FIN822 Li 11

Binary independent and dependent variables

FIN822 Li 22

Binary independent variables

y = 0 + 1x1 + 2x2 + . . . kxk + u

FIN822 Li 33

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

FIN822 Li 44

A Dummy Independent Variable

Consider a simple model with one continuous variable (x) and one dummy (d)

y = 0 + 0d + 1x + u

This can be interpreted as an intercept shift

If d = 0, then y = 0 + 1x + u

If d = 1, then y = (0 + 0) + 1x + u

The case of d = 0 is the base group

FIN822 Li 55

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 + 0d + 1x + u

FIN822 Li 66

Example of 0 > 0

x

y

{0

}0

y = (0 + 0) + 1x

y = 0 + 1x

slope = 1

d = 0

d = 1

FIN822 Li 77

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

FIN822 Li 88

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

FIN822 Li 99

Example,

Y=turnover=trading volume/outstanding sharesX=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…

FIN822 Li 1010

Interactions with Dummies

Can also consider interacting a dummy variable, d, with a continuous variable, x

y = 0 + 1d + 1x + 2d*x + u

If d = 0, then y = 0 + 1x + u

If d = 1, then y = (0 + 1) + (1+ 2) x + u

This is interpreted as a change in the slope (and intercept).

FIN822 Li 1111

y

x

y = 0 + 1x

y = (0 + 0) + (1 + 1) x

Example of 0 > 0 and 1 < 0

d = 1

d = 0

FIN822 Li 1212

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

FIN822 Li 1313

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.

FIN822 Li 1414

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.

FIN822 Li 1515

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.

FIN822 Li 16

Binary Dependent Variables Logit and Probit models

P(y = 1|x) = G(0 + x)

FIN822 Li 17

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

FIN822 Li 18

The intuition

FIN822 Li 19

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)-1/2exp(-z2/2) 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

FIN822 Li 20

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

FIN822 Li 21

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

FIN822 Li 22

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/ ∂xj = g(0 +x)j, where g(z) is dG/dz

FIN822 Li 23

For logit model

∂p/ ∂xj =j exp(0 +x(1 + exp(0 +x) ) 2

For Probit model:

∂p/ ∂xj = j (0 +x) = j (2)-1/2exp(-(0 +x2/2)

FIN822 Li 24

Interpretation (continued)

Can examine the sign and significance (based on a standard t test) of coefficients,