Some aspects of modeling conditional heteroskedasticity: Theory and applications

This dissertation concerns theoretical and empirical aspects of a class of conditionally heteroskedastic models. We apply the White's information matrix (IM) test to the linear regression model with autocorrelated errors. A special case of one component of the test is found to be identical to the Engle's Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH). Given Chesher's interpretation of the IM test as a test for parameter heterogeneity, this establishes a connection among the IM test, ARCH and parameter variation. The LM test for ARCH is interpreted as a test for the constancy of the autocorrelation coefficients. This also enables us to specify conditional heteroskedasticity in a more general and convenient way. As a result of our analysis, we propose the augmented autoregressive conditional heteroskedasticity (AARCH) model as an extension of ARCH model. In addition, we provide a clear explanation on the difference between the unconditional and conditional heteroskedasticities and suggest some new tests for higher order moments such as tests for heteroskewcity and heterokurtocity. ARCH type models introduce a specific form of heteoskedasticity into time series data analysis. It has been an econometric tradition to incorporate autocorrelation in the model to capture the time series dynamics. We attempt to study the interrelationship between autocorrelation and conditional heteroskedasticity. As the ARCH model has emerged as an important innovation for modeling the second moment of a random variable conditional on the information set, increasing concern has been directed to the formulation of conditional variance function. In order to provide a unified approach to exploring the stationarity conditions and the test statistics for various specifications of conditional heteroskedasticity, we propose a general random coefficient disturbance process in which the AR, ARCH and GARCH processes are obtained as special cases. We develop a new procedure for deriving the stationarity conditions for our general disturbance process through the vector representation of the model and discuss the interaction between autocorrelation and conditional heteroskedasticity. Also we show that the stationarity conditions for the GARCH model can be obtained in a much easier way the Bollerslev's method. Test statistics for conditional heteroskedasticity in the presence of autocorrelation and vice versa when the lagged dependent variable are included as regressors are proposed. (Abstract shortened with permission of author.)U of I OnlyETDs are only available to UIUC Users without author permissio