Chapter 13: Limited Dependent Vars. Zongyi ZHANG College of Economics and Business Administration.
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Transcript of Chapter 13: Limited Dependent Vars. Zongyi ZHANG College of Economics and Business Administration.
Chapter 13: Limited Dependent Vars.
Zongyi ZHANGCollege 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:
E(Yi|Xi) = b1 + b2Xi since E(ui) = 0. Let
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 Graph of F(X) vs X
The CDF's most commonly chosen are : 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?.
iZ sii dseZFDP 2/2
2
1)()1Pr(
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