Abstract: One of the purposes of the paper is to develop a linear regression model with an autoregressive moving average with exogenous inputs(ARMAX) error whose conditional variance follows a generalized autoregressive conditional heteroskedasticity (GARCH) process. The new model
called a modified ARMAX-GARCH model. A comparison between Bayesian and non-Bayesian (namely, maximum likelihood) estimators in the small sample of the modified model can be done replicated based on the Monte Carlo experiments. The empirical results showed the accuracy of the Bayesian estimators and confirmed that,theBayesian methodisbetterthanthe maximum likelihood (MLE)
method intermsofsmallermeansquareerrorsinparticularwhenthesamplesizeisrelativelysmall.Another
purpose of the paper is to developamethod to test the small-sample properties based on Bayesian estimators
of the modified model such as, strictstationarityandergodicityoftheconditionalvariance,-near
epoch dependent finite
unconditional variance and standard deviation in the GARCHprocess.Thus, we estimate the proposed model for the daily data of Egyptian stock
price index, and test the small-sample properties of the GARCHprocess.
The empirical results showed that,
eveniftheposteriormeansandstandarddeviations oftheGARCHprocessaresimilar,theposteriorprobabilities ofpropertiesmaynot.
Keywords and phrases: ARMAX model, GARCH model, Bayesian estimation, maximum likelihood (MLE) estimation, strict stationarity and ergodicity.
