Smooth Transition Quantile Capital Asset Pricing Models with Heteroscedasticity

Transcription

1 Comput Econ (22 4:9 48 DOI.7/s y Smooth Transition Quantile Capital Asset Pricing Models with Heteroscedasticity Cathy W. S. Chen Simon Lin Philip L. H. Yu Accepted: 9 March 2 / Published online: 2 April 2 Springer Science+Business Media, LLC. 2 Abstract Capital asset pricing model (CAPM has become a fundamental tool in finance for assessing the cost of capital, risk management, portfolio diversification and other financial assets. It is generally believed that the market risks of the assets, often denoted by a beta coefficient, should change over time. In this paper, we model timevarying market betas in CAPM by a smooth transition regime switching CAPM with heteroscedasticity, which provides flexible nonlinear representation of market betas as well as flexible asymmetry and clustering in volatility. We also employ the quantile regression to investigate the nonlinear behavior in the market betas and volatility under various market conditions represented by different quantile levels. Parameter estimation is done by a Bayesian approach. Finally, we analyze some Dow Jones Industrial stocks to demonstrate our proposed models. The model selection method shows that the proposed smooth transition quantile CAPM GARCH model is strongly preferred over a sharp threshold transition and a symmetric CAPM GARCH model. Keywords Bayesian inference CAPM GARCH Quantile regression Skewed-Laplace distribution Smooth transition JEL Classification C C22 C5 C52 C. W. S. Chen (B S. Lin Feng Chia University, Tiachung, Taiwan chenws@fcu.edu.tw P. L. H. Yu The University of Hong Kong, Pokfulam, Hong Kong

2 2 C. W. S. Chen et al. Introduction Capital asset pricing model (CAPM is a centerpiece of modern finance that describes the relationship between risk and expected return and is used in valuing risky securities and even portfolios of securities. Based on original work on portfolio theory of Markowitz (959, Sharpe (964 and Lintner (965 developed the CAPM which relates the expected return on a security or a portfolio to a measure of its risk relative to the market, which is called systematic risk and is often denoted by a beta coefficient. There is considerable evidence (see for example Banz (98 and Fama and French (992 suggesting that the beta in CAPM is not a constant but varies over time. Models exhibiting time-variation of market betas have then been proposed in the literature. For instance, Jagganathan and Wang (996 postulated that the market beta and the risk premium vary over time, and their specification worked well in explaining the cross-section of average returns on NYSE and AMEX stocks. However, Ghysels (998 argued that the betas varying at any time may be too much overdone to exploit dynamics of nonlinearities, implying that betas could be varied much slowly, possibly in a discrete manner. The aim of this paper is to propose a more general time-varying market risk model to investigate the difference in market risks under various market conditions using quantile regressions, which would allow the separate assessment of market risks at different quantiles, rather than a mean regression line. Financial data often exhibit some stylized facts such as volatility clustering, asymmetry in conditional mean and variance, mean reversion, and fat tail distributions. It is thus important to develop an appropriate model which can capture these stylized facts. To capture the dynamical features of volatility, the popular choices are the autoregressive conditional heteroscedastic (ARCH and generalized ARCH (GARCH models of Engle (982 and Bollerslev (986, which allow the conditional volatility to be predicted from its lagged terms and the past news. Both ARCH and GARCH models are widely employed for describing dynamic volatility in financial time series. Bollerslev et al. (992 advocate that a GARCH(, model would be usually sufficient for most financial data. Recently, Chen et al. (2 introduced a multi-regime CAPM GARCH model which can capture asymmetric risk through allowing market beta to change discretely between regimes, asymmetric volatility and mean equation dynamics. They confirmed that the discrete time variation of market beta exists in many Dow Jones Industrial stocks. A criticism of threshold models is their discontinuous coefficients, since the switch between regimes is a sharp transition. In response to this criticism, Bacon and Watts (97 first proposed a more gradual regime transition via a smooth continuous transition function. A smooth transition model is more general than a threshold model in the sense that it covers the sharp threshold transition function as a special case. In practice, the smooth transition function is chosen to be a logistic, exponential or any cumulative distribution function. Chan and Tong (986 applied smooth transition models to analyze nonlinear time series. Smooth transition models gained popularity following Granger and Teräsvirta (993 and Teräsvirta (994. van Dijk et al. (22 gave a comprehensive review of the smooth transition autoregressive (STAR model. Recently, Gerlach and Chen (28 further incorporated smooth transition functions into autoregressive conditional heteroskedastic models to allow for smooth

3 Smooth Transition Quantile Capital Asset Pricing Models 2 nonlinearity in mean and asymmetry in volatility. It is thus worthwhile to develop smooth transition CAPM GARCH models to study smooth nonlinearity of the market betas in CAPM. Many empirical studies found that beta coefficient can behave differently under different market conditions. Levy (974 identified different beta values under bull and bear markets. Silvapulle and Granger (2 found that the betas of Dow Jones Industrial stocks are highly unstable when there are negative large movements in the stock returns than when the market is normal or bullish. This implies that the market betas could behave differently over different quantile levels of the stock returns. Quantile regression, initially developed by Koenker and Bassett (978 and Bassett and Koenker (982, is commonly used to describe different regression relationships across quantile levels and has been widely used in many areas including human growth analysis, environmental modeling and financial risk management. Recently, Chen et al. (29 employed quantile regression to study the Granger causality of markets in Asia-Pacific region over different quantile levels. Chen and Gerlach (2 investigated quantile threshold autoregressive models with heteroscedasticity. The aim of this paper is to develop a smooth transition quantile CAPM with heteroscedasticity, nonlinear market betas and nonlinear volatility dynamics under different quantile levels. We will employ a Bayesian approach via Markov chain Monte Carlo methods (MCMC for parameter estimation. It has been well received by Chen et al. (26, 2 and many others for similar nonlinear time series models. The popular Deviance Information Criterion (DIC, suggested by Spiegelhalter et al. (22, is to determine the CAPM specifications. The remainder of this paper is set out as follows. In Sect. 2, we introduce a family of nonlinear CAPMs with heteroscedasticity, including threshold CAPM GARCH and smooth transition CAPM GARCH. In Sect. 3, we further introduce two nonlinear quantile CAPMs with heteroscedasticity and discuss how quantile regression works in estimating the models. Section 4 presents the prior distribution and Bayesian estimation methods for the models in the CAPM family. Section 5 applies the proposed methods to analyze three major US stocks and to study the nonlinear behavior of the market betas over various quantile levels. Section 6 gives concluding remarks. 2 Capital Asset Pricing Models The capital asset pricing model (CAPM measures the sensitivity of the expected excess returns on security to expected market risk premium. The basic-form of CAPM can be described by the security market line below: E(R t r f,t = β(e(r m,t r f,t, where E(R t is the expected return of the asset at time t, R m,t is the expected market portfolio return at time t and r f,t is the risk free rate at time t. E(R t r f,t and E(R m,t r f,t are called the expected risk premium and market risk premium, respectively. The coefficient β of the expected market risk premium, on the security market line, can be determined in terms of the variance of the market excess return

4 22 C. W. S. Chen et al. and the covariance between the asset and the market excess returns, i.e., β = Cov (R t r f,t, R m,t r f,t /Var(R m,t r f,t. Hence, the market β represents a measure of the risk of the asset relative to the market, and a smaller value of β of an asset indicates a lower risk of the asset as compared to the market. Based on the fitted CAPM, investors can develop appropriate portfolio investment strategy by taking into consideration the market risk in CAPM. They can also determine the fair price of an asset or a portfolio based on the CAPM. 2. Two-Regime Threshold CAPM GARCH Model Ferson and Harvey (993, 999 proposed a conditional CAPM that describes the time-varying dynamics of market betas as follows: ( E t (r t+ = φ + β t E t rm,t+, β t = b + b Z t, where E t (. is the conditional expectation using the past information up to time t; r t+ and r m,t+ are the risk-adjusted (excess asset return and market portfolio return, respectively; Z t is a known vector of exogenous factors associated with the asset at time t such as market size, earning-to-price ratio, etc. To describe a slowly changing market betas, Ghysels (998 and Akdeniz et al. (23 developed the homoscedastic threshold CAPM by choosing Z t = (I (r m,t c, I (r m,t > c, where I (. is an indicator function and c is the change point to be determined. Chen et al. (2 further extended it to a multi-regime threshold CAPM GARCH model, based on the threshold non-linearity argument of Tong (978 and Tong and Lim (98, that allows an asymmetric response in both the conditional mean and volatility equations within the models. The two-regime threshold CAPM GARCH is described as follows: r t = { φ ( + φ ( r t + β ( r m,t + a t, if r m,t d c φ (2 + φ (2 r t + β (2 r m,t + a t, if r m,t d > c a t = h t ε t, h t = ε t i.i.d. N(, ( { α ( + α ( a2 t + λ( h t, if r m,t d c α (2 + α (2 a2 t + λ(2 h t, if r m,t d > c, where d is the parameter of delay which is often to be a small integer. This model allows asymmetric behavior due to the difference in the parameters between the two regimes. 2.2 Smooth Transition CAPM GARCH Model The two-regime threshold CAPM GARCH model in ( assumes that the border between the two regimes is given by a sharp transition determined according to a threshold value on r m,t d. A more gradual transition between regimes can be obtained by replacing the sharp transition indicator function by a continuous func-

5 Smooth Transition Quantile Capital Asset Pricing Models 23 tion G i (r m,t d ; γ i, c in the mean and volatility equations, which changes smoothly fromtoasr m,t d increases. Motivated by the idea of smooth transition model (Chan and Tong 986; Granger and Teräsvirta 993, threshold CAPM GARCH model of Chen et al. (2 and the smooth transition GARCH model of Gerlach and Chen (28, we introduce the following smooth transition CAPM GARCH model: r t = φ, + φ, r t + β r m,t +G ( rm,t d ; γ, c ( φ,2 + φ,2 r t + β 2 r m,t + at a t = h t ε t, ε t i.i.d. N(, (2 where h t = α, + α, a 2 t + λ,h t +G 2 ( rm,t d ; γ 2, c ( α,2 + α,2 a 2 t + λ,2h t, G i (r m,t d ; γ i, c = ( { } (r m,t d c + exp γ i,γ>, i =, 2, s m (3 where γ and γ 2 are the smoothness parameters, and s m is sample standard deviation of r m. The smooth transition function G i (r m,t d ; γ i, c is from to. A small value of γ i represents that the curve of G i is more gradual while a large value of γ i represents that the curve of G i has fast transition around the point r m,t d = c. Figure shows a class of logistic smooth transition functions for various values of the smoothness parameter γ. It can be seen that when γ becomes very large, the smooth transition function switches to a sharp transition function and the model becomes a threshold model in (. Therefore the two-regime threshold CAPM GARCH model is a special case of our proposed smooth transition CAPM GARCH model. The new model can also capture asymmetric behavior by demonstrating the significance of the parameters in regime 2. Notice that instead of using a common γ(= γ = γ 2 specified in Gerlach and Chen (28, our model can provide separate smooth transition structures for the mean and variance equations. 3 Nonlinear Quantile Capital Asset Pricing Models It is well known that classical regression studies how the conditional mean of the response variable relates to a set of predictors whereas quantile regression studies how the median or other quantiles of the response variable relates to the predictors. Because of the recent global financial crises, many financial institutions have been getting more cautious to understand the risk of their financial investments under extreme market conditions than the normal market conditions. It is therefore more suitable to use quantile regression to study how the market betas and other risk parameters

6 24 C. W. S. Chen et al. G γ= γ=5 γ= γ=2 2 2 r m,t Fig. Effects of γ on logistic function G(r m t d ; γ,c as given in (3 with (s m, c = (, change under some pre-specified extreme quantile levels. In the following, we first briefly review how quantile regression works. Then following the work in Koenker and Zhao (996 and Chen et al. (29, we consider two nonlinear quantile CAPM GARCH models and describe how the parameter estimation of these models can be formulated as a semi-parametric quantile regression. 3. Review of Quantile Regression Consider a general dynamic regression model y t = f (φ X t + u t, where y t is the response at time t; X t is a set of regressors at time t; φ is a vector of unknown parameters; f is a known function of φ and X t ; u t is the random error with an unspecified distribution. To estimate the conditional quantile of y t at probability level (,, denoted by q (y t X t, Koenker (25 proposed a semi-parametric quantile regression model defined by q (y t X t = f (φ( X t, where h is a known function defined above and φ( is a vector of parameters depending on. Koenker and Bassett (978 and Koenker (25 suggested estimating q (y t X t by minimizing the loss function

7 Smooth Transition Quantile Capital Asset Pricing Models 25 min φ( ρ (y t q (y t X t, (4 t where the function ρ is a loss function defined by ρ (u = u ( I (u <. Let v t = y t q (y t X t. It has been shown, see, eg. Koenker and Machado (999, that the quantile regression based on the minimization of the loss function above is equivalent to the maximum likelihood estimation by assuming that the v t s are i.i.d. skewed-laplace distributed with unit scale (δ = and probability density function (pdf (SL(δ =,: f (v; δ, = ( δ exp { v } δ ( I (v <. Although the variance under the above skewed-laplace density is not one, it is not required to scale it to one as it always leads to the same minimization problem (4. However, this is not the case when there is heteroscedasticity. Let h t = Var(v t F t, where F t represents a set of the information up to time t. To cater for conditional heteroscedasticity, Chen et al. (29 suggested casting the quantile regression as a maximum likelihood estimation by assuming that ε t = v t / h t are i.i.d. skewed- Laplace distributed with unit variance (denoted by SL ( and its pdf is given by: { } g(ε t ; = exp ε t ( I (ε t < ( { } = exp ε t. I (ε t 3.2 Nonlinear Quantile CAPM GARCH Models In the following, we consider two nonlinear quantile CAPM GARCH models. Let q (r t be the -th conditional quantile of the excess return r t. (i Two-regime threshold quantile CAPM GARCH model: q (r t = { φ ( φ (2 ( + φ( (r t + β ( (r m,t, if r m,t d c( ( + φ(2 (r t + β (2 (r m,t, if r m,t d > c( (5 and h t = { α ( ( + α( (a2 t + λ( ( + α(2 α (2 (h t, if r m,t d c( (a2 t + λ(2 (h t, if r m,t d > c( (6 where a t = ( r t q (r t. Define φ j = φ ( j j,φ(,β( j (, and α j = α ( j j j,α(,λ(.let = ( φ, φ 2, α, α 2, c, d be the set of parameters used in this model.

8 26 C. W. S. Chen et al. Instead of using the sharp transition function as in (5, we can consider a smooth transition function for the conditional quantile of the returns. (ii Smooth transition quantile CAPM GARCH model: q (r t = φ, ( + φ, (r t + β (r m,t +G (r m,t d ; γ (, c((φ,2 ( + φ,2 (r t + β 2 (r m,t, (7 and h t ( = α, ( + α, (a 2 t + λ,(h t +G 2 (r m,t d ; γ 2 (, c((α,2 ( + α,2 (at 2 + λ,2(h t, (8 where a t = r t q (r t, and G i (r m,t d ; γ i (, c ( { = + exp γ i ( (r } m,t d c(,γ i >, i =, 2. s m Denote the set of all parameters under this model by 2 = (φ, φ 2, α, α 2, γ,c, d, where φ j = (φ, j,φ, j,β j, and α j = (α, j,α, j,λ, j, j =, 2. Given that the ε t (= a t / h t s are i.i.d. SL (, then the likelihood functions of the above two nonlinear quantile CAPM GARCH models are in the form: L ( i ( r, r m ( T t=s+ ht ( { T exp t=s (r t q (r t ht (( I (r t q (r t }, i =, 2, where T is the sample size, s = max(, d, the maximum number of lag-order parameters in (5 or(7, and r and r m are vectors of r t and r m,t, respectively. As advocated by Yu and Moyeed (2 and Chen et al. (29, accurate parameter estimation can be achieved by adopting a Bayesian approach. 4 Bayesian Inference In Bayesian estimation, it is required to specify prior distributions. We follow the similar settings used in Chen et al. (2 and Gerlach and Chen (28 for threshold and smooth-transition models, respectively. For our threshold quantile models in (5 6, we assume a normal prior φ j N(, j, where j is a diagonal matrix with sufficient large numbers on the diagonal. To ensure stationarity and nonnegative volatilities, the variance equation

9 Smooth Transition Quantile Capital Asset Pricing Models 27 parameters α j follow a uniform prior, p(α j I (C j, for j =, 2, where C j is a collection of α j that satisfies the following restrictions: <α ( < b, α (, λ ( < b 2, α ( + λ ( < b 3, <α (2 < b 4, α (2, λ (2, α (2 + λ (2 < (9 where b, b 2, b 3 and b 4 are chosen by the user. For example, choosing b 2, b 3 can allow an explosive first regime ( j =.Theb and b 4 are typically chosen to be proportional to the sample variance of the data (see Chen et al. (29 for a discussion of choices for these hyper-parameters. For the delay parameter d,weassumeadiscrete uniform prior, p(d = /d, d =,...,d. The prior for the threshold parameter c follows a uniform distribution on a range ( r m (l, r m (u, where r m (l and r m (u are the l and u percentiles of the threshold variable r m,t, respectively. In our smooth transition models in (7 8, we can employ the same priors used in the threshold models except φ j, α j and γ. Note that the parameters φ j s and α j s will become non-identifiable when γ goes to. To remedy this problem, Gerlach and Chen (28 suggested choosing a mixture prior formulation for the φ j s in the mean equations only and commented that such a mixture prior is not strictly necessary for the α j s in the variance equations as they will be restricted to a finite range and their posterior distribution must be proper. Here we adopt their idea and specify the prior distributions of φ j = (φ, j,φ, j,β j, j =, 2, via the mixture of two normals: φ i, j δ i, j ( ( ( δ i, j N, k 2 σ 2 + δ i, j N,σ 2, i =,, β j δ 2, j ( δ 2, j N (, k 2 σ2 2 { ifj = orγ>ξ δ i, j γ = if j = 2 and γ ξ i + δ 2, j N i =,, 2. i (,σ 2 2, and We choose k to be a small positive value such that k 2 σi 2. = ifγ ξ. Assuming prior independence, the prior for φ j is p ( φ j δ j = p ( φ, j δ, j p ( φ, j δ, j p ( β j δ 2, j, where δ j = (δ, j,δ, j,δ 2, j. To ensure stationarity and nonnegative volatilities, we assume that the parameters in α j follow a constrained uniform prior under the following restrictions: <α, < b, α,, λ, < b 2, α, + λ, < b 3, ( α, + α,2 >, α, + α,2, λ, + λ,2. α, + λ, +.5 ( α,2 + λ,2 <. ( Similar to (9, setting b 2, b 3 can allow a possibly explosive regime. Gerlach and Chen (28 indicated that the above constraints are able to ensure stationarity and

10 28 C. W. S. Chen et al. nonnegative variances. Hence we assume that the parameters α j follow a constrained uniform prior over the space bounded by ( and (. Finally, the prior of γ i is set such that ln γ i N(μ γ,σγ 2 to enforce γ. For all the parameters of or 2 except d, the posterior distributions are not of a standard form. We thus turn to use Metropolis Hastings (MH algorithms (Metropolis et al. 953; Hastings 97. The procedure of a general MH algorithm is described below. All parameters are drawn by an iterative Gibbs sampling scheme over a partition of parameter groups. We use the following groups: (i φ j or φ j, j =, 2; (ii γ j, j =, 2; (for smooth transition models only (iii α j or α j, j =, 2; (iv c; (vd. The groups were chosen in order to allow optimal mixing and improved convergence properties. Since the posterior distributions of all parameters except d are not of a standard form, we resort to MH algorithms. We employ the RW-MH algorithm for the parameters in (i, (ii and (iv above. For the parameters in α j or α, to speed up mixing and reduce the autocorrelation of the MCMC iterates generated from a RW-MH algorithm, we use an adaptive MH algorithm to simulate α j or α. In particular, after the burn-in period, we switch from the RW-MH algorithm to the IK-MH algorithm for the sampling period, as in So et al. (25. Finally, the delay parameter d in step (v can be drawn from the multinomial distribution: Pr ( d = j r, r m, j, d = L ( 2, d, d = j r, r m Pr(d = j d i= L ( 2, d, d = i r, r m Pr(d = i, j =,...,d, where j, d is the vector of all model parameters excluding d. 4. Model Selection Using Deviance Information Criterion (DIC Spiegelhalter et al. (22 proposed a Bayesian model comparison criterion, DIC, which is a generalized criterion of Akaike information criterion (AIC and Bayesian information criterion (BIC. DIC can be easily computed during the MCMC sampling and it has been shown to be well supported for model selection and comparison. Define the deviance of a model by D(θ = 2logp(r θ, where r is the set of data, θ are the unknown parameters, and p(r θis the likelihood of the data r. The DIC is decomposed into two parts, goodness of fit and model complexity. The component of Goodness of fit is measured by D = E θ r [D(θ], and the component of model complexity is measured by the estimate of the effective number of parameters which is given by

11 Smooth Transition Quantile Capital Asset Pricing Models 29 P D = E θ r [D(θ] D[E θ r (θ] = D D( θ. D tends to decrease as the dimension of the parameters θ increases, but the penalty term P D tends to increase. Hence, the DIC is calculated by DIC = D + P D = 2 D D( θ and the best model is chosen to be the one with the smallest DIC value. In this paper, we will use DIC for model comparisons of CAPMs. 5 Empirical Applications We analyze three stocks from the Dow Jones Industrial Stocks to illustrate our proposed model. We consider daily excess returns from three stocks that are heavily traded on the New York Stock Exchange and NASDAQ, while the market portfolio is the S&P 5 index. The daily rate of return on the three-month US Treasury-bill is taken as the proxy of the risk-free interest rate. First of all, it is necessary to transform the daily three-month Treasury-bill rate i t (in % into daily risk free rate r f,t (in % via the following conversion formula: ( r f,t = + i t ( 365 %. Then the excess returns on the individual stock and the market portfolio are given by ( Pt r t = ln r m,t = ln P t ( Pm,t P m,t % r f,t, % r f,t, where P t and P m,t are the stock price and the value of the market portfolio on day t, respectively. The three stocks considered are shares of Procter & Gamble Company (P&G, International Business Machines (IBM, and Intel Corporation (INTC. The P&G manufactures a wide range of articles for daily use, and is the sixth most profitable corporation in the world as of mid 2. The IBM is the fourth largest technology company and the second most valuable (after Coca-Cola by global brand. The Intel Corporation is the largest semiconductor chip maker in the world. All of them are well-known corporations and their products are widely used in the world. Our data downloaded from Datastream International consists of daily three-month Treasury bill rate, closing price of the three stocks and S&P5 index over the period from January, 2 to March 3, 2, representing of a maximum of 2,324 observations. Table shows the summary statistics of the T-bill, market and stock excess returns. The excess returns on the market portfolio and stocks range from 2.5 to 8.5%, but their mean returns are all close to zero. Their excess kurtosis are greater

12 3 C. W. S. Chen et al. Table Summary statistics for the T-bill, market and stock excess returns Mean Std. Min Max Q Median Q3 Skewness Excess kurtosis Normality test p-value T-bill < r t S&P < r t P&G < r t IBM < r t Intel <

13 Smooth Transition Quantile Capital Asset Pricing Models 3 IBM INTC 5 5 PG SP // 8//2 3/9/4 2/7/6 25/5/8 3/3/ date Fig. 2 The time series plot of stock excess returns than zero and ranges from 5.8 to 8.4, confirming that the excess equity returns are general leptokurtic. This is also evidenced by a clear rejection of normality assumption of the excess returns based on the Jarque-Bera normality test. Figure 2 shows the time series plots of the excess returns on these three stocks and S&P 5 index. It is clear that all series of returns are more volatile during the global financial crisis in Plots of excess returns on the market portfolio versus those on each of the three stocks are shown in Figs. 3, 4, and 5. The vertical and horizontal reference lines are the 2.5 and 97.5% quantiles of r m,t and r t, respectively. It can be seen that there exist positive correlations between r t and r m,t but it seems that their dependence at the extremes looks slightly different from the one at the middle region. That implies that market betas may behave differently at different extremes. It is thus worthwhile to study such relationship under different extreme market conditions. In the following, we will consider three quantile CAPM models to investigate the structure of market betas under various market conditions. Let a t = r t q (r t.

14 32 C. W. S. Chen et al. r Fig. 3 Plot of the excess returns on the market portfolio vs the excess returns on P&G r m r Fig. 4 Plot of the excess returns on the market portfolio vs the excess returns on IBM r m

15 Smooth Transition Quantile Capital Asset Pricing Models 33 r r m Fig. 5 Plot of the excess returns on the market portfolio vs the excess returns on INTC (a Quantile CAPM GARCH (Q-CAPM GARCH model: q (r t = φ ( + φ (r t + β (r m,t h t ( = α ( + α (a 2 t + λ (h t (b Two-regime threshold quantile CAPM GARCH (TQ-CAPM GARCH model: q (r t = { φ ( φ (2 ( + φ( (r t + β ( (r m,t, if r m,t d c( ( + φ(2 (r t + β (2 (r m,t, if r m,t d > c( and h t = { α ( ( + α( (a2 t + λ( ( + α(2 α (2 (h t, if r m,t d c( (a2 t + λ(2 (h t, if r m,t d > c(. (c Smooth transition quantile CAPM GARCH (STQ-CAPM GARCH model: q (r t = φ, ( + φ, (r t + β (r m,t +G (r m,t d ; γ (, c((φ,2 ( + φ,2 (r t + β 2 (r m,t,

16 34 C. W. S. Chen et al. and h t ( = α, ( + α, (a 2 t + λ,(h t +G 2 (r m,t d ; γ 2 (, c((α,2 ( + α,2 (a 2 t + λ,2(h t, where G i (r m,t d ; γ i (, c = ( { + exp γ i ( (r } m,t d c(,γ i >, i=, 2. s m We estimate these three quantile CAPM models at various quantile levels: = 25, 5,., 5,.5,.75,.9,.95,.975. Following the prior setup specified in Sect. 4, we set the maximum lag for d to be d = 3, and (l, u = (.,.9, ( μ γ = ln 5,σ 2 γ so that at least 99% of γ i lie in the interval 2 ln γ i N = ln 3 (.5, 5. Tables 2, 3, and 4 present the Bayesian estimates of the two-regime threshold quantile CAPM GARCH model for the three stocks while Tables 5, 6, and 7 present the corresponding Bayesian estimates of the smooth transition quantile CAPM GARCH model. Mean equation parameter estimates for which is not contained inside the 95% credible interval are in bold. Parameter estimates and 95% credible intervals against quantile levels are shown in Figs. 6, 7, 8, 9,, and, where the shadowed areas represent the 95% credible interval estimates. All parameters obviously change with, except the ARCH effects ( α (,α(2,α,, α, + α 2, and GARCH effects ( λ (,λ(2,λ,,λ, + λ 2,. The ARCH and GARCH effects do not seem to vary with. We summarize the results as follows:. The intercepts φ (, φ(2, φ, and φ, + φ,2 increase with, and are negative under low quantile levels and positive under high quantile levels. They are close to under =.5. φ,2 decreases as increases. 2. The effects of lagged excess returns φ (, φ, for P&G and IBM have positive effects under low quantile levels and significant negative effects under high quantile levels, but the positive effects seem marginally smaller and insignificant for P&G. For INTC, there are significantly negative effects under all quantile levels, except = 25 and Both φ (2 φ ( and φ,2 can respond to the asymmetry of the effects of lagged excess returns. We find that asymmetric behavior is significant under most quantile levels for all stocks. 4. Focusing on the estimates of market beta β (, β(2, β and β +β 2 in TQ-CAPM GARCH and STQ-CAPM GARCH, we find the following: (a P&G shows less risky than the market, and more risky at low quantile levels, except for = 25. The risks decrease as the stock returns increase, but the relationship is non-monotonic. Both the CAPMs show that P&G has the highest risk at = 5 quantile of its excess returns.

17 Smooth Transition Quantile Capital Asset Pricing Models 35 Table 2 Bayesian estimations of two-regime threshold quantile CAPM GARCH model for P&G returns over various quantile levels for S&P5 returns φ ( (.523 (.3 (.356 (928 (.36 (678 (6 (.44 (532 φ ( (477 (368 (657 (47 (546 (482 (37 (39 (233 β ( (49 (279 (695 (49 (43 (343 (282 (4 (257 φ ( (39 (222 (27 (66 (6 (85 (243 (29 (343 φ ( (2 (82 (79 (86 (88 (99 (29 (84 (24 β ( (77 (88 (82 (77 (8 (89 (7 (9 (23 φ (2 φ ( (52 (47 (78 (57 (577 (54 (434 (43 (3 β (2 β ( (443 (337 (68 (472 (446 (397 (324 (447 (324 α ( (288 (55 (28 (62 (888 (9 (65 (564 (333 α ( (56 (48 (275 (429 (653 (65 (398 (377 (234 λ ( (59 (52 (276 (456 (734 (64 (42 (384 (238 α ( (89 (69 ( (38 (26 (24 (24 (59 (82 α ( (238 (76 (88 (27 (3 (32 (224 (247 (25 λ ( (242 (78 (8 (226 (452 (35 (227 (256 (253 c (4 (79 (33 (22 (.78 (.6 (44 (249 ( H ( H ( d H ( j α ( j = α ( j is the unconditional variance of regime j. Standard errors are in parentheses. Mean β( j equation parameter estimates for which is not contained inside the 95% credible interval are in bold type

18 36 C. W. S. Chen et al. Table 3 Bayesian estimations of two-regime threshold quantile CAPM GARCH model for IBM returns over various quantile levels for S&P5 returns φ ( (76 (526 (63 (82 (.332 (.594 (.45 (.328 (.2 φ ( (253 (667 (36 (428 (889 (648 (484 (542 (448 β ( (575 (.58 (465 (446 (542 (472 (343 (35 (45 φ ( (482 (246 (28 (79 (83 (233 (253 (339 (553 φ ( (36 (69 (93 (64 (27 (87 (48 (7 (397 β ( (29 (28 (254 (92 (93 (23 (26 (3 (458 φ (2 φ ( (284 (682 (36 (458 (. (67 (56 (575 (589 β (2 β ( (64 (.75 (527 (483 (63 (58 (46 (42 (558 α ( (427 (398 (.27 (.389 (.576 (.33 (59 (329 (338 α ( (434 (25 (92 (646 (856 (673 (353 (32 (383 λ ( (435 (23 (2 (662 (.6 (72 (355 (324 (38 α ( (859 (33 (58 (233 (237 (28 (77 (263 (.262 α ( (5 (59 (22 (255 (36 (38 (24 (9 (279 λ ( ( (48 (92 (35 (48 (294 (22 (8 (283 c (5 (7 (87 (556 (.52 (93 (62 (34 (33 H ( H ( d H ( j α ( j = α ( j is the unconditional variance of regime j. Standard errors are in parentheses. Mean β( j equation parameter estimates for which is not contained inside the 95% credible interval are in bold type

19 Smooth Transition Quantile Capital Asset Pricing Models 37 Table 4 Bayesian estimations of two-regime threshold quantile CAPM GARCH model for INTC returns over various quantile levels for S&P5 returns φ ( (.456 (82 (.55 (.455 (896 (.874 (85 (.8 (.35 φ ( (27 (238 (3 (525 (275 (497 (35 (272 (22 β ( (55 (278 (383 (99 (63 (492 (457 (363 (32 φ ( (375 (549 (4 (36 (282 (29 (487 (47 (46 φ ( (42 (8 (72 (56 (54 (5 (28 (76 (8 β ( (27 (49 (32 (358 (336 (265 (323 (27 (86 φ (2 φ ( (35 (32 (348 (56 (35 (58 (444 (332 (232 β (2 β ( (622 (49 (489 (.87 (79 (556 (552 (449 (354 α ( (788 (.83 (32 (723 (.26 (.3332 (788 (.3686 (59 α ( (756 (83 (42 (446 (322 (66 (383 (326 (26 λ ( (765 (9 (45 (466 (363 (653 (393 (33 (2 α ( (796 (584 (728 (289 (9 (277 (57 (.989 (323 α ( (364 (235 (28 (9 (59 (4 (24 (37 (263 λ ( (374 (38 (32 (39 (95 (64 (23 (362 (267 c (33 (55 (224 (38 (592 (674 (27 (65 (89 H ( H ( d H ( j α ( j = α ( j is the unconditional variance of regime j. Standard errors are in parentheses. Mean β( j equation parameter estimates for which is not contained inside the 95% credible interval are in bold type

20 38 C. W. S. Chen et al. Table 5 Bayesian estimations of smooth transition quantile CAPM GARCH model for P&G returns over various quantile levels for S&P5 returns φ, (855 (.25 (825 (843 (.64 (968 (.297 (.585 (.2 φ, (488 (45 (325 (456 (566 (489 (79 (464 (583 β (48 (325 (384 (374 (549 (463 (543 (665 (524 φ, (958 (.34 (882 (939 (.633 (.24 (.423 (.65 (.443 φ, (548 (45 (374 (525 (737 (62 (864 (68 (749 β (524 (395 (43 (437 (777 (586 (642 (77 (587 α, (375 (448 (.733 (.45 (.6 (227 (.52 (.34 (.767 α, (54 (76 (98 (444 (63 (696 (74 (396 (463 λ, (6 (9 (9 (489 (762 (8 (684 (45 (474 α, (463 (57 (.675 (.344 (. (59 (.54 (.487 (2 α, ( (32 (82 (578 (644 (689 (777 (54 (54 λ, (8 (39 (8 (638 (793 (76 (769 (574 (55 c (43 (37 (27 (83 (8 (.83 (.482 (.226 (577 γ (.7254 (4.88 (.3529 (.847 (.534 (.595 (.725 (.555 ( γ (7.283 (8.469 (6.532 (4.332 (3.544 (2.644 (3.427 ( (7.84 H H d α H =, α α, β, H, 2 =, +α,2 (α, +α,2 (β, +β,2. Standard errors are in parentheses. Mean equation parameter estimates for which is not contained inside the 95% credible interval are in bold types (b IBM also has less risky than the market except for = 25, 5. Under the condition of bad news (low regime, IBM is more risky under bear market than under bull market, but the risks are more or less constant as the excess returns increase under the condition of good news (high regime.

21 Smooth Transition Quantile Capital Asset Pricing Models 39 Table 6 Bayesian estimations of smooth transition quantile CAPM GARCH model for IBM returns over various quantile levels for S&P5 returns φ, (.2 (.287 (.4 (886 (.426 (.663 (.45 (.3 (.43 φ, (264 (385 (35 (47 (857 (683 (524 (382 (46 β (647 (528 (563 (486 (55 (479 (629 (463 (424 φ, (.29 (.3 (.74 (935 (.593 (.976 (.474 (.26 (.525 φ, (3 (435 (373 (472 (.34 (862 (57 (46 (437 β (696 (586 (638 (559 (647 (645 (696 (539 (474 α, (.35 (967 (.3529 (288 (.62 (348 (.975 (.66 (458 α, (433 (577 (85 (786 (54 (49 (567 (228 (75 λ, (427 (585 (722 (88 (584 (494 (555 (229 (8 α, (274 (.4 (.359 (28 (.583 (267 (.99 (.24 (593 α, (43 (65 (8 (86 (54 (66 (59 (284 (2 λ, (56 (645 (699 (838 (563 (589 (58 (28 (2 c (629 (98 (695 (.32 (.35 (.646 (359 (228 (25 γ (4.67 (.952 (5.6 (3.654 ( (.6549 ( (.7455 (2.339 γ (8.88 ( ( ( (3.266 ( ( ( ( H H d α H =, α α, β, H, 2 =, +α,2 (α, +α,2 (β, +β,2. Standard errors are in parentheses. Mean equation parameter estimates for which is not contained inside the 95% credible interval are in bold type (c INTC is more risky than the market under all quantile levels, and more risk under bear market than under bull market. We cannot find any monotonic relationship between risks and the excess returns for INTC.

22 4 C. W. S. Chen et al. Table 7 Bayesian estimations of smooth transition quantile CAPM GARCH model for INTC returns over various quantile levels for S&P5 returns φ, (.899 (.356 (.34 (.662 (.34 (555 (.49 (64 (43 φ, (. (584 (376 (59 (384 (56 (52 (459 (438 β (. (.64 (634 (.36 (848 (85 (537 (556 (456 φ, (.373 (82 (.393 (.734 (.455 (668 (.62 (35 (37 φ, (39 (948 (43 (547 (455 (576 (573 (55 (59 β (.88 (.847 (732 (.9 (.3 (.37 (624 (64 (557 α, (.5525 (86 (.598 (.242 (938 (865 (.3269 (4 (35 α, (46 (34 (442 (9 (375 (7 (424 (273 (5 λ, (66 (64 (424 (3 (374 (7 (43 (29 (57 α, (.5526 (.3438 (.5264 (.99 (886 (749 (.328 (484 (85 α, (25 (3 (47 (2 (367 (76 (489 (366 (279 λ, (63 (276 (56 (46 (356 ( (492 (383 (278 c (.293 (.29 (986 (942 (.94 (94 (525 (56 (727 γ (.325 (37 (3.566 ( ( (2.493 (2.257 (.7979 (.679 γ ( ( ( (4.983 (4.286 (4.6 (4.944 (5.48 ( H H d α H =, α α, β, H, 2 =, +α,2 (α, +α,2 (β, +β,2. Standard errors are in parentheses. Mean equation parameter estimates for which is not contained inside the 95% credible interval are in bold type 5. The estimates of β (2 β ( and β 2 are helpful for observing the changes in risks for the three stocks between two regimes. For P&G, under the condition of good news, the risks decrease for almost all quantile levels. For IBM, the risks decrease

23 Smooth Transition Quantile Capital Asset Pricing Models 4 ( φ (2 φ α ( α (2 φ (2 φ ( ( φ (2 φ α ( α (2 β (2 β ( β ( β (2 λ ( λ (2 c Fig. 6 Quantile levels versus the corresponding parameter estimates for TQ-CAPM GARCH (P&G under low quantile levels and increase under high quantile levels as good news is obtained. For INTC, the risks increase under good news for all quantile levels. 6. The estimates of γ 2 is greater than that of γ under almost all quantile levels, except for = 5,. for IBM and =. for INTC. This means that the smooth transition in the variance equation is faster than in the mean equation. 7. Most quantile levels have asymmetric behavior in the AR term or market betas, except for =.5 for IBM. Hence, we infer that the data have asymmetric effect. Clearly, there is no monotonic relationship between risk and security returns for the three stocks. Volatility intercepts are large for extreme quantile levels and small for the middle quantiles. While the two models performed similarly for the three stocks, we employ the DIC to determine which model is more appropriate for the data. We

24 42 C. W. S. Chen et al φ, φ, φ, φ,2 β β φ, +φ, φ, +φ, β +β α, α,2 c α,..8.6 α,2.. γ λ, λ,2 γ 2 Fig. 7 Quantile levels versus the corresponding parameter estimates for STQ-CAPM GARCH (P&G compare quantile CAPM GARCH, two-regime threshold quantile CAPM GARCH and smooth transition quantile CAPM GARCH using DIC, and the DIC values are represented in Table 8. First, the quantile CAPM GARCH performs the worst for all quantile levels of these three stocks, since it cannot respond to the feature of asymmetry. The model comparisons for TQ-CAPM GARCH and STQ-CAPM GARCH are summarized as follows:. For P&G, the TQ-CAPM GARCH performs better than STQ-CAPM GARCH over the quantile levels (.5,.75,.9. STQ-CAPM GARCH is obviously superior over the extreme quantile levels, = (25, 5, For IBM, TQ-CAPM GARCH seems better than STQ-CAPM GARCH since there are five smaller DIC values in nine quantile levels, but again STQ-CAPM GARCH is better under more extreme quantiles (25,.95,.975.

25 Smooth Transition Quantile Capital Asset Pricing Models 43 ( φ (2 φ 2 2 ( α (2 α.5.3. (2 ( φ φ.6 ( φ (2 φ.3. ( α..8.6 (2 α..8.6 β (2 β ( β ( β ( ( λ..8.6 (2 λ..8.6 c Fig. 8 Quantile levels versus the corresponding parameter estimates for TQ-CAPM GARCH (IBM 3. For INTC, STQ-CAPM GARCH is much better than TQ-CAPM GARCH, especially under the levels (25,.9,.95,.975. We note that the threshold quantile CAPM GARCH model is preferred for [.5,.75] in all assets. Is summary, we infer that the smooth transition quantile CAPM GARCH model is more favored than the two-regime threshold quantile CAPM GARCH model under most quantile levels, especially under the extreme quantile levels. 6 Conclusions and Future Works In this paper, we first review the development and significance of CAPM and quantile regression. We extend the threshold CAPM GARCH and the smooth transition

26 44 C. W. S. Chen et al φ, φ,2 φ, +φ,2 α, α,2 c φ, φ,2 φ, +φ,2 α, α,2 γ β β 2 β +β 2 λ, λ,2 γ 2 Fig. 9 Quantile levels versus the corresponding parameter estimates for STQ-CAPM GARCH (IBM GARCH models of Chen et al. (2 and Gerlach and Chen (28, and introduce the smooth transition quantile CAPM GARCH model. In order to efficiently estimate the coefficients, we implement a Bayesian approach and MCMC methods. We illustrate our proposed nonlinear quantile CAPM GARCH models for the three Dow Jones Industrial stocks using a Bayesian approach. The proposed quantile CAPM GARCH model can be used to study the linear relationship between the expected returns on a security and its asymmetric market risk over various quantile levels. The STQ-CAPM GARCH model is a continuously time-varying model for market beta which also includes threshold quantile CAPM GARCH model as a special case. Empirical application shows that the proposed STQ-CAPM GARCH model captures the stylized factors in financial data, and more importantly it is more appropriate

27 Smooth Transition Quantile Capital Asset Pricing Models 45 ( φ (2 φ ( α (2 α φ (2 φ ( ( φ.. (2 φ ( α..8.6 (2 α..8.6 β (2 β (.3.. β ( β ( ( λ..8.6 (2 λ c Fig. Quantile levels versus the corresponding parameter estimates for TQ-CAPM GARCH (INTC than sharp transition CAPM GARCH to describe the stock returns under most quantile levels, especially under the extreme quantile levels. Our findings also reveal that the estimated smooth transition parameters are greater in volatility than those of mean equation. This shows that the smooth transition is more gradual in mean equation when we deal with daily asset returns. It also represents that there is no monotonic relationship between risk and security returns. The DIC values confirm that the smooth transition function in CAPM is important, especially over the extreme quantile levels ( >.975 or < 25. We conclude that the smooth transition switching is demanded between regimes in financial modeling. For future works, it is interesting to extend the STQ-CAPM GARCH model to a multi-regime smooth transition quantile CAPM GARCH, which allows multinomial smooth functions both in mean and variance equations.

28 46 C. W. S. Chen et al φ, φ,2 φ, +φ,2 α, α,2 c φ, φ,2 φ, +φ,2 α, α,2 γ β β 2 β +β 2 λ, λ,2 γ 2 Fig. Quantile levels versus the corresponding parameter estimates for STQ-CAPM GARCH (INTC A single factor, beta, is used on CAPM to compare a portfolio with the market as a whole. More generally, we can add factors to the model to give a better fit. The best known approach is the three factor model developed by Fama and French (993. We would like to consider some useful exogenous variables in previous CAPMs, e.g. market equity, book-to-market equity and earnings/price (E/P of a firm s common stock. A factor model can be expanded on the CAPM by adding size and value factors in addition to the market risk factor in CAPM. In this model, the fact that value and small cap stocks outperform markets on a regular basis can be considered. By including these two additional factors, the model adjusts for the outperformance tendency, which is thought to make it a better tool for evaluating manager performance. Further extension to multi-factor models might be considered.

29 Smooth Transition Quantile Capital Asset Pricing Models 47 Table 8 The DIC values for Q-CAPMs, TQ-CAPMs and STQ-CAPMs P&G Q-CAPM TQ-CAPM STQ-CAPM IBM Q-CAPM TQ-CAPM STQ-CAPM INTC Q-CAPM TQ-CAPM STQ-CAPM The bold values present the lowest values of DIC at each quantile level Acknowledgements We thank the editor and anonymous reviewer. Cathy Chen is supported by the grants: NSC M-35--MY2 from the National Science Council (NSC of Taiwan. Part of the work of Philip Yu, undertaken during a research visit to Feng Chia University, was supported by Mathematics Research Promotion Center, NSC. References Akdeniz, L., Altay-Salih, A., & Caner, M. (23. Time-varying betas help in asset pricing: The threshold CAPM. Studies in Nonlinear Dynamics & Econometrics, 6. Available online at: Bacon, D. W., & Watts, D. G. (97. Estimating the transition between two intersecting straight lines. Biometrika, 58, Banz, R. W. (98. The relationship between return and market value of common stocks. Journal of Financial Economics, 9, 3 8. Bassett, G., & Koenker, R. (982. An empirical quantile function for linear models with iid errors. Journal of the American Statistical Association, 77, Bollerslev, T. (986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 3, Bollerslev, T., Chou, R. Y., & Kroner, K. F. (992. ARCH modeling in finance; a review of the theory and empirical evidence. Journal of Econometrics, 52, Chan, K. S., & Tong, H. (986. On estimating thresholds in autoregressive models. Journal of Time Series Analysis, 7, Chen, C. W. S., & Gerlach, R. H. (2. Semi-parametric quantile estimation for double threshold autoregressive models with exogenous variables and heteroskedasticity. Technical report. Chen, C. W. S., Gerlach, R. H., & Lin., M. H. (2. Multi-regime nonlinear capital asset pricing models. Quantitative Finance. doi:.8/ (forthcoming. Chen, C. W. S., Gerlach, R. H., & Wei, D. C. M. (29. Bayesian causal effects in quantiles: Accounting for heteroscedasticity. Computational Statistics and Data Analysis, 53, Chen, C. W. S., Gerlach, R. H., & So, M. K. P. (26. Comparison of non-nested asymmetric heteroscedastic models. Computational Statistics and Data Analysis, 5, Engle, R. F. (982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 5,