In the case of a simple regression, where the independent variable is measured with i.i.d. error,
A) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$
B) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w 1 } ^ { 2 } }$
C) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow }
\frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$
D) $\hat { \beta } _ { 1 }\stackrel { p } { \longrightarrow }
\beta _ { 1 } + \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } }$

Question

In the case of a simple regression, where the independent variable is measured with i.i.d. error,
A) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$
B) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w 1 } ^ { 2 } }$
C) $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow }
 \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$
D) $\hat { \beta } _ { 1 }\stackrel { p } { \longrightarrow }
 \beta _ { 1 } + \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } }$

Quizplus · Accepted Answer

In a simple regression with i.i.d. error in the independent variable, the estimated coefficient $\hat{\beta}_1$ converges in probability to a fraction of the true coefficient $\beta_1$, scaled down by the ratio of the variance of the independent variable $\sigma_X^2$ to the sum of its variance and the variance of the measurement error $\sigma_w^2$. This reflects attenuation bias due to measurement error in the independent variable.

In the Case of a Simple Regression, Where the Independent