GARCH Models Estimation and Inference

Transcription

1 GARCH Models Estimation and Inference Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics / 1

2 Likelihood function The procedure most often used in estimating θ 0 in GARCH 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) /σ 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. Rossi GARCH Financial Econometrics / 1

3 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 [ σ t (θ) ] t = 1,,... The term 1 ln [ σ 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 σ t (θ) Rossi GARCH Financial Econometrics / 1

4 Likelihood function The log-likelihood function for the full sample: log L T (y T, y T 1,..., y 1 ; ψ) = T l t (y t ; ψ). 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: t=1 ψ T = arg max log L T (ψ) ψ Rossi GARCH Financial Econometrics / 1

5 Likelihood function Assuming the conditional density and the µ t (θ) and σ 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 ) T s t (y t ; ψ ) = 0 t=1 s t l t (y t, ψ) ψ is the score vector for the t-th observation. For the conditional mean and variance parameters in θ l t (y t, ψ) θ = f [z t (θ) ; η] 1 f [z t (θ) ; η] z t (θ) θ 1 [ σ t (θ) ] 1 σ t θ Rossi GARCH Financial Econometrics / 1

6 Gaussian log-likelihood 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 z t (θ) the log-likelihood is: The score is l t = 1 log (π) 1 z t (θ) 1 log ( σt ) s t = µ t (θ) ε t (θ) θ σt + 1 ( σ t (θ) ) 1 σt (θ) θ [ ] ε t (θ) (σt (θ)) 1 Several other conditional distributions have been employed in the literature to capture the degree of tail fatness in speculative prices. Rossi GARCH Financial Econometrics / 1

7 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) = Γ( 1 (ν + 1)) ( ) [1 + x νπγ ν 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): [ ] (ν+1) f (z t ; ν) = c(ν) 1 + z t ν c(ν) = Γ( 1 (ν + 1)) Γ( 1 ν) π(ν ) 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 = 3(ν ) (ν 4) Rossi GARCH Financial Econometrics / 1

8 Nonnormal distributions - Student s t log-likelihood As ν the density function converges to N(0, 1). The gamma function is defined as Γ(u) = In the EGARCH(p,q) model: 0 x u 1 e x dx, u > 0 E[ z t ] = ν Γ[(ν + 1)/] π(ν 1)Γ[ν/]. The log-likelihood: l t = 1 log (h t) + log (c(ν)) ν + 1 ( ) log 1 + z t ν Rossi GARCH Financial Econometrics / 1

9 Nonnormal distributions - GED f (z t ; υ) = υ exp [ ( ) 1 zt /λ υ] λ (1+1/υ) Γ (1/υ) [ = C(υ) exp 1 z t υ] λ < z t <, 0 < υ λ [ ( /υ) Γ (1/υ) /Γ (3/υ) ] 1/ C(υ) = υ υ is a tail-thickness parameter [ Γ(3/υ) Γ(1/υ) 3 ] 1/ l t (ψ) = log {f [z t (θ) ; υ]} 1 log [ σt (θ) ] t = 1,,... = log (C(υ)) 1 { log [ σt (θ) ] + z t υ } λ Rossi GARCH Financial Econometrics / 1

10 Nonnormal distributions - Skew-T Skew-T density: f (z t ; ν, λ) = ( bc bc ν ( ν ( ( ) ) (ν+1)/ bz t+a 1 λ for z t < a b bz t+a 1+λ ) ) (ν+1)/ for z t a b c = Γ ( ) ν+1 Γ ( ) ν π(ν ) b = 1 + 3λ a ( ) ν a = 4λc ν 1 This density is defined for < ν < and 1 < λ < +1. Rossi GARCH Financial Econometrics / 1

11 Nonnormal distributions - Skew-T This density encompasses a large set of conventional densities, allowing us to use standard ML tests: Let 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. where s is a sign dummy The log-likelihood contribution is d t = (bz t + a)(1 λs) { 1 zt < a/b s = 1 z t a/b l t = log (b) + log (c) 1 log (σ t (θ)) (ν + 1) ( log 1 + d t ) ν Rossi GARCH Financial Econometrics / 1

12 Quasi-maximum likelihood estimation Quasi-maximum likelihood estimation (QML) = the method based on the maximization of the log likelihood assuming conditional normality. Thus, we do as if the conditionally standardized process z t follows a normal distribution. Even if the normality assumption does not hold (i.e., the true distribution is not conditionally normal), the estimator - then called quasi-maximum likelihood estimator - is consistent and asymptotically normal Rossi GARCH Financial Econometrics / 1

13 Quasi-maximum likelihood estimation Under regularity conditions, the QML estimator is asymptotically normal distributed with ) T ( θn θ0 d ( N 0, A 1 BA 1) The matrices A and B are, respectively, equal to: A = 1 [ ] T E log L (θ) 0 θ θ B = 1 T E 0 [ log L (θ) θ ] log L (θ) θ 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. Rossi GARCH Financial Econometrics / 1

14 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 Consider the ARCH model with α 1 = α =... = α q = 0 σ t = ω + α 1 ɛ t α q ɛ t q The test procedure is to run the OLS regression and save the residuals: y t = x t ˆβ + ˆɛ t Regress the squared residuals (ˆɛ t ) on a constant and q lags and test TR as a χ q. Rossi GARCH Financial Econometrics / 1

15 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. 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. Rossi GARCH Financial Econometrics / 1

16 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 ( σ t ) = log ( σ 0t (δ 0z 0t ) ) + δ az at where σ 0t (δ 0 z 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 Rossi GARCH Financial Econometrics / 1

17 Test for Asymmetric Effects This form encompasses both the GARCH and EGARCH models. For the GARCH(1,1) model σ0t (δ 0z 0t ) = δ 0z 0t z at = z 0t [ 1, σ t 1, ε t 1 δ 0 [ω, β, α] δ a = [β, φ, ψ ] [ log ( ( σt 1 ) ε t 1 εt 1,, )] /π σ t 1 σ t 1 ] Rossi GARCH Financial Econometrics / 1

18 Test for Asymmetric Effects The encompassing model is log ( σt ) = log [ ω + βσt 1 + αε t 1] + β log ( σt 1 ) +φ ε ( t 1 + ψ εt 1 ) /π σ t 1 σ t 1 when α = β = 0 is an EGARCH(1,1) while with β = φ = ψ = 0 is a GARCH(1,1) 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 σ t Rossi GARCH Financial Econometrics / 1

19 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 0t δ 0 + z at δ a + u t where ( ) σ z0t σ t 0t δ 0 ( ) σ zat σ t 0t δ a Both σ t and σ t are evaluated at δ a = 0 and δ 0 δ δ 0 (the maximum likelihood a estimator of δ 0 under H 0 ) Rossi GARCH Financial Econometrics / 1

20 Test for Asymmetric Effects Let υ t be the normalized residual corresponding to observation t under the volatility model hypothesized: υ t ε t σ t The optimal form for conducting the sign bias test is: where υ t = a + b 1 S t 1 + γ z 0t + e t S t 1 = { 1 εt 1 < 0 0 otherwise the regression for the negative size bias test is: the positive size bias test statistic: υ t = a + b S t 1 ε t 1 + γ z 0t + e t υ t = a + b 3 S + t 1 ε t 1 + γ z 0t + e t S + t 1 = { 1 εt 1 > 0 0 otherwise Rossi GARCH Financial Econometrics / 1

21 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 z0t 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. Rossi GARCH Financial Econometrics / 1