Please provide a proof to show whether the coefficient estimate for the regression of x onto y is a consistent estimator of 1/?. (Hint: The proof needs the law of large numbers to show the convergence of the estimator. )

A ssume that we have i.i.d data (331,311), ..., (mmyn), where yi is the response variable and m,- is the covariate. For simplicity, let us consider the case that 2:2- is univariate. We fit a simple linear regression without intercept, 3% = 5331+ 5i: (1) where [3 is the coefcient parameter. We know that the least square estimator is derived by regressing 3; on 2:. It has the following form n I . 5) = Zi:1 $12242 ' (2) i=1$ y is a consistent estimator of 1/ B . For this question, you can start from the least square formula in (2). (Hint: The proof needs the law of large numbers to show the convergence of the estimator. )