Suppose $\epsilon_1,...,\epsilon_n$ are i.i.d random variables with normal distribution $N(0,\sigma^2)$, and for $i=1,...,n$ we have $$Y_i=\beta_0+\beta_1x_i+\beta_2z_i+\epsilon_i.$$
I have found the maximum likelihood estimator $\hat \beta_0=\frac{1}{n}\sum_iY_i$. I am asked to then use the central limit theorem to deduce the approximate distribution of $\hat \beta_0$ when $n$ is large in the cases where we no longer assume the $\epsilon_i$ are normally distributed.
I am confused as the CLT requires the $n$ random variables that are i.i.d. but the mean of each $Y_i$ is $\beta_0+\beta_1x_i+\beta_2z_i$ which are not necessarily the same . Am I missing something?
I think it's a hand-wavy application of the CLT. The $Y_i - E[Y_i] = \epsilon_i$ are presumably i.i.d. with mean zero, so $\frac{1}{\sqrt{n}}\sum_i (Y_i - E[Y_i])$ converges in distribution to some normal distribution $N(0, \sigma^2)$ as $n \to \infty$.
They probably then want you to say that for large $n$, $\frac{1}{\sqrt{n}}\sum_i (Y_i - E[Y_i])$ is approximately $N(0, \sigma^2)$, and then rearrange this to say that $\frac{1}{n} \sum_i Y_i$ is approximately normal with some mean involving the $E[Y_i]$ and $n$.