# Limited Dependent Variable Models by hongkonguniv

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```									Limited Dependent Variable Models
Course: Applied Econometrics Lecturer: Zhigang Li

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

Linear Probability Model (LPM) (Section 7.5)
y=βX+u,
 where y is a binary variable, one for success and zero for failure

 In this model, β 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.

Binary Response Models
 A latent variable model Y=1 if Y*=βX+ε>0 Y=0 if Y*=βX+ε<=0  This implies P(Y=1|X)=G(βX)  Logit Model: ε follows a logistic distribution P(Y=1|X)=eβX/(1+eβX)  Probit Model: ε follows a normal distribution P(Y=1|X)=∫-∞βXφ(v)dv  The magnitude of β 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.

Binary Response Models: Interpretation I
 The partial effect of (continuous) xj is: ∂p(X)/∂xj=G’(βX)βj=g(βX)βj
 Where g(.) is the density function of ε.

 Implications
 The effect of xj depends on the value of X.  The relative effect of xi and xj is fixed.

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/.25=1.6.  The logit slope estimates should be divided by 4 to make them roughly comparable to the LPM (Linear Probability Model) estimates.

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).

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.

A Tobit Model
 Latent variable: y*=βX+u, u|x ~ Normal (0,σ2).  Observed response: y=max(0,y*)  Likelihood of yi
 yi>1: φ[(y-βX)/σ]/σ  yi=0: P(y*<0|X)=1-Φ(βX/σ)

What if OLS is used?
 Conditional expectation E(y|y>0,x)
 E(y|y>0,x)=βX+σλ(βX/σ)  Where λ(c)=φ(c)/Φ(c) is called the inverse Mills ratio.

 Unconditional expectation E(y|x)
 E(y|x)=P(y>0|x)E(y|y>0,x)=Φ(βX/σ)[βX+σλ(βX /σ)], which is a nonlinear function of x and β.

 Simple OLS can not consistently estimate β in either of the above cases.

What if OLS is used (continued)?
 The partial effects of xj on E(y|y>0,x) and E(y|x) have the same sign as the coefficient βj, but the magnitude depends on the values of all explanatory variables and parameters.
 ∂E(y|y>0,x)/∂x=β{1-λ(βX/σ)[βX/σ+λ(βX/σ)]}  ∂E(y|x)/∂x= βΦ(βX/σ)  To make the Tobit coefficient comparable to OLS estimates, we must multiple the Tobit estimate by an adjustment factor Φ(βX/σ).

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(y>0|x) and E(y|y>0,x) depend on different parameters.

Censored Regression Model I
 y=βX+u, u|x,c ~ Normal (0,σ2)  w=min(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.

Censored Regression Model II
 An OLS regression using only the uncensored observations produces inconsistent estimators of β.

 If there is heteroskedasticity or nonnormality, the MLEs are generally inconsistent.

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(y|xβ)/F(c|xβ).

Poisson Regression Model
 Dependent variable is a count variable, which takes on nonnegative integer values: 0, 1, 2, …  Likelihood of yi
 P(y=h|x)=exp[-exp(βx)][exp(βx)]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).

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 β.  The estimated standard errors, however, may be inconsistent and need to be adjusted. (P. 576)

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.

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

Modeling Incidental Truncation
 Population model: y=Xβ+u, X exogenous  Incidental truncation: sy=sXβ+su s=1[Zγ+v>0], Z exogenous s=1 if observed; 0 otherwise  Correlation between u and v generally causes a sample selection (endogeneity) problem.

Consistency of Incidental Truncation Model
 What are we estimating with the incidentally truncated data? y=Xβ+ρλ(Zγ)  ρ=0 when u and v are uncorrelated.  λ(.) is the inverse Mills ratio  Since Z may include X, the estimate of β may be biased if the term ρλ(Zγ) is omitted from OLS regression.  Solution: Heckit method  Estimate λ,calculate λ(Zγ), and include it in the OLS regression.  It is preferred to have Z including X as a subset. Otherwise, multicollinearity problem may result.

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