You Have a Limited Dependent Variable (Y)and a Single Explanatory

question 6

Essay

You have a limited dependent variable (Y)and a single explanatory variable (X).You
estimate the relationship using the linear probability model, a probit regression, and a
logit regression.The results are as follows: $$\begin{array} { c } \hat { Y } = 2.858 - 0.037 \times X \\\quad\quad\quad\quad( 0.007 ) \\\operatorname { Pr } ( Y = 1 \mid X ) = F ( 15.297 - 0.236 \times X ) \\\\\operatorname { Pr } ( Y = 1 \mid X ) = \Phi ( 8.900 - 0.137 \times X ) \\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad( 0.058 )\end{array}$$ (a) Although you cannot compare the coefficients directly, you are told that "it can be shown" that certain relationships between the coefficients of these models hold approximately. These are for the slope: $\widehat { \beta } _ { \text {probit } } \approx 0.625 \times \widehat { \beta } _ { \text {Logit } } , \widehat { \beta } _ { \text {linear } } \approx 0.25 \times \widehat { \beta } _ { \text {Logit } }$ . Take the logit result above as a base and calculate the slope coefficients for the linear probability model and the probit regression. Are these values close?

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Definitions:

Deflation

A decrease in the general price level of goods and services in an economy over a period of time.

Cyclical Unemployment Rate

The component of overall unemployment that corresponds to the cyclical conditions of an economy, fluctuating with the business cycle.

Overall Unemployment Rate

The scope of unemployment in the labor force, focusing on those actively seeking employment.

Disinflation

A reduction in inflation rates, signaling a deceleration in the pace at which the prices of goods and services rise.