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

question 49

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: $\begin{array} { c } \hat { Y } = 2.858 - 0.037 \times X \\( 0.007 ) \\\operatorname { Pr } ( Y = 1 \mid X ) = F ( 15.297 - 0.236 \times X ) \\\operatorname { Pr } ( Y = 1 \mid X ) = \Phi ( 8.900 - 0.137 \times X ) \\( 0.058 )\end{array}$
(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: $\hat \beta _ { \text {probit } }$ ? 0.625 × $\hat \beta \text { Logit }$ , $\hat \beta \text { linear }$ ? 0.25 × $\hat \beta \text { Logit }$ 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?
(b)For the intercept, the same conversion holds for the logit-to-probit transformation. However, for the linear probability model, there is a different conversion: $\hat \beta _ { 0 , \text { linear } }$ ? 0.25 × $\hat \beta 0 , \text { Logit }$ + 0.5
Using the logit regression as the base, calculate a few changes in X (temperature in degrees of Fahrenheit)to see how good the approximations are.

---

Definitions:

Carotid Baroreceptors

Sensory nerve endings located in the carotid artery that detect changes in blood pressure and help regulate it.

Glossopharyngeal Nerve

The ninth cranial nerve, responsible for sensory and motor functions in the throat and back of the tongue.

Sympathetic Nerve

The sympathetic nervous system is part of the autonomic nervous system that prepares the body for 'fight or flight' responses during perceived threats.

Carotid Baroreceptors

Sensors located in the carotid arteries that monitor blood pressure and relay information to the central nervous system to maintain stable blood pressure.