TIME AGGREGATION OF NORMAL MIXTURE GARCH MODELS Carol Alexander and Emese Lazar ISMA Centre, School of Business, The University of Reading Whiteknights Park, PO Box 242 United Kingdom C.Alexander@ismacentre.rdg.ac.uk and E.Lazar@ismacentre.rdg.ac.uk ABSTRACT Normal mixture GARCH models capture the time- variation of variance, skewness and kurtosis that characterizes financial data. These models are more flexible and have been shown to offer a better fit than symmetric and asymmetric t-GARCH models. In this paper we give a weak definition for the normal mixture GARCH(1,1) model that is aggregating in time. This result paves the way for the analysis of the continuous time limits of such models. KEY WORDS GARCH process, normal mixture, time aggregation 1. Introduction The empirical distributions of returns to financial assets are characterized by skewness and excess kurtosis ([1], [2], [3], [4] and [5]). One of the most flexible and tractable families of non-normal distributions that are able to capture the skewness and kurtosis commonly found in financial data is the normal mixture family ([6] and [7]). The normal mixture density function is given by: ) ( ) ( 1 x p x i K i i φ η ∑ = = 1 1 = ∑ = K i i p where [p 1 , p 2 ,…, p K ] is the positive mixing law and ) , ; ( ) ( 2 i i i x x σ µ φ φ = are normal density functions. A random variable whose distribution is characterized by a density function of this form is denoted ) ,..., ; ,..., ; ,..., ( ~ 2 2 1 1 1 K K K p p NM X σ σ µ µ Normal mixtures have a wide range of interpretations ([6], [4], [8]). From a behavioral point of view two major directions can be differentiated: (i) the mixing law determines the relative frequencies of the different behavioral types in the population and (ii) it gives the probabilities that any element of the population behaves according to each of the behavioral types ([9]). Another feature of financial data that is widely accepted is the volatility clustering of returns. To capture this, the broad class of normal GARCH models was introduced by Engle (1982) and Bollerslev (1986) ([10] and [11]). They assume: ) , 0 ( ~ | , ' 2 1 t t t t t N I y σ ε ε − + = γ X t where I t-1 represents the information set available at time t-1 and the conditional variance follows a deterministic process with autoregressive characteristics. The basic GARCH(1,1) conditional variance process for the errors is defined by: 2 1 2 1 2 − − + + = t t t βσ αε ω σ , , 0 > ω , 0 , ≥ β α 1 < + β α The normal GARCH(1,1) process can account for some, but not all of the unconditional kurtosis in the data. Although the unconditional kurtosis can be greater under the t-GARCH process, where the conditional density of the errors is t-distributed ([12]) this model cannot account for the time variation in skewness and in kurtosis that characterizes the conditional distributions of financial asset returns ([13], [14], [15]) except if these are explicitly modeled as in [14], [16] and [17]. Many authors have considered simple normal mixture GARCH models, where the error term follows a normal mixture distribution ([18], [19], [20], [21], [22] and [23]). However most of these papers impose severe constraints on the dynamics of conditional skewness and kurtosis. Consequently Haas, Mittnik and Paolella (2004) introduced the general normal mixture NM(K)- GARCH(p, q) process ([24]). Alexander and Lazar (2005) provide an extensive analysis of the theoretical and empirical properties of the NM(K)- GARCH(1,1) process and show that the general NM(2)- GARCH(1,1) model offers a better fit than symmetric and asymmetric t-GARCH models ([25]). Moreover, time variation in conditional skewness and kurtosis is endogenous to the model. 437-033 210