I am studying time series with Ruey S. Tsay Financial time Series(3rd ed). In his text (page 32), he introduced Box & Pierce Portmanteau statistics
$$Q^*(m) = T\sum_{l=1}^m\hat{\rho}_l^2$$
for testing the null hypothesis $H_0 : \rho_1=\cdots=\rho_m=0$ against the alternative hypothesis $H_a : \rho_i\ne 0$ for some $i$.
where $\hat{\rho}_l$ is a sample lag-$l$ autocorrelation of the time series {$r_t$}.
When I read the statement above, I understood the necessity of the test.
But, the following statement made me confused.
"Under the assumption of the {$r_t$} is an iid sequence with the certain moment, $Q^*(m)$ is asymptotically a chi-squared variable with $m$ degrees of freedom."
Specifically, this "the assumption of the {$r_t$} is an iid sequence"
statement made me confused because if {$r_t$} is iid, it is independent and then all lag autocorrelation is zero. So, the Portmanteau test is not necessary. Because when the test statistic is used for testing the Hypothesis, then the test uses the statement above and then, automatically, since ${r_t}$ is iid, $\rho_l$ holds for all $l$.
Am I right? If not, What is the missing point of my observation?
It is better to view this as a test of independence vs dependence. The idea is as follows. Under $H_0$, we assume the sequence is iid. We know how $Q^*(m)$ behaves under this assumption. So if the value is very different from what we'd expect, this suggests that something is wrong with our assumption. In that case, we conclude that $H_0$ is likely to be false and that there is dependence. In particular, the definition of the statistic and the rejection region suggest that there is autocorrelation.