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

question 6

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 $\bar { X }$ $\stackrel { p } { \rightarrow }$ ${ } ^ { { } ^ { \mu } } \widetilde { X }$

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Definitions:

Mean

A statistical measure that represents the arithmetic average of a set of numbers.

Statistically Significant

A term used to describe a result that is unlikely to have occurred by chance alone, according to a predetermined threshold of probability.

Variation

The occurrence of differences among individuals within a population, due to genetic differences or environmental factors.

Participants

Individuals who take part in a research study or experiment, providing data for analysis.