Inconsistant coefficients in ARMA-GARCH model

[ssccffv]

I am doing an ARMA-GARCH estimation on sp500 in the most recent 30 years.

But if I get an "optimal" ARMA specification based on AIC, SIC, and then fit it into an ARMA-GARCH model, both values and significance of coefficients of ARMA factors will change badly. I found some papers saying that neglect of heteroskedasticity will cause the t-statistics and p-values of estimated coefficients invalid but the mean value won't be biased. Then I don't understand why even the values of the coefficients will change dramatically. I think "unbiased" coefficients means the value of coefficients will stay consistent. Am I right?

Another question is that I'm not really clear about the effect of variance equation on mean equation. If I don't add a volatility factor in the mean equation, why would I get different results of mean equations if I only change the variance model? e.g. I use EGARCH instead of GARCH to fit the volatility.

In short, my question is just how variance equation influences mean equation if no volatility variable's in the mean equation.

Thank you very much and I am greatly appreciated if anyone can recommend some references on this.

[trubador]

ARMA-GARCH is quite a general and a data-dependent model. Assuming there is no model specification error, such dramatic changes may well arise as a result of the empirical implementation of the model. Presample variance, backcasting MA terms, optimization parameters, etc. are the usual suspects and should be considered case-wise.