# GARCH specification when series is autocorrelated

I want to model volatility of a stock index with 1460 observations. The specification of the GARCH model is planned to be EGARCH-X where X is the external regressor, with no mean specification. Unfortunately the result of LB test on raw series of diff-in-log (daily return) proves that autocorrelation exists

Box.test(na.omit(data_xts[,1]), lag = 10, type = "Ljung")
Box-Ljung test

data:  na.omit(data_xts[, 1])
X-squared = 19, df = 10, p-value = 0.04026


while the result of ARCH-LM test from FinTS library:

ArchTest(na.omit(data_xts[,1]), lag=10)
ARCH LM-test; Null hypothesis: no ARCH effects

data:  na.omit(data_xts[, 1])
Chi-squared = 314.92, df = 10, p-value < 2.2e-16


proves that the ARCH effect exists.

Should I use the mean specification such as ARMA to model the autocorrelation which in turn leads to GARCH-in-mean model specification? I prefer not to use GARCH-in-mean because my objective is "to forecast volatility and back-testing Value-at-Risk measures" especially without a mean specification.

An ARMA(p,q)-GARCH(r,s) model looks like this: \begin{aligned} x_t &= \mu_t + u_t, \\ \mu_t &= c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \theta_1 u_{t-1} + \dots + \theta_q u_{t-q} \quad \text{("ARMA")}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2 \quad \text{("GARCH")}, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $$D$$ is some probability distribution with zero mean and unit variance.
A GARCH(r,s)-in-mean looks like this: \begin{aligned} x_t &= \mu_t + u_t, \\ \mu_t &= \dots + c\sigma_t^2 \quad \text{("GARCH in mean")}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $$D$$ is some probability distribution with zero mean and unit variance. There could be some additional terms in the equation for $$\mu_t$$, thus the dots before $$c\sigma_t^2$$.
Also, note that it is impossible not to have a mean specification. The specification need not be complicated, though; it could be just a constant: $$\mu_t\equiv\mu$$.