Question

1. Suppose the random variable A is determined by three variables, B, C, and u in the following way: A = β0 + β1B + β2C + u

Where the following relationships hold:

• Cov(A,B)=10
• Cov(B,C)=40
• Var(B)=100
• Cov(B,u) = 0 and Cov(C,u) = 0

Finally, suppose that β2 = 3.

(a) You take a random sample of observations of A and B and estimate theregression:

A = β0 + β1B + ε
which omits C. Find E[βˆ1] when your regression omits C (You can find an exact number, not an algebraic expression).

(b) Is the regression coefficient estimate above a biased estimate of the causal effect of B on A? If so why, and what is the magnitude of the bias (you can find an exact number for this).

(c) You take a random sample of observations of A, B, and C and estimate the multiple regression:

A = β0 + β1B + β2C + u
Since Cov(u,B) = 0, this multiple regression will produce an unbiased estimate of β1. What is E[βˆ1] when C is being held constant?

(d) The causal impact of B on A is negative, but paradoxically the correlation between A and B is positive. How can this be the case?

Answer 1