 # 1. Consider the following model: Yi = B1xi1 + B2xi2 + €i; i =...

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1. Consider the following model: Yi = B1xi1 + B2xi2 + €i; i = 1,2, , n (1) where E(c)) =o2Vv Derive the the least squares (OLS) estimators B1 and B2. What happens to the estimators when xi2 = kail, Vi, where k is a fixed constant? 2. Consider the following two-variable linear regression model: yi=atBrited; i =1,2,***,n (2) where E(Ei)=0Vi E(c}) =02V E(cij)=0Vitj and xi's are fixed in repeated random sampling. Consider its OLS estimator â. We derived in one of the classes that it could be expressed as a linear function of yi's: n & = " Viyi; where Vi = : 1 txxi - 2 (3) i=1 Define an arbitrary linear estimator of a: n = (4) i=1 where ai's are non-stochastic weights. We also saw in the class that, if * is an unbiased estimator, the variance of &* was: = = (5) Demonstrate that minimizing (5) subject to conditions for unbiasedness gives the OLS weights of (3) for ai. [Hint: use constrained optimization.]

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