categorised as: 1. Parametric: makes a fully parametric distributional and model form assumption. e.g. AR-GARCH model class with Gaussian errors. 2. Non-parametric: minimal assumptions made about the error distribution, nor the exact form of the dynamic specifications; e.g. historical simulation (i.e. using past sample quantiles). 3. Semi-parametric: some assumptions are made, either about the error distribution, its extremes, or the model dynamics; e.g. Extreme Value Theory, quantile 4

7 regression (CAViaR). A further class of methods, Monte Carlo (MC) simulation, overlap with and might be contained inside each of the classes above. For instance, multi-step ahead VaR estimates are not available in closed form for many models (e.g. GARCH with non-gaussian errors), whether parametric or semi-parametric. MC methods are used in these cases, or similar bootstrap-type methods could be used in a truly non-parametric situation. Naturally MCMC methods, as used in this paper, fit in this class also. A popular method for VaR estimation is RiskMetrics, proposed by J.P. Morgan in 1996, where an IGARCH(1,1) process, with no mean equation, is employed. For expositional purposes, the standard GARCH(1,1) model is: y t = µ + a t i.i.d. a t = h t ǫ t, ǫ t D(0, 1) h t = α 0 + α 1 a 2 t 1 + β 1 h t 1, with the IGARCH(1,1) Riskmetrics model a special case that sets D N(0, 1), µ = α 0 = 0 and α 1 = 1 β 1. The model is non-stationary in volatility: effectively the volatility dynamics have a unit root. Standard GARCH theory (e.g. see Tsay, 2005) allows closed form solutions to the relevant one-step-ahead forecast quantiles of y t y 1,t 1, based on parametric errors, being a fully parametric method. The RiskMetrics method sets β 1 = 0.94 (for daily data). In the sequel, the standard GARCH(1,1) above, with Gaussian errors will be labeled GARCH-n, while the same model with Student-t errors will be denoted GARCH-t. 2.4 CAViaR models for VaR Engle and Manganelli (2004) proposed various dynamic functions f(.) called conditional autoregressive Value at Risk (CAViaR) models. We initially focus on three of their specifications: Indirect GARCH(1,1)(IG): f t (β) = [β 1 + β 2 ft 1 2 (β) + β 3yt 1 2 ]1/2. (5) 5

8 It is straightforward to show that this equation is exactly equivalent to the dynamic quantile function for a GARCH(1,1) model (see Bollerslev, 1986) with an i.i.d. symmetric error distribution. The model thus allows efficient estimation for GARCH(1,1) quantiles with unspecified error distribution. This is an advantage since GARCH models are typically estimated by parametric likelihood or Bayesian methods that assume a specific error distribution. However, it is well known that GARCH models tend to over-estimate volatility in general (e.g. see Chen, Gerlach and Lin, 2008) and over-react to large return shocks (since they are squared). As such we prefer the models below: Symmetric Absolute Value (SAV): f t (β) = β 1 + β 2 f t 1 (β) + β 3 y t 1. (6) The model again responds symmetrically, about 0, to the lagged response. The two models SAV and IG are symmetric to positive and negative observations, with linear responses and parameters. To account for financial market asymmetry, via the leverage effect (Black, 1976), the SAV model was extended in Engle and Manganelli (2004)to: Asymmetric Slope (AS): f t (β) = β 1 + β 2 f t 1 (β) + (β 3 I (yt 1 >0) + β 4 I (yt 1 <0)) y t 1, (7) where the dynamic quantile function can respond differently to positive and negative responses. Such a threshold nonlinear model is similar in spirit to the GJR-GARCH model (Glosten, Jaganathan and Runkle, 1993) or EGARCH (Nelson, 1991), where asymmetry is captured by adding one parameter only, and hence the types of asymmetry captured are limited; in this case to the linear effect of the previous period s return changing, depending on its sign. We extend these models in Section 3 to capture more flexible and complete asymmetric responses, via more general threshold nonlinear forms. In general CAViaR models are semi-parametric in nature: dynamics are specified but error distributions are not. 6

9 3 Proposed nonlinear dynamic quantile family Li and Li (1996) and Brooks (2001) extended the simple GJR-GARCH model to be fully threshold nonlinear: i.e. all parameters in the volatility (and mean) equations were allowed to change between regimes, based on an observed threshold variable. These models were natural extensions of the original threshold autoregressive (TAR) model of Tong (1978) and the GARCH model, so as to allow fully threshold nonlinear dynamic volatility (and mean). So (2003) further considered an exogenous threshold variable in a double threshold DT-ARX-GARCH, (the X refers to exogenous regressor in mean) model. In this same spirit, it seems natural to extend the AS (CAViaR) model via: Threshold CAViaR (T-CAViaR): β 1 + β 2 f t 1 (β) + β 3 y t 1, z t 1 r f t (β) = (8) β 4 + β 5 f t 1 (β) + β 6 y t 1, z t 1 > r Here z is an observed threshold variable which could be exogenous, or self-exciting i.e. z t = y t and r is the threshold value, typically set as r = 0, or estimated, though empirically many estimates in the literature are not significant from zero; as such we fix r = 0 in this paper, which also makes it a direct extension of the AS CAViaR model above. Here each parameter in the dynamic quantile function can respond differently to positive and negative responses. We call this the T-CAViaR family and it includes the SAV (r = ) and AS (r = 0, β 4 =β 1 and β 5 =β 2 ) CAViaR models as special cases. We choose to focus on the SAV-type CAviaR model here, since the GARCH family of models has been shown to lack flexibility and also to over-react to large shocks in the market. However, a corresponding T-IG model could be specified as: Threshold Indirect-GARCH (T-IG): [ β1 + β 2 ft 1 2 f t (β) = (β) + β ] 1/2 3yt 1 2, zt 1 r [ β4 + β 5 ft 1 2 (β) + β ] 1/2 6yt 1 2, zt 1 > r (9) We limit focus in this paper to the T-CAViaR model (8). These models are nonparametric in their error specifications, as in (1), but simply extend the existing forms for the dynamic function f(.) to be fully threshold nonlinear: they are semi-parametric overall. 7

10 Dynamic models typically have constraints or restrictions on the parameters for stationarity (or positivity of dynamic variances). However, such restrictions are difficult to locate for CAViaR models and we choose not to set any in this paper, as in Engle and Manganelli (2004). 4 Bayesian methods Bayesian methods generally require the specification of a likelihood function and a prior distribution for model parameters. The likelihood function for the T-CAViaR is completely specified by (4) and (8). We now specify the prior distribution. 4.1 Prior and Posterior densities We choose the prior to be uninformative over the possible region for the regression-type parameters β, plus we choose the Jeffreys prior for the scale parameter τ. The joint prior is thus π(β, τ) τ 1. which is equivalent to a flat prior on β over the real line, in six dimensions, and on log(τ) over the real line. Using (4), (8) and these priors, the joint posterior density for β, τ y is: p(β, τ y, f) p(β, τ)l α (β, τ; y, f) n τ (n+1) exp{ τ 1 (y t f t (β)) [ α I (,0) (y t f t (β)) ]. (10) t=2 The posterior is not in a form permitting direct analysis or inference. We thus turn to computational MCMC methods for estimation and inference. 4.2 MCMC methods MCMC methods simulate parameters in groups, or blocks, in turn from each of their conditional posterior distributions. We choose the following sampling scheme: 8

11 1. β y, 2. τ β, y, to simulate a dependent Monte Carlo sample from the joint posterior in (10). The conditional posterior in 2., for τ β, y, is: p(τ β, y, f) π(τ)l α (β, τ; y, f) τ (n+1) exp{ τ 1 D}, which is in the form of an inverse gamma density, with n D = (y t f t (β)) [ α I (,0) (y t f t (β)) ]. t=2 Thus, τ β, y IG(n, D), (IG here denotes inverse gamma) and 2. can be simulated from directly. Note that usually interest is only in β and step 2. above can be completely ignored without consequence. The posterior for β y is a marginal density. We obtain it via: p(β y) = p(β, τ y)dτ [D] n, (11) using the form of inverse gamma density and the fact that it integrates to 1. This marginal posterior density has a non-standard form, we thus employ the Metropolis and Metropolis- Hastings (MH) (Metropolis et al., 1953; Hastings, 1970) methods to draw the MCMC iterates for β y. 4.3 Adaptive sampling using Metropolis methods To speed convergence and to allow optimal mixing properties, we employ an adaptive MCMC algorithm for β y that combines a random walk Metropolis (RW-M) and an independent kernel (IK-) MH algorithm. We extend the sampling scheme from Chen and So (2006), who used Gaussian proposal densities. For the burn-in period iterations, a Student-t proposal distribution, with low degrees of freedom (we used df = 5), is employed in a RW-M algorithm. The scale 9

12 matrix, which might initially be chosen as e.g. diagonal with suitable positive values, is subsequently tuned to achieve optimal acceptance rates between 15% and 50%, as recommended in Gelman et al. (1996). After the burn-in period, the sample mean vector and sample var-cov matrix are formed using these M iterates of β. These quantities are subsequently employed in the sampling period (iterations M + 1 to N) as the mean and scale matrix for a Student-t proposal distribution (again with ow degrees of freedom) in an IK-MH algorithm. This adaptive proposal updating procedure will speed mixing in the posterior distribution, over that for the simple RW-M method, as long as the burn-in period has covered the posterior distribution. See Chen and So (2006) for more details of the Gaussian proposal case, but the Student-t proposals will further assist in achieving coverage and mixing, over the Gaussian, for both the burn-in and sampling periods. We extensively examine trace plots and autocorrelation function (ACF) plots from multiple runs of the MCMC sampler, from differing starting points, for each element of β, so as to confirm convergence and to infer adequate coverage (e.g. see Figure 1). Our simulation results also point to the favourable convergence properties of this sampling scheme. 5 Simulation Study We now summarise the results from a simulation study. We wish to examine several facets of our proposed model and MCMC methods for CAViaR models in general. Further, we note here that since CAViaR models assume no error distribution and only specify dynamics at particular quantiles, we need to choose a specific parametric dynamic model and error distribution, in order to simulate data to test the CAViaR models and methods. We choose to focus on the popular and well-known GARCH family of models for this purpose. The simulation study examined: 1. Whether the MCMC method accurately estimated parameters and estimated/forecasted known dynamic quantiles. 10

13 2. A comparison between the MCMC method and the standard numerically optimised quantile estimator (2). 3. How 1. and 2. might change between differently specified T-CAViaR models. 5.1 Simulation Set-up and Results We simulate from a range of GARCH models with Student-t errors. We consider three models and fit the matching TCAViaR model to each data set, using both the proposed MCMC method in Section 5 and the standard optimisation routine in Matlab software to numerically find the estimate of (2). The Matlab code was adapted, and extended for full TCAViaR models, from code written by Simone Manganelli and freely available at The first model we consider is a simple GARCH-t process for the conditional standard deviation, specified as: Model 1 y t = a t, a t = σ t ε t, ε t i.i.d. t 6, σ 2 t = a2 t σ2 t 1. In this case the one-step-ahead α level quantiles (VaRs) are given as: ν 2 V ar α,t+1 = t ν (α)σ t+1, (12) ν where we consider the usual, under Basel II risk management guidelines, levels of α = 0.01, 0.05, and ν is the degrees of freedom, here equal to 6. For data from this model we fit the SAV model (6), i.e. the T-CAViaR model with β 4 = β 5 = β 6 = 0 and r =. The combined Model 1 formula for sigma t and 12 form a specific case of the SAV model. It is straightforward to show that: β 1 (α) = 0.1t 6 (α) 4 6 = 0.082t 6(α); β 2 = 0.85; and β 3 (α) = 0.1t 6 (α) 4 6 = 0.082t 6(α), giving the true values in Table (1). 400 datasets are simulated from Model 1, each with n = 2000 observations. For each dataset, the total MCMC sample size was N = 25000, with a burn-in of M = 5000 iterations. Initial MCMC iterates were set as β = ( 0.5, 0.5, 0.5); τ does not require an 11

14 initial value, while initial values are only required for β since a Metropolis accept-reject method is used. A grid of different starting values were chosen and then the simplex method (Matlab s fminsearch function) was used to find multiple estimators, one for each combination of grid starting values, that minimised (2): the best estimates, that gave the minimum value of (2), were then chosen. Results are summarised in Table (1), which for the MCMC method shows the average of the 400 posterior mean estimates for each parameter, the standard deviation from the true value for these 400 posterior means and a 95% credible interval for the 400 posterior mean estimates. Further, the same summaries are shown for the true simulated VaR value at t = n + 1, the MCMC 1 stepahead forecasted VaR estimate for t = n + 1 and the difference between the forecast and true VaRs at that t = n + 1 = The same information is shown for the traditional quantile forecasts of V ar n+1. Finally, the measures mean absolute deviation (MAD), median absolute deviation (MedAD) and root mean square error (RMSE) were used to assess the competing MCMC and traditional quantile estimates of the VaR inside the sample of n=2000. For the traditional quantile estimator, the average parameter estimate is shown, as are the standard deviation of the parameter estimates from the true value and a 95% confidence interval over the 400 parameter estimates. All parameters have been estimated similarly by each estimation method, no surprise under the diffuse prior for the Bayesian method. The traditional quantile method seems to give marginally less biased estimates, i.e. slightly closer to the true value in mean. Such biases, e.g. intercept biased upwards in magnitude, are commonly observed for GARCH-type models; see e.g. XXX and Chen, Gerlach and So (2005). However, all MCMC parameter estimates are much closer to their respective true values in standard deviation or squared error at both α levels. These differences in standard error between MCMC and the traditional quantile estimator are very large, especially for β 1 and β 2. They may explain the subsequent better estimates and forecasts of VaR by the MCMC method discussed below. The possible sources for the differences between the the Bayesian and classical methods used here are discussed in Section 5.2. Forecasted VaR results are almost completely in favour of the MCMC method. Combining across the 400 datasets, for α = 0.01 the MCMC forecasts are closer in mean and 12

15 Table 1: Summary statistics for CAViaR-SAV model fitted to simulated data from Model 1: GARCH-t. MCMC α=1% α=5% par. True Mean std. 95% C.I. True Mean std. 95% C.I. β (-1.95,-0.02) (-0.75,-0.04) β ( 0.29, 0.94) ( 0.53, 0.92) β (-0.68,-0.09) (-0.34,-0.08) V ar n (-5.61,-2.33) (-3.48,-1.44) V ar n (-5.26,-2.53) (-3.25,-1.57) V ar n+1 V ar n (-0.72, 0.89) (-0.33, 0.48) MAD ( 0.08, 0.53) ( 0.03, 0.24) MedAD ( 0.07, 0.46) ( 0.03, 0.19) RMSE ( 0.06, 0.66) ( 0.02, 0.31) Quantile β (-5.36, 0.003) (-0.61,-0.011) β (-0.63, 1.00) ( 0.60, 0.98) β (-0.68, 0.03) (-0.32,-0.02) V ar n (-5.70,-2.24) (-3.32,-1.17) V ar n (-5.26,-2.53) (-3.25,-1.57) V ar n+1 V ar n (-1.18, 0.95) (-0.33, 1.08) MAD ( 0.09, 0.67) ( 0.03, 0.27) MedAD ( 0.08, 0.55) ( 0.03, 0.23) RMSE ( 0.07, 0.87) ( 0.03, 0.38) 13

16 standard deviation to the true simulated VaRs at t = n + 1, and have smaller squared error from the true VaR values. At α = 0.05 the MCMC forecasts are closer in mean and squared error to the true VaRs, but the standard deviation of the traditional VaR forecasts is slightly closer to the true VaR standard deviation. Combining across the 400 datasets, the MCMC estimates of VaR inside the sample were more accurate than the traditional VaR estimates under all three measures: MAD, MedAD and RMSE, on average. In addition, the three accuracy measures had smaller standard deviations across the data sets, indicating less variation in the VaR estimates from the truth from sample to sample for the MCMC method. The second model we consider is a nonlinear GJR-GARCH process for the conditional standard deviation, specified as: Model 2 y t = a t, a t = σ t ε t, ε t i.i.d. t 6 σ t = ( I t 1 ) a t σ t 1 where the indicator I t a t 0. The one-step-ahead α level quantiles (VaRs) are again given by (12). It is straightforward to show that β 1 (α) = 0.1t 6 (α) 4 6 = 0.082t 6(α); β 2 = 0.85; β 3 (α) = 0.05t 6 (α) 4 6 = 0.041t 6(α); and β 4 (α) β 3 (α) = 0.15t 6 (α) 4 6 = 0.122t 6(α), giving the true values in Table (??). We simulate 500 datasets from this model and fit the AS model in (7) using the MCMC method and by numerically optimising as in (2) ; n = Results are summarised in Table (??) which for the MCMC method shows the average posterior mean estimate for each parameter, the standard deviation from the true value and a 95% credible interval for the posterior mean estimates. lso for MCMC the same summaries are shown for the true VaR at t = n + 1, the 1 step-ahead forecasted VaR estimate for t = n + 1 and the difference between estimated and true VaR at that time point. Finally, the measures mean absolute deviation (MAD), median absolute deviation (MedAD) and root mean square error (RMSE) were used to assess the MCMC estimates of the VaR inside the sample of n=2000. For the traditional quantile estimator, the average optimal estimate is shown, as are the standard deviation of the estimates from the true value and a 95% confidence 14

17 interval over the 400 parameter estimates. Also shown are the forecast accuracy and in sample estimation accuracy for this VaR estimator. All parameters have again been estimated roughly equally by each estimation method. At α = 0.01 MCMC estimates for β 1, β 2 and β 3 β 4 are closer to their true values in both mean and squared deviation. Traditional quantile estimates of β 3, β 4 are closer to their true value in mean, but again the MCMC estimates are closer in squared deviation. The situation is reversed at α = 0.05, with the quantile estimates closer in mean and standard error to their true values for β 1, β 3, the MCCM estimates achieving this for β 3 β 4 only. Forecasted VaR results also differ over α = 0.01, Combining across the 400 datasets, for α = 0.01 the MCMC forecasts are closer in mean and standard error to the true simulated VaRs at t = n+1. At α = 0.05 the MCMC forecasts are closer in standard error to the truth, while the traditional estimator is closer in mean. Combining across the 400 datasets, the MCMC estimates of VaR inside the sample were more accurate under all three measures: MAD, MedAD and RMSE, on average. In addition, the three accuracy measures had smaller standard deviations across the data sets, indicating less variation in the VaR estimates from sample to sample for the MCMC method. as: The final model we consider is a full nonlinear threshold T-GARCH-t model, specified Model 3 y t = a t, a t = σ t ε t, ε t i.i.d. t 6, σ 2 t = a 2 t σ2 t 1, if a t 1 0, a 2 t σ2 t 1, if a t 1 > 0. The one-step-ahead α level quantiles (VaRs) are again given by (12). We simulate 500 datasets from this model and fit the full T-CAViaR model in (8) using the MCMC method and by numerically optimising as in (2); n = Results are summarised in Table (??): Here the GARCH model implies a high positive value for β 2 and β 5, with β 2 > β 5, but only marginally; β 1, β 4 < 0 with β 1 < β 4 and β 3, β 6 < 0 where marginally β 3 < β 6. While all parameter values are as expected, it seems equally difficult for both estimation 15

18 Table 2: Summary statistics for CAViaR-AS model fitted to simulated data from Model 2. MCMC α=1% α=5% par. True Mean std. 95% C.I. True Mean std. 95% C.I. β (-0.96,-0.06) (-0.45,-0.07) β ( 0.61, 0.94) ( 0.72, 0.91) β (-0.63, 0.08) (-0.25, 0.01) β (-1.00,-0.21) (-0.53,-0.19) β 3 β (-0.00, 0.85) ( 0.07, 0.39) V ar n (-9.29,-2.55) (-5.68,-1.59) V ar n (-9.55,-2.73) (-5.90,-1.68) V ar n+1 V ar n (-1.06, 1.17) (-0.40, 0.48) MAD ( 0.15, 0.85) ( 0.06, 0.32) MedAD ( 0.12, 0.63) ( 0.05, 0.25) RMSE ( 0.17, 1.14) ( 0.07, 0.46) Quantile β (-0.98,-0.04) (-0.41,-0.06) β ( 0.56, 0.96) ( 0.73, 0.92) β (-0.64, 0.10) (-0.25, 0.02) β (-1.03,-0.16) (-0.52,-0.17) β 3 β (-0.03, 0.85) ( 0.07, 0.40) V ar n (-9.61,-2.53) (-5.67,-1.60) V ar n (-9.55,-2.73) (-5.90,-1.68) V ar n+1 V ar n (-1.43, 1.18) (-0.41, 0.46) MAD ( 0.15, 0.97) ( 0.06, 0.32) MedAD ( 0.11, 0.72) ( 0.05, 0.24) RMSE ( 0.16, 1.34) ( 0.07, 0.45) 16

19 Table 3: Summary statistics for the T-CAViaR model fitted to simulated data from Model 3. MCMC α=1% α=5% Quantile par. Mean std. 95% C.I. Mean std. 95% C.I. β (-2.22, 0.43) (-1.10, 0.21) β ( 0.30, 1.16) ( 0.43, 1.12) β (-0.86, 0.19) (-0.33, 0.03) β (-1.96, 0.36) (-1.24, 0.18) β (0.19, 1.03) (0.24, 0.99) β (-0.66, 0.11) (-0.34, 0.01) β 1 -β (-1.96, 1.94) (-1.05, 1.00) β 2 -β (-0.50, 0.75) (-0.43, 0.63) β 3 -β (-0.73, 0.69) (-0.25, 0.27) V ar t (-5.24, -2.08) (-3.10, -1.38) V ar t (-5.02, -2.26) (-3.11, -1.40) V ar t+1 V ar t (-0.96, 0.86) (-0.49, 0.48) MAD (0.18, 0.64) (0.08, 0.27) MedAD (0.13, 0.48) (0.06, 0.21) RMSE (0.21, 0.99) (0.11, 0.36) β (-2.64, 0.53) (-1.99, 0.63) β ( 0.14, 1.17) ( 0.05, 1.33) β (-0.97, 0.22) (-0.34, 0.06) β (-3.39, 0.30) (-2.31, 0.23) β (-0.16, 1.02) (-0.25, 1.04) β (-0.74, 0.15) (-0.34, 0.03) β 1 -β (-2.32, 3.08) (-1.61, 2.65) β 2 -β (-0.56, 1.08) (-0.80, 1.37) β 3 -β (-0.73,0.77) (-0.28, 0.29) V ar t (-5.12, -2.15) (-3.92, -1.33) V ar t (-5.02, -2.26) (-3.11, -1.40) V ar t+1 V ar t (-1.04, 0.96) (-1.81, 0.58) MAD (0.20, 0.75) (0.09, 0.32) MedAD (0.15, 0.59) (0.07, 0.26) RMSE (0.24, 1.01) (0.11, 0.44) 17

20 methods to distinguish the differences in parameter values between regimes. This is most surprising for the intercepts β 1, β 4 with high standard errors for β 1 β 4. However, again the MCMC has the lowest standard errors for these differences, while the alternative quantile estimation method incorrectly finds on average that β 1 β 4 > 0. The MCMC estimates consistently display a lower standard error for all parameter estimates. Combining across the 500 datasets, the MCMC estimates of VaR inside the sample were more accurate under all three measures, on average. In addition, the three accuracy measures had smaller standard deviations across the data sets, indicating less variation in the VaR estimates from sample to sample. Finally, VaR at time n = 2001 was again forecast and the MCMC method gave forecasts that were less biased and had smaller standard error, compared to the true VaR, across the 500 datasets. 5.2 Discussion of Simulation Results The results here consistently, if marginally, favored the MCMC estimator, moreso in estimation and forecasting of VaR, but also consistently in terms of squared error of parameter estimation. This is an interesting result given that asymptotically, Bayesian estimation under diffuse priors and classical estimates should yield the same results. The observed differences in parameter and VaR estimation between methods could be due to a number of factors. Firstly, the classical estimator might be expected to be exactly repeatable: i.e. if the same data set is estimated, using the same simplex method, from the same parameter starting positions, exactly the same estimates should be produced. This is not the case for the MCMC method, which is subject to Monte Carlo error, a natural disadvantage over the traditional quantile estimator. That is, running the MCMC method repeatably from the same starting position will give slightly different estimates because of Monte Carlo error, though qualitatively the repeated estimates will be the same, as shown in Figure 1. Thus the MCMC estimates will never equate exactly with the traditional quantile estimates that minimise (2). Secondly, in finite samples differences may be due to the inherent difference in estimation paradigms. The Bayesian approach here, as is standard, takes a posterior mean estimate, not the posterior mode that would the same as the maximum likelihood estimate under a diffuse prior. Alternatively, the 18

21 traditional approach minimises (2), effectively this is a maximum likelihoood estimator under the skewed Laplace error distribution, as in Yu and Moyeed (2001, 2005). In finite samples where the likelihood and posterior may not be symmetric or Gaussian in the parameters, the posterior mean and posterior mode/maximum likelihood estimates could be expected to differ, at least marginally. We note that while the sample size is n=2000 here, estimation here is concerned with low quantiles of the data, so the effective sample size for estimation may be somewhat lower than n = 2000, and would surely lie somewhere in the range (α n, n), which is (100, 2000) for α = 0.05 and (20, 2000) for α = 0.01 and n = This is because the quantile function (2) weights each error y t f t (β) by α for positive errors and 1 α for negative errors. For α = 0.05 and n = 2000 then, 1900 errors or sample observations would receive the very small weight of α = 0.05, while only 100 of the errors or sample points would receive the much higher weight of 1 α = It should be clear that in quantile estimation with alpha not close to 0.5, that effective sample size is much smaller than n. As such, posterior means and posterior modes/maximum likelihood estimates should not be expected to be exactly equivalent, even for quite large sample size n in these cases. What the results seem to show, is that the Bayesian posterior mean parameter estimate, even though it is subject to the extra Monte Carlo error in this estimate, is often closer in squared error to the true parameter value for CAViaR models, while exhibiting very slightly more bias, than the traditional quantile estimator. Further, this lower standard error is likely influencing the result that, on average, the MCMC estimates and forecasts of the VaR are closer in mean and squared error to the true VaR. Naturally this results holds only for the GARCH models used in this study and for n = Why has this lower square error for MCMC result occurred? One simple answer may be that the posterior mean is the result of a numerically approximated integral over the entire posterior distribution, while the function (2) is effectively numerically minimised using numerically approximated derivatives in a search routine. Surely numerical integration in general, where accuracy is fairly easily controlled, is likely to be a more accurate process/method than numerical differentiation, especially adding the factors of low effective sample size and resulting non-gaussian likelihoods/posteriors. 19

22 6 Testing VaR models A common non-test criterion to compare VaR models is the rate of violation, defined as the proportion of days for which the actual return is more extreme than the forecasted VaR level, over the forecast period. The violation rate is: VRate = n+m t=n+1 I(y t < VaR t ) m, where n is the learning sample size and m is the forecast or testing sample size. It is clearly desirable for a forecast model s VRate to be close to the nominal level α. We used VRate to compare models in a non-testing, more adhoc, fashion as follows. We employed the ratio VRate/α, to help compare the competing models, where models with VRate/α 1 are most desirable. However, as in Wong and So (2003), we note that when VRate < α, risk and loss estimates are conservative (higher than actual), while alternatively, when VRate > α, risk estimates are lower than actual and financial institutions may not allocate sufficient capital to cover likely future losses. Here solvency outweighs profitability and for models where VRate/α 1, lower rates are preferred; e.g. VRate/α = 0.9 is preferred to VRate/α = 1.1. We further consider two hypothesis-testing methods for evaluating and testing the accuracy of VaR models: the unconditional coverage test of Kupiec (1995), which is a likelihood ratio test that the true violation rate equals the nominal quantile level; and the conditional coverage test of Christoffersen (1998), which is a joint test, combing a likelihood ratio test for independence of violations and the unconditional coverage test. These tests are quite standard now and we refer readers to the original papers for details. 7 Empirical Results We considered ten daily international stock market indices: the S&P 500 (US); FTSE 100 (UK); CAC 40 (France); Dax 30 (Germany); Milan MIBTel Index (Italy); Toronto SE 300 (Canada); AORD All ordinaries index (Australia); Nikkei 225 Index (Japan); TSEC weighted index (Taiwan) and the HANG SENG Index (Hong Kong). The data were obtained from Datastream International and covered the period from January 1,

23 Table 4: Summary statistics: stock index returns for ten stock markets. Japan France Germany Italy U.K. Mean Variance Skewness (0.21) (0.51) (0.11) (<0.0001) (0.0003) Excess kurtosis (<0.0001) (<0.0001) (<0.0001) (<0.0001) (<0.0001) Canada U.S. Taiwan Hong Kong Australia Mean Variance Skewness (<0.0001) (0.010) (0.78) (<0.0001) (<0.0001) Excess kurtosis (<0.0001) (<0.0001) (<0.0001) (<0.0001) (<0.0001) Note: P-values based on asymptotic normality are listed, under a null hypothesis of 0. 21

24 to January 19, The log return data series were generated by taking logarithmic differences of the daily price index, y t = (ln(p t ) ln(p t 1 )) 100, where P t is the price index at time t. The full sample was divided into a learning sample: January 1, 2001 to January 10, 2005; and a forecast or testing sample: the 500 trading days from January 11, 2005 to mid-january, Small differences in end-dates across markets occurred due to different market-specific non-trading days. Table?? shows summary statistics from the full sample of these market indices including sample mean, variance, skewness and kurtosis. Based on the observed skewness and kurtosis results, all ten return series display the standard properties of asset return data: they are heavy-tailed and mostly negatively skewed. 7.1 TCAViaR model estimation results Tables??-?? show parameter estimates for the full T-CAViaR model applied to the full series from each market. The MCMC burn-in sample size was again iterations, followed by a sampling period of iterations for estimation and inference. Figure?? displays trace plots of the iterations from five different MCMC runs, each with a different randomly selected starting position for the parameters, for the Australian market at α = Each of the six parameters shown (β 1,...,β 6 ) has at least one starting MCMC value on each side of its reported posterior mean in Table??. Convergence to the same posterior distribution is clear in all five runs for each parameter, in each case well before the end of the burn-in sample. The inefficiency factors R (see Gelman et al, 2005, pg 296) for each parameter over these five runs were 1.029, 1.034, 1.019, 1.041, and 1.006; all quite close to 1 and highlighting clear and efficient convergence for the proposed sampling scheme. The estimation results in tables??-?? show that the 10 markets roughly fit into 3 categories in terms of the parameter estimates obtained. Consider first the estimates at α = Japan, France, Germany, Canada, US and Australia display similar posterior mean estimates across parameters: with a significantly negative intercept (β 1 < 0), 22

25 Table 5: Parameter estimates from T-CAViaR model fit to returns from major market indices. α=1% α=5% parameter Mean std. 95% C.I. Mean std. 95% C.I. Japan β (-0.470,-0.272) (-0.108,0.011) β ( 0.739, 0.805) (0.871,0.937) β (-0.412,-0.344) (-0.233,-0.177) β (-0.056,0.077) (-0.086,0.023) β (0.902,0.954) (0.890,0.948) β (-0.235,-0.175) (-0.134,-0.063) β 1 -β (-0.531,-0.232) (-0.124,0.089) β 2 -β (-0.207,-0.102) (-0.072,0.039) β 3 -β (-0.212,-0.137) (-0.149,-0.067) France β (-0.235,-0.160) (0.024,0.083) β (0.878,0.918) (0.958,0.986) β (-0.287,-0.220) (-0.259,-0.205) β (-0.042,0.046) (-0.129,-0.071) β (0.891,0.921) (0.892,0.926) β (-0.157,-0.077) (0.006,0.059) β 1 -β (-0.277,-0.129) (0.096,0.212) β 2 -β (-0.029,0.020) (0.036,0.088) β 3 -β (-0.172,-0.107) (-0.307,-0.224) Germany β (-0.266,-0.109) (-0.048,0.054) β (0.851,0.908) (0.935, 1.011) β (-0.353,-0.265) (-0.211,-0.141) β (-0.102,0.014) (-0.107,-0.038) β (0.903,0.942) (0.912,0.950) β (-0.076,-0.024) (0.007,0.053) β 1 -β (-0.274,-0.008) (-0.010,0.157) β 2 -β (-0.086,0.002) (-0.011,0.097) β 3 -β (-0.315,-0.204) (-0.247,-0.171) Italy β (0.208,0.354) (-0.027,0.027) β (1.045,1.091) (0.980,1.010) β (-0.495,-0.444) (-0.179,-0.131) β (-0.589,-0.451) (-0.078,-0.036) β (0.689,0.743) (0.894,0.926) β (0.011,0.057) (-0.006,0.052) β 1 -β (0.665,0.937) (0.009,0.101) β 2 -β (0.304,0.394) (0.060,0.113) β 3 -β (-0.530,-0.476) (-0.222,-0.143) UK β (0.061,0.137) (-0.008,0.065) β (0.994,1.040) (0.947,0.985) β (-0.318,-0.213) (-0.308,-0.249) β (-0.200,-0.127) (-0.112,-0.048) β (0.827, 0.883) (0.883,0.928) β (-0.098,0.0001) (0.036,0.093) β 1 -β (0.192,0.334) (0.041,0.175) β 2 -β (0.126,0.208) (0.021,0.098) β 3 -β (-0.268,-0.167) (-0.384,-0.301) 23

26 Table 6: Parameter estimates from T-CAV model fit to returns from major market indices. α=1% α=5% Parameter Mean std. 95% C.I. Mean std. 95% C.I. Canada β (-0.186,-0.095) (-0.064,0.013) β (0.915,0.960) (0.926,0.985) β (-0.151,-0.105) (-0.177,-0.120) β (-0.022,0.044) (-0.067,-0.004) β (0.943,0.989) (0.898,0.947) β (-0.022,0.054) (-0.054,0.005) β 1 -β (-0.225,-0.075) (-0.061,0.078) β 2 -β (-0.072,0.017) (-0.014,0.078) β 3 -β (-0.182,-0.111) (-0.164,-0.086) US β (-0.180,-0.067) (-0.011,0.072) β (0.887,0.954) (0.996,1.052) β (-0.174,-0.112) (-0.095,-0.056) β (0.036,0.135) (-0.113,-0.043) β (0.983,1.030) (0.905,0.954) β (-0.074,0.009) (0.019,0.053) β 1 -β (-0.317,-0.103) (0.033,0.182) β 2 -β (-0.130,-0.032) (0.045,0.142) β 3 -β (-0.151,-0.081) (-0.141,-0.087) Taiwan β (-0.290,0.249) (-0.161,-0.044) β (0.825,0.980) (0.873,0.934) β (-0.501,-0.380) (-0.277,-0.200) β (-0.964,-0.295) (-0.084,0.018) β (0.721,0.889) (0.867,0.934) β (0.030,0.098) (-0.084,-0.032) β 1 -β (0.148,1.136) (-0.170,0.032) β 2 -β (-0.032,0.223) (-0.047,0.053) β 3 -β (-0.583,-0.437) (-0.223,-0.138) HK β (-0.188,0.214) (-0.169,-0.040) β (0.836,0.969) (0.872,0.938) β (-0.559,-0.367) (-0.277,-0.198) β (-1.144,-0.718) (-0.089,0.029) β (0.672,0.792) (0.865,0.938) β (0.063,0.113) (-0.087,-0.033) β 1 -β (0.703,1.252) (-0.189,0.035) β 2 -β (0.090,0.266) (-0.056,0.061) β 3 -β (-0.653,-0.433) (-0.225,-0.139) Australia β (-0.275,-0.211) (-0.267,-0.126) β (0.751,0.793) (0.723,0.870) β (-0.538,-0.460) (-0.252,-0.157) β (-0.030,0.030) (-0.033,0.042) β (0.947,1.023) (0.865,0.943) β (0.079,0.148) (-0.084,-0.011) β 1 -β (-0.297,-0.188) (-0.305,-0.096) β 2 -β (-0.238,-0.160) (-0.209,-0.005) β 3 -β (-0.659,-0.565) (-0.211,-0.102) 24

27 medium positive persistence (β 2 ), well away from 1, and a large negative effect of the previous day s return (β 3 < 0), all in the negative regime; while exhibiting small insignificant intercepts (β 4 0), higher positive persistence (β 5 > β 2 ) and smaller, mostly still negative and significant, lagged return effect (0 > β 6 > β 3 ). For these markets, dynamic quantiles become more extreme (negative) following negative returns and are positively but not too strongly persistent (except US), more so following positive returns. In contrast, the markets in Italy and the UK have significantly positive intercepts (β 1 > 0) and very strong positive persistence (β 2 > 1), with estimates above 1, significantly for Italy, all in the negative regime. The effects (β 3 < β 6 < 0) are similar to the first market grouping. In the positive regime, intercepts are now significantly negative (β 4 < 0), with lower positive persistence (β 5 < β 2 ). For these two markets, dynamic quantiles still become more extreme (negative) following negative returns but now are strongly positively persistent, more so following negative returns. Finally, Taiwan and Hong Kong have insignificant intercepts in the negative regime, as well as very strong lagged return effects (β 3 << 0). In the positive regime, intercepts are very high and negative (β 4 << 0) but lagged return effects are smaller and significantly positive ( β 6 < β 3 ; β 6 > 0). Quantile persistence in these markets is medium and positive. Tables??-?? also show estimates and inferences for differences between corresponding parameters in the positive and negative regimes. These estimates indicate the level and direction of asymmetries between the two regimes. All markets show clear evidence of significant asymmetry in response to negative and positive returns, in their dynamic quantiles. The markets fell into the same three classes in their nonlinear behaviour. Firstly, the lagged return effect is consistently and significantly asymmetric across all ten markets. Dynamic quantiles become more extreme (negative) following negative returns than positive returns. For the first group above, each parameter difference is negative, but usually only the intercept and the lagged return effect are significant. For the second group, intercepts and persistence are significantly higher in the negative regime; this result also obtains for the third group. 25