Question

1. (30 marks) Friedman's Permanent Income Hypothesis states that Y is non-stohastic (a) (10 marks) Derive the OLS estimator . (Clearly setup the minimization problem and solve for the estimator) (b) (10 marks) Derive Var(k) (e) (10 marks) Prove that A is unbiased mathematically. (State the assumptions you use and where do they apply)
Answer all three questions step by step thanks

Answer 1

C_pi = k Y_pi + u_i u_i ~ N (0, r^2) We need to minimize sigma u^2_i sigma u^2_i = sigma (C_i - ku_pi)^2 For unknown k^^Partialdifferential sigma u^2_i/Partialdifferential k^^= -2 sigma (C_pi - k y_pi) (y_pi) = -2 sigma C_pi middot y_pi - sigma k y^2_pi For k^^we have Partialdifferential sigma u^2_i/Partialdifferential F = 0 sigma C_pi middot y_pi = k^^sigma y^2_pi k^^= sigma C_pi middot y_pi/sigma y^2_pi OLS estimator k^^ Vertical-bar k = sigma C_pi middot y_pi/sigma y^2_pi k^^= sigma(k y_pi + u_i)y_pi/sigma y^2_pi = k sigma y^2_pi/sigma y^2_pi + sigma u_i y_pi/sigma y^2_pi As y_pi is non-stochastic k^^= k + y_pi sigma u_i/sigma y^2_pi Therefore E(k^^) = k + y_pi sigma E()/sigma y^2_pi \r\n E(k^^) = k rightarrow unbiased iff E(u_i) = 0 Now Variance k^^E(k^^- E(k^^))^2 E(k^^) = k + sigma u_i (y_pi/sigma y^2_pi) Let sigma u_i notequalto 0: y_pi/sigma y^2_pi