Consider the heterogeneous regression model Yi = β0i + β1iXi + ui, where β0i and β1i are random variables that differ from one observation to the next. Suppose that E(ui|Xi) = 0 and (β0i, β1i) are distributed independently of Xi.
a. Let β1OLS denote the OLS estimator of β1 given in Equation (17.2). Show that

where E(β1) is the average value of β1i in the population.
b. Suppose that

where θ0 and θ1 are known positive constants. Let β1WLS denote the weighted least squares estimator. Does

Explain?

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BOLS L, E(B1),