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The question is that I want to know whether there is difference in the applying of sign bias test in detecting the exist of asymmetric effects and the adequacy of symmetric GARCH model.

In the definition of sign bias test, we need to do the t test for the coefficient $\beta$ in the regression equation $\hat{\epsilon}_t^2 = \alpha + \beta S_{t-1}^- + z_t$, where $\hat{\epsilon}_t$ is the estimated residual, $S_{t-1}^-$ is the dummy function that $\hat{\epsilon}_{t-1} < 0 $ and 0 otherwise and $z_t$ is the noise.

My quiz is that how do I get the estimated $\hat{\epsilon}$. In the original paper by Engle and Ng http://www.finance.martinsewell.com/stylized-facts/volatility/EngleNg1993.pdf

It explained that $\hat{\epsilon}_t$ is from the equation that $\hat{\epsilon}_t = y_t - \mu_t$. My understanding is that, if $\hat{\epsilon}_t$ is from the that equation, then it is only with the purpose to check the exist of asymmetric effects. But some material explain $\hat{\epsilon}_t$ is estimated from a symmetric GARCH model, for example, GARCH(1,1), if we want to show whether a symmetric GARCH model is adequate in describing the asymmetric effects. I am confused about it.

Thanks in advance.

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