1、第7章、ARCH模型和GARCH模型研究內容：研究隨時間而變化的風險。（回憶：Markowitz均值方差投資組合選擇模型怎樣度量資產的風險）本章模型與以前所學的異方差的不同之處：隨機擾動項的無條件方差雖然是常數，但是條件方差是按規律變動的量。波動率的聚類性（volatility clustering）：一段時間內，隨機擾動項的波動的幅度較大，而另外一定時間內，波動的幅度較小。如圖，§1、ARCH模型1、條件方差多元線性回歸模型：條件方差或者波動率（Condition variance，volatility）定義為其中是信息集。2、ARCH模型的定義Engle（1982）提出ARCH模

2、型（autoregressive conditional heteroskedasticity，自回歸條件異方差）。ARCH(q)模型： （1）的無條件方差是常數，但是其條件分布為 （2）其中是信息集。方程（1）是均值方程（mean equation）ü ：條件方差，含義是基于過去信息的一期預測方差方程（2）是條件方差方程（conditional variance equation），由二項組成ü 常數ü ARCH項：滯后的殘差平方習題： 方程（2）給出了的條件方差，請計算的無條件方差。證明：利用方差分解公式：Var(X) = VarYE(X|Y) + EYVar

3、(X|Y)由于，所以條件均值為0，條件方差為。那么，推出，說明3、ARCH模型的平穩性條件在ARCH(1)模型中，觀察參數的含義：當時，當時，退化為傳統情形，ARCH模型的平穩性條件：（這樣才得到有限的方差）4、ARCH效應檢驗ARCH LM Test：拉格朗日乘數檢驗建立輔助回歸方程此處是回歸殘差。原假設：H0：序列不存在ARCH效應即H0：可以證明：若H0為真，則此處，m為輔助回歸方程的樣本個數。R2為輔助回歸方程的確定系數。Eviews操作：先實施多元線性回歸view/residual/Tests/ARCH LM Test§2、GARCH模型的實證分析從收盤價，得到收益率數據序

4、列。series r=log(p)-log(p(-1)點擊序列p，然后view/line graph1、檢驗是否有ARCH現象。首先回歸。取2000到2254的樣本。輸入ls r c，得到Dependent Variable: RMethod: Least SquaresDate: 10/21/04 Time: 21:26Sample: 2000 2254Included observations: 255VariableCoefficientStd. Errort-StatisticProb. C0.0004320.0010870.3971300.6916R-squared0.000000

5、Mean dependent var0.000432Adjusted R-squared0.000000 S.D. dependent var0.017364S.E. of regression0.017364 Akaike info criterion-5.264978Sum squared resid0.076579 Schwarz criterion-5.251091Log likelihood672.2847 Durbin-Watson stat2.049819問題：這樣進行回歸的含義是什么？其次，view/residual tests/ARCH LM test，得到ARCH Test

6、:F-statistic5.220573 Probability0.000001Obs*R-squared44.68954 Probability0.000002Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 10/21/04 Time: 21:27Sample(adjusted): 2010 2254Included observations: 245 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C0.0001

7、105.34E-052.0601380.0405RESID2(-1)0.1415490.0652372.1697760.0310RESID2(-2)0.0550130.0658230.8357660.4041RESID2(-3)0.3377880.0655685.1516970.0000RESID2(-4)0.0261430.0691800.3778930.7059RESID2(-5)-0.0411040.069052-0.5952600.5522RESID2(-6)-0.0693880.069053-1.0048540.3160RESID2(-7)0.0056170.0691780.0811

8、930.9354RESID2(-8)0.1022380.0655451.5598060.1202RESID2(-9)0.0112240.0657850.1706190.8647RESID2(-10)0.0644150.0651570.9886130.3239R-squared0.182406 Mean dependent var0.000305Adjusted R-squared0.147466 S.D. dependent var0.000679S.E. of regression0.000627 Akaike info criterion-11.86836Sum squared resid

9、9.19E-05 Schwarz criterion-11.71116Log likelihood1464.875 F-statistic5.220573Durbin-Watson stat2.004802 Prob(F-statistic)0.000001得到什么結論？2、模型定階：如何確定q實施ARCH LM test時，取較大的q，觀察滯后殘差平方的t統計量的pvalue即可。此處選取q3。因此，可以對殘差建立ARCH(3)模型。3、ARCH模型的參數估計參數估計采用最大似然估計。具體方法在GARCH一節中講解。如何實施ARCH過程：由于存在ARCH效應，所以點擊estimate，在me

10、thod中選取ARCH得到如下結果Dependent Variable: RMethod: ML - ARCHDate: 10/21/04 Time: 21:48Sample: 2000 2254Included observations: 255Convergence achieved after 13 iterationsCoefficientStd. Errorz-StatisticProb. C-0.0006400.000750-0.8528880.3937 Variance EquationC9.24E-051.66E-055.5693370.0000ARCH(1)0.2447930

11、.0826402.9621420.0031ARCH(2)0.0814250.0774281.0516240.2930ARCH(3)0.4578830.1096984.1740430.0000R-squared-0.003823 Mean dependent var0.000432Adjusted R-squared-0.019884 S.D. dependent var0.017364S.E. of regression0.017535 Akaike info criterion-5.495982Sum squared resid0.076872 Schwarz criterion-5.426

12、545Log likelihood705.7377 Durbin-Watson stat2.042013為了比較，觀察將q放大對系數估計的影響Dependent Variable: RMethod: ML - ARCHDate: 10/21/04 Time: 21:54Sample: 2000 2254Included observations: 255Convergence achieved after 16 iterationsCoefficientStd. Errorz-StatisticProb. C-0.0006010.000751-0.7999090.4238 Variance E

13、quationC9.38E-051.60E-055.8807410.0000ARCH(1)0.2620090.0902562.9029590.0037ARCH(2)0.0419300.0705180.5945960.5521ARCH(3)0.4521870.1084884.1680760.0000ARCH(4)-0.0219200.050982-0.4299560.6672ARCH(5)0.0376200.0443940.8474080.3968R-squared-0.003550 Mean dependent var0.000432Adjusted R-squared-0.027830 S.

14、D. dependent var0.017364S.E. of regression0.017603 Akaike info criterion-5.483292Sum squared resid0.076851 Schwarz criterion-5.386081Log likelihood706.1198 Durbin-Watson stat2.042568觀察：說明q選取為3確實比較恰當。4、ARCH模型是對的嗎？如果ARCH模型選取正確，即回歸殘差的條件方差是按規律變化的，那么標準化殘差就會服從標準正態分布，即不會有ARCH效應了。為什么？請思考。對q為3的ARCH模型做LM test

15、，發現沒有了ARCH效應。注意，雖然是同一個檢驗名稱，但是ARCH過程后是對標準化殘差進行檢驗。注意觀察被解釋變量或者依賴變量是什么？ARCH Test:F-statistic0.238360 Probability0.992099Obs*R-squared2.470480 Probability0.991299Test Equation:Dependent Variable: STD_RESID2Method: Least SquaresDate: 10/21/04 Time: 21:56Sample(adjusted): 2010 2254Included observations: 24

16、5 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C1.1023710.2649904.1600430.0000STD_RESID2(-1)-0.0385450.065360-0.5897410.5559STD_RESID2(-2)-0.0038040.065308-0.0582520.9536STD_RESID2(-3)-0.0573130.065303-0.8776490.3810STD_RESID2(-4)-0.0103250.065277-0.1581690.8745STD_RESID2(-

17、5)0.0035370.0652800.0541850.9568STD_RESID2(-6)-0.0074200.065274-0.1136700.9096STD_RESID2(-7)0.0633170.0652640.9701650.3330STD_RESID2(-8)-0.0121670.065293-0.1863400.8523STD_RESID2(-9)-0.0106530.065278-0.1631940.8705STD_RESID2(-10)-0.0202110.065228-0.3098450.7570R-squared0.010084 Mean dependent var1.0

18、07544Adjusted R-squared-0.032221 S.D. dependent var2.112747S.E. of regression2.146514 Akaike info criterion4.409426Sum squared resid1078.160 Schwarz criterion4.566625Log likelihood-529.1546 F-statistic0.238360Durbin-Watson stat2.000071 Prob(F-statistic)0.992099方程整體是不顯著的。還可以觀察標準化殘差ARCH建模以后，procs/make

19、 residual series/可以產生殘差和標準化殘差，以分別下是殘差和標準化殘差。可以看出沒有了集群現象。還可以觀察波動率（條件方差）的圖形。對比r和殘差的圖形，發現條件方差的起伏與波動率的大小一致。ARCH建模以后，procs/make garch variance series/ 得到結論：ARCH模型確實很好描述了股票市場收益率的波動性。可以觀察系數之和小于1，滿足平穩性條件。§3、GARCH模型當q較大時，采用Bollerslov（1986）提出的GARCH模型（Generalized ARCH）1、模型定義條件方差方程ü 均值üü ：過去

20、的條件方差（也即預測方差，forecast variance）注意：均值方程中若沒有解釋變量（即只有常數，如R C），則R2沒有直觀定義了，因此可為負）2、GARCH(p, q) 模型的穩定性條件計算擾動項的無條件方差：GARCH是協方差穩定的，因此是經典回歸。3、GARCH模型的參數估計采用極大似然估計GARCH模型的參數。下面以GARCH(1, 1)為例。由GARCH(1, 1)模型可以得到yt的分布為由正態分布的定義公式，得到yt的pdf為第t個觀察樣本的對數似然函數值為其中注意yi和yj之間不相關，因而是獨立的。似然函數為取對數就得到了所有樣本的對數似然函數。其中條件方差項以非線性方式

21、進入似然函數，所以不得不使用迭代算法求解。4、模型的選擇兩條原則：1） 若ARCH(q)中q太大，比如q大于7時，則選擇GARCH(p, q)2） 使用AIC和SC準則，選擇最優的GARCH模型3） 對于金融時間序列，一般選擇GARCH(1, 1)就夠了。回顧AIC和SC定義：1）AIC準則（Akaike information criterion）AIC越小越好，結合如下兩者：K（自變量個數）減少，模型簡潔LnL增加，模型精確2）SC準則（Schwaz criterion）習題1：通貨膨脹率有ARCH效應嗎？Greene P572點擊數據文件usinf_greene_p572。進行回歸ls

22、inflation c inflation(-1)Dependent Variable: INFLATIONMethod: Least SquaresDate: 11/19/04 Time: 10:37Sample(adjusted): 1941 1985Included observations: 45 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C2.4328590.8163452.9801840.0047INFLATION(-1)0.4932130.1311573.7604660.0005R

23、-squared0.247477 Mean dependent var4.740000Adjusted R-squared0.229976 S.D. dependent var4.116784S.E. of regression3.612519 Akaike info criterion5.450114Sum squared resid561.1625 Schwarz criterion5.530410Log likelihood-120.6276 F-statistic14.14110Durbin-Watson stat1.612442 Prob(F-statistic)0.000507檢驗

24、ARCH效應ARCH Test:F-statistic0.215950 Probability0.953308Obs*R-squared1.231192 Probability0.941850Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 11/19/04 Time: 10:46Sample(adjusted): 1946 1985Included observations: 40 after adjusting endpointsVariableCoefficientStd. Errort-Statisti

25、cProb. C9.2705227.4255671.2484600.2204RESID2(-1)-0.0311620.170116-0.1831840.8557RESID2(-2)-0.0068860.170151-0.0404690.9680RESID2(-3)0.1162610.1695050.6858880.4974RESID2(-4)0.0185450.1706200.1086940.9141RESID2(-5)0.1279060.1686430.7584390.4534R-squared0.030780 Mean dependent var12.28323Adjusted R-squ

26、ared-0.111753 S.D. dependent var34.15088S.E. of regression36.00858 Akaike info criterion10.14287Sum squared resid44085.00 Schwarz criterion10.39620Log likelihood-196.8574 F-statistic0.215950Durbin-Watson stat1.034796 Prob(F-statistic)0.953308習題2：通貨膨脹率有ARCH效應嗎？Lin的數據集 點擊usinf文件series dp=100*D(log(p)l

27、s dp c dp(-1) dp(-2) dp(-3)Dependent Variable: DPMethod: Least SquaresDate: 11/19/04 Time: 10:10Sample(adjusted): 1951:1 1983:4Included observations: 132 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C0.1099070.0634051.7334100.0854DP(-1)0.3935830.0844324.6615360.0000DP(-2)0.

28、2030930.0894522.2704050.0249DP(-3)0.3020730.0841853.5882140.0005R-squared0.696825 Mean dependent var1.021373Adjusted R-squared0.689719 S.D. dependent var0.711412S.E. of regression0.396277 Akaike info criterion1.016428Sum squared resid20.10054 Schwarz criterion1.103785Log likelihood-63.08423 F-statis

29、tic98.06599Durbin-Watson stat1.970959 Prob(F-statistic)0.000000ARCH Test:F-statistic0.969524 Probability0.439318Obs*R-squared4.892009 Probability0.429201Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 11/19/04 Time: 10:13Sample(adjusted): 1952:2 1983:4Included observations: 127 af

30、ter adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C0.1081900.0353023.0646480.0027RESID2(-1)-0.0808320.090353-0.8946190.3728RESID2(-2)0.1079060.0884931.2193650.2251RESID2(-3)0.0811910.0888310.9139960.3625RESID2(-4)0.1107450.0884331.2522990.2129RESID2(-5)0.0312480.0887380.3521340.72

31、54R-squared0.038520 Mean dependent var0.147634Adjusted R-squared-0.001211 S.D. dependent var0.236307S.E. of regression0.236450 Akaike info criterion-7.13E-05Sum squared resid6.764921 Schwarz criterion0.134300Log likelihood6.004525 F-statistic0.969524Durbin-Watson stat1.990016 Prob(F-statistic)0.4393

32、18Dependent Variable: DPMethod: ML - ARCHDate: 11/19/04 Time: 10:16Sample(adjusted): 1951:1 1983:4Included observations: 132 after adjusting endpointsConvergence achieved after 25 iterationsCoefficientStd. Errorz-StatisticProb. C0.1113020.0645121.7252820.0845DP(-1)0.3783170.0961983.9326910.0001DP(-2

33、)0.1883850.0862412.1844010.0289DP(-3)0.3237310.0983453.2917880.0010 Variance EquationC0.2924650.0491875.9459390.0000ARCH(1)-0.0297610.047805-0.6225630.5336GARCH(1)-0.8733240.267371-3.2663330.0011R-squared0.696453 Mean dependent var1.021373Adjusted R-squared0.681883 S.D. dependent var0.711412S.E. of

34、regression0.401250 Akaike info criterion1.051145Sum squared resid20.12519 Schwarz criterion1.204021Log likelihood-62.37558 F-statistic47.79960Durbin-Watson stat1.938286 Prob(F-statistic)0.000000附錄：Matlab的GARCH工具箱ARMAX(R,M,Nx)/GARCH(P,Q)模型: / 1.=資產的收益率序列 =沖擊過程 =的條件方差:2. GARCH(0,Q)óARCH(Q)3.is th

35、e forecast of the next periods variance, given the past sequence of variance forecastsand past realizations of the variance itself. The Default Model: / 對金融收益率時序,(1)帶漂移的隨機游走足夠了(2)GARCH(1,1),GARCH(2,1), GARCH(1,2)足夠了結構接口Spec = garchset('Parameter1', Value1, 'Parameter2', Value2, .) 創建

36、Spec = garchset(OldSpec, 'Parameter1', Value1, .) 修正OldSpec例：spec=garchset; spec=garchset(spec, 'C', 0, 'AR', 0.6 0.2, 'MA', 0.4);GARCH建模1. 收益率時序的ARMAX/GARCH參數估計Coeff,Errors,LLF,Innovations,Sigma,Summary=garchfit(Spec, Series)/(Spec, Series, X) Series-收益率序列y, 最后為最新數據

37、Spec-結構描述, garchsetX-多種資產的收益率回歸矩陣，每列為一回歸解釋變量，最后一行為最新數據Coeff-估計系數, Errors-系數的標準差, LLF-log-likelihood函數值,Innovations-, Sigma-2. SigmaForecast,MeanForecast,SigmaTotal,MeanRMSE=garchpred(Spec,Series,NumPeriods)NumPeriods-預測步數. *SigmaForecast-的預測值. *MeanForecast-的預測值.SigmaTotal-對為 MeanRMSE-預測的標準誤差.3. GAR

38、CH過程模擬 Innovations,Sigma,Series=garchsim(Spec)/(Spec,NS,NP,Seed,X)NS-樣本個數default 100. NP-樣本路徑的個數default 1. Seed-隨機數種子default 0 Innovations-NS*NP沖擊矩陣. Sigma- Series-NS*NP收益率矩陣, 每列為單獨的實現y.例co,er,L,in,si=garchfit(xyz); e,s,y=garchsim(co,800); garchplot(e,s,y)GARCH沖擊推斷從推斷與:Innovations,Sigma,LogLikelihoo

39、d=garchinfer(Spec,Series)/(Spec,Series,X)例eInferred, sInferred = garchinfer(coeff, y); Statistics and Tests1. Akaike Bayesian信息準則AIC,BIC=aicbic(LogLikelihood,NumParams,NumObs)NumParams-參數個數 NumObs-收益率時序長度AIC=-2*LogLikelihood+2*NumParams BIC=-2*LogLikelihood+NumParams*Log(NumObs)例n21=garchcount(coeff

40、21); n11=garchcount(coeff11)=4; %參數個數AIC,BIC=aicbic(LLF21,n21,2000);AIC,BIC=aicbic(LLF11,n11,2000);%AIC, BIC沒有顯著增加, 說明GARCH(1,1)足夠了6. Likelihood ratio hypothesis test.H, pValue, Ratio, CriticalValue=lratiotest(BaseLLF, NullLLF, DoF, Alpha)例spec11=garchset('P',1,'Q',1);co11,er11,LLF11

41、,in11,si11,su11=garchfit(spec11,xyz);spec21=garchset('P',2,'Q',1);co21,er21,LLF21,in21,si21,su21=garchfit(spec21,xyz);%LLF21越大越好H,p,St,CV=lratiotest(LLF21,LLF11, 1, 0.05); %H=0說明GARCH(1,1)足夠了*此不對，要對spec11給初值。2. H, pValue, ARCHstat, CriticalValue = archtest(Residuals)/(Residuals, Lags

42、, Alpha)H0: 樣本余差時序為i.i.d.正態沖擊(i.e.無ARCH/GARCH效應).Residuals比如來自回歸的余差 Lags-default is 1即ARCH(1). H=0 接受H0 如residuals=randn(100,1);H,P,Stat,CV=archtest(residuals,1 2 4',0.10)Create synthetic residuals, 檢驗1 2 4階ARCH效應. %注意GARCH(P,Q)基本相當于ARCH(P+Q)7. 偏自相關PartialACF, Lags, Bounds=parcorr(Series)/(Serie

43、s , nLags , R , nSTDs)Series最后一數據為最新 nLags偏ACF的個數，默認為minimum20 , length(Series)-1.R若為AR(R)過程, lags>R時偏ACF為0. nSTDs-default is nSTDs = 2 (i.e.95%置信區間).Bounds-AR(R)過程的邊界.如: randn('state',0) x = randn(1000,1); y = filter(1,1 -0.6 0.08,x); % Create a stationary AR(2) process. parcorr(y , , 2)

44、 % Inspect the P-ACF with 95% confidence.3.自相關 ACF, Lags, Bounds = autocorr(Series)/(Series , nLags , M , nSTDs)M - MA(M)過程, lags > M, ACF為0.Example: randn('state',0) x = randn(1000,1); y = filter(1 -1 1 , 1 , x); % Create an MA(2) process. autocorr(y , , 2) % Inspect the ACF with 95% con

45、fidence.4. 交叉自相關XCF, Lags, Bounds = crosscorr(Series1 , Series2)/(Series1 , Series2 , nLags , nSTDs)Bounds2個序列完全不相關的XCF 的上下界.如: randn('state',100) x=randn(100,1); y=lagmatrix(x , 4); % Delay it by 4 samples. y(isnan(y) = 0; % Replace NaN's with zeros. crosscorr(x,y) % It should peak at t

46、he 4th lag.5. Ljung-Box-Pierce Q-檢驗H0: 模型后的innovations(i.e. residuals)沒有序列相關, QChi-Square分布 L 2 Q = N(N+2)Sum(r(k)/(N-k) k=1 N=樣本長度, r2(k)樣本自相關. H, pValue, Qstat, CriticalValue = lbqtest(Series)/(Series , Lags , Alpha)Series-is either the sample residuals derived from fitting a model to an observed time series, or the standardized residuals obtained by dividing the sample residuals by the conditional standard deviations.Lagsdefault minimum20 ,