Ljung-Box Test for Serial Correlation in Returns

Consider a zero-mean return series $\{r_t\}_{t=1}^{n}$. Define the sample autocorrelations: $\hat{\rho}(h) = \frac{\sum_{t=h+1}^{n} r_t \, r_{t-h}}{\sum_{t=1}^{n} r_t^2}$ (i) Write down the Ljung-Box portmanteau statistic $Q(m)$ and state its asymptotic null distribution under the assumption that $\{r_t\}$ is i.i.d. noise. (ii) Discuss the bias-variance tradeoff in choosing the number of lags $m$. How does conditional heteroskedasticity (e.g., GARCH effects) affect the validity of the standard Ljung-Box test? Propose a HAC-adjusted variant that remains valid under heteroskedasticity.