1. THE EXPONENTIAL GARCH MODEL

2. To allow for asymmetric effects between positive and negative asset returns, he considers the weighted innovation where θ and γ are real constants Both  t and |  t | − E(|  t |) are zero-mean iid sequences with continuous distributions

3. Therefore E[g(  t )] = 0 The asymmetry of g(  t ) can easily be seen by rewriting it as

4. An EGARCH(m, s) model can be written as where α 0 is a constant B is the back-shift (or lag) operator such that Bg(  t ) =g(  t−1 )

5. The use of g(  t ) enables the model to respond asymmetrically to positive and negative lagged values of a t

6. To better understand the EGARCH model, let us consider the simple model with order (1, 0)

7. Specifically, we have

8. Example We consider the monthly log returns of IBM stock from January 1926 to December 1997 for 864 observations. An AR(1)-EGARCH(1, 0) model is entertained and the fitted model is

12. For the 2-step ahead forecast