A: Consider the single Linear Regression Model:
Y = βX + U
without an intercept (β is a scalar constant; X is a scalar random variable). Imagine that you have taken a random sample from this population. If you estimate a regression using OLS, but you do not include an intercept, do you know if the mean of the residuals is zero? Explain.
B: Prove or Show:
i. Statistical Interpretation of Linear Regression:
a. U is mean independent of X. That is, E[U|X] = 0
b. Orthogonality conditions E[U] = 0 and E[XU] = 0 are satisfied
c. Cov(X_j, U) = 0 where X_j is a regressor and U is a statistical term. This is true regardless of whether E[Y|X] is linear
ii. Suppose E[XX'] is finite. As long as X is not perfectly collinear, E[XX'] is invertible.