Chapter 12: Limited Dependent Vars.

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Chapter 12 Limited Dependent Vars.
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• 1. Linear Probability Model

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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

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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

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Linear Probability Model

• We can estimate such a model by OLS.
• However, e 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).

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Derivation

• Expected value of above
• E(YiXi) b1 b2Xi since E(ui) 0.
• Let
• Pi probability that Yi1 (the event occurs)
• Then 1-Pi is the probability Yi0
• Then by definition of a mathematical expectation
• E(YiXi) 0(1-Pi) 1(Pi) Pi

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Derivation

• So E(YiXi) b1 b2Xi Pi
• So the conditional expectation is like a
  conditional probability.

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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.

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Problems with LPM

• However we cant 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 will tend
  to be N

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Problems with LPM

• Error term is heteroskedastic
• Though the E(ui) 0, the errors are not
  homoscedastic.
• var(ui) E(YiXi)1-E(YiXi)
• var (ui) Pi(1- Pi)
• This is heteroskedastic because the conditional
  expectation of Y, depends on the value taken by
  X.

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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

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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

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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.

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Problems with LPM

• Or use probit or logit model that guarantees that
  the estimated probabilities will fall between
  these limits.
• Graph

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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

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• 2. CDF

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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

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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

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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

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• 3. Probit

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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,

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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

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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 ZigtZi an individual will join, otherwise will
  not.

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Introduction

• Assume Zi is distributed normally with the same
  mean and variance as Zi.
• What's the probability that ZigtZi
• In other words, what's the probability that this
  individual will join?.

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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.

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Introduction

• How do we estimate Zi?
• Use the inverse of the cumulative normal
  function,
• Zi F-1 (Pi) b1 b2Xi