Generalized Autoregressive Score asymmetric Laplace Distribution and Extreme Downward Risk Prediction
GG ENERALIZED A UTOREGRESSIVE S CORE ASYMMETRIC L APLACE D ISTRIBUTION AND E XTREME D OWNWARD R ISK P REDICTION
A P
REPRINT
Hong Shaopeng ∗ School of StatisticsCapital University of Economics and TradeBeiJing, ChinaOctober 14, 2020 A BSTRACT
Due to the skessed distribution, high peak and thick tail and asymmetry of financial return data,it is difficult to describe the traditional distribution. In recent years, generalized autoregressivescore (GAS) has been used in many fields and achieved good results. In this paper, under theframework of generalized autoregressive score (GAS), the asymmetric Laplace distribution (ALD)is improved, and the GAS-ALD model is proposed, which has the characteristics of time-varyingparameters, can describe the peak thick tail, biased and asymmetric distribution. The model isused to study the Shanghai index, Shenzhen index and SME board index. It is found that: 1) thedistribution parameters and moments of the three indexes have obvious time-varying characteristicsand aggregation characteristics. 2) Compared with the commonly used models for calculating VaRand ES, the GAS-ALD model has a high prediction effect. K eywords Generalized Autoregressive Score · Asymmetric Laplace Distribution · VaR · ES As a special kind of economic system, the financial market has emerged with a series of unique macro laws, whichare usually called "stylized facts". For example, the return distribution is sharp and asymmetrical. Due to Knightuncertainty, static parameter models usually cannot accurately describe the characteristics of financial returns, anddynamic parameter models are needed to characterize the distribution of returns.The structure of this paper is as follows: The first part reviews the research results of asymmetric Laplacian distributionand generalized autoregression at home and abroad, the second part builds the Laplacian distribution based on thegeneralized autoregressive score, and the third part uses generalized autoregressive The Plass distribution conductsempirical research on China’s Shanghai Stock Index, Shenzhen Index and SME Board Index. The fourth part isconclusions and recommendations.
In risk management, VaR and ES have become more reliable risk measurement indicators, and how to calculate VaR andES more effectively has always been a hot issue in risk management. The core of calculating VaR and ES is to describethe distribution of its rate of return. However, as a special kind of economic system, the financial system has manyunique and typical facts. For example, financial returns have typical facts such as volatility clustering, autocorrelation,spikes and fat tails. It is unreasonable to assume that the rate of return is normally distributed. Bollerslev (1987) [1] a r X i v : . [ q -f i n . R M ] O c t PREPRINT - O
CTOBER
14, 2020proposed to use the t distribution to characterize the fat tail characteristics of the return distribution. Often financialmarket returns show right-skewed characteristics, and normal distribution and t-distribution, as distributions with zeroskewness, cannot be described. Hansen (1994)[2] proposed to use the biased t distribution to simultaneously characterizethe fat tail and biased characteristics of the return distribution. However, in the financial market, the asymmetry of gainsand losses makes the distribution of returns appear asymmetry in the left and right tails. Zhu Dongming and Galbraith(2010)[3] proposed two asymmetric t distributions with higher degrees of freedom than the biased t distribution basedon the biased t distribution. These two t distributions can better characterize the asymmetry of the return rate bycontrolling the asymmetry of the left and right tails. Keming Yu and Zhang (2005) [4] proposed an asymmetric Laplacedistribution (ALD) controlled by three parameters. Compared with other complex distributions (such as generalizederror distribution), ALD has fewer parameters, and each order moment has a clear display expression.Due to the superiority of ALD, a large number of scholars have conducted research on ALD. Liu Jianyuan andLiu Qiongsun (2007)[5] used ALD to fit the returns of China’s stock market, and compared the symmetric Laplacedistribution, the normal distribution and the asymmetric Laplace distribution to calculate the effectiveness of VaR. ThePlass distribution has the best fitting result. Du Hongjun and Wang Zongjun (2013) [6] used the asymmetric Laplaciandistribution to calculate the dynamic risk VaR and CVaR. The asymmetric Laplacian distribution can well describethe characteristics of bias, spikes and fat tails. Liu Pan and Zhou Ruomei (2015) [7] used AEPD, AST and ALDdistributions to describe the typical facts of financial asset returns, and found that the ALD distribution is significantlybetter than other distributions in calculating the long VaR value of the bottom quantile. . Based on ALD, Tang Jianyanget al. (2018) [8] proposed the AR-GJR-GARCH model with ALD error distribution, which has a better performance incalculating VaR and ES than traditional models that measure VaR and ES.
Creal (2012)[9] proposed the Generalized Autoregressive Scoring Model (GAS) as a framework for time-varyingparameter modeling, which has been widely used in recent years. Such as GAS-EWMA (Lucas, 2016[10]), beta-t-EGARCH (Harvey and Sucarrat, 2014[11]), GAS-Copula (Oh and Patton, 2016[12]), GAS-factor Copula (leaf fiveFirst Class, 2018[13]), GAS-D-Vine-Copula (Zou Yumei, 2018[14]), GJR-GARCH-skew t-GAS-copula (Xiao-Li Gong,2019[15]) and GAS-EGARCH -EGB2 (Yao Ping et al., 2019[16]). The above models have performed well in the fieldof risk management.It can be seen from the literature review that the asymmetric Laplacian distribution has a good performance in fitting therate of return, which can characterize the typical facts of the peak, thick tail, bias and asymmetry of the rate of return,and are similar. Compared with the normal distribution and the t distribution, the asymmetric Laplacian distribution hasa better performance in calculating VaR and ES. However, it is impossible to describe the time-varying nature of eachorder moment of the rate of return, and there are defects in the empirical study. In recent years, the GAS model hasbeen proposed as a time-varying parameter modeling framework. A large number of documents have proved that usingthe GAS framework to make the model dynamic can improve the performance of the model. In summary, this articleconsiders expanding the asymmetric Laplace distribution under the GAS framework to make its parameters dynamic,and strive to better describe the distribution of returns and more accurate predictions of extreme risks.The innovations made in this paper are as follows: 1) Use the GAS model to extend the ALD distribution to theGAS-ALD distribution. 2) Apply GAS-ALD to my country’s securities market to dynamically measure conditionalmean, conditional volatility, conditional skewness and conditional kurtosis. 3) Compare the GAS-ALD model withmainstream VaR and ES models. On the one hand, it enriches the calculation methods of VaR and ES, on the otherhand, it can provide investors with investment basis. It has theoretical and practical significance.
The generalized autoregressive scoremodel is a data-driven model. It is a unified framework for time-varying parametermodeling. Compared with other models, GAS has the advantage that it makes full use of distributed information toconstruct a time-varying parameter evolution process.Suppose the conditional density of the rate of return y at time t is: y t | y t − ∼ f ( y t ; θ t ) (1)Among them, y t − = ( y , . . . , y t − ) , which means all the information of y in the past moment, θ t Is a time-varyingparameter set. From the formula (1), it can be seen that the conditional distribution only depends on the information at2
PREPRINT - O
CTOBER
14, 2020the past moment and the parameter set. The core of the GAS model is the parameter evolution process of time-varyingparameters ((2) ). θ t +1 ≡ κ + As t + B θ t (2)Among them, κ is a constant matrix, A and B are coefficient matrices, then the score s t is the driving mechanism,usually defined as (3) and (4) , (5) and (6). I t ( θ t ) is the Fisher information matrix of the time-varying parameter set. γ is usually in { , , } , used to adjust the score to calculate the final update item s t , this article selects γ = 0 , that is, s t = ∇ t . It can be seen that the parameter evolution in the GAS model is divided into two parts, the first part is drivenby the score function, and the second part is the AR(1) process. s t = S t ( θ t ) ∇ t ( y t , θ t ) (3) ∇ t ( y t , θ t ) ≡ f ( y t ; θ t ) ∂θ t (4) S t ( θ t ) ≡ I t ( θ t ) − γ (5) J t ( θ t ) ≡ E t − (cid:104) ∇ t ( y t , θ t ) ∇ t ( y t , θ t ) (cid:62) (cid:105) (6) The probability density function of the asymmetric Laplace distribution is as (7): f ( x ; µ, σ, p ) = pσ (1 + p ) (cid:40) exp (cid:0) − p σ ( x − µ ) (cid:1) , x ≥ µ exp (cid:16) σ p ( x − µ ) (cid:17) , x < µ (7)The expressions of the moments of ALD are as follows: E ( X ) = µ + σ √ p − p ) (8) Var(X) = (cid:20) σ √ (cid:18) − p (cid:19)(cid:21) + σ (9)Skewness = 2 1 /p − p (1 /p + p ) / (10)Kurtosis = 6 − /p + p ) (11) From the (1), rewrite the (7) to make its parameters µ , σ and p time-varying: y t | y t − ∼ f ( y t ; µ t , σ t , p t ) (12) f ( y t ; µ t , σ t , p t ) = p t σ t (1 + p t ) (cid:40) exp (cid:0) − p t σ ( y t − µ t ) (cid:1) , x t ≥ µ t exp (cid:16) σ t p t ( y t − µ t ) (cid:17) , y t < µ t (13)The likelihood function of the asymmetric Laplace distribution is as follows: ln f = ln p t + ln (cid:0) (cid:1) − ln ( σ t ) + (cid:26) − p t σ ( y t − µ t ) , y t ≥ µ σ p t ( y t − µ t ) , y t < µ t (14)Set the parameter set to θ t = { µ t , σ t , p t } . Calculate the score function: ∇ t ( y t , µ t ) = ∂f ( y t ; µ t ) ∂µ t = (cid:26) p t σ t , y t ≥ µ t − σ t p t , y t < µ t (15)3 PREPRINT - O
CTOBER
14, 2020 − − x f ( x ) AL(0,1,0.5)AL(0,1,1)AL(0,1,0.1)AL(0,2,1)AL(0,5,1)AL(5,1,1)
Figure 1: ALD probability density ∇ t ( y t , σ t ) = ∂ log f ( y t ; σ t ) ∂ σ t = − σ t + (cid:26) p t (y t − µ ) σ − , y t ≥ µ t − t ( y t − µ ) σ − , y t < µ t (16) ∇ t ( y t , p t ) = ∂ log f ( y t ; p t ) ∂κ t = 1 p t + 2 p mathrmt p + (cid:26) − σ t ( y t − µ ) , y t ≥ µ t − σ t ( y t − µ ) p − , y t < µ t (17) ∇ t = [ ∇ t ( y t , µ t ) , ∇ t ( y t , σ t ) , ∇ t ( y t , p t )] (18)The formula (2) can be written as: (cid:32) µ t +1 σ t +1 p t +1 (cid:33) = (cid:32) κ κ κ (cid:33) + (cid:32) a a
00 0 a (cid:33) (cid:32) ∇ t ( y t , µ t ) ∇ t ( y t , σ t ) ∇ t ( y t , p t ) (cid:33) + (cid:32) b
00 0 b (cid:33) (cid:32) µ t σ t p t (cid:33) (19)(12), (13) and (19) are the GAS-ALD models. Compared with other models, the model under the GAS framework can accurately estimate the parameter θ t after givingthe past information and parameter vector φ = κ, A, B . Record y T as the sample T in y t , and φ can be estimate byMLE(20) (cid:98) φ ≡ argmax φ L ( φ ; y T ) S.T. L ( φ ; y T ) = log f ( y ; θ ) + T (cid:88) t =2 log f ( y t ; θ T ) (20)When t=1, θ = ( I − B ) − κ .The following uses the BFGS algorithm to solve the likelihood function to estimate (cid:98) φ . When making a step forward prediction, we can use the GAS-ALD model to estimate the distribution of the return rate y t +1 , thereby predicting VaR and ES one step forward. y t +1 | y t ∼ f ( y t +1 ; µ t +1 , σ t +1 , p t +1 ) (21)4 PREPRINT - O
CTOBER
14, 2020 (cid:32) µ t +1 σ t +1 p t +1 (cid:33) = (cid:32) κ κ κ (cid:33) + (cid:32) a a
00 0 a (cid:33) (cid:32) ∇ t ( y t , µ t ) ∇ t ( y t , σ t ) ∇ t ( y t , p t ) (cid:33) + (cid:32) b
00 0 b (cid:33) (cid:32) µ t σ t p t (cid:33) (22)The essence of VaR is a quantile, which can be easily obtained by using the inverse function of the distribution function.When using the GAS-ALD model, since the parameters are time-varying, the dynamic VaR can be obtained. V aR t +1 ( α ) ≡ F − ( α ; θ t +1 ) (23)Because VaR does not satisfy the subadditivity, we use ES to supplement VaR. ES, as the average part exceeding VaR,can be obtained using the formula (24). Same as VaR, the result of ES is also dynamic. ES t +1 ( α ) ≡ α (cid:90) V aR t +1 −∞ zdF ( z, θ t +1 ) (24) For the statistical test of VaR, this article mainly uses UC test[17] , CC test[18] and DQ test[19] . The above methodsare relatively mature, so I will not repeat them in this article. Because ES is generally too large relative to VaR, themethod of testing the backtest effect of VaR cannot be used to test the backtest effect of ES. In this paper, the bootstraptest proposed in the Mcneil et al.(2000) [20] is used to evaluate ES efficacy. The main idea of this test is to constructmultiple bootstrap residual samples and use them to calculate test statistics and p-values.
The main steps of the test are as follows[21, 22]:First define the excess residual e t : e t = x t − ES qt σ / t (25)Among them, x t is the true rate of return exceeding VaR, and σ t is the sum of variance of x t − ES qt .Then define the initial residual sequence l t : l t = e t − z z (cid:88) η =1 e η (26)Among them, z is the number of samples exceeding the residual e t . Assuming that the mean of l t is l and the standarddeviation is σ l , define the test statistic: δ ( l ) . δ ( l ) = lσ l (27)As you can see, δ ( l ) is the standardized residual.To get the distribution of δ ( l ) and the value of p , we need to use the bootstrap method to generate new samples. That is,generate M random numbers with uniform distribution within { , , . . . , M } , and construct a new sample with theresidual sequence l t corresponding to the random number, repeat N times (N=1000 in the following), N new bootstrapsamples can be obtained.For each bootstrap sample i ( i = 1 , , . . . , M ), use equation (19) to calculate δ i ( l ) , denoted as { δ ( l ) , δ ( l ) , . . . , δ N ( l ) } ,without loss of generality, the δ ( l ) of the initial sample can be recorded as δ ( l ) . Using the bootstrap method, theempirical distribution of δ ( l ) can be obtained.Since e t is often a right-skewed distribution, the selected hypothesis H of the bootstrap test is E ( e t ) > , and the nullhypothesis is H isE ( e t ) = 0 . Calculate the sample proportion of { δ ( l ) , δ ( l ) , . . . , δ N ( l ) } greater than δ ( l ) , whichis the P value of the test. The larger the value of P, the more unable to reject the null hypothesis, that is, the model isconsidered to have better ES estimation accuracy. 5 PREPRINT - O
CTOBER
14, 2020
When using VaR and ES for risk management, not only need to be concerned about the accuracy of VaR and ES forforecasting, but also need to be concerned about the shape of the entire left tail region of the return distribution. Theremust be a more accurate understanding of all losses. This article intends to use two risk loss functions to evaluate theprediction of the model.
Considering the loss function of VaR, this article chooses to use the quantile loss function ((28) formula), which gives ahigher weight than VaR. [23] QL αt = ( α − I αt )( y t − VaR αt ) (28) Since there is no separate evaluation of ES loss function , referring to the literature, this paper uses the FZL joint lossfunction to evaluate the prediction accuracy of ES and VaR for extreme risks.[24, 25]
FZL αt = 1 α ES αt I αt ( y t − VaR αt ) + VaR αt ES αt + log ( − ES αt ) − (29) The indexes selected in this article are Shanghai Stock Index (000001.ss), Shenzhen Index (399001.sz) and Small andMedium-sized Board Index (399005.sz). The sampling time is 3000 days from July 6, 2007 to November 1, 2019. Datasource: wind database. Use R language for data processing.
First, the daily logarithmic rate of return r t = lnp t − lnp t − is calculated, and the basic statistics such as the mean andvariance of the corresponding sample rate of return are calculated(table 1 . From Table 1, we can see that the meanvalues of the three samples are relatively close to 0; the variances are roughly the same, and the variance of the smalland medium board index is the largest; from the perspective of skewness kurtosis and the JB test p-value, all threesamples have peak fat Typical facts with tails and right skewed, and none of them obey normal distribution. Fromthe kurtosis point of view, the excess kurtosis of the three samples are all greater than 0, and the excess kurtosis ofthe Shanghai Composite Index even reached 4.5072, which implies that we need to use a distribution that can betterdescribe the peak and thick tail to describe the yield.Table 1: Basic statisticsmean variance skewness excess kurtosis JB-p000001.ss -0.0001 0.0003 -0.5256 4.5072 0.0000399001.sz -0.0001 0.0003 -0.4660 2.9263 0.0000399005.sz 0.0001 0.0004 -0.5475 2.6491 0.0000Figure 2 draws a time series diagram of index trend and rate of return. It can be clearly seen that the return ratesequence has volatility clustering. Fit the GAS-ALD model to the data in the sample period. The results are as follows. It can be seen that among theparameters of the three indices, a i and b i ( i = 1 , , ) are significantly different from 0, indicating that the parametersare time-varying.(table 2 ) JB-p: P value of Jarque-Bera test, used to test whether a set of samples can be considered as coming from a normal population From left to right: Shanghai Stock Exchange Index Trend and Return, Shenzhen Index Trend and Return, Small and Medium-sized Board Index Trend and Return K-S test line reports as p value, ***: significant at 99% level PREPRINT - O
CTOBER
14, 2020
D t ( )
D t ( )
D t ( ) − D t ( ) − D t ( ) − D t ( )
Figure 2: Index trend and returnTable 2: Parameter estimation resul000001.ss 399001.sz 399005.sz κ κ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ κ a − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ a − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ a − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ b . ∗∗∗ . ∗∗∗ . ∗∗∗ b . ∗∗∗ . ∗∗∗ . ∗∗∗ b . ∗∗∗ . ∗∗∗ . ∗∗∗ K-S test 0.737 0.751 0.426In order to test the fitting status of the GAS-ALD model to the distribution of returns, this paper uses the GAS-ALDmodel for probability integral transformation of the returns, and then performs the K-S test. The data after the probabilityintegral transformation of the original data accepts the null hypothesis of the same distribution as the uniform distributionon (0,1). This shows that the GAS-ALD model can better fit the distribution of returns.
This article draws the time-varying trajectories of three parameters (Figure 3, Figure 4 and Figure 5) It can be seen thatthe parameter trends of the three index samples are basically the same. σ t has the strongest aggregation, and there aretwo obvious peaks. And noticed that after the peak of σ t appears, it is often accompanied by a rapid decline in p t and µ t . The three pictures from top to bottom are: µ t , σ t , p t . PREPRINT - O
CTOBER
14, 2020 − D t ( )
D t ( )
D t ( )
Figure 3: Shanghai Composite Index Dynamic Parame-ters − D t ( )
D t ( )
D t ( )
Figure 4: Shenzhen Index Dynamic Parameters − µ σ κ Figure 5: Small and medium board index DynamicParametersCalculate conditional mean, conditional volatility, conditional skewness and conditional kurtosis based on dynamicparameters. The results are shown in Figure 6, Figure 7 and Figure 8 Conditional volatility and conditional kurtosishave the strongest aggregation effect. The second is the mean and kurtosis. It shows that each moment has obvioustime-varying characteristics and aggregation characteristics. From the perspective of linkage, every sudden increase involatility will result in a sharp drop in skewness and an increase in kurtosis. This shows that when the volatility rises,there will be a more obvious sharp peak and thick tail right deviation, which will lead to increased tail loss. And itcan be seen that the change trends of the moments of the three indices are roughly the same, which shows that thecorrelation of the moments of the three indices is relatively large. The four graphs from top to bottom are: conditional mean, conditional volatility, conditional skewness and conditional kurtosis PREPRINT - O
CTOBER
14, 2020 − − − − Figure 6: Shanghai Composite Index Moment Results − − − D t ( )
Figure 7: Shenzhen Index Moment Results − − − D t ( )
Figure 8: Small and medium board Index Moment Results
Consider three time points for comparative analysis. The time points selected in this article are 2008-08-18, 2011-03-10and 2015-08-21, which represent the period of the subprime mortgage crisis, the rising period, and the period of leveragestock disasters, respectively. Plot the return distribution of each index at each time point. As shown 10, 11 and 12 . Figure 9: Time selection The green line is the asymmetric Laplace distribution density obtained by the full sample fitting, and the black line, red line andblue line are the asymmetric Laplace distribution density at a specific time point PREPRINT - O
CTOBER
14, 2020It can be found first. During the rising period, the distribution of the return rate is sharp and thin compared to thefull sample period, indicating that most of the return is concentrated near 0, and the value of VaR and ES can beappropriately lowered at this time. During the subprime mortgage crisis and leveraged stock crisis, compared with thefull sample, the tail thickness increased, indicating that the probability of loss at this time is much greater than that ofthe full sample and the rising period. At this time, the risk exposure faced by investors will greatly increase. In addition,when a leveraged stock disaster occurs, the thickness of the tails of the three index yield distributions is greater thanthat of the subprime mortgage crisis, indicating that the risk exposure faced by investors during the leveraged stockdisaster is greater than the risk exposure faced by the subprime mortgage crisis. This is because in 2015, a large numberof investors increased their investment leverage through borrowing, resulting in losses that they faced when the stockmarket plummeted far beyond their own capabilities. − − − − − ALD(0,0.034,1.002)2011 − − − ALD(0.001,0.013,1)2015 − − − ALD(0,0.039,1.001)ALD(0,0.016,1)
Figure 10: Time distribution of Shanghai stock indexreturn rate − − − − − ALD(0,0.032,1.002)2011 − − − ALD(0.001,0.015,1)2015 − − − ALD(0,0.044,1.001)ALD(0,0.02,1)
Figure 11: Time distribution of Shenzhen index return − − − − − ALD(0,0.33,1.002)2011 − − − ALD(0.001,0.012,1)2015 − − − ALD(0.001,0.042,1)ALD(0,0.18,1)
Figure 12: Time distribution of small and medium boardindex returns
This article uses three types of models as a comparison. The first type is historical simulation method and fixedparameter ALD distribution. Among them, the historical simulation method uses short-term (25-day) sample quantiles,and the second type is other distributions under the GAS framework. Normal distribution (GAS-norm), t distribution(GAS-t) and partial t distribution (GAS-skst). The third type is GARCH model. This article uses APARCH model withpartial t distribution error term (APARCH-skst) As a control group.We selects the first 1500 days of the sample as the training group, rolls forward forecast 1500 days, re-estimates themodel every 5 days, and calculates the VaR and ES of 1%.
Table 3 reports the results of the VaR backtest. Combining the UC, CC, and DQ test results, GAS-norm and GAS-tperformed the worst. This may be due to the fact that the normal distribution and t distribution cannot describe theasymmetry and biased characteristics of their returns. This shows that when calculating VaR, the model needs to beable to describe the asymmetry and biased characteristics of the rate of return. The GAS-ALD model has the bestperformance in the UC, CC and DQ tests of the Shanghai Stock Index and the Small and Medium-sized Board Index;the GAS-ALD model has the best performance in the UC and CC tests of the Shenzhen Index, and the DQ test isslightly worse. The GAS-skst model, which shows that the Shenzhen Index is more suitable to use the GAS-skst modelto calculate the VaR of 1%.Table 4 reports the ES backtest results. It was found that GAS-norm, GAS-t and GAS-skst performed poorly incalculating ES, and all rejected the null hypothesis. The ES backtest of the Shanghai Composite Index shows that the10
PREPRINT - O
CTOBER
14, 2020GAS-ALD model is slightly worse than APARCH-skst; the Shenzhen Index and the small and medium-sized boardindex have the best performance in the ES backtest of the GAS-ALD model.Table 3: VaR Backtesting
Table 4: ES BacktestingES-1% 000001.ss 399001.sz 399005.szHistorical Simulation 0.014 0.006 0.001Fixed Parameter ALD 0 0.247 0.032GAS-norm 0 0 0GAS-t 0 0 0GAS-skst 0 0 0GAS-ALD 0.271 0.527 0.533APARCH-skst 0.658 0.373 0.339
Use the predicted VaR and ES results to calculate the QL loss function and FZL loss function. The results are shown inTable 5 .For the QL loss function, the GAS-ALD model has the smallest average QL in the Shanghai Stock Index and theShenzhen Index, indicating that among the compared models, GAS-ALD has better prediction accuracy for VaRprediction, while in the small and medium board In the index, the average QL of APARCH-skst is the smallest, followedby the average QL of GAS-ALD. This article believes that this is because APARCH-skst cannot dynamically describethe changes in distribution, but it can describe the asymmetric effect of the return rate. There are often obviousasymmetric effects in the small and medium board index, which makes APARCH-skst’s prediction accuracy for VaRSlightly higher than the GAS-ALD model.For the FZL loss function, the GAS-ALD model has the smallest average FZL value, indicating that the GAS-ALDmodel has better ES prediction accuracy than the control model.Table 5: Loss Function000001.ss 399001.sz 399005.szLoss Function QL FZL QL FZL QL FZLHistorical Simulation 7.13 -2.629 7.428 -2.594 7.5 -2.611Fixed Parameter ALD 7.198 -2.627 7.501 -2.597 7.399 -2.589GAS-norm 5.599 -2.921 6.693 -2.621 6.812 -2.582GAS-t 5.852 -2.768 7.081 -2.491 7.29 -2.392GAS-skst 6.301 -2.711 7.74 -2.444 7.749 -2.436GAS-ALD 5.543 -2.994 6.631 -2.736 6.716 -2.706APARCH-skst 5.453 -2.984 6.839 -2.833 6.167 -2.614 The QL loss function results are all multiplied by , and the reported value is the average PREPRINT - O
CTOBER
14, 2020
In this paper, by improving the asymmetric Laplace distribution so that its parameters are all time-varying, the GAS-ALD model is proposed and used to conduct empirical research on China’s Shanghai Stock Exchange Index, ShenzhenIndex and Small and Medium-sized Board Index. The following conclusions:1) The distribution of logarithmic returns of Shanghai Composite Index, Shenzhen Index and Small and Medium-sizedBoard Index conforms to the GAS-ALD model, and its conditional mean, conditional volatility, conditional skewnessand conditional kurtosis are all time-varying and clustering. If the normal distribution is used Waiting for the staticdistribution will lead to failure to capture the time-varying characteristics of the moment.2) UC test, CC test, DQ test and bootstrap test are used to test the risk prediction ability of GAS-ALD model. Comparedwith GAS-norm and GAS-t, which are equal to the distribution under the GAS framework and APARCH-skst, theGAS-ALD model has a better feasibility on VaR and ES of 1%. And the use of QL loss function and FZL loss functionshows that the GAS-ALD model has better prediction accuracy.In summary, GAS-ALD can better describe the distribution of peaks and thick tails, biased, and left-to-right asymmetry,and has good accuracy in the prediction of VaR and ES. These conclusions can provide some help in risk measurementfor participants in the capital market.As a means of describing the distribution, GAS-ALD can be used as the marginal distribution of returns to fit Copula,and to measure the VaR and ES of the portfolio. In the future we will involve more research in this area.
References [1] Bolerslev T.A conditional heteroskedastic time series model for speculative prices and rates of return[J]. TheReview of Economics and Statistics, 1987, 69(3):542-547[2] Bruce E. Hansen. Autoregressive Conditional Density Estimation[J]. International Economic Review, 1994,35(3):705-730.[3] Dongming Zhu,John W. Galbraith. A generalized asymmetric Student-t distribution with application to financialeconometrics[J]. Journal of Econometrics,2010,157(2).[4] Yu K ,Zhang J. A Three-Parameter Asymmetric Laplace Distribution and Its Extension[J]. Communications inStatistics Theory and Methods, 2005, 34(9-10):1867-1879.[5] Liu Jianyuan, Liu Qiongsun. Research on VaR based on asymmetric Laplace distribution[J].Statistics andDecision,2007(18):33-35.[6] Du Hongjun, Wang Zongjun. Dynamic risk measurement based on Asymmetric Laplace distribution[J].Statisticsand Decision,2013(23):15-18.[7] Liu Pan, Zhou Ruomei. The typical fact description and VaR measurement of financial asset return under thedistribution of AEPD, AST and ALD[J]. China Management Science, 2015, 23(2):21-28.[8] Tang Jianyang, Li Yi, Wan Chuang, et al. VaR and ES metrics based on asymmetric Laplace distribution[J].SystemEngineering,2018,36(09):30-40. To[9] Creal D, Koopman S J, André Lucas. Generalized Autoregressive Score Models with Applications[J]. Journal ofApplied Econometrics, 2013, 28(5):777-795.[10] Lucas,André, Zhang X.Score-driven exponentially weighted moving averages and Value-at-Risk forecasting[J].International Journal of Forecasting, 2016, 32(2):293-302.[11] Harvey A, Sucarrat G. EGARCH models with fat tails, skewness and leverage[J]. Computational Statistics & DataAnalysis, 2012, 76(76):320-338.[12] Dong Hwan Oh, Andrew J. Patton. Time-Varying Systemic Risk: Evidence from a Dynamic Copula Model ofCDS Spreads[J]. Ssrn Electronic Journal, 2013, 36(2).[13] Ye Wuyi, Tan Keqi, Miao Baiqi. Inter-industry systemic risk analysis based on dynamic factor Copulamodel[J].China Management Science,2018,26(03):1-12.[14] Zou Yumei, Fan Jingya. Application of time-varying D-vine based on generalized autoregressive scoremodel[J].Mathematical Statistics and Management,2018,37(01):74-82.[15] Xiao-LiGong,Xi-Hua Liu,Xiong Xiong. Measuring tail risk with GAS time varying copula,fat tailed GARCHmodel and hedging for crude oil futures[J]. Pacific-Basin Finance Journal,2019,55.12
PREPRINT - O