Question

1. (30 marks) Friedman's Permanent Income Hypothesis states that Y is non-stohastic p2 where Cpi and Ypi are permanent consumption and income, respectively (a) (10 marks) Derive the OLS estimator k. (Clearly setup the minimization problem and solve for the estimator) (b) (10 marks) Derive Var(k) (c) (10 marks) Prove that k is unbiased mathematically. (State the assumptions you use and where do they apply)

Please solve the problem step by step. Thank you!

Answer 1

C_pi = kY_pi + u_i u_i ~ N (0, r^2) we need to minimize Sigma u^2_i Sigma u^2_i = Sigma (c_i e - k Y_pi)^2 For unknown k^^partial differential Sigma u^2_i/partial differential k^^= -2 Sigma (C_p_i - k Y_p_i) (Y_p_i) = -2 Sigma C_p_i Y_p_i - Sigma k Y^2_p_i For k^^we have partial differential Sigma C^2_i/partial differential k^^= 0 Sigma C_p_i Y_p_i = k^^Sigma y^2_p_i k^^= Sigma c_p_i y_p_i/Sigma y^2_p_i OLS estimator k^^of k = Sigma C_p_i Y_p_i/Sigma Y^2_P_i k^^= Sigma (k Y_p_i + u_i)Y_p_i/Sigma Y^2_p_i = k Sigma Y^2_p_i/Sigma Y^2_p_i + Sigma u_i Y_p_i/Sigma Y^2_p_i As Y_p_i is non - stochastic k^^= k + Y_p_i Sigma u_i/Sigma Y^2_p_i E (k^^) = k + Y_p_i Sigma E(u_i)/Sigma Y^2_p_i \r\n E(v^^) = k rightarrow unbiased iFF E (u_i) = 0 now variance of k^^E (k^^- E (k^^))^2 E(k^^) = k + Sigma u_i (Y_p_i/Sigma Y^2_p_i) Let Sigma u_i notequalto 0 Y_p_i/Sigma P^2_p_i = Psi_i E(k^^) - k = Var (k^^) = E [k^^- E (i^^)]^2 = E [Psi_i Sigma u_i]^2 = E [Psi^2_i (Sigma u_i)^2] = E (Psi^2_i) E (Sigma u_i)^2 E [(u_1 + u_2 + ... u_n)^2]