2. Volatility Models. 2.1 Background. Below are autocorrelations of the log-index.

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1 2. Volatility Models 2.1 Background Example 2.1: Consider the following daily close-toclose SP500 values [January 3, 2000 to March 27, 2009] Below are autocorrelations of the log-index. Obviously the persistence of autocorrelations indicate that the series is integrated (see definition below). The autocorrelations of the return series suggest that the returns might be stationary SP500 Daily Returns and Index Values SP500 Index Autocorrelations Index (log) ACF Lag SP500 Return Autocorrelations Years Return (%) ACF Lag: ac p-val Lag 1 2

2 Definition 2.1: Time series y t, t =1,...,T is covariance stationary if Definition 2.3: Times series y t is said to be integrated of order 1, if it is of the form (1) E[y t ] = μ, for all t Cov[y t,y t+k ] = γ k, for all t Var[y t ] = γ 0 (< ), for all t (3) (1 L)y t = δ + ψ(l)u t, denoted as y t I(1), where (4) ψ(l) =1+ψ 1 L + ψ 2 L 2 + ψ 3 L 3 + Any series that are not stationary are said to be nonstationary. Definition 2.2:Time series u t is a white noise process if E[u t ] = μ, for all t such that j=1 ψ j <, ψ(1) 0, roots of ψ(z) =0 are outside the unit circle [or the polynomial (4) is of order zero], and u t is a white noise series with mean zero and variance σu 2. (2) Cov[u t,u s ] = 0, for all t s Var[u t ] = σ 2 u We denote u t WN(μ, σ 2 u). <, for all t. Remark 2.1: Usually it is assumed in (2) that μ =0. Remark 2.2: A WN-process is obviously stationary. 3 4

3 Remark 2.3: If a time series process is of the form of the right hand side of (4), i.e., (5) x t = δ + ψ(l)u t, where ψ(l) satisfies the conditions of Def 1.3, it can be shown that x t is stationary. We denote x t I(0), i.e, integrated of order zero. Remark 2.4: The assumption ψ(1) 0 is important. It rules out for example trend stationary series (6) y t = α + βt + ψ(l)u t. Because E[y t ]=α + βt, y t is nonstationary. However, (7) (1 L)y t = β + ψ(l)u t, where (8) ψ(l) =(1 L)ψ(L). Now, although, (1 L)y t is stationary, however, ψ(1) = (1 1)ψ(1)=0, Example 2.2: (Example 2.1 continued) Below are results for ARMA(1, 1). R (stats::arima): AR(1) > arfit <- arima(x = spr, order = c(1,0,q)); > print(arfit); > tsdiag(arfit) Coefficients: ar1 ma1 intercept s.e sigma^2 estimated as 1.954: log likelihood = , aic = T = 2320, R2 = Residuals: LB p-value which does not satisfy the rule in Definition 1.3, and hence a trend stationary series is not I(1). 5 6

4 Residual autocorrelations for ARMA(1, 1): Standardized Residuals Time ACF of Residuals The residual autocorrelations and related Q-statistics indicate some, although small autocorrelation left to the series. The autocorrelations of the squared residuals strongly suggest that there is still left time dependency into the series. ARMA(1,1) residuals squared ACF Lag p values for Ljung-Box statistic p value ACF Lag The dependency is nonlinear by nature, however lag Lag ac p-val

5 Also a casual analysis with rolling k = 22-day ( month of trading days) mean volatility (annualized standard deviation) (9) (10) t = k,...t m t = 1 k s t = 2521 k t u=t k+1 t u=t k+1 r u, (r u m t ) 2, SP500 daily returns and 22-day rolling average Because squared observations are the building blocks of the variance of the series, the results suggest that the variation (volatility) of the series is time dependent. This leads to the so called ARCH-family of models. Note: Volatility not directly observable!! Methods: Return (%) a) Implied volatility b) Realized volatility c) Econometric modeling (stochastic volatility, ARCH) SP500 absolute returns and 22-day rolling volatility Volatility (%p.a) sqrt(252) x abs ret 22 day volat Day The inventor of this modeling approach is Robert F. Engle (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50,

6 2.2. Conditional Heteroscedasticity ARCH-models The general setup for ARCH models is (11) y t = x tθ + u t with x t =(x 1t,x 2t,...,x pt ), θ =(θ 1,θ 2,...,θ p ), t =1,...,T, and (12) u t F t 1 N(0,σ 2 t ), where F t is the information available at time t (usually the past values of u t ; u 1,...,u t 1 ), and Furthermore, it is assumed that ω>0, α i 0 for all i and α α q < 1. For short it is denoted u t ARCH(q). This reminds essentially an AR(q) process for the squared residuals, because defining ν t = u 2 t σ2 t, we can write u 2 t = ω + α 1u 2 t 1 + α 2u 2 t α qu 2 t q + ν t. (14) Nevertheless, the error term ν t is time heteroscedastic, which implies that the conventional estimation procedure used in AR-estimation does not produce optimal results here. σ 2 t = Var(u t F t 1 )=ω + α 1 u 2 t 1 + α 2u 2 t α qu 2 t q. (13) 11 12

7 Properties of ARCH-processes Consider (for the sake of simplicity) ARCH(1) process (15) σt 2 = ω + αu 2 t 1 with ω > 0 and 0 α < 1 and u t u t 1 N(0,σt 2). (a) u t is white noise (i) Constant mean (zero): (16) E[u t ]=E[E t 1 [u t ]]=E[0]=0. }{{} =0 Note E t 1 [u t ] = E[u t F t 1 ], the conditional expectation given information up to time t 1. (ii) Constant variance: Using again the law of iterated expectations, we get (18) Var[u t ] = E[u 2 t ]=E [ E t 1 [u 2 t ]] = E[σ 2 t ]=E[ω + αu2 t 1 ] = ω + αe[u 2 t 1 ]. = ω(1 + α + α α n ) + α n+1 E[u 2 t n 1 ] }{{} 0, as n = ω ( lim n ni=0 α i) = ω 1 α. (iii) Autocovariances: Exercise, show that autocovariances are zero, i.e., E[u t u t+k ]=0 for all k 0. (Hint: use the law of iterated expectations.) The law of iterated expectations: Consider time points t1 <t 2 such that F t1 F t2, then for any t>t 2 (17) E t1 [E t2 [u t ]] = E [E[u t F t2 ] F t1 ] = E[u t F t1 ]=E t1 [u t ]

8 (b) The unconditional distribution of u t symmetric, but nonnormal: (i) Skewness: Exercise, show that E[u 3 t ]=0. (ii) Kurtosis: Exercise, show that under the assumption u t u t 1 N(0,σt 2 ), and that α< 1/3, the kurtosis (19) ω 2 E[u 4 t ]=3 (1 α) 2 1 α 2 1 3α 2. Hint: If X N(0,σ 2 ) then E[(X μ) 4 ]=3(σ 2 ) 2 =3σ 4. is Because (1 α 2 )/(1 3α 2 ) > 1 we have that ω 2 (20) E[u 4 t ] > 3 (1 α) 2 =3[Var(u t)] 2, we find that the kurtosis of the unconditional distribution exceed that what it would be, if u t were normally distributed. Thus the unconditional distribution of u t is nonnormal and has fatter tails than a normal distribution with variance equal to Var[u t ]= ω/(1 α)

9 (c) Standardized variables: Estimation of ARCH models Write z t = u t (21) σt 2 then z t NID(0, 1), i.e., normally and independently distributed. Thus we can always write (22) u t = z t σ 2 t, where z t independent standard normal random variables (strict white noise). This gives us a useful device to check after fitting an ARCH model the adequacy of the specification: Check the autocorrelations of the squared standardized series. Given the model (23) y t = x tθ + u t with u t F t 1 N(0,σt 2), we have y t {x t, F t 1 } N(x t θ,σ2 t ), t =1,...,T. Then the log-likelihood function becomes (24) with l(θ) = T t=1 l t (θ) l t (θ) = 1 2 log(2π) 1 2 log σ2 t 1 2 (y t x tθ) 2 /σt 2, (25) where θ =(θ,ω,α). In practice more fat tailed (and/or skewed) conditional distributions are assumed

10 Most popular conditional distributions: Student t: (26) l t = 1 2 log ( π(ν 2)Γ(ν/2) 2 Γ((ν+1)/2) 2 ) 1 2 log σ2 t (ν+1) 2 log (1+ (y t x tθ) 2 σ 2 t (ν 2) Γ( ) the gamma function, ν>2 the degrees of freedom. Generalized error distribtuion (GED): (27) l t = 1 2 log ( Γ(1/r) 3 Γ(3/r)(r/2) 2 ) 1 2 log σ2 t ( Γ(3/r)(yt x tθ) 2 σ 2 t Γ(1/r) ), ) r/2, r the tail parameter, r = 2 a normal distribution, r<2 fat-tailed. Skewed-Student: (28) ( ) l t = 1 2 log π(ν 2)Γ(ν/2)2 (ξ+1/ξ) 2 1 4s 2 Γ((ν+1)/2) 2 2 log σ2 t ) log (1+ (m+s(y t x tθ)) 2 ξ 2I t, σ 2 t (ν 2) where { 1, if (yt x I t = tθ) m/s (29) 1, if (y t x tθ) < m/s ξ is the asymmetry parameter (ξ = 1, symmetric). Also skewed normal and skewed ged are available in some versions. More details:

11 The maximum likelihood (ML) estimate ˆθ is the value maximizing the likelihood function, i.e., (30) l(ˆθ) = max l(θ). θ The maximization is accomplished by numerical methods. Non-Gaussian series are estimated often by quasi maximum likelihood (QML), i.e., as if the error were conditionally normal. Under fairly general conditions the QML estimator of ˆθ is consistent and asymptotically normal: (31) T (ˆθ θ) N(0, Jθ 1 I θ Jθ 1 ), where [ ] lt (θ) (32) J θ = E θ θ, and (33) [ ] lt (θ) l t (θ) I θ = E θ θ. Remark 2.5: If the conditional ditribution is Gaussian, J θ = I θ

12 Generalized ARCH models In practice the ARCH needs fairly many lags. Usually far less lags are needed by modifying the model to (34) σ 2 t = ω + αu2 t 1 + βσ2 t 1, with ω>0, α>0, β 0, and α + β<1. The model is called the Generalized ARCH (GARCH) model. Usually the above GARCH(1,1) is adequate in practice. Note again that defining ν t = u 2 t σ2 t, we can write (35) u 2 t = ω +(α + β)u2 t 1 + ν t βν t 1 a heteroscedastic ARMA(1,1) process. Applying backward substitution, one easily gets (36) σ 2 t = ω 1 β + α an ARCH( ) process. j=1 β j 1 u 2 t j Thus the GARCH term captures all the history from t 2 backwards of the shocks u t. Econometric packages call α (coefficient of u 2 t 1 ) the ARCH parameter and β (coefficient of σt 1 2 ) the GARCH parameter

13 Imposing additional lag terms, the model can be extended to GARCH(r, q) model (37) σ 2 t = ω + r j=1 [c.f. ARMA(p, q)]. β j σ 2 t j + q αu 2 t i i=1 Example 2.3: MA(1)-GARCH(1,1) model of SP500 returns estimated with conditional normal, u t F t 1 N(0,σt 2), and GED, u t F t 1 GED(0,σt 2) The model is (38) r t = μ + u t + θu t 1 σ 2 t = ω + αu 2 t 1 + βσ2 t 1. Nevertheless, as noted above, in practice GARCH(1,1) is adequate SP500 Returns

14 > fgarch::garchfit(spr ~arma(0,1) + garch(1,1), data = spr, cond.dist = "ged", trace = F) Conditional Distribution: norm Coefficient(s): mu ma1 omega alpha1 beta Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(> t ) mu ma *** omega *** alpha e-15 *** beta < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized: AIC: BIC: (computed from fit@llh ) > fgarch::garchfit(spr ~arma(0,1) + garch(1,1), data = spr, cond.dist = "ged", trace = F) Conditional Distribution: ged Coefficient(s): mu ma1 omega alpha1 beta1 shape Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(> t ) mu ** ma *** omega * alpha e-11 *** beta < 2e-16 *** shape < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized: AIC: BIC: (computed from fit@llh ) Goodness of fit (AIC, BIC) is marginally better for GED. The shape parameter estimate is < 2 (statistically significantly) indicated fat tails. Otherwise the coefficient estimates are about the same

15 Residual Diagnostics: Autocorrelations of squared standardized residuals: Normal: GED: Lag LB p-value LB p-value JB = , df = 2, p-value = (dropping the outlier JB = 43.96) The autocorrelations of the squared standardized residuals pass (approximately) the white noise test. Nevertheless, the normality of the standardized residuals is strongly rejected. Usually this affects mostly to stantard errors. Common practice is to use some sort of robust standard errors (e.g. White) SP500 Return residuals [MA(1)-GED-GARCH(1,1)] Conditional Volatility Volatility [sqrt(252*h_t)] Days 29 30

16 The variance function can be extended by including regressors (exogenous or predetermined variables), x t,init (39) σ 2 t = ω + αu 2 t 1 + βσ2 t 1 + πx t. Note that if x t can assume negative values, it may be desirable to introduce absolute values x t in place of x t in the conditional variance function. Example 2.4: Monday effect in SP500 returns and/or volatility? Pulse (additive) effect on mean and innovative effect on volatility y t = φ 0 + φ m M t + φ(y t 1 φ m M t 1 )+u t (40) σt 2 = ω + πm t + αu 2 t 1 + βσ2 t 1, where M t = 1 if Monday, zero otherwise. For example, with daily data a Monday dummy could be introduced into the model to capture the non-trading over the weekends in the volatility

17 SAS: conditional t-distribution proc autoreg data = tmp; model sp500 = mon/ nlag = 1 garch = (p=1, q=1) dist = t; hetero mon; run;... The AUTOREG Procedure GARCH Estimates SSE Observations 2321 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE Normality Test Pr > ChiSq <.0001 Standard Approx Variable DF Estimate Error t Value Pr > t Variable Label Intercept mon AR ARCH ARCH <.0001 GARCH <.0001 TDFI <.0001 Inverse of t DF HET E E Degrees of freedom estimate: 1/ ARCH-M Model The regression equation may be extended by introducing the variance function into the equation (41) y t = x t θ + λg(σ2 t )+u t, where u t GARCH, and g is a suitable function (usually square root or logarithm). This is called the ARCH in Mean (ARCH-M) model [Engle, Lilien and Robbins (1987) ]. The ARCH-M model is often used in finance, where the expected return on an asset is related to the expected asset risk. The coefficient λ reflects the risk-return tradeoff. No empirical evidence of a Monday effect in returns or volatility. 33 Econometrica, 55,

18 Example 2.5: Does the daily mean return of SP500 depend on the volatility level? Model AR(1)-GARCH(1, 1)-M with conditional t-distribution (42) y t = φ 0 + φ 1 y t 1 + λ σ 2 t + u t σ 2 t = ω + αu 2 t 1 + βσ2 t 1, where u t F t 1 t ν (t-distribution with ν degrees of freedom, to be estimated). proc autoreg data = tmp; model sp500 = / nlag = 1 garch = (p=1, q=1, mean = sqrt) dist = t; /* t-dist */ run; The AUTOREG Procedure GARCH Estimates SSE Observations 2321 MSE Uncond Var. Log Likelihood Total R-Square SBC AIC MAE AICC MAPE Normality Test Pr > ChiSq <.0001 Standard Approx Variable DF Estimate Error t Value Pr > t Intercept AR ARCH ARCH <.0001 GARCH <.0001 DELTA TDFI <.0001 Inverse of t DF The volatility term in the mean equation (DELTA) is not statistically significant neither is the constant, indicating no discernible drift in SP500 index (again ˆν =1/ )

19 Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model A stylized fact in stock markets is that downward movements are followed by higher volatility than upward movements. A rough view of this can be obtained from the cross-autocorrelations of z t and z 2 t, where z t defined in (21). Example 2.6: Cross-autocorrelations of z t and z 2 2 from Ex 2.5 Cross-correlations of z and z^2 Threshold ARCH, TARCH (Zakoian 1994, Journal of Economic Dynamics and Control, , Glosten, Jagannathan and Runkle 1993, Journal of Finance, ) is given by [TARCH(1,1)] (43)σ2 t = ω + αu2 t 1 + γu2 t 1 d t 1 + βσ 2 t 1, where ω>0, α, β 0, α + 1 2γ + β<1, and d t =1,ifu t < 0 (bad news) and zero otherwise. ACF Lag The impact of good news is α and bad news α + γ. Thus, γ 0 implies asymmetry. Some autocorrelations signify possible leverage effect. Leverage exists if γ>

20 Example 2.7: SP500 returns, MA(1)-TARCH model (EViews, Dependent Variable: SP500 Method: ML - ARCH (Marquardt) - Generalized error distribution (GED) Sample (adjusted): 1/04/2000 3/27/2009 Included observations: 2321 after adjustments Convergence achieved after 28 iterations MA Backcast: 1/03/2000 Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) + C(6)*GARCH(-1) ============================================================== Variable Coefficient Std. Error z-stat Prob C MA(1) ============================================================== Variance Equation ============================================================== C RESID(-1)^ RES(-1)^2*(RES(-1)<0) GARCH(-1) ============================================================== R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================== Inverted MA Roots.07 ============================================================== The EGARCH model Nelson (1991) (Econometrica, ) proposed the Exponential GARCH (EGARCH) model for the variance function of the form (EGARCH(1,1)) (44) log σt 2 = ω + β log σ2 t 1 + α z t 1 + γz t 1, σt 2 where z t = u t / is the standardized shock. Again, the impact is asymmetric if γ 0, and leverage is present if γ<0. Statistically significant positive asymmetry parameter estimate indicates presence of leverage

21 Example 2.8: MA(1)-EGARCH(1,1)-M estimation results. Dependent Variable: SP500 Method: ML - ARCH (Marquardt) - Generalized error distribution (GED) Sample (adjusted): 1/04/2000 3/27/2009 Included observations: 2321 after adjustments Convergence achieved after 25 iterations MA Backcast: 1/03/2000 Presample variance: backcast (parameter = 0.7) LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(5) *RESID(-1)/@SQRT(GARCH(-1)) + C(6)*LOG(GARCH(-1)) ============================================================= Variable Coefficient Std. Error z-statistic Prob C MA(1) ============================================================= Variance Equation ============================================================= C(3) C(4) C(5) C(6) ============================================================= GED PARAM ============================================================= R-squared Mean dependent var Adj R-sq S.D. dependent var S.E. of regr Akaike info criterion Sum sq resid Schwarz criterion Log lik Hannan-Quinn criter F-statistic Durbin-Watson stat P(F-stat) ============================================================= Inverted MA Roots.07 ============================================================= Power ARCH (PARCH) Ding, Granger, and Engle (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance. PARC(1,1) (45)σ δ t = ω + βσδ t 1 + α( u t 1 γu t 1 ) δ, where γ is the leverage parameter. γ>0 implies leverage. Again Again statistically significant leverage

22 Example 2.9: R: fgarch::garchfit MA(1)-APARCH(1,1) results for SP500 returns. R parametrization: MA(1), mu and theta gfa <- fgarch::garchfit(sp500r~arma(0,1) + aparch(1,1), + data = sp500r, cond.dist = "ged", trace=f) The leverage parameter ( gamma1 ) estimates to unity. ˆδ = ( ) does not deviate significantly from unity (standard deviation process). Mean and Variance Equation: data ~ arma(0, 1) + aparch(1, 1) [data = sp500r] Conditional Distribution: ged Error Analysis: Estimate Std. Error t value Pr(> t ) mu ma *** omega *** alpha e-11 *** gamma < 2e-16 *** beta < 2e-16 *** delta e-09 *** shape < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized:

23 Integrated GARCH (IGARCH) 2.3 Predicting Volatility Often in GARCH ˆα+ˆβ 1. Engle and Bollerslev (19896). Modelling the persistence of conditional variances, Econometrics Reviews 5, 1 50, introduce integrated GARCH with α + β =1. (46) σ 2 t = ω + αu2 t 1 +(1 α)σ2 t 1. Close to the EWMA (Exponentially Weighted Moving Average) specification Predicting with the GARCH models is straightforward. Generally a k-period forward prediction is of the form σt k 2 = E [ ] [ ] (48) t u 2 t+k = E u 2 t+k F t k =1, 2,... (47) σ 2 t = αu2 t 1 +(1 α)σ2 t 1 favored often in by practitioners. Unconditional variance does not exist [more details, see Nelson (1990). Stationarity and persistence in in the GARCH(1,1) model. Econometric Theory 6, ]

24 Because (49) u t = σ t z t, where generally z t i.i.d(0, 1). Thus in (48) E t [u 2 t+k ] = E t[σ 2 t+k z2 t+k ] = E t [σ 2 t+k ]E t[z 2 t+k ] (z t are i.i.d(0, 1)) = E t [σt+k 2 ]. (50) This can be utilize to derive explicit prediction formulas in most cases. ARCH(1): (51) (52) where (53) σ 2 t+1 = ω + αu2 t σ 2 t k = ω(1 αk 1 ) 1 α + α k 1 σ 2 t+1 Recursive fromula: (54) σ 2 t k = = σ 2 + α k 1 (σ 2 t+1 σ2 ), σ 2 = Var[u t ]= ω 1 α σ 2 t+1 for k =1 ω + ασ 2 t k 1 for k>

25 GARCH(1,1): IGARCH: (55) σ 2 t+1 = ω + αu2 t + βσ2 t. σ t k 2 = ω(1 (α+β)k 1 ) +(α + β) 1 (α+β) k 1 σt+1 2 (56) = σ 2 +(α + β) k 1 (σt+1 2 σ2 ), where (57) σ 2 = ω 1 α β. (58) (59) σ 2 t+1 = ω + αu2 t +(1 α)σ 2 t. σ 2 t k =(k 1)ω + σ t

26 TGARCH: (60) σ 2 t+1 = ω + αu2 t + γu2 t d t + βσ 2 t. σ 2 t k = ω(1 (α+1 2 γ+β)k 1 ) 1 (α+γ/2+β) (61) with (62) +(α γ + β)k 1 σ 2 t+1 = σ 2 +(α γ + β)k 1 (σ 2 t+1 σ2 ) σ 2 = ω 1 α 1 2 γ β. EGARCH and APARCH prediction equations are a bit more involved. Recursive formulas are more appropriate in these cases. The volatility forecasts are applied for example in Value At Risk computations

27 Evaluation of predictions unfortunately not that straightforward! (see, Andersen and Bollerslev (1998) International Economic Review. 53