Question

Exercise 7.15 Take the linear model E(el ) with n observations and xi is scalar (real-valued). Consider the estimator Find the asymptotic distribution of /n (8- 8) as n

Answer 1

$Assume~x_i's~are~non-stochastic,~y_i's~are~independent,\\ ~Var(e_i|x_i)=Var(e_i)=\sigma^2~\forall~i~and~E(e_i|x_i)=E(e_i)=0~\forall~i. \\E(\hat{\beta})=\frac{1}{\sum_{i=1}^nx_i^4}\sum_{i=1}^nx_i^3E(y_i)=\frac{\beta}{\sum_{i=1}^nx_i^4}\sum_{i=1}^nx_i^4=\beta\\ Var(\hat{\beta})=\frac{1}{(\sum_{i=1}^nx_i^4)^2}\sum_{i=1}^nx_i^6Var(y_i)=\frac{\sigma^2\sum_{i=1}^nx_i^6}{(\sum_{i=1}^nx_i^4)^2}\\ From~central~limit~theorem,\\ \sqrt{n}(\hat{\beta}-\beta)\overset{L}{\rightarrow}Normal\left(0,~\frac{n\sigma^2\sum_{i=1}^nx_i^6}{(\sum_{i=1}^nx_i^4)^2} \right )$