Econometrics

Week 10

Institute of Economic StudiesFaculty of Social Sciences

Charles University in Prague

Fall 2012

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Recommended Reading

For the todayLimited Dependent Variable ModelsChapter 17 (pp. 529 – 565).

In the next weekAdvanced Time Series TopicsSelected topics from Chapter 18

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Today’s Talk

Limited Dependent Variables (LDV) are simply dependentvariables whose range of values is substantially restricted.For example binary dependent variable (values 0 or 1).These kind of variables need a special treatment.In econometrics, we often use several type of models forthese variables we are going to discuss today:

LogitProbitTobitCensored and Truncated Regression Models

We will focus on their cross-sectional applications, while itcan be used as well in panel data or time series data.Main disadvantage of these models is that they are difficultto interpret.

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Binary Dependent Variables

Recall the linear probability model (LPM):

P (y = 1|x) = β0 + xβ

The main problem is that:Fitted probabilities can be less than zero or greater than 1.Partial effect of any explanatory variable is constant.

These limitations may be overcome by modeling theprobability as some general function, instead of assumptionof linearity in parameters:

Binary Response Models

P (y = 1|x) = G(β0 + β1x1 + . . .+ βkxk) = G(β0 + xβ),

where x is full set of explanatory variables, G is a functiontaking values strictly between zero and one: 0 < G(z) < 1.

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The Logit Model

While G can is general nonlinear function, mostapplications use 2 cases.First common case is to choose G to be a logisticfunction which is cdf for a standard logistic randomvariable:

The Logit Model

G(z) = exp(z)/ [1 + exp(z)] = Λ(z)

Taking values between zero and one for all real values z.Sometimes we refer to this model as a logistic regression.

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The Probit Model

Another common choice of G is the standard normalcumulative distribution function (cdf)

The Probit Model

G(z) = Φ(z) =∫ z

−∞φ(z)dv,

where φ(z) is the standard normal density:

φ(z) = (2π)−1/2 exp(−z2/2)

Both G functions have very similar shapes, they areincreasing in z, most quickly around 0.

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The Logit and Probit Models

Figure: Red Line is logistic function, black line standard normal

-3 -2 -1 0 1 2 30.0

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1.0

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The Logit and Probit Models cont.

There is no real reason to prefer one model to another.Traditionally, logit was used more often as the logisticfunction leads to easier computation of the model.In economics, assumption of standard normal distributionis more realistic, thus probit is preferred by economists.There is no useful interpretation of coefficients βj .

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The Logit and Probit Models Estimation

Due to its nonlinear nature, we have to use MaximumLikelihood Estimation (MLE).Assuming we have a random sample of size n. To obtainMLE, we need the density of yi given xi first:

f(y|xi;β) = [G(xiβ)]y[1−G(xiβ)](1−y),

where y = 0, 1

MLE of βTo obtain MLE of β, we need to maximize followinglog-likelihood:

L(β) =n∑

i=1

yi log[G(xiβ)] + (1− yi) log[1−G(xiβ)],

where if G(.) is the standard logic cdf, β̂ is the logit estimator, ifG(.) is the standard normal cdf, β̂ is the probit estimator.

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The Logit and Probit Models Estimation cont.

The general theory of conditional MLE for random samplesimplies that, under very general conditions, the MLE isconsistent, asymptotically normal andasymptotically efficient.Thus we can derive asymptotic standard errors forestimates easily.And we can use them to test H0 : β = 0.We can also simply test multiple restrictions in the logicand profit models.Easiest way is to use Likelihood Ratio (LR) test:

LR = 2(Lu − Lr) a∼ χ2q

where u is unrestricted and r restricted model and we haveq restrictions.In this way we can simply test the importance of variables.If we drop a variable from the model and log-likelihoodsignificantly decreases, we know that it is significant for themodel. 10 / 23

Interpretation of the Logit and Probit Estimates

The most difficult aspect of these models in presenting andinterpreting results.In general, we care about the effect of x on P (y = 1|x).βj coefficients give us the sign of the partial effect of eachxj on the response probability, that is∂p/∂x = g(β0 + xβ)βj .We can compare significance (H0 : βj = 0) and signs ofcoefficients.Coparison of magnitudes is more complicated, as we needto calculate the derivatives at the means.STATA will do this in the probit case.

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Interpretation of the Logit and Probit Estimatescont.

To measure goodness-of-fit, we can not simply use R2.One possibility is a pseudo R2 based on the log-likelihood:1− Lu/Lr.We can also look at the percent correctly predictedmeasure – if predict a probability > 0.5 then that matchesy = 1 and vice versa.percent correctly predicted is in fact percentage oftimes the predicted yi matches actual yi (which is zero orone).

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The Tobit Model

Another kind of limited dependent variable is one where wehave roughly continuous variables over strictly positivevalues but zero otherwise.

The Tobit Model

y∗ = β0 + xβ + u, u|x ∼ N(0, σ2)y = max(0, y∗)

Latent variable y∗ satisfies the classical linear modelassumptions (normal, mohoskedastic distribution withlinear conditional mean).

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The Tobit Model cont.

To estimate β and σ, we use MLE again.

MLE of Tobit modelTo estimate the parameters of the Tobit model, we need tomaximize log-likelihood obtained by summing followingfunction across all i:

`i(β, σ) = 1(yi = 0) log[1− Φ(xiβ/σ)]+1(yi > 0) log ((1/σ)φ[(yi − xiβ)/σ]) ,

where φ is standard normal density function.

Notice how log-likelihood functions has two parts

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Interpreting the Tobit Model

Again, we obtain parameter estimates from softwarepackage, but they are not so straightforward to interpret aswith OLS case.β estimates are effects of x on y∗, but we are interested toexplain y.Basically, we can estimate two expectations:

E(y|x) = P (y > 0|x)E(y|y > 0, x) = Φ(xβ/σ)E(y|y > 0, x)E(y|y > 0, x) = xβ + σλ(xβ/σ),where λ(c) = φ(c)/Φ(c) is so called Inverse Mills ratio.

Expected value of y conditional on y > 0 is equal to xβ andsome strictly positive term.

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Interpreting the Tobit Model cont.

The Tobit model rely heavily on normality andhomoskedasticity in the underlying latent variable.In OLS, we can deal with heteroskedasticity for example bycomputing heteroskedasticity robust standard errors.But in Tobit, in case of heteroskedasticity we never knowwhat MLE is actually estimating.

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Censored and Truncated Regression Models

Model with similar structure as the Tobit model iscensored regression model.These two are sometimes interchanged, but they aredifferent.Unlike the Tobit, the censored regression arises due to thedata censoring.We assume underlying dependent variable to be normallydistributed, but it is censored below or above a certainvalue due to the way we collect data.A truncated regression model arises when we exclude asubset of population in a sampling scheme.In other words, we do not have random sampling, but havesome rule to sample the data.a

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Censored Regression Models

Generally, censored regression model can be definedwithout distributional assumptions.We will study it under assumption of normal distribution.

The Censored Normal Regression Model

yi = β0 + xiβ + ui, ui|xi, ci ∼ N(0, σ2)wi = min(yi, ci),

with censoring value ci.

While we only observe wi = min(yi, ci) in right censoreddata,in left censored data, we observe wi = max(yi, ci).

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Censored Regression Models cont.

A good example of censored data is some type of surveys.If we ask respondents about their wealth, but they areallowed to respond with “more than 1.000.000 CZK”, thenwe observe actual wealth of people below this threshold,but not above.Censoring threshold ci in this case is constant for all i.In many situations, censoring threshold changes withindividuals i.We are able to estimate the censored regression modelusing MLE.

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Truncated Regression Models

Similar to censored model, but in this case, we do notobserve any information about certain part of population.For example, if it is not possible to survey part ofpopulation for any reason.

The Truncated Normal Regression Model

y = β0 + xβ + u, u|x ∼ N(0, σ2)

This model would satisfy the CLM assumptions.BUT, we do not have random sample (we observe only partof population).Thus OLS suffers from selection bias.Again, we have to use MLE.

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Sample Selection Correction

Truncated regression is a special case of general problem:nonrandom sample selection.We can also thing about it as omitted variable bias. Whatis omitted now is how was the selection of the sample made.One way how to correct for it and obtain consistentestimates of β from E(y|z, s = 1) = xβ + ρλ(zγ),where z is set of instrumental variables:zγ = γ0 + γ1z1 + . . .+ γmzm.

Sample Selection CorrectionUsing all observations, estimate a probit model of selectionindicator si on zi.Compute the inverse Mills ratio λ̂i = λ(ziγ̂).Using the selected sample (observations where si = 1), runregression of yi on xi and λ̂i.

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Sample Selection Correction

We can use a simple test for selection bias using thisprocedure.We can test H0 : ρ = 0 under which there is no sampleselection problem.Use t statistic on λ̂i.When ρ 6= 0, OLS errors are not correct.

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Thank you

Thank you very much for your attention!

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