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questions - LECTURE 1 1 Let y = E(y |x be the conditional...

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LECTURE 1 1. Let ˆ y = E ( y | x ) be the conditional expectation of y given x . Prove that E { ( y ˆ y ) 2 } ≤ E { ( y π ) 2 } , where π = π ( x ) is any other function of x . Show that E x ( y ˆ y ) = 0 and give an interpretation of this condition. Demonstrate that, if E ( y | x ) is a linear function of x , then we have E ( y | x ) = E ( y ) + β x E ( x ) , where β = C ( x, y ) /V ( x ). 2. A simple model of the economy comprises two equations in income y , consumption c and investment i : y = c + i, and c = α + βy + ε. Also, s = y c or s = i , where s is savings. Demonstrate that the probability limit of the ordinary least-squares estimate of β the marginal propensity to consume, which is attained as the sample size increases, is given by plim ˆ β = β + σ 2 (1 β ) m ii + σ 2 = βm ii + σ 2 m ii + σ 2 . When could the bias in this estimator be safely ignored and under which circumstances is it liable to become severe? LECTURE 2 3. Let P = X ( X X ) 1 X . Show that the conditions P = P = P 2 , which indicate that P is a symmetric idempotent matrix, are equivalent to the condition that P ( I P ) = 0. Prove that, if Xb = Py , then y Xb y , where γ is any vector of the appropriate order. Show that, if y ( Xβ, σ 2 I ), then b = ( X X ) 1 X y has E ( b ) = β and D ( b ) = σ 2 ( X X ) 1 . Prove that, if Ay = β has E ( β ) = β , then V ( q ˆ β ) V ( q β ), where q is an arbitrary non stochastic vector. 4. Derive the least-squares estimator of β in the restricted regression model ( y ; | = r, σ 2 I ) under the assumption that the matrix X has full column rank. Show that the restricted estimator has the same structure as the conditional expectation of the vector y given the value of a vector x with which it is jointly distributed: E ( y | x ) = E ( y ) + C ( y, x ) D 1 ( x ) { x E ( x ) } .
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