In the Simple, One-Explanatory Variable, Errors-In-Variables Model, the OLS Estimator

question 41

Essay

In the simple, one-explanatory variable, errors-in-variables model, the OLS estimator for
the slope is inconsistent.The textbook derived the following result $$\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 } .$$
Show that the OLS estimator for the intercept behaves as follows in large samples:
$$\hat { \beta } _ { 0 } \stackrel { p } { \rightarrow } \beta _ { 0 } + \mu _ { \bar { X } } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 } ,$$
where $\widetilde { \widetilde { X } } \stackrel { p } { \rightarrow } \mu _ { \tilde { X } }$