Assume that we have i.i.d data (x1,y1),...,(xn,yn), where yi is the response variable and xi is the covariate. For simplicity, let us consider the case that xi is univariate. We fit a simple linear regression without intercept,yi = ?xi + ?i (1),where ? is the coefficient parameter. We know that the least square estimator is derived byregressing y on x. It has the following form (2): see the attached file

A few years ago, a professor asked me the following question about the model (1). What happens if we regress x on y (rather than y on x)? Would the least square estimate for the regression of x onto y be a good estimate of 1/??

Please provide a proof to show whether the coefficient estimate for the regression of x onto y is a consistent estimator of 1/?. 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.)(as you see the formula in (2), we assume that the mean of x and y are 0)

(2)