Econometrics
Prove that the variance of the sample mean, Var(x ̅), equals σ_x^2 / n. State clearly any assumptions that you need to make.
Can you prove that MSE(β ̂) = Var(β ̂) + Bias2 ?
Hint: Use MSE(β ̂) = E(β ̂ – β)2
Var(β ̂) = E(β ̂2) – E2(β ̂)
and Bias2 = E2(β ̂ − β).
Can two variables share a deterministic relationship, and yet be uncorrelated? For instance, suppose that X follows a continuous, uniform distribution between −1 and 1:
X ~ U[−1, 1].
What does E(X) equal? Explain.
Let Y = X2. Clearly Y and X are not independent; in fact, a deterministic relationship exists between them (if you know one, you know the other). Show, however, that Corr(X,Y) = 0. Hint: Use E(XY) = E(X3¬).
Can you explain why there is no correlation despite an obvious relationship?
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