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STAT 6104 - Financial Time Series Chapter 9 - Heteroskedasticity Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 43

Agenda 1 Introduction 2 AutoRegressive Conditional Heteroskedastic Model (ARCH) 3 Genearlized ARCH (GARCH) model 4 Estimation and Testing for ARCH 5 Foreign exchange rates example Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 2 / 43

Introduction Heteroskedasticity = non-constant variance Regression Context: where Y = Xβ + e, w 1 0 0 Var(e) = σ 2 0 w 2 0...... 0 0 w n Generalized least squares is used instead of ordinary least squares to account for the heterogenity of e Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 3 / 43

Introduction Variation is small for a number of successive periods of time and large for some other successive periods Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 4 / 43

Heteroskedasticity in time series e.g. log Assets price series {Y t }, i.e. Y t = log(p t ), where P t = asset price at time t Usually, X t = Y t = ɛ t is a good model i.e. Y t ARIMA(0, 1, 0) Empirically, ɛ t is found to have different variance over time Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 5 / 43

Stylized facts for log asset series X t In reality 1 X t is heavy-tailed (probability of taking a very small/large value is high) 2 {X t } is not serially correlated 3 {X 2 t } is serially correlated 4 Volatility cluster: large (small) changes in {X t } tend to be followed by large (small) changes Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 6 / 43

AutoRegressive Conditional Heteroskedastic Model (ARCH) A very popular model for volatility Model: ARCH(p) X t = σ t ɛ t, ɛ iid N (0, 1) p σt 2 = α 0 + α i Xt i 2 Meaning of conditional heteroskedastic model Let F t 1 = σ(x t 1, X t 2,...) be the σ-field (information) generated by all past information up to time t 1 i=1 E[X 2 t F t 1 ] = σ 2 t E[ɛ 2 t F t 1 ] = σ 2 t = α 0 + p α i Xt i 2 i=1 Conditional variance depends on previous values of X 2 t s Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 8 / 43

AutoRegressive Conditional Heteroskedastic Model (ARCH) ARCH(p) model: X t = σ t ɛ t, ɛ iid N (0, 1) p σt 2 = α 0 + α i Xt i 2 i=1 Restriction for the parameter set (α 0, α 1,..., α p ) α i 0, i = 0, 1,..., p Variance must be positive α 1 +... α p < 1 Must sure σ 2 t is stationary, otherwise σ 2 t explodes Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 9 / 43

Example 9.1 (Cont ) As α 1 < 1 and n Xt 2 n = α 0 α j 1 ɛ2 t ɛ 2 t j j=1 X t is causal but nonlinear and we have two methods to find E[X 2 t ] 1 E[X 2 t ] = α 0 j=0 αj 1 (E[ɛ 2 t ] = 1, t) 2 Take expectation on both sides on X 2 t = ɛ 2 t (α 0 + α 1 X 2 t 1), then E[X 2 t ] = α0 1 α 1 E[X 2 t ] = α 0 E[ɛ 2 t ] + α 1 E[X 2 t 1] as E[Xt 2 ] = E[Xk 2 ] for all pairs of t, k Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 10 / 43

Example 9.1 (Cont ) Properties of ARCH(1) X t is causal and non-linear E[X t ] = E{E(X t F t 1 )} = E[σ t E{ɛ t F t 1 }] = 0 Var(X t ) = E[X 2 t ] = α 0 1 α 1 Cov(X t+h, X t ) = E[X t+h X t ] = E[E{X t+h X t F t 1 }] = E[X t E{X t+h F t 1 }] = 0 E[X 2 t F t 1 ] = σ 2 t = α 0 + α 1 X 2 t 1 E[X 2 t ] = α 0 1 α 1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 11 / 43

Example 9.1 (Cont ) Relation between ARCH(1) and AR(1) ARCH : X t = σ t ɛ t, σ 2 t = α 0 + α 1 X 2 t 1 Rewrite Xt 2 = σt 2 + Xt 2 σt 2 = α 0 + α 1 Xt 1 2 + σt 2 (1 ɛ 2 t ) = α 0 + α 1 X 2 t 1 + ν t If X t ARCH(1), then X 2 t AR(1) with sequence ν t = σ 2 t (1 ɛ 2 t ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 12 / 43

Example 9.1 (Cont ) Properties of ν t = σ 2 t (1 ɛ 2 t ) E[ν t ] = E[E{σ 2 t (1 ɛ 2 t ) F t 1 }] = E[σ 2 t ]E[1 ɛ 2 t ] = 0 Var(ν t ) = E[ν 2 t ] = E[σ 4 t (1 ɛ 2 t ) 2 ] = 2E[σ 4 t ] (ɛ t N (0, 1) E[σ 4 t ] = 3) Cov(ν t, ν t+h ) = E[E{σt 2 σt+h(1 2 ɛ 2 t )(1 ɛ 2 t+h) F t+h 1 }] = 0 Thus, {ν t } is indeed a white noise. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 13 / 43

Example 9.1 (Cont ) ARCH : X t = σ t ɛ t, σt 2 = α 0 + α 1 Xt 1 2 AR(1) representation: Xt 2 = α 0 + α 1 Xt 2 2 + ν t, where ν t = σt 2 (1 ɛ 2 t ). As Xt 2 = σt 2 ɛ 2 t σt 2, (E[ɛ 2 t ] = 1) The AR(1) representation can be regarded as the description of the convolution of the variance of the process {X t } The correlation structure of Xt 2 model follows the same calculation of AR(1) Var(Xt 2 ) = Var(ν t) 1 α1 2, Cov(Xt 2, Xt+h) 2 = α h 1 Var(ν t) 1 α1 2, Corr(Xt 2, Xt+h) 2 = α h 1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 14 / 43

Example 9.1 (Cont ) To completely determine the covariance strcuture of {X 2 t } it remains to compute Var(ν t ) = 2E[σ 2 t ] Square both sides of σt 2 = α 0 + α 1 Xt 1 2 and take expectation E[σ 2 t ] = α 2 0 + 2α 0 α 1 E[X 2 t 1] + α 2 + 1E[X 4 t 1] = α0 2 α 0 + 2α 0 α 1 3α 1 α 1E[σ 2 t 4 ] 1 = α0 2(1 + α 1) (1 α 1 )(1 3α1 2) Note that the 4 th moment E[σ 2 t ] exists only if α 2 1 < 1 3 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 15 / 43

The Usefulness of ARCH(1) Model Stylized fact 1: X t is heavy-tailed Usually the heavier the tail, the larger the 4 th moment E[X 4 t ] = E[σ 4 t ]E[ɛ 4 t ] = 3α 0 (1 + α 1 ) (1 α 1 )(1 3α 2 1 ) For larger α 0 and small α 1, E[X 4 t ] can be very large In practice, the α 1 found is around 0.9 in many financial time series Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 16 / 43

The Usefulness of ARCH(1) Model Stylized fact 2: Correlated structure {X t } is not serially correlated {X 2 t } is non-negatively serially correlated This means Cov(X t, X t+h ) = 0, for h 0 Corr(X 2 t, X 2 t+h ) = α h 1 > 0 as α 1 > 0 by assumption Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 17 / 43

The Usefulness of ARCH(1) Model Stylized fact 3: Volatility cluster Large (small) changes in {X t } tend to be followed by large (small) changes Well explained by the equation σ 2 t = α 0 + α 1 X 2 t 1, as α 1 > 0 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 18 / 43

Genearlized ARCH (GARCH) model GARHC(p, q) model X t = σ t ɛ t, ɛ t N (0, 1) σt 2 = p α 0 + β i σt i 2 + i=1 Parameter set: (α 0, α 1,..., α q, β 1,..., β p ) - more flexibility for modelling Constraints α 1 +... + α q + β 1 +... + β p < 1 q α j Xt j 2 j=1 α j 0, j = 0,..., q, β i 0, i = 1,..., p, Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 20 / 43

GARCH Example 9.2: GARCH(1,1) is the most commonly used GARCH model in practice X t = σ t ɛ t, σ 2 t α 0 + α 1 X 2 t 1 + β 1 σ 2 t 1 Condition to ARMA(1,1) model: X 2 t = σ 2 t + X 2 t σ 2 t = α 0 + α 1 X 2 t 1 + β 1 σ 2 t 1 + X 2 t σ 2 t = α 0 + α 1 X 2 t 1 β 1 (X 2 t 1 σ 2 t 1) + β 1 X 2 t 1 + X 2 t σ 2 t = α 0 + (α 1 + β 1 )X 2 t 1 + ν t β 1 ν t 1, ARMA(1, 1)with white noise {ν t }, where ν t = X 2 t σ 2 t = σ 2 t (ɛ 2 t 1) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 21 / 43

GARCH and ARMA models In general, we have Theorem 1 If X t is a GARCH(p,q) process, then Xt 2 is an ARMA(m,p) process with noise ν t = σt 2 (ɛ 2 t 1), where m = max{p, q} Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 22 / 43

I-GARCH model For GARCH model with σt 2 = α 0 + α 1 Xt 1 2 + β 1σt 1 2 one necessary condition is to be stationary, α 1 + β 1 < 1 In practice, α 1 + β 1 1 is observed in many real datasets When α 1 + β 1 = 1, the process in non-stationary and is known as Integrated GARCH (I-GARCH) process. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 23 / 43

I-GARCH model E[σ 2 t+1 F t 1 ] = E[α 0 + α 1 X 2 t + β 1 σ 2 t F t 1 ] = α 0 + (α 1 + β 1 )σ 2 t = α 0 + σ 2 t E[σ 2 t+1 F t 1 ] = E[E{σ 2 t+2 F t } F t 1 ] = E[α 0 + σ 2 t+1 F t 1 ] Similarly, = 2α 0 + σ 2 t E[σ 2 t+j F t 1 ] = jα 0 + σ 2 t Thus, I-GARCH in non-stationary The volatility is persistent: For any j > 0, E[σ 2 t+j F t 1 ] depends on σ 2 t For GARCH with α 1 + β 1 = 1, E[σt+j 2 F t 1 ] depends on σt 2 through ρ j σt 2 which dminishes. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 24 / 43

Further generalization of GARCH model t-garch Use t-distribution instead of Normal for noise sequence ɛ t Capture more heavy-tail features e-garch Use the exponential function to model the asymmetry Others: n-garch, GARCH.M,... Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 25 / 43

Estimation and Testing for ARCH ARCH: X t = σ t ɛ t, Conditional probability density: σ 2 t = α 0 + α 1 X 2 t 1 +... + α px 2 t p f (x t F t 1 ) = 1 2πσt 2 e 1 2σ t 2 xt 2 Given σ 2 t, we have X 2 t N (0, σ 2 t ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 27 / 43

Estimation and Testing for ARCH Likelihood of ARCH(p): f (x n,..., x 1 x 0 ) = f (x n x n 1,... x 0 )... f (x 2 x 1, x 0 )f (x 1 x 0 ) log{f (x n,..., x 1 x 0 )} = log f (x t F t 1 ) = n n 2 log(2π) + 1 2 log σ2 t 1 2 Substitute σt 2 = α + p i=1 α ixt 1 2, the MLE can be obtained by maximizing the log-likelihood numerically. t=1 n xt 2 σ 2 t=1 t Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 28 / 43

Estimation and Testing for ARCH Likelihood of GARCH(p,q): In fact, the same argument holds: log{f (x n,..., x 1 x 0 )} = n n 2 log(2π) + 1 2 log σ2 t 1 2 t=1 The only difference is that σ 2 t is not observed explicitly Use recursion to obtain the approximate version of σ 2 t, σ 2 t σ 2 1 = α 0 σ 2 2 = α 0 + α 1 X 2 1 + β 1 σ 2 1 σ 2 3 = α 0 + α 1 X 2 2 + α 2 X 2 1 + β 1 σ 2 2 + β 2 σ 2 1. σ t 2 = q p α 0 + α i Xn i 2 + β j σ n j 2 i=1 j=1 n xt 2 σ 2 t=1 t Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 29 / 43

Testing for GARCH model To select the order p for ARCH model Use the AR representation and use (P)ACF plot to identify p More difficult for GARCH GARCH(p, q) corresponds to ARMA(p, q), m = max{p, q}. Information on q is lost Difficult to detect the order of ARMA using (P)ACF plot Use AIC/BIC Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 30 / 43

Testing for GARCH model Goodness of fit Protmanteau test for residuals: n Q = n(n + 2) r 2 (j)/(n j) j=1 r(j) is the ACF of the residuals of the GARCH model Q χ 2 n m m is the number of parameters in the model Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 31 / 43

Testing for GARCH model Testing for heterogeneity Lagrange multiplier test (LM) Theorem 2 If X t ARCH(p), then for the fitted regression equation p ˆX t 2 = α 0 + ˆα i Xt i 2 then R 2 satisfies nr 2 χ 2 p. Under H 0 : No heterogeneity, if H 0 is rejected, X t is heteroskedastic. i=1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 32 / 43

Foreign exchange rates example Weekly echange rates of US dollar and British pound between 1980-1988 exchange.dat Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 34 / 43

Foreign exchange rates example Heavy tail phenomenon in the data ex.s<-scan("d://exchange.dat") dex<-diff(ex.s) hist(dex,prob=t) library(mass) lines(density(dex,width=0.03),lty=3) x<-seq(-0.1,0.1,0.01) lines(x,dnorm(x,mean(dex),sqrt(var(dex))),lty=1) leg.names<-c("kerel Density", "Normal Density") legend("topleft",leg.names,lty=c(3,1)) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 35 / 43

Foreign exchange rates example Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 36 / 43

Foreign exchange rates example Exploratory data analysis ex.s<-scan("d://exchange.dat") dex<-diff(ex.s) par(mfrow=c(3,2)) ts.plot(ex.s) acf(ex.s) ts.plot(dex) acf(dex) ts.plot(dex*dex) acf(dex*dex) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 37 / 43

Foreign exchange rates example Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 38 / 43

Foreign exchange rates example Test for heteroskedasticity Lagrange Multiplier test ex.s<-scan("d://exchange.dat") dex<-diff(ex.s) x1<-lag(dex); x1<-c(0,x1[1:468]) x2<-lag(x1); x2<-c(0,x2[1:468]) x3<-lag(x2); x3<-c(0,x3[1:468]) x4<-lag(x3); x4<-c(0,x4[1:468]) z1<-x1*x1; z2<-x2*x2; z3<-x3*x3; z4<-x4*x4 lm.1<-lm(dex*dex z1+z2+z3+z4) summary(lm.1) t<-470*0.04026 >1-pchisq(t,4) [1] 0.0008140926 Conclusion: p-value = 0.0081 < 0.05, the presence of heterogeneity is confirmed Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 39 / 43

Foreign exchange rates example > summary(lm.1) Call: lm(formula = dex * dex z1 + z2 + z3 + z4) Residuals: Min 1Q Median 3Q Max -0.0028480-0.0006073-0.0004331 0.0001337 0.0187001 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 5.113e-04 9.319e-05 5.486 6.76e-08 *** z1 4.693e-02 4.614e-02 1.017 0.30963 z2 3.328e-02 4.575e-02 0.727 0.46729 z3 1.379e-01 4.576e-02 3.013 0.00273 ** z4 1.099e-01 4.615e-02 2.381 0.01766 * --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 0.00151 on 464 degrees of freedom Multiple R-squared: 0.04026, Adjusted R-squared: 0.03199 F-statistic: 4.866 on 4 and 464 DF, p-value: 0.0007485 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 40 / 43

Foreign exchange rates example #install.packages(tseries)# >library(tseries) >dex.mod<-garch(dex,order=c(1,1)) >summary(dex.mod) # -------------- Call: garch(x = dex, order = c(1, 1)) Model: GARCH(1,1) Residuals: Min 1Q Median 3Q Max -4.4816-0.6675 0.0000 0.5410 4.3634 Coefficient(s): Estimate Std. Error t value Pr(> t ) a0 3.069e-05 1.356e-05 2.262 0.02369 * a1 6.638e-02 2.040e-02 3.254 0.00114 ** b1 8.932e-01 3.339e-02 26.748 < 2e-16 *** # -------------- Fitted model is: σt 2 = 3.2 10 5 + 0.89σt 1 2 + 0.067X t 1 2. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 41 / 43

Foreign exchange rates example ex.s<-scan("d://exchange.dat") dex<-diff(ex.s) library(tseries) dex.mod<-garch(dex,order=c(1,1)) e<-dex.mod$residuals e<-e[2:469] par(mfrow=c(2,2)) ts.plot(e) acf(e) acf(e*e) qqnorm(e) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 42 / 43

Foreign exchange rates example Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 43 / 43