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

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 $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ Show that the OLS estimator for the intercept behaves as follows in large samples: $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \beta _ { 0 } + \mu \widetilde { \mathrm { X } } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { \mathrm { X } } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ where $\bar { X }$ $\stackrel { p } { \rightarrow }$ ${ } ^ { { } ^ { \mu } } \widetilde { X }$