Univariate GARCH(1,1): Which mean eq spec is right?

[garchrookie]

Let's say I try to estimate a univariate GARCH(1,1) with two different specifications of the mean equation.

Specification 1: Mean equation: y_t = Constant + e_t, Variance equation = GARCH(1,1)

Specification 2: Mean equation: y_t = Constant + y_t-1+e_t-1 + e_t , Variance equation = GARCH(1,1)

(Note: the mean equation 1 only contains a constant term and the mean equation 2 is a ARMA(1,1) model.)

Q1: When I change the mean equation from Spec (1) to Spec (2), all residuals or error terms will change accordingly? And, the two variance GARCH(1,1) equations provide completely different estimates?

Q2. If so, how do I know which mean equation specification is appropriate?

P.S. I am a self learner and I have not attended any GARCH class or lecture. I apologize if my question is too simple or stupid. Thanks a lot.

[startz]

Let's say I try to estimate a univariate GARCH(1,1) with two different specifications of the mean equation.

Specification 1: Mean equation: y_t = Constant + e_t, Variance equation = GARCH(1,1)

Specification 2: Mean equation: y_t = Constant + y_t-1+e_t-1 + e_t , Variance equation = GARCH(1,1)

(Note: the mean equation 1 only contains a constant term and the mean equation 2 is a ARMA(1,1) model.)

Q1: When I change the mean equation from Spec (1) to Spec (2), all residuals or error terms will change accordingly? And, the two variance GARCH(1,1) equations provide completely different estimates?

Q2. If so, how do I know which mean equation specification is appropriate?

P.S. I am a self learner and I have not attended any GARCH class or lecture. I apologize if my question is too simple or stupid. Thanks a lot.

Equation (1) is nested within equation (2). You can probably do a likelihood ratio test. An F-test on the coefficients of y_t-1 and the MA(1) term is probably also okay.