#StackBounty: #least-squares #asymptotics #average Asymptotic dist of an average involving OLS coefs?

Bounty: 50

Suppose that we have iid sample of size $n$. i.e., the random vector $(Y_{i}, X_{1i}, X_{2i}, X_{3i})$ is iid from $1,ldots,n$. And suppose the following relationship is true:

$$
Y_i = beta_0 + beta_1X_{1i} + beta_2X_{2i} + beta_3X_{1i}X_{2i} + epsilon_i
$$

Suppose for simplicity that $X_{1i}$ and $X_{2i}$ are uniformly distributed from 0 to 1, and are correlated. Let’s assume further that $epsilon_i$ is normally distributed and independent of $X_{1i}$ and $X_{2i}$.

Let the OLS estimators be $hat{beta}_0, hat{beta}_1, hat{beta}_2$.

Let $Z_i$ be

$$
Z_i = 1hat{beta}_0 + 2X_{1i}hat{beta}_1 + 3X_{2i}hat{beta}_2 + 4hat{beta}3*X{1i}*X_{2i}
$$

How do I find the asymptotic distribution of $bar{Z}=frac{1}{n}sum_{i=1}^n Z_i$?

I cannot apply a CLT since the $Z_i$ are correlated with each other because of the $hat{beta}$. In addition to solving this particular case, any reference to theory I can study related to this would be helpful. I do not have an advanced statistical theory knowledge.

I would like to derive a non-degenerate asymptotic distribution, i.e., something like $sqrt{n}(bar{Z} – E(Z_i))$.


Get this bounty!!!

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