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 $\widetilde { \widetilde { X } } \stackrel { p } { \rightarrow } \mu _ { \tilde { X } }$

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Definitions:

Independent Variable

In an experiment, the variable that is manipulated or changed by the researcher to observe its effect on the dependent variable.

Dependent Variable

The variable being tested and measured in an experiment, which is influenced by the independent variable.

Experimental Results

The data and observations collected during an experiment, which are analyzed to draw conclusions about the hypothesis tested.

Centimeter

A unit of length in the metric system, equal to one hundredth of a meter, commonly used to measure small distances or heights.