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 

• V ar(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 the regression: A = β0 + β1B + e which omits C. Find E[βˆ 1] 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. 

(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[βˆ 1] 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?

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