Consider the basic panel data setup in which the population regression model is given by Yit = Bo + B1Xit + ai + Uit (1) for i E {1, 2, ..., n},t E {1, 2, ...,T} and where we assume Ea;] = 0 (because we have included an intercept) and E(uit |Xit] = 0). (a) (4 points) Show that Y; = Bo + B1X; + Qi +j (2) where the overbar represents the average over time for a given individual. (b) (6 points) Suppose that is uncorrelated with X; but Cov(Xit, Q;) = 0 Xa for all i,t. Show that Cov(X;,a +;) = 0 Xa (e) (5 points) Let @ be the OLS estimate from estimating equation (2). This is sometimes referred to as the "between estimator". Show that X plim = 31+ Var(X;) (d) (2 points) Assume further that the Xit for all t are uncorrelated with constant variance of Show that TdXa o plim1 = Be + (e) (3 points) If the explanatory variables are indeed uncorrelated across time, what does the result in part (d) suggest about whether the inconsistency in the between estimator is smaller when there are more time periods? Consider the basic panel data setup in which the population regression model is given by Yit = Bo + B1Xit + ai + Uit (1) for i E {1, 2, ..., n},t E {1, 2, ...,T} and where we assume Ea;] = 0 (because we have included an intercept) and E(uit |Xit] = 0). (a) (4 points) Show that Y; = Bo + B1X; + Qi +j (2) where the overbar represents the average over time for a given individual. (b) (6 points) Suppose that is uncorrelated with X; but Cov(Xit, Q;) = 0 Xa for all i,t. Show that Cov(X;,a +;) = 0 Xa (e) (5 points) Let @ be the OLS estimate from estimating equation (2). This is sometimes referred to as the "between estimator". Show that X plim = 31+ Var(X;) (d) (2 points) Assume further that the Xit for all t are uncorrelated with constant variance of Show that TdXa o plim1 = Be + (e) (3 points) If the explanatory variables are indeed uncorrelated across time, what does the result in part (d) suggest about whether the inconsistency in the between estimator is smaller when there are more time periods