Suppose you have the following specification, where e may be correlated with r: 1 = BX + e. 1. You have instruments z and ( which are mutually uncorrelated. What are their necessary properties to provide consistent IV estimators , and 8-? Derive the asymptotic distributions of these estimators. 2. Calculate the optimal IV estimator as a linear combination of B, and Be. 3. You notice that . and p. are not that close together. Give a test statistic which allows you to decide if they are estimating the same parameter. If the test rejects, what assumptions are you rejecting?Consider the consumption function (11.1) where C is aggregate consumption at t, and Y, is aggregate income at f. The ordinary least squares (OLS) estimation applied to (11.1) may give an inconsistent estimate of the marginal propensity to consume (MPC) A. The remedy suggested by Haavelmo lies in treating the aggregate income as endogenous: Y = Gith+G (11.2) where I, is aggregate investment at t, and Gr is government consumption at t, and both variables are exogenous. Assume that the shock e, is mean zero IID across time, and all variables are jointly stationary and ergodic. A sample of size 7 containing Yr. Ct, It, and G, is available. 1. Show that the OLS estimator of A is indeed inconsistent. Compute the amount and direction of this inconsistency. 2. Econometrician A intends to estimate (o, A)' by running 2SLS on (11.1) using the instru- mental vector (1, It, G,)'. Econometrician B argues that it is not necessary to use this rela- tively complicated estimator since running simple IV on (11.1) using the instrumental vector (1, It + Gr)' will do the same. Is econometrician B right? 3. Econometrician C regresses Yr on a constant and Cr, and obtains corresponding OLS esti- mates (Go, fc)'. Econometrician D regresses Y on a constant, Cy, It, and G, and obtains corresponding OLS estimates (do, c. or, be)'. What values do parameters fc and op con- sistently estimate?Consider the model where E (u, v)' |x =0 and V [(u, v)'|x]= E, with E unknown. The triples {(2;, 2, #1)} consti- tute a random sample. 1. Show that or, * and E are identified. Suggest analog estimators for these parameters. 2. Consider the following two stage estimation method. In the first stage, regress z on r and define & = mr, where a is the OLS estimator. In the second stage, regress y in 2 to obtain the least squares estimate of a. Show that the resulting estimator of a is inconsistent. 3. Suggest a method in the spirit of 2SLS for estimating a consistently