Limited Dependent Variable Models

1
Limited Dependent Variable Models

• Course Applied Econometrics
• Lecturer Zhigang Li

2
Limited Dependent Variable Models

• Examples
• Discrete dependent variable models
• Binary dependent variable models
• Corner solution response models
• Censored and truncated variables models
• Count variable (nonnegative integer values) models

3
Linear Probability Model (LPM)(Section 7.5)

• yßXu,
• where y is a binary variable, one for success and
  zero for failure
• In this model, ß measures the change in the
  probability of success when x changes.
• Shortcomings
• Predictions (probability of success) can be less
  than zero or greater than one.
• Probability of success is linearly related to
  independent variables for all values.
• Heteroskedasticity must be present.

4
Binary Response Models

• A latent variable model
• Y1 if YßXegt0
• Y0 if YßXelt0
• This implies
• P(Y1X)G(ßX)
• Logit Model e follows a logistic distribution
• P(Y1X)eßX/(1eßX)
• Probit Model e follows a normal distribution
• P(Y1X)?-8ßXf(v)dv
• The magnitude of ß is not meaningful because the
  latent variable Y does not has a well-defined
  unit of measurement. Nevertheless, we may measure
  the effect of X on the probability for Y to be
  one.

5
Binary Response Models Interpretation I

• The partial effect of (continuous) xj is
• ?p(X)/?xjG(ßX)ßjg(ßX)ßj
• Where g(.) is the density function of e.
• Implications
• The effect of xj depends on the value of X.
• The relative effect of xi and xj is fixed.

6
Binary Response Models Interpretation II

• Probit g(0).4
• Logit g(0).25
• Linear probability model g(0)1
• To make the logit and probit slope estimates
  comparable, we can multiply the probit estimates
  by .4/.251.6.
• The logit slope estimates should be divided by 4
  to make them roughly comparable to the LPM
  (Linear Probability Model) estimates.

7
Binary Response Models Evaluation

• A rough measure of the performance of the binary
  models is called percent correctly predicted,
  i.e. the percentage of times the predicted yi
  matches the actual yi.
• It is important to note that one should report
  the percentage correctly predicted for each
  outcome (0 and 1).

8
Tobit Model for Corner Solution Reponses

• Corner Solution Response A variable is zero for
  a nontrivial fraction of the population but is
  roughly continuously distributed over positive
  values.
• E.g., monthly earning
• A linear model is conceptually wrong because it
  predicts negative values for the dependent
  variable.

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A Tobit Model

• Latent variable yßXu, ux Normal (0,s2).
• Observed response ymax(0,y)
• Likelihood of yi
• yigt1 f(y-ßX)/s/s
• yi0 P(ylt0X)1-F(ßX/s)

10
What if OLS is used?

• Conditional expectation E(yygt0,x)
• E(yygt0,x)ßXs?(ßX/s)
• Where ?(c)f(c)/F(c) is called the inverse Mills
  ratio.
• Unconditional expectation E(yx)
• E(yx)P(ygt0x)E(yygt0,x)F(ßX/s)ßXs?(ßX/s),
  which is a nonlinear function of x and ß.
• Simple OLS can not consistently estimate ß in
  either of the above cases.

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What if OLS is used (continued)?

• The partial effects of xj on E(yygt0,x) and
  E(yx) have the same sign as the coefficient ßj,
  but the magnitude depends on the values of all
  explanatory variables and parameters.
• ?E(yygt0,x)/?xß1-?(ßX/s)ßX/s?(ßX/s)
• ?E(yx)/?x ßF(ßX/s)
• To make the Tobit coefficient comparable to OLS
  estimates, we must multiple the Tobit estimate by
  an adjustment factor F(ßX/s).

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A Tobit Model Specification Issues

• The Tobit model relies crucially on normality and
  homoskedasticity. If any of the assumptions fail,
  then it is hard to know what the Tobit MLE is
  estimating.
• Nevertheless, for moderate departures from the
  assumptions, the Tobit model may provide good
  estimates.
• In a Tobit model, xj has similar effects on both
  the selection decision and the magnitude
  decision. This restriction may be unrealistic and
  can be tested. (See pp.573.)
• This problem may be solved with two-part models,
  in which P(ygt0x) and E(yygt0,x) depend on
  different parameters.

13
Censored Regression Model I

• yßXu, ux,c Normal (0,s2)
• wmin(y,c)
• Note that u is independent of c.
• With censored data, OLS is simply wrong due to
  endogeneity resulted from nonrandom measurement
  errors.
• With corner solution data, OLS is right on the
  average.

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Censored Regression Model II

• An OLS regression using only the uncensored
  observations produces inconsistent estimators of
  ß.
• If there is heteroskedasticity or nonnormality,
  the MLEs are generally inconsistent.

15
Truncated Regression Models

• In a truncated regression model, we do not
  observe any information about a certain segment
  of the population (therefore we have a nonrandom
  sampling of dependent variables). In a censored
  regression model, we still have some information
  on censored observations.
• OLS tends to flatten the estimated line relative
  to the true regression line in the whole
  population.
• Likelihood of yi is f(yxß)/F(cxß).

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Poisson Regression Model

• Dependent variable is a count variable, which
  takes on nonnegative integer values 0, 1, 2,
• Likelihood of yi
• P(yhx)exp-exp(ßx)exp(ßx)h/h!
• As with the probit, logit, and Tobit models, we
  cannot directly compare the magnitudes of the
  Poisson estimates of an exponential function with
  the OLS estimates of a linear model. Some rough
  comparison is possible after some adjustment (see
  section 17.3).

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Issues with the Poisson Model

• Poisson distribution may be a too strict
  assumption on the error term.
• All moments of the Poisson distribution are
  determined by the mean.
• Fortunately, whether or not the Poisson
  distribution holds, we still get consistent,
  asymptotically normal estimates of the ß.
• The estimated standard errors, however, may be
  inconsistent and need to be adjusted. (P. 576)

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Nonrandom Sample Selection (in dependent
variables)

• Truncated regression is a special case of
  nonrandom sample selection.
• Incidental Truncation
• We do not observe y because of the outcome of
  another variable.
• For example, wage offers are observed only for
  those who are working. Labor force participation
  may be affected by some unobserved variables that
  also affect wage offer. This would produce biased
  estimates in the wage offer equation.

19
Consistency of OLS with Selected Sample

• If sample selection is entirely random, then OLS
  estimates are unbiased.
• If sample selection depends on the explanatory
  variables and additional random terms that are
  independent of x and u, then OLS is also
  consistent.
• If sample selection is correlated with error
  term, then OLS is inconsistent.
• Truncated data
• Incidental truncation

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Modeling Incidental Truncation

• Population model yXßu, X exogenous
• Incidental truncation
• sysXßsu
• s1Z?vgt0, Z exogenous
• s1 if observed 0 otherwise
• Correlation between u and v generally causes a
  sample selection (endogeneity) problem.

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Consistency of Incidental Truncation Model

• What are we estimating with the incidentally
  truncated data?
• yXß??(Z?)
• ?0 when u and v are uncorrelated.
• ?(.) is the inverse Mills ratio
• Since Z may include X, the estimate of ß may be
  biased if the term ??(Z?) is omitted from OLS
  regression.
• Solution Heckit method
• Estimate ?,calculate ?(Z?), and include it in the
  OLS regression.
• It is preferred to have Z including X as a
  subset. Otherwise, multicollinearity problem may
  result.