Parametric and Semi-parametric models of Value-at-Risk: On the way to bias reduction

Transcription

1 Parametric and Semi-parametric models of Value-at-Risk: On the way to bias reduction Yan Liu and Richard Luger Emory University October 24, 2006 Abstract Even if a GARCH model generates unbiased variance forecasts, due to a Jensen s inequality effect, non-linear transformations of these forecasts will be biased. The estimation of Value-at-Risk (VaR) uses the square root of the variance and is subject to this non-linear transformation bias. This paper proposes a two-stage estimation procedure to reduce the VaR estimation bias in conventional GARCH models. Instead of forecasting conditional variances, the first-stage model forecasts the standard deviation directly to avoid the non-linear transformation bias. We further construct a second-stage model via quantile regression by including some instrument variables from a new Monte Carlo test. We illustrate the use of this two-stage model in international stock indices, foreign exchange rates and individual stocks. The empirical results support the effectiveness of this two stage model. JEL classification: Key Words: Risk management; GARCH; Conditional quantile 1

2 1 Introduction There are three major types of risks in financial markets, Credit Risk, Liquidity Risk and Market Risk. For trading purpose, market risk is the most important risk to be considered. Value at Risk (VaR) was introduced by J.P.Morgan (1996) and has become the standard measure to quantify market risk. VaR is generally defined as the maximum possible loss for a given position or portfolio within a known confidence interval over a specific time horizon. The measure can be used by the financial institutions to assess their risks or by a regulatory committee to set margin requirements. Both purposes will lead to the same VaR measure, even though the concepts are different. In other words, VaR is used to help the financial institutions stay in business after a catastrophic event. The concept of VaR is easy, while the estimation of VaR is non-trivial. There are a variety of approaches to estimate the VaR. They range from parametric (Riskmetrics, GARCH, etc.) to semi-parametric (Extreme Value Theory, CAViaR, etc.) and non-parametric (Historical Simulation and its variants, etc.). In practice, people face the important issue of how to choose the best model among so many candidates. Since different methodologies can yield different VaR measures for the same portfolio, there has to be some models leading to significant error in risk measurement. The risk of choosing inappropriate model is called model risk and is an important question left to the risk manager. As the result, the VaR model comparison becomes a very important issue. Abundant literature concern about this problem, such as Christoffersen etc. (1998, 2001, 2004), Sarma etc. (2003), Lopez(1998), etc. A well specified VaR model should be able to generate a iid Bernoulli hit sequence {I t I t 1,..., I 1 } T t=1 with coverage p. Christoffersen(1998) noticed this fact and proposed a conditional coverage test. In addition, the conditional expectation of {I t p} T t=1 must always be equal to zero given any information known at t 1. Engle and Manganelli(2004) developed a dynamic quantile test based on this fact. However, their papers didn t illustrate how to apply their tests for the model specification. The most common parametric method for the VaR estimation is the Variance-Covariance method and the most popular model of this family is the Autoregressive Conditional Heteroscedasticity (ARCH) introduced by Engle(1982) and extended by Bollerslev(1986). Harris and Guermat(2005) shows even if a volatility model generates unbiased variance 2

3 forecasts, non-linear transformations of these forecasts will be biased estimators because of Jensens inequality. The VaR forecast using GARCH are the product value of the selected percentile of the return distribution and the square root of the forecasted conditional variance, a non-linear transformations of the variance forecasts. Thus conventional GARCH model might be a good volatility model but would be an unsatisfied model for the VaR estimation. This paper proposes a Two-stage VaR model to correct the VaR bias. This Two-stage VaR model jointly apply the parametric volatility model and the semi-parametric quantile regression to make a two-step bias reduction. In the first-stage, instead of modeling the variance process, we propose to model the standard deviation directly to avoid the nonlinear transformation bias of conventional GARCH model in the VaR estimation. Based on the generalized conditional coverage test, we are able to select some informative variables from a list of candidate variables and then construct the second-stage model via quantile regression. The rest of paper is organized as follows. Section 2 shows the definition of Value-at-Risk. In section 3, we illustrate the non-linear transformation bias of traditional GARCH model and introduce the conditional standard deviation models to estimate the VaR. Section 4 proposes a generalzied conditional coverage test. Section 5 develops the semi-parametric quantile regression model as the second-stage VaR model. The two-stage VaR model is applied to the stock index, individual stock and exchange rate data in section 6 where we compare the performance with conventioanl GARCH and PARCH. Section 7 concludes. 2 Definition of VaR Consider the return on an asset or portfolio in period t denoted by r t. The VaR measure with loss probability p, V ar t t 1 (θ, p), is defined by Pr ( ) r t < V ar t t 1 (θ, p) Ω t 1 = p, (1) where Ω t represents information available at time t and θ is a set of parameters associated with the VaR model. The VaR places an upper bound on losses in the sense that losses will exceed the VaR threshold with only a small probabilility p. Typical values of the loss probability range from 1 to 5 percent, depending on the time horizon (here normalized to one.) 3

4 3 Parametric Approaches 3.1 GARCH Models Parametric models in the location-scale class are based on the assumption that returns can be represented as r t = µ t + σ t ε t, (2) where µ t and σ t are Ω t 1 -measurable and ε t has a zero-location unit-scale probability distribution that may have additional shape parameters (such as the degrees of freedom associated with the t-distribution.) A very popular formualtion for σ t is the GARCH(p, q) model: p q σt 2 = α 0 + α i ε 2 t i + β j σt j, 2 (3) i=1 j=1 where the error terms ε t follow either a standard normal or a t-distribution. In practice, the order is often taken as (p, q) = (1, 1). The benchmark measure advocated in Morgan s (1996) RiskMetrics sets the conditional mean, µ t, in (2) to a constantand specifies the variance as σt 2 = (1 α)ε 2 t 1 + ασt 1, 2 (4) where α is simply set to 0.94 for daily data. The RiskMetrics specification in (4) is seen to be a special case of (3). Given forecasts of the conditional mean and variance, ˆµ t t 1 and ˆσ t t 1 2, the VaR is then computed as V ar t t 1 (ˆθ, p) = ˆµ t t 1 + F 1 (p) ˆσ t t 1 2, (5) where F 1 is the inverse of the cummumlative distribution function of the error terms ε t. Abundant literature apply the GARCH family models for the VaR estimation, such as Angelidis, Benos and Degiannakis (2004), Burns (2002), Giot and Laurent (2004), So and Yu (2006), etc. Suppose that a volatility model is well specified in that it generates unconditionally unbiased forecasts; i.e., E[ˆσ t t 1 2 ] = E[σ2 t ] = σ 2. From (5) it is clear that the VaR is a nonlinear transformation of the variance, in this case a square-root transformation. More generally, consider a nonlinear transformation of the variance, g( ), which is applied to the forecast of the variance. 4

5 The unconditional bias of g(ˆσ t 2 ) is given by B t = E(g(ˆσ t 2 ) g(σt 2 )). In general, B t will not be zero because of Jensen s inequality. That effect can be approximated by taking second-order Tatlor series expansions around σ 2 to find that g(ˆσ t 2 ) g(σ 2 ) + g (σ 2 )(ˆσ t 2 σ 2 ) g (σ 2 )(ˆσ t 2 σ 2 ) 2 (6) and g(σ 2 t ) g(σ 2 ) + g (σ 2 )(σ 2 t σ 2 ) g (σ 2 )(σ 2 t σ 2 ) 2, (7) where g ( ) and g ( ) are the first and second derivatives with respect to σ 2. By taking the unconditional expectation of (6) and (7), an approximation of the unconditional bias of g(ˆσ 2 t ) is found to be B t 1 2 g (σ 2 ) [ var(ˆσ 2 t ) var(σ 2 t ) ] (8) The terms on the right-hand side of (8) can each be estimated. g(σ 2 t ) = σ 2 t In the case of VaR, so that g (σ 2 ) = (1/4)σ 3. That term can be estimated by first using the sample variance of the demeaned returns r t ˆµ t to get ˆσ 2, and then forming (1/4)ˆσ 3. Similarly, the sample variance of ˆσ t can be used to estimate var(ˆσ 2 t ). Finaly, an estimate of var(σ 2 t ) can be based on the realized volatility a well-known quantity that is constructed from high-frequency intraday returns. Given the times a = t 0 < t 1 <... < t m = b at which asset prices p(t) are observed, the realized volatility can be defined following Hansen and Lunde (2005) by RV Ξ [a,b] = m {p(t i ) p(t i 1 )} 2, (9) i=1 for a specific partition Ξ = {t 0,..., t m }. If the intraday returns have a zero mean and are uncorrelated, the realized volatility in (9) is an unbiased and consistent estimator of the daily variance σ 2 t (Andersen et al. 2001). A model-free measure of daily variance may then be obtained by summing high-frequency squared returns. If the sampling frequency is too high, however, a bias estimator may result from microstructure effects such as bid-ask bounces, price discreteness, and non-syncronous trading. As a tradeoff between these two biases, Andersen et al. (2001) suggest using 5- minute returns to compute the realized volatility. In order to illustrate the bias in VaR, we consider a sample of stock returns for General Motors (GM), International Business Machines (IBM), and Boeing (BA). The sample covers 5

6 Table 1. Estimated Unconditional Bias: In-Sample Forecasts interval ˆσ 1 4 σ 3 Bias E(σ t ) Bias/E(σ t ) GM % % % % BA % % % % IBM % % % % the period January 3, 2003 to December 31, Volatility forecasts are based on a GARCH(1,1) model and various intraday intervals from 1 to 15 minutes are considered in the estimation of realized volatility. Table 1 reports the estimated unconditional bias for the one-day in-sample forecasts of VaR. The bias is positive for the three considered stocks, ranging from 20.52% to 85.22%. For each stock, the bias does not vary much across the different intraday time intervals. The magnitude of these biases is comparable to those in Harris and Guermat(2005). They report estimates of the unconditional bias of the VaR (based on a GARCH(1,1) specification) that range from 9.9% to 61.7%. 3.2 PARCH Models The bias in VaR arising from the nonlinear transformation of the volailtiy forecast can be avoided by modeling directly the conditional standard deviation: σ t = α 0 + p q α i ε t i + β j σ t j. (10) i=1 j=1 6

7 Taylor (1986) and Schwert (1989) consider such a specification for the dynamics of the conditional standard deviation. The models in (3) and (11) are nested in the more general Power ARCH (PARCH) specification of Ding, Granger, and Engle (1993) defined as σ λ t = α 0 + p q α i ( ε t i ) λ + β j σt j, λ (11) i=1 j=1 where λ plays the role of a Box-Cox transformation of the conditional standard deviation. He and Teräsvirta (1999a,b) study the properties of the APARCH model. The PARCH model in (11) provides a way to estimate and test the value of λ. It is important to note, however, that although a particular value of λ might be preferred on the basis of, say, a comparison of the attained values of the likelihood function, that in itself does not mean that the resulting VaR forecasts will be unbiased or efficient. In the next section, we propose a general test of VaR forecast efficiency. 4 Testing VaR Forecasts Let Ξ in sequentially index the in-sample observations and Ξ out the out-of-sample ones. Correspondingly, let ˆθ t, t Ξ in represent the parameter estimates used to generate the time-(t + 1) VaR forecast for t + 1 Ξ out. We will say that the VaR forecasts are efficient with respect to Ω t if E [ ( ) s r t+1 V ar t+1 t (ˆθ t, p) Ω t ] = p, t Ξ in, t + 1 Ξ out, (12) where s(x) = 1 if x < 0, and s(x) = 0 if x 0. The definition of efficiency in (12) follows Giacomini and White (2005) and Giacomini and Komunjer (2005) in that it depends on the parameter estimates rather than on population values as in Christoffersen (1998). This represents a paradigm shift from evaluating a forecasting model to evaluating a forecasting method, which includes the model as well as the estimation procedure. The moment condition in (12) states that no information available to the risk manager at time t should help predict whether time t + 1 s return falls above or below the VaR forecast reported at time t. using the techniques of Campbell and Dufour (1995). This type of orthogonality condition can readily be tested An analogue of the t-statistic is 7

8 given by the sign statistic S Ξout (g) = t+1 Ξ out s [( ) ] r t+1 V ar t+1 t (ˆθ t, p) g t, (13) where g t = g t (Ω t ), t Ξ in, is a sequence of measurable functions of the information set Ω t. Suppose that the instruments {z t, z t,...} are contained in the information set Ω t. The functions g t (z t, z t 1,...) allow one to consider various (possibly nonlinear) transformations of the instruments, provided that g t only depends on information contained in Ω t. The sign statistic S Ξout can be used to evaluate the out-of-sample performance of a VaR forecasting model. When no instruments are included and g t 1 = 1, (13) becomes a test of unconditional unbiasedness. Serial correlation among the VaR violations can be detected by including instruments like z t = r t V ar t t 1 (ˆθ t 1, p). In this respect, the proposed tests are comparable to the dynamic quantile tests of Engle and Manganelli (2004). Inference based on the dynamic quantile tests relies on asymptotic theory. The question of the reliability of such procedures in finite samples naturally arises. On the contrary, the exact finite-sample null distribution of the sign statistics in (13) is characterized as follows. Proposition 1. Let V ar t+1 t (ˆθ t 1, p) be a sequence of VaR forecasts such that Pr[r t+1 = V ar t+1 t (ˆθ t, p)] = 0 for t Ξ in and t + 1 Ξ out. Suppose further that Pr[g t = 0] = 0 for t Ξ in. Then, under condition (12), the statistic S Ξout (g) follows a binomial distribution with probability of success p and number of trials T out, where T out denote the cardinality of Ξ out. The proof of Proposition 1 follows directly from Campbell and Dufour (1995). This distributional result holds under very general assumptions. In particular, none of the parametric assumptions (stationarity, finite moments,...) required for the dynamic quantile tests are required. Even for relatively small sample sizes, the standard normal distribution provides a very good approximation to the standardized version S Ξ out (g) = S Ξ out (g) T out p Tout p(1 p). (14) Suppose that one has several sequences of instruments or transformations yielding S Ξ out (g i ), i = 1,..., n. In order to combine these tests, consider the statistic S Ξ out = max 1 i n S Ξ out (g i ). (15) 8

9 Given that the individual statistics share the same marginal distribution, the criterion in (15) is equivalent to choosing the statistic with the smallest two-sided p-value. This type of criterion is derived from the logical equivalence that (12) is true if and only if it holds true for each considered instrument or transformation. This method of combining tests was suggested by Tippett (1931) and Wilkinson (1951) in the case of independent test statistics. It is clear, however, that the exact dependence structure among the resulting statistics S Ξ out (g i ), i = 1,..., n, may very well be intractable. Nevertheless, the following distributional result shows how to combine inference based on the multiple sign tests. Proposition 2. Let V ar t+1 t (ˆθ t 1, p) be a sequence of VaR forecasts such that Pr[r t+1 = V ar t+1 t (ˆθ t, p)] = 0 for t Ξ in and t + 1 Ξ out. Suppose further that Pr[g t,i = 0] = 0 for t Ξ in and i = 1,..., n. Then, under (12) and conditional on g i, i = 1,..., n, the statistic S Ξ out is distributed like S Ξ out = max S Ξout(g i ) T out p (16) 1 i n Tout p(1 p) with S Ξout (g i ) = s [B t+1 g i,t ], for i = 1,..., n, (17) t+1 Ξ out where B t+1 are mutually independent Bernoulli variables such that B t+1 = 1 with probability p and B t+1 = 0 with probability 1 p. The result in Proposition 2 can be used to construct conditional tests of the efficiency condition in (12). Step 1 is to generate a sequence of Bernoulli draws B 1,..., B Tout compute the statistics in (17). Note that the same realized sequence B 1,..., B Tout and is used n times when computing S Ξout (g i ) for i = 1,..., n. Those n values of S Ξout (g i ) are then used in Step 2 to compute (16). A distribution of simulated values of S Ξ out can be generated by repeating Steps 1 and 2, say, M times, and a one-sided p-value for the original value S Ξ out is then computed as P ( S Ξ out ) = 1 M M I( S Ξ out,j > SΞ out ), j=1 where I( ) is the indicator function and S Ξ out,j, j = 1,..., M, are the simulated values of the test statistic. It is important to emphasize that the simulated statistics are computed holding the g s constant in the same order used to compute the original statistic. Finally, note that the Monte Carlo procedure proposed by Dwass (1957) can be used to obtain a precise p-value even when M is small; see also Dufour and Khalaf (2001). 9

10 5 Semi-Parametric Approach The proposed test procedure can be used to backtest a VaR model. For example, a sequence of VaR forecasts V ar t+1 t (ˆθ t, p) from a parametric model could be backtested for forecast efficiency with respect to g t,1,..., g t,n. In the event that the VaR forecasts are found to violate condition (12), one might want to reconsider the model specification before generating VaR forecasts over a subsequent period. Redefine Ξ in to be a sequential index of all the observations available at the time of backtesting, and Ξ out will now refer to future observations from this point in time. Following the approach of Chernozhukov and Umantsev (2001), one could obtain (insample) efficiency gains with a semi-parametric VaR specification of the form f t t 1 (β, p) = β 0 + β 1 V ar t t 1 (ˆθ t 1, p) + n γ i g t 1,i, (18) where the vector β has dimension n + 2. The model in (18) extends the approach in Taylor (1999), where the conditional quantile is specified as a linear function of (transformations of) estimated volatilities. The regressors here are taken as the instruments or transformations with respect to which the estimated VaR s are found to be inefficient. i=1 In this sense, the semi-parametric approach can be seen as an attempt to rehabilitate the original VaR model. The specification in (18) can also be interpreted as a version of Engle and Manganelli s (2004) conditional autoregressive VaR (CAViaR) model. A generic CAViaR specification takes the form f t t 1 (β, p) = β 0 + q β j f t j t j 1 (β, p) + j=1 r γ k l(x t k ), (19) where l( ) is a function of a finite number of lagged values of observables. The parameters β can be estimated over Ξ out by the quantile regression 1 min β T in k=1 t Ξ in [p s(r t f t (β, p))] [r t f t (β, p)]. (20) In the case of CAViaR, the problem in (20) is complicated by the fact that (19) is nonlinear in the parameters. On the contrary, the semi-parametric regression quantile model is entirely based on observables and is linear in the parameters. Further, the VaR forecasts on the right-hand side of (18) may even be those from a CAViaR model à la Engle and Manganelli. Linear regression quantiles were introduced by Koenker and Bassett (1978) 10

11 and there are several statisitcal software programs for the point estimation of regresison quantile parameters. Let ˆβ t denote the argument that solves (20) for the semi-parametric specification. With these parameter estimates, semi-parametric VaR forecasts are obtained as f t+1 t ( ˆβ t, p) = ˆβ 0 + ˆβ n 1 V ar t+1 t (ˆθ t, p) + ˆγ i g t,i, (21) for t+1 Ξ out, t Ξ in. Note that, although not necessary, the parametric VaR forecasts in (21) are based on updated parameter estimates; i.e., θ can be reestimated over the redefined set Ξ in before computing ˆβ t. Of course, the VaR forecasts based on the semi-parametric model can themselves be backtested over the new set Ξ out. The next section illustrates this approach. i=1 6 Application 6.1 A Monte Carlo Example Under the following specification of our Monte Carlo experiment, we illustrate how to apply our methodology and how powerful the Generalized conditional coverage test are in rejecting inferior VaR models under the realistic finite-sample conditions. Imagine a univariate time series generated by the following process r t = bµ t + σ t ε t, ε t i.i.d.n(0, 1) (22) σ t = α 0 + α r t 1 + βσ t 1 (23) where µ t is the rolling standard deviation with rolling window 500. The parameters are specified as {b = 1, α 0 = 1e 6, α = 0.1, β = 0.8}. The total number of observations is 2500 and the first 500 observations are dropped. From the 501 st to 1500 th observations are the in-sample data for the VaR model estimation. The remaining 1,000 observations are the out-of-sample data for the backtesting. At the first stage, we consider a volatility model based on the conditional standard deviation model (23) and estimate the VaR by Eqn. (27), where Φ 1 is correctly specified as the inverse CDF of Normal distribution with probability θ. The first stage VaR model is correctly specified with respect to the volatility process. However, the mean process is misspecified since we don t include the rolling standard deviation in the VaR calculation. 11

12 We apply the generalized conditional coverage test to backtest this VaR forecast. We include two instrumental variables: (1) the conditional variance forecast from the conventional GARCH(1,1) model; and (2) the rolling standard deviation. If this test is valid, we expect the joint test should reject the null hypothesis that the model is well specified. In addition, we expect the individual test of the conditional variance forecast should fail to reject the null, while the individual test of the rolling standard deviation should reject the null. There are 1,000 Monte Carlo simulations for the model estimation and test statistics calculation and 1,000 bootstrapping simulations for the critical value estimation. Finally all the test statistics and the critical values are taken average. The simulation results are given by the following table. Obviously the simulation results prove that our expectation is correct. Since the joint test rejects the null hypothesis, we need include some instrumental variable into the second-stage model to reduce the bias. The individual test results indicate the rolling standard deviation is a good candidate. Thus we include the rolling standard deviation and first stage VaR forecasts into the second-stage model (??). We notice the parameter for the first-stage VaR is fixed as unit and there is no intercept in the model. We apply the second stage model for the second round simulation. The specification of the data generation process and simulation repetition remain the same. The results are reported in the following table. The generalized conditional coverage test fails to reject the null which indicate the second-stage model is a well specified model. The parameter estimates of the volatility model are biased because we assume the conditional mean is zero in the mean equation which is misspecified. However, in the second-stage model, we include this missing information and reduce the VaR bias from this mis-specification. p 1 p 2 p ˆθ ˆα0 ˆα ˆβ First-stage e Second-stage p 1: p-value for the conditional variance forecast p 2: p-value for the rolling standard deviation p : joint test p-value 12

13 ˆθ: instrument parameter estimate in the second-stage model ˆα 0, ˆα, ˆβ: volatility model estimates in the first-stage model 6.2 Power of the test In order to get some evidence on the power of the generalized conditional coverage test in rejecting inappropriate VaR forecast, we extend the above Monte Carlo experiment. The data generation process follows above specification and the first-stage model is still (23). Obviously this VaR model is not well specified since it discards the rolling standard deviation in the mean equation. Figure (1) shows the power of the generalized conditional coverage test to reject the misspecified first-stage model when data is generated by Eqn. (22) and (23). I have let the sample size, n, vary between 250 and 2,000 in increments of 250 and the parameter of rolling standard deviation, b, vary between 0.25 and 2.0 in increments of The Monte Carlo repetition and Bootstrapping repetition are both 1,000. The shape of the power plots illustrates that, given a certain sample size, the power of the test increases with the increasing of parameter b. When b is larger, the first-stage model becomes more misspecified since it discards the information from the rolling standard deviation, which makes the null easier be rejected. Figure (1) also illustrates the sample size is critical for the test. When the sample size increases, the power increases respectively. In general the test is powerful since the power of the test is above 70% under the most of settings. 6.3 Empirical Application In this section, we implement our two-stage VaR model on real data. The performance of the Two-stage VaR model is compared to that of conventional GARCH and PARCH VaR models. We have interest on VaR estimation of the international stock index return, including S&P 500 Index return (SP), HANG SENG Index return (HSI), All Ordinaries Index return (AOI), FTSE 100 Index return (FTSE) and Nikkei 225 Index return (N225). In order to check the robustness of our method, we also consider an individual stock General Motors stock return (GM) and an exchange rate USD/GBP exchange rate return (UU). We compute the daily return as 100 times the difference of the log of the prices. S&P 500 Index return and General Motors stock return are from Simone Manganelli s dataset 13

14 prepared for Engle and Manganelli (2004). All of the other returns are from Yahoo Finance website. Table 1 gives the summary statistics of our data. From the table we recognize that All of the return series are weakly autocorrelated with first order autocorrelation less than In addition, all of the return series experience heavy tailed with kurtosis larger than 3. We notice US stock market are more heavy tailed and more volatile, shown from the standard deviation and kurtosis of S&P 500 and GM. Nearly all of the return series are negative skewed. An interesting finding is the mean return and the standard deviation are positive correlated, which is an evidence of the theory of high risk, high return. We use the first 1, 000 observations to estimate the volatility model and the next 1, 000 observations (in-sample data) to estimate the second-stage model. The rest of the observations are for the out-of sample testing. We estimate 1% and 5% one-day ahead VaRs, using the two-stage model described in section 4. We use the conditional standard deviation(1,1) model as the first-stage model. For the comparison purpose, we also report the results of the VaR forecast from conventional GARCH(1,1) and PARCH(1,1). In the volatility model, t-distribution innovation is used. The three models we considered are given by: F irst stage : σ t = α 0 + α 1 r t 1 + β 1 σ t 1 (24) GARCH(1, 1) : σt 2 = α 0 + α 1 a 2 t 1 + β 1 σt 1 2 (25) P ARCH(1, 1) : σt λ = α 0 + α 1 ( r t 1 ) λ + β 1 σt 1 λ (26) With the volatility forecasts from these three models, we apply the following equation to calculate the VaR forecast. Q t = V ar t = Φ 1ˆσ t where Φ 1 is the inverse CDF of t-distribution with probability p. The instruments used in the generalized conditional coverage test are the standard deviation forecasts from model (24), the variance forecasts from model (25) and the volatility forecasts from model (26). The instruments rejecting the generalized conditional coverage test are chosen as input in the second-stage model given by: q f t (θ) = V ar t + θ i Zi,t 1( ˆβ), (27) where V ar t is the VaR estimation from the first stage model (24), Zi,t 1( ˆβ) is the included instrumental variable based on the generalized conditional coverage test. 14 i=1

15 If the joint test statistic of first-stage model does not reject the null, we will keep the first-stage model as the desired model and don t need apply the second-stage model (27) % VaR Table 2 presents the test results for 5% VaR estimation. The testing periods are separated into two periods, in-sample period and out-of-sample period. {p 1, p 2, p 3} are the individual test p-value for the three instruments. p is the joint test p-value and Vio(%) is the violation ratio. If the joint test of first-stage model fails to reject null, no result will be reported for the second-stage model. The term in-sample is only applied to the second-stage model and is in fact out-of-sample for GARCH, PARCH and first-stage model. For S&P 500 index return, in the in-sample period, none of the three VaR models pass the test. All of the three individual tests of the first-stage model reject the null, thus we include all of three instruments into the second-stage model. After the estimation of the second-stage model, we apply this model to the in-sample data and pass the test. Since the second-stage model is a quantile regression model, the violation ratio is very close to the correct coverage 5%. In the out-of-sample period, the second-stage model has very good performance since it pass the test and the violation ratio is close to the 5%. In this period, GARCH(1,1) also pass the test. For HANG SENG index return, in the in-sample period, no model has a chance to pass the test. Based in the individual tests, we include the variance forecast from GARCH model as the instrument in the second-stage model. The including of the instrument is proved to be successful in the out-of-sample period. Among all of the VaR models, the second-stage model is the only model passes the test. All Ordinaries index return has a different story. All three models perform well in the first sub-period. This indicates we don t need use the second-stage model. Thus we keep using the first-stage model in the second sub-period. PARCH and our first-stage model pass the test in the second sub-period. For FTSE 100 index return, GARCH and first-stage model pass the test in the in-sample period, which implies we can continue using the first-stage model. Surprisingly, all three models fail the test in the second sub-period. It seems there are some structural breaks between these two periods. We observe the violation ratio differs a lot for all three models between two periods. 15

16 Nikkei 225 index return support our first-stage model. Although all three models pass the test in the first sub-period, only our model pass the test in the second sub-period. Our model also generates violation ratio close to the 5%. USD/GBP exchange rate return also strongly supports our model. From the in-sample test, we get to know the standard deviation from our model should be included as an input into our second-stage model. As a promising result, our second-stage model is the only model pass the test in both periods. We notice the violation ratios of the second-stage model are amazingly close to 5% in both periods. Finally, all three models generate satisfactory VaR foreacst for General Motors stock return since all models pass the test in both sub-periods. In summary, Two-stage model has superior performance in the 5% VaR estimation using our data. Compared to GARCH and PARCH, Two-stage model performs consistently well in both sub-periods and all considered return series. One observation is that the violation ratio is informative but should not be the only criteria in the VaR model evaluation % VaR Table 3 presents the test results for 1% VaR estimation. The performance of Two-stage model becomes mixed compared to the 5% VaR estimation. For the S&P 500 index return, only GARCH and second-stage model pass the in-sample test. All three models fail to pass the test. It looks hard to generate satisfactory 1% VaR estimation for the S&P 500 index return. The first-stage model passes the first sub-period test for HANG SENG index return and we keep using it in the second sub-period. However, our model fail to pass the test in this period. We notice GARCH passes the test in both period. AOI, N225 and GM have similar testing results. All three models pass the test in both periods. FTSE and UU have similar pattern in the generalized conditional coverage test. All three models pass the test in the first sub-period. However, none of them passes the test in the second sub-period. Obviously the test results are pretty mixed for the 1% VaR estimation. An interesting observation is that all of the volatility models have good performance in the first sub-period but might not be in the second sub-period. This phenomenon suggests the behavior of the 16

17 very extreme tail of the return distribution might change over time. Thus the volatility models maybe performs well in the first sub-period but not in the second sub-period. 7 Conclusion Due to the existence of the non-linear transformation bias in the VaR estimation using GARCH model, we propose a new Two-stage VaR model based on our generalized conditional coverage test. In order to eliminate the non-linear transformation bias, the first-stage model starts from a parametric conditional standard deviation model and is then tested by the generalized conditional coverage test. If the first-stage model passes the test, we will keep using this model since it is well specified. If the model fails to pass the test, we will incorporate some additional variables, selected based on the test, into the second-stage model, a semi-parametric quantile regression model. Simulation results indicate that our generalized conditional coverage test is powerful. Applications to real data further prove that the Two-stage model can generate satisfactory VaR estimates for the stock index return. As a robustness check, the Two-stage VaR model also performs well for the individual stock return and exchange rate series. For the 5% VaR estimation, Two-stage model performs consistently better than GARCH and PARCH. For the 1% VaR estimation, the results are more complicated and need further research in the future. 17

18 References Angelidis, T., A. Benos and S. Degiannakis, (2004). The Use of GARCH Models in VaR Estimation, Statistical Methodology, 1(2), Bao, Y. and A. Ullah (2004). Bias of a Value-at-Risk estimator, Finance Research Letters, 4(1), Burns, P., (2002). The Quality of Value at Risk via Univariate GARCH, working paper, Burns Statistics Limited Bollerslev, Tim (1986). Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics 31, Christoffersen, P. (1998). Evaluating Interval Forecasts, International Economic Review, 1998, Volume 39, Christoffersen, P., J. Hahn and A. Inoue (2001). Testing and Comparing Value-at-Risk Measures, Journal of Empirical Finance, 2001, Volume 8, Christoffersen, P. and D.Pelletier (2004).Backtesting Value at Risk: A Duration-Based Approach, Journal of Financial Econometrics, 2004, Volume 2, Ding, Zhuanxin, C. W. J. Granger and R. F. Engle, (1993). A long memory property of stock market returns and a new model, Journal of Empirical Finance 1, Engle, Robert F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kindom Inflation, Econometrica 50(4), Engle, R. and S. Manganelli (2004). CAViaR: Conditional Value at Risk by Quantile Regression, Journal of Business&Economic Statistics, 2004, 22(4), Giot, P. and S. Laurent, (2004). Modeling daily Value-at-Risk using realized volatility and ARCH type models, Journal of Empirical Finance, 11(3), Harris, R.D.F. and C. Guermat (2005). Bias in the Estimation of Non-Linear Transformations of the Integrated Variance of Returns, forthcoming, Journal of Forecasting. Lopez J. A. (1998), Methods for evaluating value-at-risk estimates. Federal Reserve Bank of New York research paper n Riskmetrics (1996). Technical Document. Technical report. J.P. Morgan Sarma, M., S. Thomas, and A. Shah (2003), Selection of Value at Risk models, Journal of Forecasting, 22(4),

19 Schwert, G.W. (1989). Why does stock market volatility change over time?, Journal of Finance, 44, So, M.K.P. and Yu, P.L.H. (2006). Empirical analysis of GARCH models in Value at Risk estimation, Journal of International Financial Markets, Institutions & Money, 16(2), forthcoming Taylor, S.J.(1986). Modeling Financial Time Series, John Wiley and Sons Ltd. 19

20 power b n Figure 1: Power of the General conditional coverage test 20

21 Table 1. Summary Statistics The table reports summary statistics for S&P 500 Index return (SP), HANG SENG Index return (HSI), All Ordinaries Index return (AOI), FTSE 100 Index return (FTSE), Nikkei 225 Index return (N225), USD/GBP exchange rate return (UU) and General Motors stock return (GM). We compute the daily return as 100 times the difference of the log of the prices. S&P 500 Index return and General Motors stock return are from Simone Manganelli s dataset prepared for Engle and Manganelli (2004). All of the other returns are from Yahoo Finance website. ρ is the first order autocorrelation. Description Period obs. Mean S.D. Skew. Kurt. ρ SP S&P 500 Index (US) 04/07/86-04/07/ HSI HANG SENG Index (HK) 01/02/90-12/31/ AOI All Ordinaries Index (Aus) 01/02/90-12/31/ FTSE FTSE 100 Index (UK) 01/02/90-12/31/ N225 Nikkei 225 Index (Japan) 01/02/90-12/31/ UU USD/GBP 01/02/90-12/31/ GM General Motors 04/07/86-04/07/

22 Table 2. Estimates and Relevant Statistics for the 5% VaR Models In-sample Out-of-sample p 1 p 2 p 3 p Vio(%) p 1 p 2 p 3 p Vio(%) SP GARCH PARCH First-stage Second-stage HIS GARCH PARCH First-stage Second-stage AOI GARCH PARCH First-stage Second-stage FTSE GARCH PARCH First-stage Second-stage N225 GARCH PARCH First-stage Second-stage UU GARCH PARCH First-stage Second-stage GM GARCH PARCH First-stage Second-stage 22

23 Table 3. Estimates and Relevant Statistics for the 1% VaR Models In-sample Out-of-sample p 1 p 2 p 3 p Vio(%) p 1 p 2 p 3 p Vio(%) SP GARCH PARCH First-stage Second-stage HIS GARCH PARCH First-stage Second-stage AOI GARCH PARCH First-stage Second-stage FTSE GARCH PARCH First-stage Second-stage N225 GARCH PARCH First-stage Second-stage UU GARCH PARCH First-stage Second-stage GM GARCH PARCH First-stage Second-stage 23