Download Chapter 8: The Labor Market

Chapter 13:
Limited Dependent Vars.
Zongyi ZHANG
College of Economics and Business
Administration
1. Linear Probability
Model
Introduction

Sometimes we have a situation where
the dependent variable is qualitative in
nature


It takes on two (or more) mutually
exclusive values
Examples:


Whether or not a person is in the labor force
Union membership
Linear Probability Model

Examine choice of whether an individual
owns a house.


Yi = b1 + b2Xi + ui
where



Yi = 1 if family owns a house
Yi = 0 if family does not own a house
Xi = family income
Linear Probability Model

We can estimate such a model by OLS.


However, we don't get good results.
This is called a linear probability model
because E(Yi| Xi) is the conditional
probability that the event (buying a
house) will occur given Xi (family
income).
Derivation

Expected value of above:


Let



E(Yi|Xi) = b1 + b2Xi since E(ui) = 0.
Pi = probability that Yi=1 (the event occurs)
Then 1-Pi is the probability Yi=0
Then by definition of a mathematical
expectation:

E(Yi|Xi)= 0(1-Pi) + 1(Pi) = Pi
Derivation

So E(Yi|Xi)= b1 + b2Xi = Pi

So the conditional expectation is like a
conditional probability.
Problems with LPM

Error term is not normally distributed
but follows a binomial probability
distribution


For OLS we do not require that the error
term is distributed normally.
But we do assume this for the purposes of
hypothesis testing.
Problems with LPM

However we can’t assume normality for the
error term here

Ui takes on only two values:




When Yi = 1 then ui = 1 - b1 - b2Xi
Yi = 0 then ui = - b1 - b2Xi
So ui is not normally distributed, but follows a
binomial distribution.
Note that the OLS point estimates still
remain unbiased.

As n rises the estimators will tend to be ~ N
Problems with LPM

Error term is heteroskedastic



Though the E(ui) = 0, the errors are not
homoscedastic.
var(ui) = E(Yi|Xi)[1-E(Yi|Xi)]
var (ui)= Pi(1- Pi)

This is heteroskedastic because the conditional
expectation of Y, depends on the value taken
by X.
Problems with LPM

What does this imply?

With heteroskedasticity, OLS estimators are
unbiased but not efficient


They do not have minimum variance.
We correct the heteroskedasticity 

Transform data with weight = Pi(1- Pi)
This eliminates the heteroskedasticity
Problems with LPM

In practice we don't know the true
probability - so estimate it:



a. Run OLS on original model.
b. Get predicted Yi and construct wi =
predictedYi*(1-predictedYi)
c. Do OLS regression on transformed data
Problems with LPM

Probabilities falling outside 0 and 1 is
main problem with LPM.


Although in theory P(Yi| Xi) would fall
between 0 and 1, there is no guarantee
that predicted probabilities in the linear
model will
We can estimate by OLS and see if
estimated probabilities lie outside these
bounds, then assume them to be at 0 or 1.
Problems with LPM


Or use probit or logit model that
guarantees that the estimated probabilities
will fall between these limits.
Graph
Problems with LPM

LPM assumes that probabilities increase
linearly with the explanatory variables

Each unit increase in an X has the same
effect on the probability of Y occurring
regardless of the level of the X.

More realistic to assume a smaller effect at
high probability levels.

Probit and Logit make this assumption
2. CDF
Introduction

Probit and Logit have a S shaped
probability function


As X increases, probability of Y increases,
but never steps outside the 0-1 interval
The relationship between the probability of
Y and X is nonlinear

It approaches zero at slower and slower rates
as X gets small
Introduction


It approaches one at slower and slower rates
as X gets large.
The S-shaped curve can be modeled by
a cumulative distribution function (CDF).


The CDF of a random variable X:
F(X) = P(X  x)

CDF measures the probability that X takes a
value of less than or equal to a given x
Introduction


The CDF's most commonly chosen are :



Graph of F(X) vs X
The logistic function - logit;
The cumulative normal - probit
Logit and probit quite different models,
different interpretation.

Logit distribution has flatter tails

Approaches the axes more slowly
3. Probit
Introduction

Suppose the decision to join union
depends on some unobserved index Zi
"the propensity to join" for each
individual.

Don't observe the "propensity to join"


Just observe union or not.
So we only observe dummy variable D,
Introduction

Defined as:




D = 0 if a worker is nonunion.
D = 1 if a worker is union member
Behind this "observed" dummy variable is the
"unobserved" index
Assume Z depends on explanatory
variables such as wage.

So Zi = b1 + b2Xi

where Xi is the wage of the i'th individual
Introduction

Each individual's Z index can be expressed a
function of some intercept term and wage with
attached coefficient


Reality: many X's, not just wage
Suppose there's a critical level or
threshold level of the Z, -- Zi*,

If Zi>Zi* an individual will join, otherwise
will not.
Introduction


Assume Zi* is distributed normally with
the same mean and variance as Zi.
What's the probability that Zi>Zi*

In other words, what's the probability that this
individual will join?.
Pi  Pr( D  1)  F ( Z i ) 
1
2

Zi

e
s2 / 2
ds
Introduction

Pi, the probability of joining, is
measured by the area under the
standard normal curve from - to Zi.

Individuals are at different points along this
function

Have different critical values pushing them into
joining, depending on characteristics.
Introduction

How do we estimate Zi?

Use the inverse of the cumulative normal
function,

Zi =F-1 (Pi) = b1 +b2Xi