GARCH Models Estimation and Inference

Università di Pavia GARCH Models Estimation and Inference Eduardo Rossi

Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function constructed under the auxiliary assumption of an i.i.d. distribution for the standardized innovation z t θ). z t θ) ε t θ)/ t θ) z t θ) i.i.d.0,1) z t θ) f z t θ);η) η is the nuisance parameter, η H R k. Let y T,y T 1,...,y 1 ) be a sample realization from an ARCH model, and ψ θ,η ), the combined m+k) 1 parameter vector to be estimated for the conditional mean, variance and density functions. Eduardo Rossi c - Time series econometrics - CIdE 011

Likelihood function The log-likelihood function for the t-th observation is then given by l t y t ;ψ) = log{f [z t θ);η]} 1 log[ t θ) ] t = 1,,... The term 1 ln[ t θ) ] is the Jacobiam that arises in the transformation from the standardized innovations, z t θ) to the observables y t f y t ;ψ) = f z t θ);η) J, where J = z t y t = 1 t θ) Eduardo Rossi c - Time series econometrics - CIdE 011 3

Likelihood function The log-likelihood function for the full sample: logl T y T,y T 1,...,y 1 ;ψ) = T l t y t ;ψ). t=1 The maximum likelihood estimator for the true parameters ψ 0 θ 0,η 0), say ψ T is found by the maximization of the log-likelihood: ψ T = argmax ψ logl Tψ) Eduardo Rossi c - Time series econometrics - CIdE 011 4

Likelihood function Assuming the conditional density and the µ t θ) and t θ) functions to be differentiable for all ψ Θ H Ψ, the MLE ψ is the solution to where S T y T,y T 1,...,y 1 ; ψ ) s t l ty t,ψ) ψ T s t y t ; ψ ) = 0 t=1 is the score vector for the t-th observation. For the conditional mean and variance parameters in θ l t y t,ψ) = f [z t θ);η] 1 f [z t θ);η] z tθ) 1 [ t θ) ] 1 t Eduardo Rossi c - Time series econometrics - CIdE 011 5

Likelihood function where f [z t θ);η] f z tθ);η) z t ) z t θ) = ε t θ) = t and ) y t µ t θ) t where = µ t t 1 = µ t t ) 1/ t ε tθ) t t θ) ) 1/ 1 t θ) ) 3/ t ε tθ). ε t θ) y t µ t θ). Eduardo Rossi c - Time series econometrics - CIdE 011 6

Likelihood function In practice the solution to the set of m+k non-linear equations is found by numerical optimization techniques. With the normal distribution: f [z t θ);η] = π) 1/ exp the log-likelihood is: { z tθ) l t = 1 logπ) 1 z tθ) 1 log t ) } Eduardo Rossi c - Time series econometrics - CIdE 011 7

Gaussian likelihood - the score It follows that the score vector takes the form: z t s t = z t 1 t θ) ) 1 t θ) ) = z t µ t t θ) ) 1/ 1 1 t θ) ) 1 t θ) ) = ε tθ) µ t θ) t θ) 1/ + 1 t 1 = µ tθ) t θ) ) 1 t θ) ε t θ) t θ) + 1 t θ) t θ) ) 3/ t ε tθ) t θ) ) 3/ t θ) ε t θ) t θ)) 1 t θ) ) ε t θ) t θ) t θ) ) 1 Eduardo Rossi c - Time series econometrics - CIdE 011 8

Gaussian likelihood - the score s t = µ tθ) ε t θ) t + 1 t θ) ) 1 t θ) [ ] ε t θ) t θ)) 1 Several other conditional distributions have been employed in the literature to capture the degree of tail fatness in speculative prices. Eduardo Rossi c - Time series econometrics - CIdE 011 9

Nonnormal distributions - Student s t density The X t ν, with E[X] = 0 and Var[X] = ν ν has pdf f X x;ν) = Γ1 ν +1)) ) νπγ ν [1+ x ν ] ν+1) the standardized r.v. Z = ν /ν)x has standardized Student s t density, f Z z t ;ν) = ν/ν )f X ν/ν )z;ν): fz t ;ν) = cν) [ ] ν+1) 1+ z t ν cν) = Γ1 ν +1)) Γ 1 ν) πν ) ν degrees of freedom, with ν >. The condition for a finite moment of order n is n < ν. In particular the kurtosis is finite when ν > 4 and then k = 3ν ) ν 4) Eduardo Rossi c - Time series econometrics - CIdE 011 10

Nonnormal distributions - Student s t density As ν the density function converges to N0,1). The gamma function is defined as Γu) = 0 x u 1 e x dx, u > 0 In the EGARCHp,q) model: E[ z t ] = ν Γ[ν +1)/] πν 1)Γ[ν/]. The log-likelihood: l t = 1 logh t)+logcν)) ν +1 log ) 1+ z t ν Eduardo Rossi c - Time series econometrics - CIdE 011 11

Nonnormal distributions - GED f z t ;υ) = υexp[ ) 1 zt /λ υ] λ 1+1/υ) Γ1/υ) = Cυ) exp [ 1 z t λ υ] < z t <,0 < υ λ [ ] 1/ /υ) Γ1/υ)/Γ3/υ) [ Γ3/υ) ] 1/ Cυ) = υ Γ1/υ) 3 υ is a tail-thickness parameter l t ψ) = log{f [z t θ);υ]} 1 log[ t θ) ] t = 1,,... = logcυ)) 1 { log [ t θ) ] + z t υ } λ Eduardo Rossi c - Time series econometrics - CIdE 011 1

Nonnormal distributions - Skew-T Skew-T density: fz t ;ν,λ) = bc bc 1+ 1 ν 1+ 1 ν ) ) ν+1)/ bzt +a 1 λ for z t < a b bzt +a 1+λ ) ) ν+1)/ for z t a b c = Γ ) ν+1 Γ ν ) πν ) b = 1+3λ a ) ν a = 4λc ν 1 This density is defined for < ν < and 1 < λ < +1. Eduardo Rossi c - Time series econometrics - CIdE 011 13

Nonnormal distributions - Skew-T This density encompasses a large set of conventional densities, allowing us to use standard ML tests: 1. if λ=0, the Skew-t reduces to the traditional Student s t distribution.. If λ=0 and ν= we have the normal density. Let d t = bz t +a)1 λs) where s is a sign dummy 1 z t < a/b s = 1 z t a/b Eduardo Rossi c - Time series econometrics - CIdE 011 14

Nonnormal distributions - Skew-T The log-likelihood contribution is l t = logb)+logc) 1 log t θ)) ν +1) log ) 1+ d t ν Eduardo Rossi c - Time series econometrics - CIdE 011 15

Quasi-maximum likelihood When θ = α,β ) where α are the conditional mean parameters and β are the conditional variance parameters, the score takes the form: s t = l t α l t β where l t α = µ tα) α l t β = 1 ε t α) t β) t β) ) 1 t β) β [ ] ε t α) t β)) 1. Eduardo Rossi c - Time series econometrics - CIdE 011 16

Quasi-maximum likelihood Under regularity conditions, the QML estimator is asymptotically normal distributed with T θn θ 0) d N 0,A 1 BA 1) The matrices A and B are, respectively, equal to: [ ] loglθ) A = 1 T E 0 B = 1 [ loglθ) T E 0 ] loglθ) Eduardo Rossi c - Time series econometrics - CIdE 011 17

Quasi-maximum likelihood The matrices A and B are not, in general, equal when specification errors are present. Thus comparing estimates of the matrices A and B can be useful for detecting specification errors. Eduardo Rossi c - Time series econometrics - CIdE 011 18

Quasi-maximum likelihood The second derivatives matrix of the t-th log-likelihood function is equal to: l t = 1 t θ) ) t θ) t θ) 1 ε t θ) t θ) t θ) t θ)) 3 + 1 ε t θ) t θ) t θ)) µ tθ) + ε tθ) t θ) t θ) ) 1 t θ) t θ) µ t θ) t θ) ) 1 µ t θ) ε tθ) t θ) t θ)) µ t θ). ε t θ) t θ)) µ t θ) Eduardo Rossi c - Time series econometrics - CIdE 011 19

Quasi-maximum likelihood given that E t 1 [ that A t [ ] l t = E 0 = E 0 [ 1 ε t θ) t θ)) 1/ ] t θ) ) t θ) [ ] ε = 0 and E t θ) t 1 t θ)) t θ) + t θ) ) 1 µ t θ) = 1, we have ] µ t θ) Eduardo Rossi c - Time series econometrics - CIdE 011 0

Quasi-maximum likelihood The information matrix for time t is [ ] lt l t B t = E 0 [ = E 0 µ t θ) [ µ t θ) ε t θ) t θ) + 1 ε t θ) t θ) + 1 t θ) t θ) ε t θ) t θ)) 1 ε t θ) t θ)) 1 t θ) t θ) t θ) ) 1 t θ) ) 1 ] ] Eduardo Rossi c - Time series econometrics - CIdE 011 1

Quasi-maximum likelihood B t = E 0 where E 0 [ 1 1 4 1 t θ) t θ)) t θ) + t θ) ) 1 µ t θ) 1 t θ) t θ)) 3 [ M 3t θ) = E t 1 ε 3 t θ) ] K t θ) = E [ t 1 ε 4 t θ) ] t θ)) K t θ) 1) µ t θ) µ t θ) + + µ tθ) t ) ] θ) M 3t θ). Eduardo Rossi c - Time series econometrics - CIdE 011

Quasi-maximum likelihood Whenever it is possible to decompose the parameter vector in θ = α,β ), the hessian matrix for the t-th observation is: E[ t β) ) ] 1 µ t α) µ t α) 0 A t = α α [ 1 0 E t θ) ) t β) β t β) β ] Eduardo Rossi c - Time series econometrics - CIdE 011 3

Quasi-maximum likelihood The information matrix depends on M 3t and K t : B t = E E [ t θ) ) 1 µ t α) α 1 t ) 3 α) t β) β ] µ t α) α µ t α) α M 3t θ) E E 4 1 t ) 3 α) 1 t ) θ) µ t α) t β) α β M 3t θ) t β) t β) β β K t θ) 1) Eduardo Rossi c - Time series econometrics - CIdE 011 4

Quasi-maximum likelihood When the true conditional distribution is normal, i.e. z t N0,1) and M 3t θ) = 0 and K t θ) = 3, the blocks B 1,t = 0 B 1,t = 0 B,t = E [ 1 t β) t θ)) β t β) β and the expressions for A t and B t coincide. ] The asymptotic variance-covariance matrices of the QML estimator β T reduces to: Var asy[ )] T βt β = [ 1 T E t [ 1 t θ) ) t β) β t β) β ] ] 1. Eduardo Rossi c - Time series econometrics - CIdE 011 5

Asymptotic results For GARCH1,1): Lee and Hansen 1994) and Lumsdaine 1996) proved that the local QMLE is consistent and asympotically normal, assuming that Elogα 1 z t +β 1 )) < 0 necessary and sufficient condition for strict stationarity). Lee and Hansen 1994) required that E t τ [zt +k ] < uniformly with k > 0. Lumsdaine 1996) required that E[z 3 t ] <. Lee and Hansen 1994) showed that the global QMLE is consistent if ǫ t is covariance stationary. Eduardo Rossi c - Time series econometrics - CIdE 011 6

Asymptotic results For GARCHp,q): Ling and Li 1997) proved that the local QMLE is consistent and asympotically normal if E[ǫ 4 t] <. Ling and McAleer 00) proved the consistency of the global QMLE under only the second-order moment condition. Ling and McAleer 00) derived the asymptotic normality of the global QMLE under the 6th moment condition. Eduardo Rossi c - Time series econometrics - CIdE 011 7

Testing for ARCH disturbances Test for the presence of ARCH effect. This can be done with a LM test. The test is based upon the score under the null and information matrix under the null. The null hypothesis is α 1 = α =... = α q = 0 Consider the ARCH model with t = z t α) where ) is a differentiable function. z t = 1, ǫ t 1,..., ǫ ) t q α = α 1,...,α q ) where ǫ t are the OLS residuals. Eduardo Rossi c - Time series econometrics - CIdE 011 8

Testing for ARCH disturbances Under the null, t is a constant t = 0. The derivative of t with respect to α is t α = z t where = z t α) is the scalar derivative of z t α). The log-likelihood function is logl T = T l t α) = t=1 T t=1 1 the derivative of l t with respect to α is: l t α = z t ] [ ǫ t t t 1 [ ln t ) ǫ + t t ] Eduardo Rossi c - Time series econometrics - CIdE 011 9

Testing for ARCH disturbances the score under the null is logl T 0 = α 0 where and f 0 = [ ǫ 1 0 Z = z 1,...,z T) t z t ǫ t 0 ) ǫ 1,..., T 0 ) 1 = 0 Z f 0 )] 1 is a q +1) T) matrix. The second derivatives matrix is l t α α = z t ] z t [ ǫ t t 4 t 1 + z t [ z t + ǫ t t t 4 ) ǫ = t z tz t + 1 ) z tz t t t t ] Eduardo Rossi c - Time series econometrics - CIdE 011 30

Testing for ARCH disturbances This yields the information matrix under the null: A αα,0 = 1 [ ] T E logl T α α 0 = 1 [ [E ]] T E l t α α Φ t 1 0 = 1 [ E [ ]] T E l t α α Φ t 1 0 [ = 1 E ) ǫ [ T E t t t z tz t + 1 ) ]] t z tz t Φ t 1 0 { = 1 T 1 ) ) E[z } T tz t ]+ E[z tz t ] = = 1 t=1 0 ) 1 T 0 T E[z tz t ]. t=1 0 Eduardo Rossi c - Time series econometrics - CIdE 011 31

Testing for ARCH disturbances The LM statistic is given by ξ LM = 1 ) loglt 0 A 1 loglt αα,0 T α α 0 ) the LM statistic is ξ LM = f 0 Z = f 0 Z 0 [ 1 ) T ] 1 E[z tz t ] 0 t=1 T 1 E[z tz t ]) Z f 0 / t=1 it can be consistently estimated by 0 Z f 0 ξ LM = f0 ZZ Z) 1 Z f 0. Eduardo Rossi c - Time series econometrics - CIdE 011 3

Testing for ARCH disturbances When we assume normality plim ) f 0 f 0 /T asymptotically equivalent statistic would be =. Thus an ξ = Tf0 ZZ Z) 1 Z f 0 f 0 f 0 ) = TR where R is the squared multiple correlation between f 0 and Z. Since adding a constant and multiplying by a scalar will not change the R of a regression, this is also the R of the regression of ǫ t on an intercept and q lagged values of ǫ t. The statistic will be asymptotically distributed as chi square with q degrees of freedom when the null hypothesis is true. The test procedure is to run the OLS regression and save the residuals. Regress the squared residuals on a constant and q lags and test TR as a χ q. This will be an asymptotically locally most powerful test. Eduardo Rossi c - Time series econometrics - CIdE 011 33

Test for Asymmetric Effects Engle and Ng 1993) put forward three diagnostic tests for volatility models: 1. the Sign Bias Test. the Negative Size Bias Test 3. the Positive Size Bias Test. Eduardo Rossi c - Time series econometrics - CIdE 011 34

Test for Asymmetric Effects These tests examine whether we can predict the squared normalized residual by some variables observed in the past which are not included in the volatility model being used. If these variables can predict the squared normalized residual, then the variance model is misspecified. The sign bias test examines the impact of positive and negative return shocks on volatility not predicted by the model under consideration. The negative size bias test focuses on the different effects that large and small negative return shocks have on volatility which are not predicted by the volatility model. The positive size bias test focuses on the different impacts that large and small positive return shocks may have on volatility, which are not explained by the volatility model. Eduardo Rossi c - Time series econometrics - CIdE 011 35

Test for Asymmetric Effects To derive the optimal form of these tests, we assume that the volatility model under the null hypothesis is a special case of a more general model of the following form: log t) = log 0t δ 0z 0t ) ) +δ az at where 0tδ 0z 0t ) is the volatility model hypothesized under the null, δ 0 is a k 1) vector of parameters under the null, z 0t is a k 1) vector of explanatory variables under the null, δ a is a m 1) vector of additional parameters, z at is a m 1) vector of missing explanatory variables: H 0 : δ a = 0 Eduardo Rossi c - Time series econometrics - CIdE 011 36

Test for Asymmetric Effects This form encompasses both the GARCH and EGARCH models. For the GARCH1,1) model 0tδ 0z 0t ) = δ 0z 0t z 0t [ 1,t 1,ε ] t 1 δ 0 [ω,β,α] δ a = [β,φ,ψ ] z at = [ log t 1 ) ε t 1,, t 1 εt 1 )] /π t 1 Eduardo Rossi c - Time series econometrics - CIdE 011 37

Test for Asymmetric Effects The encompassing model is log t ) = log [ ω +βt 1 +αε t 1] +β log t 1 ) ) +φ ε t 1 t 1 +ψ εt 1 t 1 /π when α = β = 0 is an EGARCH1,1) while with β = φ = ψ = 0 is a GARCH1,1) model. The null hypothesis is δ a = 0. Let υ t be the normalized residual corresponding to observation t under the volatility model hypothesized: υ t ε t t Eduardo Rossi c - Time series econometrics - CIdE 011 38

Test for Asymmetric Effects The LM test statistic for H 0 : δ a = 0 is a test of δ a = 0 in the auxiliary regression υ t = z 0tδ 0 +z atδ a +u t where z 0t z at 0t 0t t δ 0 t δ a ) ) Both t and t are evaluated at δ a = 0 and δ 0 δ δ 0 the maximum a likelihood estimator of δ 0 under H 0 ) Eduardo Rossi c - Time series econometrics - CIdE 011 39

Test for Asymmetric Effects The derivatives are t = [ 0tδ 0z 0t )e δ δ 0 δ 0 a z at ] = 0t δ 0 e δ a z at t = [ 0tδ az 0t )e δ δ a δ a t δ 0 t δ a δ0 = δ 0,δ a =0 δ0 = δ 0,δ a =0 a z at ] = 0t δ 0 = 0t = 0tδ 0z 0t )e δ a z at z at e δ a z at δ0 = δ 0 δ 0 z 0t ) z at Eduardo Rossi c - Time series econometrics - CIdE 011 40

Test for Asymmetric Effects If the parameters restrictions are met, the right-hand side variables should have no explanatory variables power at all. Thus, the test is often computed as ξ LM = TR where R is the squared multiple correlation of auxiliary regression, and T is the number of observations in the sample. ) Under the encompassing model, t evaluated under the null is δ a equal to 0tz at, hence zat = z at. The regression actually involves regressing υt on a constant z0t and z at. The variables in z at are S t 1, St 1 ε t 1 and S t 1 + ε t 1. Eduardo Rossi c - Time series econometrics - CIdE 011 41

Test for Asymmetric Effects The optimal form for conducting the sign bias test is: υ t = a+b 1 S t 1 +γ z 0t +e t where S t 1 = 1 ε t 1 < 0 0 otherwise the regression for the negative size bias test is: υ t = a+b S t 1 ε t 1 +γ z 0t +e t the positive size bias test statistic: υt = a+b 3 S t 1 + ε t 1 +γ z0t +e t S t 1 + = 1 ε t 1 > 0 0 otherwise Eduardo Rossi c - Time series econometrics - CIdE 011 4

Test for Asymmetric Effects The t-ratios for b 1, b and b 3 are the sign bias, the negative size bias, and the positive size bias test statistics, respectively. The joint test is the LM test for adding the three variables in the variance equation under the maintained specification: υ t = a+b 1 S t 1 +b S t 1 ε t 1 +b 3 S + t 1 ε t 1 +γ z 0t +e t The test statistics is TR. If the volatility model is correct then b 1 = b = b 3 = 0, γ = 0 and e t is i.i.d. If z 0t is not included the test will be conservative; the size will be less than or equal to the nominal size, and the power may be reduced. Eduardo Rossi c - Time series econometrics - CIdE 011 43