σ t σ t σt nikkei HP nikkei4csv H R nikkei4<-readcsv("h:=y=ynikkei4csv",header=t) (1) nikkei header=t nikkei4csv 4 4 nikkei nikkei4<-dataframe(n

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1 R 1 R R R tseries fseries 1 tseries fseries R Japan(Tokyo) R library(tseries) library(fseries) 2 t r t t 1 Ω t 1 E[r t Ω t 1 ] ɛ t r t = E[r t Ω t 1 ] + ɛ t ɛ t 2 iid (independently, identically distributed) ɛ t = σ t z t, σ t > 0, z t iid N (0, 1) R Ver241 a-matsumo@osipposaka-uacjp 1 fseries tseries fseries GARCH garchfit 2 [2000] z t z t z t 1

2 σ t σ t σt nikkei HP nikkei4csv H R nikkei4<-readcsv("h:=y=ynikkei4csv",header=t) (1) nikkei header=t nikkei4csv 4 4 nikkei nikkei4<-dataframe(nikkei4) (2) nikkei<-nikkei4[,4] (3) dnikkei dnikkei <-diff(log(nikkei)) (4) dnikkei 21 ARCH(q) ARCH(:Auto-regressive Conditional Heteroskedasticity) σt 2 ɛ 2 t 1, ɛ 2 t 2, ARCH(q) : σ 2 t = ω + q α j ɛ 2 t j, ω > 0, j α j 0 j=1 tseries ARCH garch ARCH(2) arch2 arch2<-garch(dnikkei, order=c(0,2)) (5) summary(arch2) 3 3 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** Warning: 2

3 Call: garch(x = dnikkei, order = c(0, 2)) Model: GARCH(0,2) Residuals: Min 1Q Median 3Q Max Coefficient(s): Estimate Std Error t value Pr(> t ) a0 1366e e < 2e-16 *** a1 1157e e e-15 *** a2 8335e e e-08 *** --- Signif codes: 0 *** 0001 ** 001 * Diagnostic Tests: Jarque Bera Test data: Residuals X-squared = , df = 2, p-value < 22e-16 Box-Ljung test data: SquaredResiduals X-squared = 11212, df = 1, p-value = ARCH(2) σ 2 t = ɛ 2 t ɛ 2 t 2, Std Error: ( ) (001452) (001487) 22 GARCH(p, q) GARCH(:Generalized Auto-regressive Conditional Heteroskedasticity) σt 2 ɛ 2 t 1, ɛ 2 t 2, σt 1, 2 σt 2, 2 GARCH(p, q) : σ 2 t = ω + p β i σt i=1 q α j ɛ 2 t j, ω > 0, i β i 0, j α j 0 j=1 GARCH garch GARCH(1,1) garch11 garch11<-garch(dnikkei, order=c(1,1)) (6) summary(garch11) Call: garch(x = dnikkei, order = c(1, 1)) Model: GARCH(1,1) Residuals: Min 1Q Median 3Q Max Coefficient(s): Estimate Std Error t value Pr(> t ) a0 3886e e e-16 *** 3

4 a1 6483e e e-16 *** b1 9134e e < 2e-16 *** --- Signif codes: 0 *** 0001 ** 001 * Diagnostic Tests: Jarque Bera Test data: Residuals X-squared = , df = 2, p-value < 22e-16 Box-Ljung test data: SquaredResiduals X-squared = 10566, df = 1, p-value = 0304 GARCH(1,1) σ 2 t = ɛ 2 t σ 2 t 1, Std Error: ( ) (00049) ( ) 23 ARCH,GARCH GARCH(1,1) summary(garch11) GARCH(1,1) Diagnostic Tests: Jarque Bera Test data: Residuals X-squared = , df = 2, p-value < 22e-16 Box-Ljung test data: SquaredResiduals X-squared = 10566, df = 1, p-value = 0304 tseries GARCH Jarque Bera Test: p Ljung-Box Test: 1 4 p 5 n<-length(dnikkei) nparam<-length(coef(garch11))-1 L<-30 garchbox<-rep(0,l) garchboxsq<-rep(0,l) for(i in 1:L){ box<-boxtest(garch11$resid,lag=i,type="ljung-box") boxsq<-boxtest(garch11$resid^2,lag=i,type="ljung-box") garchbox[i]<-box$pvalue 4 GARCH ɛ t 5 HP acf pacf plot=false 4

5 garchboxsq[i]<-boxsq$pvalue } par(mfrow=c(3,4),mar=c(4,4,4,05),las=1) plot(garch11$resid,type="l",main="residual") plot(garchbox,type="b",ylim=c(0,1),main="ljung-box p-value") abline(h=005,lty=2,col="red") abline(v=25) acf(garch11$resid[nparam:n],main="acf") pacf(garch11$resid[nparam:n],main="pacf") cpgram(garch11$resid,main="cumulative Periodgram") qqnorm(garch11$resid,main="normal QQ plot") qqline(garch11$resid) plot(garch11$resid,type="n",bty="n",xlab="",ylab="",xaxt="n",yaxt="n") plot(garch11$resid^2,type="l",main="residual Square") plot(garchboxsq,type="b",ylim=c(0,1),main="ljung-box p-value") abline(h=005,lty=2,col="red") abline(v=25) acf(garch11$resid[nparam:n]^2,main="acf") pacf(garch11$resid[nparam:n]^2,main="pacf") Ljung-Box p (30 ) p p 1 3 ACF 35 ) 5% PACF( QQ Ljung-Box p 30 p 3 2 ACF 35 ) 3 3 PACF 35 5

6 3 31 EGARCH(p, q) GJR(p, q) ARCH GARCH ARCH GARCH EGARCH GJR GJR(p, q) : GJR(p, q) ɛ t

7 D t 1 σ 2 t = ω + p β i σt i 2 + i=1 q (α j ɛ 2 t j + γ j D t j ɛ2 t j), ω > 0, i, j β i, α j, γ j 0, j=1 EGARCH(p, q) : σt 2 ɛ 2 t j EGARCH ɛ t j σ t j z t j := ɛ t j /σ t j ln(σ 2 t ) = ω + p β i ln(σt i) 2 + i=1 q { α j θzt j + γ( z t j E z t j ) } j=1 R fseries garchoxinterface fseries garchoxinterface Step1: Ox Ox ( PC ) H:=YOxMetrics4=YOx OxMetrics Ox H:=YOx 6 : omori/ox/indexhtm Step2: Ox Ox G@RCHVer42 H:=YOx=Ypackages laurent/g@rch/site/indexhtml Step3 : GarchOxModelingox Modified GarchOxModelling file H:=YOx=Ylib Step4 : Step3 A corrected version of the garchoxfit function Modified GarchOxModelling file txt garchoxfit Rtxt 6 H:=YOx 7

8 R 7 16 command = "C:=Y=YOx=Y=Ybin =Y=Yoxlexe C:=Y=YOx=Y=Ylib=Y=YGarchOxModellingox" C:=Y H:=Y Ox H R source("garchoxfit Rtxt") : garchoxinterface ARMA ARMA(1,0) AR(1) GJR(1,1) m1 m1<-garchoxfit(formulamean= arma(1,0), formulavar= gjr(1,1),series=dif) (7) 8 ******************** ** SPECIFICATIONS ** ******************** Dependent variable : X Mean Equation : ARMA (1, 0) model No regressor in the mean Variance Equation : GJR(1, 1) model No regressor in the variance The distribution is a Gauss distribution Strong convergence using numerical derivatives Log-likelihood = Maximum Likelihood Estimation (StdErrors based on Second derivatives) Coefficient StdError t-value t-prob Cst(M) AR(1) Cst(V) ARCH(Alpha1) GARCH(Beta1) GHR(Gamma1) No Observations : 3920 No Parameters : 6 Mean (Y) : Variance (Y) : Skewness (Y) : Kurtosis (Y) : Log Likelihood : Alpha[1]+Beta[1]: Warning : To avoid numerical problems, the estimated parameter Cst(V), and its stderror have been multiplied by 10^4 The sample mean of squared residuals was used to start recursion The positivity constraint for the GARCH (1,1) is observed This constraint is alpha[l]/[1 - beta(l)] >= 0 7 R garchoxfit Rtxt R R C Program Files garchoxfit 8 GARCH CPU 15GHz 512MB PC 8

9 The unconditional variance is The conditions are alpha[0] > 0, alpha[l] + beta[l] < 1 and alpha[i] + beta[i] >= 0 => See Doornik & Ooms (2001) for more details The condition for existence of the fourth moment of the GARCH is observed The constraint equals and should be < 1 => See Ling & McAleer (2001) for details Estimated Parameters Vector : ; ; ; ; *************** ** FORECASTS ** *************** Number of Forecasts: 15 Horizon Mean Variance e e e e e e e e e e e-005 Forecasts errors measures cannot be computed because there are not enough out-of-sample observations) Elapsed Time : 4031 seconds (or minutes) *********** ** TESTS ** *********** Statistic t-test P-Value Skewness e-023 Excess Kurtosis Jarque-Bera NaN Information Criterium (to be minimized) Akaike Shibata Schwarz Hannan-Quinn Q-Statistics on Standardized Residuals --> P-values adjusted by 1 degree(s) of freedom Q( 10) = [ ] Q( 15) = [ ] Q( 20) = [ ] H0 : No serial correlation ==> Accept H0 when prob is High [Q < Chisq(lag)] Q-Statistics on Squared Standardized Residuals --> P-values adjusted by 2 degree(s) of freedom Q( 10) = [ ] Q( 15) = [ ] Q( 20) = [ ] H0 : No serial correlation ==> Accept H0 when prob is High [Q < Chisq(lag)] ARCH 1-2 test: F(2,3913)= [04201] ARCH 1-5 test: F(5,3907)= [05868] ARCH 1-10 test: F(10,3897)= [08387] Diagnostic test based on the news impact curve (EGARCH vs GARCH) Test P-value Sign Bias t-test Negative Size Bias t-test Positive Size Bias t-test Joint Test for the Three Effects

10 Joint Statistic of the Nyblom test of stability: Individual Nyblom Statistics: Cst(M) AR(1) Cst(V) ARCH(Alpha1) GARCH(Beta1) GJR(Gamma1) Rem: Asymptotic 1% critical value for individual statistics = 075 Asymptotic 5% critical value for individual statistics = 047 Adjusted Pearson Chi-square Goodness-of-fit test # Cells(g) Statistic P-Value(g-1) P-Value(g-k-1) Rem: k = 5 = # estimated parameters r t = r t 1 + ɛ t, Std Error: ( ) ( ), σ 2 t = σ 2 t 1 + ( D t 1 )ɛ2 t 1, Std Error: (000802) (000998) (00076) (00336) AR(1) EGARCH(1,1) m2 m2<-garchoxfit(formulamean= arma(1,0), formulavar= egarch(1,1),series=dif) 32 AIC AIC garch AIC AIC AIC := 2 log L + 2n log L n GARCH(p, q) n = p + q for(p in 1:3){ for(q in 1:3){ tmp<-garch(dif,order=c(p,q)) likelihood<-tmp$nlikeli 9 HP 10

11 nparam<-length(coef(tmp)) aic<- -2*likelihood+2*nparam cat("order1",p,"order2",q,"aic",aic, "\n")}} 9 AIC 4 Doornik, JA (2002), Object-Oriented Matrix Programming Using Ox, 3rd ed London: Timberlake Consultants Press and Oxford: wwwdoornikcom (2004),, tseries fseries Ox garchoxfit 11

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y i = Z i δ i +ε i ε i δ X y i = X Z i δ i + X ε i [ ] 1 δ ˆ i = Z i X( X X) 1 X Z i [ ] 1 σ ˆ 2 Z i X( X X) 1 X Z i Z i X( X X) 1 X y i σ ˆ 2 ˆ σ 2 = [ ] y i Z ˆ [ i δ i ] 1 y N p i Z i δ ˆ i i RSTAT