*Bounty: 50*

*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_1*X_{1i} + beta_2*X_{2i} + beta_3*X_{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 = 1*hat{beta}_0 + 2*X_{1i}*hat{beta}_1 + 3*X_{2i}*hat{beta}_2 + 4*hat{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 $sqrt{n}(bar{Z} – E(Z_i))$.