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Consider the linear regression model, y = Xβ + e, where as usual y and e are of dimension n × 1, X is n × k and β is k × 1. Additionally, the error term is correlated with the data such that E(e|X) = γ column vector not equal to zero. Let the corresponding OLS estimator be βˆ = ((X′X)^-1)(X′y). Assume conditionality on X.

I have solved all the other parts of the question. But I am not able to understand what does the below question mean and how I show the proof . Q. Does this regression model produces biased predictions of the outcome variable?

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Predictions are $\hat{y}=X\hat{\beta}$; and $$\begin{align}E[\hat{y}|X]&=E[X\hat{\beta}|X]=E[X(X^TX)^{-1}X^T(X\beta+\epsilon)|X]\\&=E[X\beta+X(X^TX)^{-1}X^T\epsilon|X]\\&=X\beta+\underbrace{X(X^TX)^{-1}X^T}_{A}E[\epsilon|X]\\&=X\beta+A\gamma\end{align}$$ Normally, you should have $X\beta$ only, given the data, i.e. you should be expecting your calculated outcome to be the outcome calculated with the true parameters, $\beta$, however the right term, $A\gamma$, is an additional term that lead to bias.

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