L6 - Binary Choice Models

Question 1

What is the linear equation for a Binary Choice Model?

Answer 1

• the continuous scale of y* is not observed by the researcher
  
  • WE only observe the binary outcome
    
    • the decision-maker may see this continuum y*

Question 2

What does the Binary choice model look like on a graph?

Answer 2

• If your y* was observed we would have a linear regression model
• This graph should the relationship between the y* and the probability that y=1 –> for a specific value of x
  
  • shaded –> Pr(y=1|X)
  • unshaded –> Pr(y=0|X)
• This can be interpreted as
  
  • Pr ( y = 1| x) = Pr(y* > 0|X) –> NOT SURE IF RIHT
  • y* = α+βx+ε > 0
    
    • Moving α+βx to the right hand side and subbing in we get
  • Pr(y=1│x)=Pr(ε > -[α+βx] | x)
    
    • The probability depends on the distribution of the error ε

Question 3

How is the relationship between the linear Binary Choice model linked to the nonlinear probability model?

Answer 3

• Pr(y = 1 | x) = F(α + βx)
  
  • it is actually a cumulative distribution function of your error term
• The S-shaped sigmoid curve relates to the shaded distribution functions of the linear model on the right –> for each individual value of x
  
  • That is because y* is not observed so we only have a distribution of error terms for each level of the explanatory variable

Question 4

What is one important distinction between linear regression model and non-linear model to do with the variance of the error term?

Answer 4

• One important distinction between linear regression models and non-linear models is that under OLS we can determine the variance of the error term
  
  • For discrete choice models, because the models reveal less information (you do not observe the continuum of y*) you have to assume the variance of the return –> this assumption will affect the shape of the distribution
    
    • e.g. fatter/thinner tails —> but wont effect the proportion of the shaded area
• There we assume the variance of the error terms –> will only consider two types of models (Binary Probit and Binary Logit models)

Question 5

How does the normalisation of the error term affect the parameters of our Binary Choice equation?

Answer 5

• Same as say setting the variance of the error term
• So we have an original equation of utility
  
  • Say we multiply this equation by a constant λ
    
    • λ U⁰_nj becomes U¹_nj
    • The variance of the error term is now multiplied by λ²
• Given that Var(ε⁰_nj )=σ² is not observed how can I make this equal to 1 (in the case of the Probit model?)
  
  • divide the variance by σ², given that σ²=λ², then we must divide the original utility equation by σ, so that λ=(1/σ)
  • This will now give us a Binary-Probit model, but when we estimate the equation we will not estimate beta, but actually beta/σ –>, therefore, cannot directly interpret the magnitude of the estimated coefficients in this model –> as σ is still unknown
• On the same line for Logit models –> if the variance of the error term is equal to roughly 1.6, we divide by σ² and then multiple by π^2/6 = 1.6 . Therefore λ in the original equal equals –> sqrt(1.6)/σ

Question 6

How do we test of well does a model fits the discrete choice model?

Answer 6

• LL Ratio index
  
  • constructed by estimating the model with no parameters in it just with a constant term and extracting the LL function of that and then subtracting the LL of our full model and multiplying of -2
    
    • If greater than the chi-squared statistics we reject the null that our parameters are jointly statistically insignificant
• Pseudo R-squared
  
  • constructed as 1 minus the LL ratio of our the LL of our full model - the LL of a model with no parameters and just a constant term
  • This cannot be interpreted on its own, we need different model specifications (by adding different variables if needed), then the higher the pseudo- R-squared the better the model
    
    • has no explanatory power on its own

Question 7

What are the density and Cumulative density functions of the error term in the Binary Logit Model?

Answer 7

Question 8

What is the Maximum Likelihood Estimation formula?

Answer 8

Question 9

What does maximum likelihood Estimation look like on a graph?

Answer 9

Question 10

What is interesting to note about the difference between the Probit and Logit model coefficient estimates?

Answer 10

• Due to the normalisation of the variance, the Logit model’s coefficient estimate will always be a multiple of sqrt(1.6) greater than the Probit’s estimates

Question 11

What does the partial effect of the Logit and Probit explanatory variables tell us?

Answer 11

• The percentage change in probability of the likelihood of the binary choice outcome
• Although the coefficient estimate between the two models are different the partial effects are nearly identical (the probability of the shaded areas doesn’t change between models)
  
  • e.g. probability of going to the hospital against income has the estimated partial effect of -0.09844
  • So when income increased by £1 the probability of going to see the doctor reduced by 9.844%
• For binary choice models, the partial effect will refer to the percentage of the base category
  
  • Say the ‘gender’ explanatory =1 for females and =0 for males and has the partial effect of 0.14026
  • We say that females are 14.026% more likely to go to the doctors than males

Question 12

Where do you find the measure of fit on stata?

Answer 12