Let be the continuous random variables: x , y , z , whose mathematical expectations exist and are finite. Prove the following laws of conditional expectations (Hint: start from the definition of expectation of a continuous random variable function, noting that if x is a random variable, then E(y x) is a function of x).
• E[E(yx)]=E(y)
• E[E(y|x,z)|x=E(y|x)
2) Let be the discrete random variables: x , y , z , whose mathematical expectations exist and are finite.Prove the following laws of conditional expectations:
• E[E(yx)]=E(y)
• E[E(y|x,z)|x=E(y|x)
3) Suppose you are interested in obtaining an estimator of the variable yi from xi , where xi is a vector of variables correlated with yi .
• Let m(xi) be an unrestricted function of xi. Show that the Conditional Expectation Function E(yi | xi ) , is the best unrestricted estimator of yi .
• Suppose now that E(yi|xi) is a constraint function exi, such that E(yi|xi) is linear in a parameter vector b: E(yi |xi )=xi ‘b. Where b is of dimension kx1. Show that the Ordinary Least Squares estimator is the best predictor of b given a sample of size n>>k .
• Suppose now that E ( yi | xi ) is not linear. What is the best linear approximation of E ( yi | xi ) given a sample of size n >>k ?
4) According to Hayashi’s textbook, the strict exogeneity assumption says that the expectation of the ith error conditional on the entire information matrix X is equal to zero: E (ui | X ) . This implies that the errors are
orthogonal to the entire information matrix: E (ui X ). Suppose now the following auto-regressive time series model of order 1(AR(1)): yt = β yt−1+ut,t=0,1,2,3,…,T,such that it satisfies E(uiyt−1 )=0.
Explain in detail why the assumption of strict exogeneity does not hold in this case.
