BILINEAR GARCH TIME SERIES MODELS Mahmoud Gabr, Mahmoud El-Hashash Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt Department of Mathematics and Computer Science, Bridgewater State University, Bridgewater, MA, USA Abstract In this paper the class of BL-GARCH (Bilinear General AutoregRessive Conditional Heteroskedasticity) models is introduced. The proposed model is a modification to the BL-GARCH model proposed by Storti and Vitale (2003). Stationary conditions and autocorrelation structure for special cases of these new models are derived. Maximum likelihood estimation of the model is also considered. Some simulation results are presented to evaluate our algorithm. Keywords : Time series, ARCH models, GARCH models, Bilinear models, weak dependence, . 1. Introduction p Xt q P Q ai Xt i c j et j i 1 bij X t i e t j j 1 (1) et i 1j 1 where {et } is a set of independent random variables. We define the model (1) as a bilinear time series model BL (p,r,m,k) and the process {Xt} as a bilinear process. In econometrics, a vast literature is devoted to the study of autoregressive conditionally heteroskedastic (ARCH) models for financial data. One of the best-known model is the GARCH model (Generalized Autoregressive Conditionally Heteroskedastic) introduced by [3] Engle (1982) and [1] Bollerslev (1986). The classical GARCH(p,q) model is given by the equations 2 A lot of time series encountered in empirical applications are nonlinear and non-stationary. Their structures such as means and variances may vary over time. The problem of nonlinear time series identification and modeling has attracted considerable attention for the past 30 years in diverse fields such as financial econometrics, biometrics, socioeconomics, transportation, electric power systems, and aeronautics which exhibit nonlinear process. A good nonlinear model should be able to capture some of the nonlinear phenomena in the data. Moreover, it should also have some intuitive appeal. Therefore a number of wide classes of nonlinear time series models have been proposed, investigated and studied. One of these classes which has received a great deal of attention is that of bilinear models. Bilinear time series models and its statistical and probabilistic properties have been extensively studied by [7] Granger and Andersen (1978), [14]Subba Rao (1981), [5] Gabr (1992) and comprehensively surveyed by [15] Subba Rao and Gabr (1984) and [11] Pham (1993). A class of non-linear model, called a bilinear class, may be regarded as a plausible non-linear extension of ARMA, rather than of the AR model. Bilinear models incorporate cross-product terms involving lagged values of the time series and of the innovation process. The model may also incorporate ordinary AR and MA terms. The general form of a bilinear time series {Xt , t 0, 1, 2,...} denoted by BL(p, q, P, Q) is defined by ε t =σ t Zt , h t =σ t ht 2 1 t 1 0 2 q t q q 0 1h t 1 pht p p i 2 t i i 1 j ht j (2) j 1 where 0 >0, i ≥0, j ≥0, q≥0, p≥0 are model parameters and {Zj, j=1, 2, 3, …} are independent identically distributed (i.i.d.) random variables with zero mean and variance 1. The variables εt, σt, Zt in (2) are usually interpreted as financial (log) returns (εt), their volatilities or conditional standard deviations (σt), and so-called innovations or shocks (Zt), respectively; the innovations are supposed to follow a certain fixed distribution (e.g., standard normal). Later, a number of modifications of (4.1) were proposed, which account for asymmetry, leverage effect, heavy tails and other” stylized facts”. Under some additional conditions, similarly as in the case of ARMA models, the GARCH model can be written as ARCH(∞) model i.e., ht can be represented as a moving average of the past squared returns 2 s , s < t, with exponentially decaying coefficients (see [1] Bollerslev, 1986) and absolutely summable exponentially decaying autocovariance function. For instance, the GARCH(p, q) process of (2) can be written as t Zt , t ht 1 (1) where (B) 1 1 0 1 (B) ht 2 t, Xt (B) 2t i Xt i 0 i Xt i 0 i 1 (4) i 1 p 1B p B and B stands for the back-shift operator, BkXt = Xt−k . This leads to the ARCH(∞) representation; t Zt , t ht b0 bi ht 2 t 2 t i (3) i 1 1 where {Zt, t=1, 2, 3, …} are i.i.d. random variables, with zero mean and variance 1, and j , j , j ≥ 0 are real (not necessary nonnegative) coefficients. Equation (4) appears naturally when studying the class of processes with the property that the conditional mean μt = E(Xt /Xs, s < t) is a linear combination of Xs, s < t, and the conditional variance 1 (1) with b0 0 and with positive exponentially decaying weights bi, i ≥ 1 defined by the generating function (y) / 1 Zt i (y) bi y . It is interesting to h 2t is the square of a linear combinations of Xs, s < t, as it is in the case of (4): i.e. i 1 note that the non-negativity of the regression coefficients αj, βj in (2) is not necessary for non-negativity of bj in the corresponding ARCH(∞) representation, see [10] Nelson and Cao (1992). 2 Clearly, if E(Zt / εs , s < t) = 0, E( Z t /εs , s < t) =1 then εt has 2 conditional mean zero and a random conditional variance t , i.e. 2 E(εt / εs , s < t) = 0, var( t /εs , s < t) = t h t The general framework leading to the model (2) was introduced by [12] Robinson (1991) in the context of testing for strong serial correlation and has been subsequently studied by [8] Kokoszka and Leipus (2000) in the change-point problem context. The class of ARCH(∞) models includes the finite order ARCH and GARCH models of [3] Engle (1982) and [2]Bollerslev (1986). 2. The Bilinear ARCH Models Formally, the classes AR, ARCH, LARCH (at least, their finite memory counterparts ARMA, GARCH, ARCH) all belong to the general class of bilinear model (1). [6] Giraitis and Surgailis (2002) studied the heteroscedastic bilinear equation 2 t = Var(Xt /Xs, s < t) t E X t / Xs ,s t i Xt i 0 i 1 2 h 2t 2 t var X t / Xs ,s t i Xt i 0 i 1 Clearly, the case j ≡ 0, j ≥ 1 gives the linear AR(∞) equation, while j ≡ 0 (j ≥ 0) results in the Linear ARCH (LARCH) model, introduced by [12] Robinson (1991), defined by the equation t t Zt , ht ht t cj t j j 1 The main advantage of LARCH is that it allows modeling of long memory as well as some characteristic asymmetries (the “leverage effect”). Both these properties cannot be modeled by the classical ARCH(∞) with finite fourth moment. The coefficients ci satisfy c j ~ k jd 1 for some 0 <d < ½ , k> 0 which implies the condition c2j j 1 Neither α nor the cj are assumed positive and, unlike in (4.3), 2 t ), is a linear combination of the past values of εt, σt (not rather than their squares. [4] Engle and Ng (1993) introduced a nonlinear asymmetric GARCH model which captures asymmetry by means of interactions between past returns and volatilities In the simple (p=1,q=1) case the conditional variance equation is given by 2 t h 2t 0 2 t p h 2t a1 ( t j 1h t 1 ) 2 b1h 2t 1 (5) with the model becoming asymmetric when the coefficient 1 is equal to zero. [13] Starti and Vitale (2003) have generalized From (6) and (8), we can see that the two models contain exactly the same number of terms, although the number of parameters required by each model is different. In fact the positivity of the parameters h 2t p aj 0 2 t j j1 cj t jh t (6) j cj b j h 2t j 2 a j bj cj t jh t j aj b j ht t j 2 j i 2 i , lie outside the unit circle. The Bilinear GARCH process (8) can be rewritten as h 2t p 0 j Zt j j h2 j t j 0 j 1 (9) which is a random coefficient autoregressive representation for h 2t where ( j Zt j j) 2 (10) the expectation of (9) is given by Hence for, α0 > 0, a sufficient condition for ht > 0, in (6), is p E h 2t j 2 j 2 j p E h 2t j j 1 Since, h2 t E jE 0 j 1 E E h 2t j j 1 0 (7) E 0 j 1 p Model (6) with the condition (7) leads us to introduce a simpler reduced parameter Bilinear GARCH model in the form; p 2 j ht j E 0 given by for j=1, 2, …, p jh t Taking in consideration the properties of {Zt }, t jh t j 2 4a j bj p 2 j ht j 1 j j=1, 2, …, p c2j of model (8). Proof j1 the advantage of being characterized by a more flexible parametric structure In this model leverage effects are explained by the interactions between past observations and volatilities To see the positivity of the conditional variance in equation (6), we can write 2 t j iu 2 i where i where a j , b j ,c j j=1, 2, ..., p are constants. This model has aj j i 1 j j1 , The Bilinear GARCH process (8) is stationary in wide sense if and only if the roots ui of the polynomial p p b j h 2t j Theorem (u) 1 t Zt , t 2 t (8) jh t j j t j j 1 this model to the following BL-GARCH model p 2 0 number of parameters in (8) is less than that in (6) by p parameters. Moreover, we do not need the condition of t Zt , t t Zt , t E 2 t/ t 1 E 2 t h 2t j The sequence of variances converges to the constant if it follows that, 2 1 p Yt (11) j Yt j 0 j 1 where Yt 2 1 1 suffices for wide sense stationarity. Under normality assumption E( 4t ) E(E( 4t / E 2t . Letting B be the backward shift operator defined by Bk Yt Yt k 3E 0 2 2 1 t 1 2 2 1 ht 1 2 1 1 t 1h t 1 3E 0 2 2 1 t 1 2 2 1 ht 1 2 1 1 t 1h t 1 , equation (11) can be rewritten as, 0 Yt p 1 jB Therefore Yt in (11) converges to a finite value if and only if p all the roots ui of the polynomial (u) 1 iu i lie outside i 1 the unit circle which completes the proof. The simplest but often very useful Bilinear GARCH process is that of order 1 given by / t 1 ~ N(0, h 2t ) 2 2 1 1) (15) (1 12 12 )(1 3 14 14 6 12 12 ) The necessary and sufficient condition for the existence of the fourth moment is 4 3 14 6 12 12 1 (16) 1 The coefficient of Kurtosis is (12) E( 0 1h t 1 1 t 1 2 3 02 (1 E( 4t ) where ht 2 From which we obtain, j j 1 t 3E(h 4t ) t 1 )) 2 E (13) 3 1 4 t) 2 t 2 1 3 4 1 2 2 1 1 4 1 6 2 2 2 1 1 (2.78) with 0 . The unconditional variance is 0 E( 2 t) E[E( 2 t / In fact it is typically found that the GARCH (1,1) model yields an adequate description of many financial time series data , see, for example , [2] Bollerslev,Chou, and Kroner (1992). A data set which requires a model of order greater than GARCH (1, 2) or GARCH (2, 1) is very rare. A series of size N=300 is generated from the simple BLGARCH model , t 1 )] E(h 2t ) E( 0 0 2 2 1 t 1 2 2 1 E( t 1 ) 0 2 2 1 h t 1 2 1 1 t 1h t 1 ) 2 2 1 E(h t 1 ) 2 2 1 E( t 1 ) 2 1 0 2 1 E( 2 2 1 E( t 1 ) With 2 t 1) The series { } is a sequence of i.i.d. N(0, 1). The initial which implies that E( 2t ) 1 0 2 1 2 1 (14) values are chosen as the series { } and { respectively and . The graph of } are presented in figures (1) and (2) ( Let 0 1 have the observations x ... p 1 m 1 ... p) and suppose that we ,...,x 0 ,x1,...,x n for the time series {xt }. Under a reasonable assumption that we have known the σ-field σ { m 1,..., 1, 0 } , we can obtain the joint conditional density function of x1,..., x n given the σ-field {x m 1,..., x0 , m 1,..., 1, 0 } as follows f (x1,..., xn / x0 ,..., x m 1, 0 ,..., m 1 ) = f (x 2 ,..., x n / x1 , x0 ,..., x m 1, 0 ,..., f (x1 / x0 ,..., x m 1, 0 ,..., m 1 ) m 1) …….. n = f (xi / x t 1 ,..., x m 1 , t 1 ,..., m 1) t 1 Figure (1) n 1 = t 1 exp 2 ht 2 t 2h 2t Thus the MLE ˆ of the parameter vector is the value of which maximizes the logarithm likelihood function L( ) ln f (x1,..., x n / x0 ,..., x m 1, 0 ,..., n ln(h t ) = t 1 2 t 2h 2t m 1) n ln(2 ) 2 n = Qt ( ) C t 1 Using the recursive Newton-Raphson iteration algorithm, the MLE ˆ can be obtained by the following iteration: (k 1) where (k) (k) H 1 ( (k) ) G( (k) ) is the set of estimates obtained at the kth stage of iteration. G( ) is the gradient vector of partial derivatives, Figure (2) L( ) G( ) 3. MLE of BL-GARCH Parameters We now consider the maximum likelihood estimation of the parameters in the BL-GARCH model (8). L( ) .... 1 2p 1 and H( ) is the Hessian matrix of second order partial derivatives, 2 H( ) L( ) i j The first order partial derivatives are given by where are calculated recursively from the equations: The second order derivatives are given by: where Note that the estimated the Hessian matrix Ĥ( ) may be singular and some numerical problems may arise. One common way to deal with this problem is the LevenbergMarquardt procedure [9] (Marquardt(1963)). 4. Monte Carlo Simulation The Newton-Raphson with Marquordt algorithm, described in the previous section were tried successfully on many sets of data simulated from several stationary BL-GARCH models. We shall consider here the following model with and The series { } is a sequence of i.i.d. N(0, 1). The initial values are chosen as and . The Newton-Raphson algorithm is applied at the above model with sample size N=300 and replicate simulations 100 times. The results from the MonteCarlo study shows, clearly, that the mean of each parameter estimates is close the true value. The standard deviations of the estimates are small indicating that the estimators are consistent. Parameter [6] Giraitis, L. and Surgailis, D. (2002) ARCH-type bilinear models with double long memory. Stoch. Process. Appl. 100, 275–300. [7] Granger, C.W.J. and Andersen, A.P. (1978) An Introduction to bilinear time series Models. Gottengen: Vendenhoek and Ruprechet. [8] Kokoszka, P. and Leipus, R. (2000) Change-point estimation in ARCH models. Bernoulli 6, 513–539. [9] Maquardt, D. (1963) “An algorithm for least squares estimation of nonlinear parameters”. J. Soc. Ind. Appl. Aath., pp 431-441. [10] Nelson, D. B. and Cao, C. Q. (1992) Inequality constraints in the univariate GARCH model. Journal of Business & Economic Statistics, 10, 229–235. N=100 estimates True value 0.8 0.5 0.4 Mean 0.809 0.481 0.376 S.D. 0.087 0.092 0.095 References [1] Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327. [2] Bollerslev,Chou, and Kroner (1992) Arch Modeling in Finance: A Review of the Theory and Empirical Evidence. Journal of Econometrics, April-May 1992, pp. 5-59. [3] Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987– 1008. [4] Engle , R.F. and V.Ng (1993) Measuring and testing the Impact of News on Volatility. Journal of Finance 48, 1749-1778. [5] Gabr, M.M. (1992) Recursive estimation of Bilinear Time Series Models”. Commun. Statist.: Theory & Meth., 21(8), 2261-2277. [11] Pham, D.T. (1993) Bilinear Times Series Models. In Dimension Estimation and od-els (ed. H. Tong), 191-223. Singapore: World Scienti.c Publishing Co. [12] Robinson, P. M. (1991). 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