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

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?