Citing the vignette of "rugarch" package in R,
The sign bias test of Engle and Ng (1993) <...> tests the presence of
leverage effects in the standardized residuals (to capture possible
misspecification of the GARCH model), by regressing the squared
standardized residuals on lagged negative and positive shocks as
follows:
$$ \hat z_t^2 = c_0 + c_1 I\{\hat\varepsilon_{t-1}<0\} + c_2
> I\{\hat\varepsilon_{t-1}<0\} \cdot \varepsilon_{t-1} + c_3
> I\{\hat\varepsilon_{t-1}\geqslant 0\} \cdot \varepsilon_{t-1} + u_t $$
where $\hat\varepsilon_t$ are the estimated residuals from the GARCH
process. The Null Hypotheses are $H_0: \ c_i = 0$ (for
$i = 1, 2, 3$), and that jointly $H_0: \ c_1 = c_2 = c_3 = 0$. <...> [If
rejected,] using instead a model such as the apARCH would likely
alleviate these effects.
If one, several or all the hypotheses are rejected, the idea is to use a model that allows for asymmetric effects such as GJR-GARCH, APARCH, TARCH. (Hence, you may try the ones you have not tried yet.)
However, if you are unable to obtain a satisfactory model that allows for asymmetry, perhaps using a relatively good model without assymetry will be a lesser evil than using a poor asymmetric model. The model choice is essentially an empirical question. You could try doing out-of-sample performance evaluation to select the model that suits your needs best.
References
