1. (a) Let Y = X^3 + u where X and u are independent standard normals. Compute E
ID: 3227818 • Letter: 1
Question
1.
(a) Let Y = X^3 + u where X and u are independent standard normals. Compute
E (Y | X = x):
(b) Suppose we have iid observations (Xi; Yi) i=1 to n from the above model. We run the following
regression Yi = alpha + Xi(beta) + error to obtain the OLS estimator What would you expect the value of
OLS to be when n approaches infinity? (hint: EX = EX3 = 0; EX2 = 1; and EX4 = 3)
2.
Let Y=1+2X+u where X=Z1+Z2, u=Z1+Z2; Z1 and Z2 are independent standard normals. We have iid observations (Xi; Yi)n i=1 from this model.
(a) Suppose we run the following regression Yi = alpha + Xi(beta) + error1i to obtain the OLS estimator beta hat
OLS: What would you expect the value of beta hat OLS to be when n approaches inifinity? Please give a numerical answer.
(b) Suppose we run the following regression Xi = alpha + Yi(beta) + error2i to obtain the OLS estimator beta hat OLS: What would you expect the value of beta hat to be when n approaches inifinity? Please give a numerical answer.
Explanation / Answer
1a) When X = x , Y = x^3 + u
E(Y / X = x) = E(x^3 + u) = E(x^3) + E(u) = x^3 + 0
E(Y / X = x) = x^3
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