Properties and Estimation of GARCH(1,1) Model

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1 Metodološki zvezki, Vol., No., 005, Properties Estimation of GARCH1,1 Model Petra Posedel 1 Abstract We study in depth the properties of the GARCH1,1 model the assumptions on the parameter space under which the process is stationary. In particular, we prove ergodicity strong stationarity for the conditional variance squared volatility of the process. We show under which conditions higher order moments of the GARCH1,1 process exist conclude that GARCH processes are heavy-tailed. We investigate the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH1,1 model. A bounded conditional fourth moment of the rescaled variable the ratio of the disturbance to the conditional stard deviation is sufficient for the result. Consistent estimation asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix. 1 Introduction Financial markets react nervously to political disorders, economic crises, wars or natural disasters. In such stress periods prices of financial assets tend to fluctuate very much. Statistically speaking, it means that the conditional variance for the given past VarX t X t 1, X t,... is not constant over time the process X t is conditionally heteroskedastic. Econometricians usually say that volatility σ t = VarX t X t 1, X t,... changes over time. Understing the nature of such time dependence is very important for many macroeconomic financial applications, e.g. irreversible investments, option pricing, asset pricing etc. Models of conditional heteroskedasticity for time series have a very important role in today s financial risk management its attempts to make financial decisions on the basis of the observed price asset data P t in discrete time. Prices P t are believed to be nonstationary so they are usually transformed in the so-called log returns X t = log P t log P t 1. Log returns are supposed to be stationary, at least in periods of time that are not too long. Very often in the past it was suggested that X t represents a sequence of independent identically distributed rom variable, in other words, that log returns evolve 1 Faculty of Economics, University of Zagreb, Zagreb, Croatia

2 44 Petra Posedel like a rom walk. Samuelson suggested modelling speculative prices in the continuous time with the geometric Brownian motion. Discretization of that model leads to a rom walk with independent identically distributed Gaussian increments of log return prices in discrete time. This hypothesis was rejected in the early sixties. Empirical studies based on the log return time series data of some US stocks showed the following observations, the so-called stylized facts of financial data: serial dependence are present in the data volatility changes over time distribution of the data is heavy-tailed, asymmetric therefore not Gaussian. These observations clearly show that a rom walk with Gaussian increments is not a very realistic model for financial data. It took some time before R. Engle found a discrete model that described very well the previously mentioned stylized facts of financial data, but it was also relatively simple stationary so the inference was possible. Engle called his model autoregressive conditionally heteroskedastic- ARCH, because the conditional variance squared volatility is not constant over time shows autoregressive structure. Some years later, T. Bollerslev generalized the model by introducing generalized autoregressive conditionally heteroskedastic - GARCH model. The properties of GARCH models are not easy to determine. GARCH1,1 process Definition.1 Let Z n be a sequence of i.i.d. rom variables such that Z t N0, 1. X t is called the generalized autoregressive conditionally heteroskedastic or GARCHq,p process if X t = σ t Z t, t Z.1 where σ t is a nonnegative process such that σ t = α 0 + α 1 X t α qx t q + β 1σ t β pσ t p, t Z. α 0 > 0, α i 0 i = 1,..., q β i 0 i = 1,...,p..3 The conditions on parameters ensure strong positivity of the conditional variance.. If we write the equation. in terms of the lag-operator B we get σ t = α 0 + αbx t + βbσ t,.4 where αb = α 1 B + α B α q B q βb = β 1 B + β B β p B p..5

3 Properties Estimation of GARCH1,1 Model 45 If the roots of the characteristic equation, i.e. 1 β 1 x β x... β p x p = 0 lie outside the unit circle the process X t is stationary, then we can write. as σ t = α 0 1 β1 + αb 1 βb X t = α0 + δ i Xt i.6 i=1 where α 0 = α 0 1 β1, δ i are coefficients of B i in the expansion of αb[1 βb] 1. Note that the expression.6 tells us that the GARCHq,p process is an ARCH process of infinite order with a fractional structure of the coefficients. From.1 it is obvious that the GARCH1,1 process is stationary if the process σ t is stationary. So if we want to study the properties higher order moments of GARCH1,1 process it is enough to do so for the process σ t. The following theorem gives us the main result for stochastic difference equations that we are going to use in order to establish the stationarity of the process σ t. Theorem. Let Y t be the stochastic process defined by or explicitly Y t = Y 0 Y t = A t + B t Y t 1, t N,.7 t B j + j=1 t m=1 A m t j=m+1 B j, t N..8 Suppose that Y 0 is independent of the i.i.d. sequence A t, B t. Assume that t 1 Then E ln + A < E ln B < 0..9 a Y t D Y for some rom variable Y such that it satisfies the identity in law where Y A, B are independent. Y = A + BY,.10 b Equation.10 has a solution, unique in distribution, which is given by Y D = m=1 m 1 A m j=1 B j..11 The right h side of.11 converges absolutely with probability 1.

4 46 Petra Posedel c If we choose Y 0 D = Y as in.11, then the process Y t t 0 is strictly stationary. Now assume the moment conditions E A p < E B p < 1 for some p [1,. d Then E Y p <, the series in.11 converges in pth mean. e If E Y 0 p <, then Y t converges to Y in pth mean, in particular E Y t p E Y p as t. f The moments EY m are uniquely determined by the equations m m EY m = E B k A m k EY k, m = 1,..., p.1 k where p denotes the floor function. In the next theorem we present the stationarity of the conditional variance process σ t. Theorem.3 Let σt be the conditional variance of GARCH1,1 process defined with.1.. Additionally, assume that that σ 0 is independent from Z t. Then it holds a the process σ t is strictly stationary if E [ ln α 1 Z 0 + β 1] < 0.13 σ 0 D = α 0 m 1 m=1 j=1 β1 + α 1 Z j 1.14 the series.14 converges absolutely with probability 1. b Assume that σ t is strictly stationary let σ = σ 0, Z = Z 1. Let E β 1 + α 1 Z p < 1 for some p [1,. Then E σ m < for some 1 m p. For such integer m it holds E [ σ m] = [ 1 E β 1 + α 1 Z m 1 m] 1 Proof: From. we have or m E α 1 Z k + β 1 α0 m k k E [ σ k ] <..15 σ t = α 0 + α 1 X t 1 + β 1σ t 1, σ t = α 0 + α 1 Z t 1 + β 1 σ t 1

5 Properties Estimation of GARCH1,1 Model 47 that represents a stochastic difference equation Y t = A t + B t Y t 1, where Y t = σ t, A t = α 0 B t = α 1 Z t 1 +β 1. From the assumptions of the theorem we have that E ln + A < E ln B = E [ ln β 1 +α 1 Z t 1] < 0. So, from Theorem. we have that σ t is strictly stationary with unique marginal distribution given by.14 this shows the first statement of the theorem. Additionally, suppose that E β 1 + α 1 Z p < 1. In that case we have E B p = E β 1 + α 1 Z p < 1 for some p [1, so from part f of Theorem. it follows.15. Example.4 Let X t be GARCH1,1 process.let µα 1, β 1, p = E α 1 Z + β 1 p, p [1,. In that case, it follows from Theorem.3 that a necessary condition for the existence of the stationary moment of order m, 1 m p, of a GARCH1,1 process is given by µα 1, β 1, p < 1. In the special case of m = it follows that the stationary fourth moment of the GARCH1,1 process exists if µα 1, β 1, = a j α j j 1β m j 1 < 1, that is equivalent to j=0 β 1 + α 1β 1 + 3α 1 < 1. From the recursive formula given in the Theorem. in the case of m = 1 m = we obtain E Xt = E Z t E σ α 0 t = 1 α 1 β 1 E Xt 4 = E Z 4 t E σ 4 t ] 1 = 3 [ α0 + E Xt α0 α 1 + β 1 ] [1 β1 α 1β 1 3α1 [ ] = 3 α0 + α0 α 1 + β 1 [1 β1 1 α 1 β α 1β 1 3α1 [ 1 = 3α0 1 + α ] 1 + β 1 [1 ] β1 α 1 β 1 3α α 1 β 1 = 3α 01 + α 1 + β 1 [ 1 α 1 β 1 1 β 1 α 1 β 1 3α 1 ] 1. Since the marginal kurtosis is given by k = EX4 t [ EX t ], ] 1

6 48 Petra Posedel from the previous calculus it immediately follows that k = 31 + α 1 + β 1 1 α 1 β 1. 1 β1 α 1 β 1 3α1 A little calculus shows 3Varσt = E [ ] Xt 4 3 E X t 3α0 = 1 + α [ 1 + β 1 1 α 1 β 1 1 β1 α 1β 1 3α1 3 α 0 1 α 1 β 1 ] = 3α 0 1 α 1 β 1 α 1 1 β1 α.16 1β 1 3α1. Since from the assumptions we have that α 0 > 0, 1 α 1 β 1 > 0 1 β1 α 1 β 1 3α1 < 1, it follows that all the factors in.16 are positive so we conclude that the GARCH1,1 process has the so-called leptokurtic distribution. 3 Estimation of the GARCH1,1 model Although in this section we assume that Z t are i.i.d. sequence of rom variables, the results we shall present can also be shown for the Z t strictly stationary ergodic sequence of rom variables. In that case, the assumptions for the process Z t are little modified but the main part of the calculus we present here also holds for not such strong assumptions. 3.1 Description of the model the quasi-likelihood function Suppose we observe the sequence Y t such that Y t = C 0 + ε 0t, t = 1,...,n, where we assume that ε 0t is GARCH1,1 process, exactly ε 0t = Z t σ 0t, F t = σ {ε 0s, s t}, where Z t is a sequence of i.i.d. rom variables σ0t = ω 01 β 0 + α 0 ε 0t 1 + β 0σ0t 1 a.s. 3.1 From Theorem. we have that the strict stationary solution of 3.1 is given by σ 0t = ω 0 + α 0 β k 0 ε 0t 1 k a.s. if it holds E [ ln β 0 +α 0 Z ] < 0. The process is described with the vector of parameters θ 0 = C 0, ω 0, α 0, β 0.

7 Properties Estimation of GARCH1,1 Model 49 The model for the unknown parameters θ = C, ω, α, β is given by Y t = C + ε t, t = 1,..., n, σ t θ = ω1 β + αε t 1 + βσ t 1 θ, t =,..., n with the initial condition σ 1θ = ω. With that kind of notation we have the following expression for the process of conditional variance: Let us define the compact space t σt = ω + α β k ε t 1 k. Θ = { θ : C l C C d, 0 < ω ω d, 0 < α l α α d, 0 < β l β β d < 1 } { θ : E [ ln β + αz ] < 0 }. Additionally, assume that θ 0 Θ so it immediately follows that α 0 > 0 β 0 > 0. Inference for GARCH1,1 process usually assumes that Z t are i.i.d. rom variables such that Z t N0, 1 so the likelihood function is easy to determine. Assuming that the likelihood function is Gaussian, the log-likelihood function is of the form ignoring constants L T θ = 1 T T t=1 l t θ, where l t θ = ln σt θ + ε t. σt θ Since the likelihood function does not need to be Gaussian, in other words, the process Z t does not need to be the Gaussian white noise, L T is called the quasi-likelihood function. 3. Consistency of the quasi-maximum likelihood estimator Although a finite data set is available in practice, this is not enough to determine good properties of an estimator. We shall see in this section how useful results can be obtained taking into consideration the strictly stationary model for the conditional variance that we have previously defined. We shall note it in the following way θ = ω + α β k ε t 1 k, ε t = Y t C, to avoid confusion with the original conditional variance process σ t. In that case the quasi-likelihood function is given by L ut θ = 1 T T t=1 l ut θ, where l ut θ = ln θ + ε t. θ

8 50 Petra Posedel Additionally, we are going to show that the stationary the non-stationary model are not far away in some sense. So, all the calculus is done using the stationary model then connecting the two models. Let us define σεt θ = ω + α β k ε 0t 1 k. The process is a strictly stationary model of the conditional variance which assumes an infinite history of the observed data. The process σεt is in fact identical to the process except that it is expressed as a function of the true innovations ε 0t instead of the residuals ε t. We suppose that the following conditions on the process Z t hold: 1 Z t is a sequence of i.i.d. rom variables such that EZ t = 0; Z t is nondegenerate; 3 for some δ > 0 exists S δ < such that E [ Z +δ t 4 E [ ln β 0 + α 0 Z t ] < 0; 5 θ 0 is in the interior of Θ; 6 if for some t holds then c i = c i σ 0t = c 0 + ] Sδ < ; c k ε t k i σ0t = c 0 + k=1 for every 1 i <. k=1 c k ε t k We call the conditions 1 6 elementary conditions. The proof for the following result for the case of the general GARCHq, p process can be found in [5]. Proposition 3.3 If the elementary conditions hold, there are not two different vectors ω, α, β, C ω, α, β, C such that σ 0t = ω + α Y t 1 C + β σ 0t 1 σ 0t = ω + α Y t 1 C + βσ 0t 1. The following lemma would be very helpful for the results we shall provide. The proof can be found in [10]. Lemma 3.4 Uniformly on θ B 1 σ εt θ σ ut θ Bσ εt θ a.s. where αd α B = β d 1 d. Cd C l max, 1 + Cd C l 1 β d

9 Properties Estimation of GARCH1,1 Model 51 Although we are not going to discuss the rational moments of the process σ 0t, we will still mention that, under the elementary conditions, there exists 0 < p < 1 such that E σ 0t p <. 3. The proof for such a result can be found in [13], Theorem 4. The following lemma gives us the basic properties of the process σ ut the likelihood function l ut. Lemma 3.5 If the elementary conditions hold i The process σ ut θ is strictly stationary ergodic; ii The process l ut θ the processes of its first second derivatives with respect to θ are strictly stationary ergodic for every θ in Θ; iii For some 0 < p < 1 for every θ Θ it holds E σ ut θ p H p <. Proof: The statement 1 follows from Theorem.3. Since l ut θ = ln θ + ε t, θ l ut ε ω = t σ ut σ 1 ut θ 1 ω θ, 3.3 where l ut ε α = t σ 1 ut θ α l ut ε C = t σ 1 ut θ C 1 σ utθ, σ utθ ε t σ utθ 3.5 l ut ε β = t σ 1 ut θ 1 β θ, 3.6 σ ut ω = 1 + β σ ut 1 ω, 3.7 σ ut α = ε t 1 + β σ ut 1 α, 3.8 C = αε t 1 + β σ ut 1 C 3.9 σ ut β = σ ut 1 + β σ ut 1 β, 3.10 it follows that the process l ut θ processes of its first second derivatives are measurable functions of strictly stationary ergodic process ε t so they are also strictly

10 5 Petra Posedel stationary ergodic. Finally, let 0 < p < 1 from 3.. Then it follows from Lemma 3.4 E σ ut θ p B p E σεt θ p = B p E ω + α β k ε 0t 1 k p [ B p ω p + α p β kp E ε0t k 1 ] p. Since ε 0t 1 k α 1 0 σ 0t for every k, using 3. it follows E p B [ω p p + α p β kp 1 α p E ] σ p 0t 0 [ = B p ω p + αp α p E ] σ p 1 0t 0 1 β p B [ω p p d + αp d α p E ] σ p 1 0t 0 1 β p H p <. d Some nontrivial calculus give us the following result. Lemma 3.6 Under the elementary conditions it holds LuT θ L T θ 0 a.s. when T. sup θ Θ Finally, we want to find additional constraints for the expression σ 0t σ ut its inverse uniformly on θ Θ. We will do so by splitting the parameter space. Let R l = RK 1 l α l < 1 where Rψ = + ψp [ ] 1 < 1, for ψ > 0, P = 1 0, 1 + ψ +δ δ S δ δ S δ define in the elementary conditions K l = ω d + α d <. Let η l η d be ω 0 α 0 positive constants such that 1 η l < β 0 1 R 1 l 1 η d < β 0 1 R 1, where R 0 = Rα 0 < 1. For 1 r 1 define constants subspaces β rl = β 0 R 1 r l + η l < β 0 β rd = β 0 η d R 1 r 0 > β 0, Θ r l = {θ Θ : β rd β β 0 } Θ r d = {θ Θ : β 0 β β rd } We will need r to be 1 in Lemma 4.. Our aim is to find the minimal r so that all the statements presented bellow hold for every θ Θ r. 0

11 Properties Estimation of GARCH1,1 Model 53 Θ r = Θ r l Θr d. The values η l η d will depend on constants R l R 0 which are functions of the parameter space Θ. Observe that we can choose Θ = Θ rmax Θ r, for all 1 r 1. Now we are able to present the result about the convergence in probability of the unconditional likelihood process. Lemma 3.7 Under the elementary conditions for every θ Θ 1 it holds: ε 1 E t H θ 1 C d C l + BH c where H c = ω 0 + α 0 <. α l η l In this case it holds L ut θ P Lθ when T, where Lθ = E lut θ Proof: It is straightforward to show that σ 0t σεtθ r H c. Hence, using Lemma 3.4 g = C 0 C we have the following. ε E t ε0t + g = E [ ] [ ε = BE 0t 1 + ge E ] [ ] g ε σεt 0t F t 1 + E [ ] [ ] ε BE 0t + g σ = BE 0t + g σ εt B σ 0t σ εt 1 + g σ εt so ε E t BH c + g Cd C l BH c + H 1 that proves the first statement. Additionally, we have E lut θ = E ln θ + ε t θ E ln σ ut θ + E ε t σ utθ.

12 54 Petra Posedel But, for x 1 0 < p < 1 it holds the inequality ln x < 1 p xp, so we have E ln σ ut θ ln ωl + E ln σ ut θ [ ln ωl 1 σ p ] + p E ut θ = ln ωl + 1 p E [ σ p utθ ] since σ ut θ 1. Finally, using Lemma 3.5 we have E l ut θ <. Since l ut θ is strictly stationary ergodic, it follows L ut θ = 1 T T P l ut θ 1 E[ l ut θ ] = Lθ, θ Θ 1. t=1 The convergence in probability that we have presented in Lemma 3.7 is not a sufficient condition for the consistency of the quasi-maximum likelihood estimator. It is necessary that the convergence we have previously obtained holds uniformly. In order to obtain that, it is sufficient to find an upper bound for the score vector of the log-likelihood function l ut θ uniformly on θ. The details regarding the explicit forms of the upper bounds can be found in [10]. Let A = traa 1 A r = E A r 1 r be the Euclidean norm of a matrix or a vector the L r norm of a rom matrix or a vector respectively. Now we are going to present the local consistency of the quasi-maximum likelihood estimator. Let us define ˆθ T = arg max θ Θ 3 L T θ. ˆθ T is the parameter value that maximizes the likelihood function on the set Θ 3 Θ. Theorem 3.8 Under the elementary conditions ˆθ T P θ 0 when T.

13 Properties Estimation of GARCH1,1 Model 55 4 Asymptotic normality of the quasi-maximum likelihood estimator In this section we present the asymptotic distribution of the quasi-maximum likelihood estimator QMLE. In order to do so, we need stronger conditions on the process Z t than the elementary conditions we have given in the previous section. In fact, we pretend that the fourth moment of the rom variable Z t is finite. We are going to call the following condition additional condition. E Z 4 0 K <. We do not present the proof for the following results as this would require long nontrivial calculus. Lemma 4.1 Under the elementary conditions under additional condition it holds i E l ut θ l ut θ <, for every θ Θ 1 ; ii Let 1 T D l t θ 0 N0, A 0, where A 0 = E l ut θ 0 l ut θ 0. T t=1 B T θ = 1 T T l t θ Bθ = E l ut θ. t=1 Lemma 4. Suppose the elementary conditions the additional condition to hold. Then i E sup θ Θ 1 l ut θ < ; ii For i = 1,, 3, 4, E sup iii θ Θ 1 l ut θ θ i <, where θ i is the i-th element of θ; P sup B T θ Bθ 0 Bθ is a continuous function on Θ 1. θ Θ 1 The following result presents one of the classical results in asymptotic analysis it will be the basic tool for our further considerations. The details regarding the proof can be found in [9, p. 185]. Theorem 4.3 Let X T be a sequence of rom m n matrices let Y T be a P sequence of rom n 1 vectors such that X T C Y D T Y Nµ, Ω when T. Then the limiting distribution of X T Y T is the same as that of CY ; that is X T Y T D NCµ, CΩC when T. The following result assures that B 0 is a regular matrix.

14 56 Petra Posedel Lemma 4.4 Suppose that the joint distribution of ε t, ε t, σ ut is nondegenerate. Then for every θ Θ the matrix [ ] σ E ut σ 4 θ θ ut is positive definite. Finally, we have all the necessary results for studying the asymptotic behavior of the parameter estimator. In fact, using the results presented above, the following theorem can be proved. Theorem 4.5 Suppose the elementary conditions the additional condition to hold. Then T ˆθT θ 0 D N0, V 0, where V 0 = B 1 0 A 0B 1 0, B 0 = Bθ 0 = E l ut θ 0 A 0 is defined in Lemma 4.1. Notice that A 0 = 1 EZ B 0. So, in the case in which Z t is a sequence of rom variables such that Z t N0, 1 we would have EZ0 4 1 = A 0 = B 0. Let ˆB T = B T ˆθ T. In the case of maximum likelihood estimator, ˆBT would be the stard estimator of the covariance matrix. But in a more general case of quasi-maximum likelihood estimator, the asymptotic covariance matrix is B0 1 A 0 B0 1 according to Theorem 4.5. Since this is not equal to B0 1, ˆB T would not be a consistent estimator of that value. Let us define A T θ = 1 T l t θ l t θ T t=1 Â T = A T ˆθ T Aθ = E l ut θ l ut θ. The following result presents the consistency of the covariance matrix estimator. Lemma 4.6 Suppose the elementary conditions the additional condition to hold. Then i sup AT θ Aθ P 0 Aθ is continuous on Θ 1 ; θ Θ 1 ii ˆV T = 1 ˆB T Â ˆB 1 P T T B0 1 A 0B0 1. Lemma 4.6 completes our characterization of classical properties of the QMLE for GARCH1, 1 model. We show that the covariance matrix estimator is consistent for the asymptotic variance of the parameter estimator.

15 Properties Estimation of GARCH1,1 Model 57 References [1] Amemiya, T. 1985: Advanced Econometrics. Cambridge: Harvard University Press. [] Anderson, T.W. 1971: The Statistical Analysis of Time Series. New York: Wiley. [3] Basrak, B., Davis R.A., Mikosch, T. 00: Regular variation of GARCH processes. Stochastic. Process. Appl., 99, [4] Basrak, B., Davis R.A., Mikosch, T. 00: A characterization of multivariate regular variation. Ann. Appl. Probab., 1, [5] Berkes, I., Horváth, L., Kokoszka, P. 003: GARCH process: structure estimation. Bernoulli, 9, [6] Billingsley, P. 1968: Convergence of Probability Measures. New York: Wiley. [7] Billinglsley, P. 1979: Probability Measure. New York: Wiley. [8] Embrechts, P., Klüppelberg, C., Mikosch, T. 1997: Modelling Extremal Events. Berlin: Springer-Verlag. [9] Hamilton, J.D. 1994: Time Series Analysis. Princeton: Princeton University Press. [10] Lee, S.W. Hansen, B.E. 1994: Asymptotic theory for the GARCH1,1 quasimaximum likelihood estimator. Econometric Theory, 10, 9-5. [11] Mikosch, T. amd Stărică, C. 1999: Change of structure in financial time series, long range dependence the GARCH model. IWI Preprint, 5. [1] Mikosch, T. Straumann, D. 003: Stable limits of martingale transforms with application to the estimation of GARCH parameters. Working Paper, no [13] Nelson, D.B. 1990: Stationarity persistance in the GARCH1,1 model. Econometric Theory, 6, [14] Straumann, D. 003: Estimation in Conditionally Heteroscedastic Time Series Models, Ph.D. Thesis, University of Copenhagen. [15] Williams, D. 1991: Probability with Martingales. Cambridge: Cambridge University Press.

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