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:

Quartet

In nuclear magnetic resonance (NMR) spectroscopy, a signal that is split into four distinct peaks, indicating the presence of three equivalent protons in the neighboring environment.

Doublet

A pattern in spectroscopy indicating two adjacent peaks, often associated with the spin of a nucleus affecting a neighboring nucleus.

NMR Spectrum

A representation of the nuclear magnetic resonance absorption by nuclei in a sample.

Isobutyl Acetate

An organic compound and ester formed from the reaction of isobutanol with acetic acid, commonly used as a solvent and in flavorings.