# #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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