1. FIN822 Li11 Binary independent and dependent variables

2. FIN822 Li22 Binary independent variables y =  0 +  1 x 1 +  2 x 2 +...  k x k + u

3. FIN822 Li33 Dummy Variables A dummy variable is a variable that takes on the value 1 or 0 Examples: male (= 1 if are male, 0 otherwise), NYSE (= 1 if stock listed in NYSE, 0 otherwise), etc. Dummy variables are also called binary variables

4. FIN822 Li44 A Dummy Independent Variable Consider a simple model with one continuous variable (x) and one dummy (d) y =  0 +  0 d +  1 x + u This can be interpreted as an intercept shift If d = 0, then y =  0 +  1 x + u If d = 1, then y = (  0 +  0 ) +  1 x + u The case of d = 0 is the base group

5. FIN822 Li55 Example, Y=turnover=trading volume/outstanding shares X=size of firms=log (stock price*outstanding shares) D=1 for NYSE stocks, 0 otherwise (AMEX and NASDAQ) If you assume the slope is the same for both groups, then regress y =  0 +  0 d +  1 x + u

6. FIN822 Li66 Example of  0 > 0 x y { 00 } 00 y = (  0 +  0 ) +  1 x y =  0 +  1 x slope =  1 d = 0 d = 1

7. FIN822 Li77 Dummies for Multiple Categories We can use dummy variables to deal with something with multiple categories Suppose everyone in your data is either a HS dropout, HS grad only, or college grad To compare HS and college grads to HS dropouts, include 2 dummy variables hsgrad = 1 if HS grad only, 0 otherwise; and colgrad = 1 if college grad, 0 otherwise

8. FIN822 Li88 Multiple Categories (cont) Any categorical variable can be turned into a set of dummy variables The base group is represented by the intercept; if there are n categories there should be n – 1 dummy variables

9. FIN822 Li99 Example, Y=turnover=trading volume/outstanding shares X=size of firms=log (stock price*outstanding shares) D=1 for NYSE stocks, 0 otherwise (AMEX and NASDAQ) If you assume the slope and intercept are different for both groups, you could run two separate regression for each group, or…

10. FIN822 Li10 Interactions with Dummies Can also consider interacting a dummy variable, d, with a continuous variable, x y =  0 +  1 d +  1 x +  2 d*x + u If d = 0, then y =  0 +  1 x + u If d = 1, then y = (  0 +  1 ) + (  1 +  2 ) x + u This is interpreted as a change in the slope (and intercept).

11. FIN822 Li11 y x y =  0 +  1 x y = (  0 +  0 ) + (  1 +  1 ) x Example of  0 > 0 and  1 < 0 d = 1 d = 0

12. FIN822 Li12 Caveats A typical use of a dummy variable is when we are looking for a program effect For example, we may have individuals that received job training and wish to test the effect of training. We need to remember that usually individuals choose whether to participate in a program, which may lead to a self-selection problem

13. FIN822 Li13 Self-selection Problems If we can control for everything that is correlated with both participation and the outcome of interest then it’s not a problem Often, though, there are unobservables that are correlated with participation. For example, people’s skill before the training. Low skill people may be more likely to take the training. In this case, the estimate of the program effect is biased.

14. FIN822 Li14 Self Selection Problem: An example Suppose one professor finds that his evening class students perform better that his daytime class. Does that mean the professor teaches better at night or students learn better at night? No. It turns out that students participating in night classes might be more hard working. They care to come at night, sometimes after a daytime work, etc.

15. FIN822 Li15 Self-selection Problems If we can control (control means adding more explanatory variables) for everything that is correlated with both participation (the dummy variable) and the outcome (y), then it’s not a problem.

16. FIN822 Li16 Binary Dependent Variables Logit and Probit models P(y = 1|x) = G(  0 + x  )

17. FIN822 Li17 Binary Dependent Variables A linear probability model can be written as P(y = 1|x) =  0 + x  A drawback to the linear probability model is that predicted values are not constrained to be between 0 and 1 An alternative is to model the probability as a function, G(  0 + x  ), where 0<G(z)<1

18. FIN822 Li18 The intuition

19. FIN822 Li19 The Probit Model One choice for G(z) is the standard normal cumulative distribution function (cdf) G(z) =  (z) ≡ ∫  (v)dv, where  (z) is the standard normal, so  (z) = (2  ) -1/2 exp(-z 2 /2) This case is referred to as a probit model Since it is a nonlinear model, it cannot be estimated by our usual methods Use maximum likelihood estimation

20. FIN822 Li20 The Logit Model Another common choice for G(z) is the logistic function G(z) = exp(z)/[1 + exp(z)] =  (z) This case is referred to as a logit model, or sometimes as a logistic regression Both functions have similar shapes – they are increasing in z, most quickly around 0

21. FIN822 Li21 Probits and Logits Both the probit and logit are nonlinear and require maximum likelihood estimation No real reason to prefer one over the other Traditionally saw more of the logit, mainly because the logistic function leads to a more easily computed model

22. FIN822 Li22 Interpretation of Probits and Logits In general we care about the effect of x on P(y = 1|x), that is, we care about ∂p/ ∂x For the linear case, this is easily computed as the coefficient on x For the nonlinear probit and logit models, it’s more complicated: ∂p/ ∂x j = g(  0 +x  )  j, where g(z) is dG/dz

23. FIN822 Li23 For logit model ∂p/ ∂x j =  j exp(  0 +x  (1 + exp(  0 +x  ) ) 2 For Probit model: ∂p/ ∂x j =  j  (  0 +x  ) =  j (2  ) -1/2 exp(-(  0 +x  2 /2)

24. FIN822 Li24 Interpretation (continued) Can examine the sign and significance (based on a standard t test) of coefficients,