regression with a binary dependent variable

Question 1

when Y is binary what is the linear regression model called and why?

Answer 1

it is called the linear probability model because pr(Y=1|X) = B0+B1X

Question 2

what is the predicted value for the linear probability model?

Answer 2

the predicted value is a probability

Question 3

what is B1 equal to in the linear probability model?

Answer 3

B1= the difference in probability that Y =1 associated with a unit difference in x

Question 4

what is the formula for B1 in the linear probability model?

Answer 4

B1= [Pr(Y=1|X=x+change in x) -Pr(Y=1|X=1)]/change in X

Question 5

what are the advantages of the linear probability model>

Answer 5

simple to estimate and interpret
inference is the same as for multiple regression ( need heteroskedacity-robust standard errors)7

Question 6

what are the disadvantages of the linear probability model>

Answer 6

a LPM says that the changes in the predicted probability for a given change in X is the same for all values of X but that doesnt make sense
also LPM predicted probabilities can be <0 or >1

Question 7

how can the disadvantages of the linear probability model be solved?

Answer 7

the disadvantages can be solved by using a nonlinear probability model such as probit regression or logit regression

Question 8

what is the probit regression?

Answer 8

the probit regression models the probability that Y=1 using the cumulative standard normal distribution function Φ(z), evaluated at z=B0 +B1X

Question 9

what is the equation of the probit regression model?

Answer 9

Pr(Y=1|X) =Φ(B0+B1X) where Φ is the cumulative normal distribution and z=B0+B1X

Question 10

why use the cumulative normal probability distribution?

Answer 10

it provides an S shape which gives us what we need: Pr(Y=1|X) is increasing in X for B1>0 and 0 ≤Pr(Y=1|X)≤1 for all X
it is also easy to use as the probabilies are tabulated in the cumulative normal tables
it also has a relatively straightforward interpretation - B1 is the change in Z value for a unit change in X

Question 11

what is the equation of the probit regression with multiple regressors?

Answer 11

Pr(Y=1|X1,X2) =Φ(B0+B1X+B2X2) where Φ is the cumulative normal distribution and z=B0+B1X1+B2X2

Question 12

what is the B1 for probit regression with multiple regressors?

Answer 12

β1 is the effect on the z-score of a unit change in X1, holding constant X2 (when a causal interpretation is justified)

Question 13

what is the logit regression model?

Answer 13

Logit regression models the probability of Y = 1, given X, as the cumulative standard logistic distribution function

Question 14

what is the equation of the logit regression model>

Answer 14

Pr(Y=1|X ) = F(β0+β1X)
where F is the cumulative logistic distribution function:
F(β0+β1X) = 1/ (1+e^[-(B0+B1X)

Question 15

how is the non linear least squares different to the OLS?

Answer 15

the non linear least squares extends the idea of the OLS to models in which the parameters enter nonlinearly

Question 16

what is the minimisation problem for the non linear Least sqaures

Answer 16

min_b0,b1 Σ[Y_i - Φ(β0 + β1X_i)]^2

Question 17

how do we solve the minimisation problem for the nonlinear least squares?

Answer 17

calculus doesnt give us an answer
it is solved numerically by a computer

Question 18

what is the likelihood function?

Answer 18

the likelihood function is the conditional density of Y1,..,Yn given X1,….,Xn treated as a function of unknown parameters B0 and B1

Question 19

what is the maximum likelihood estimator (MLE) ?

Answer 19

MLE is the value of (B0,B1) that maximises the likelihood function. it is the value which best describes the full distribution of data

Question 20

in large samples, what is the maximum likelihood estimator?

Answer 20

it is consistent, normally distributed and efficient (has the smallest variance of all consistent estimators)

Question 21

what are the measures of fit for logit and probit?

Answer 21

1) the fraction correctly predicted = fraction of Ys for which the predicted probability is >50% when Y_i=1 or is <50% when Y_i=0
2) the pseudo R^2 measures the improvement in the value of the log likelihood, relative to having no X’s. the pseudo R^2 simplifies to the R^2 in the linear model with normally distributed errors

Question 22

in large samples, what are the features of the probit likelihood with one X?

Answer 22

estimator of B0_MLE and B1_MLE are consistent, normally distributed and asymptotically efficient