cds m phil econometrics limited dependent variable models ... · 10/05/2011 · limited dependent...

3/3/2014

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CDS M Phil Econometrics Vijayamohan

CDS M Phil Econometrics

Vijayamohanan Pillai N

13-Mar-14

CDS Mphil Econometrics Vijayamohan

Limited Dependent Variable Limited Dependent Variable Models: Models: TobitTobit


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IntroductionIntroduction

Limited Dependent Variable Models:Limited Dependent Variable Models:

Truncation and CensoringTruncation and Censoring

MaddalaMaddala, G. 1983. , G. 1983. Limited Dependent and Limited Dependent and Qualitative Variables in EconometricsQualitative Variables in Econometrics. . Cambridge University Press.Cambridge University Press.


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TruncationTruncation

A A truncated distribution truncated distribution is the part of an is the part of an untruncateduntruncated distribution that is above or distribution that is above or below some specified value.below some specified value.

If a continuous random variable If a continuous random variable xx has has pdfpdff(x) f(x) and and aa is a constant, then the density of the is a constant, then the density of the truncated RV istruncated RV is

)ax(obPr

)x(f)ax|x(f

>=>

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Truncated standard

normal distribution

for a = – 0.5, 0, and 0.5

a = a = –– 0.5 0.5

a = 0 a = 0

a = 0.5 a = 0.5


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TruncationTruncation

Truncation occurs when some observations on Truncation occurs when some observations on both the dependent variable and both the dependent variable and regressorsregressorsare lost. are lost.

For example, income may be the dependent For example, income may be the dependent variable and only lowvariable and only low--income people are income people are included in the sample. included in the sample.

In effect, truncation occurs when the sample In effect, truncation occurs when the sample data is drawn from a subset of a larger data is drawn from a subset of a larger population.population.

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CensoringCensoring

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censoring occurs when the value of an censoring occurs when the value of an observation is only partially known.observation is only partially known.

One of the earliest attempts to One of the earliest attempts to analyseanalyse a a statistical problem involving censored data: statistical problem involving censored data:

Daniel Bernoulli's 1766 analysis Daniel Bernoulli's 1766 analysis

of smallpox morbidity and mortality data to of smallpox morbidity and mortality data to demonstrate the efficacy of vaccination.demonstrate the efficacy of vaccination.


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Censored Regression ModelCensored Regression Model

Censoring occurs when data on the Censoring occurs when data on the dependent variable is lost (or limited) dependent variable is lost (or limited)

but not data on the but not data on the regressorsregressors..

When the dependent variable is censored, When the dependent variable is censored, values in a certain range are all transformed values in a certain range are all transformed to (or reported as) a single value.to (or reported as) a single value.


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Censored Regression ModelCensored Regression Model

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For example, For example,

people of all income levels may be included in people of all income levels may be included in the sample, the sample,

but for some reason but for some reason

the income of highthe income of high--income people income people

may be topmay be top--coded as, say, Rs100,000. coded as, say, Rs100,000.

A defect in the sampleA defect in the sample


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An ExampleAn ExampleA labor supply model estimates the relationship between hours worked by employees and characteristics of employees such as age, education and family status.

For people who are unemployed, it is not possible to observe the number of hours they would have worked had they had employment.

Still we know age, education and family status for those observations.

Another ExampleAnother Example


Suppose we are interested in finding out Suppose we are interested in finding out the amount of money a HH spends on a the amount of money a HH spends on a house in relation to sociohouse in relation to socio--economic economic variables.variables.

Many HHs may not have purchased Many HHs may not have purchased house:house:

Zero expenditure for themZero expenditure for them


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Another ExampleAnother Example

•• Suppose we are interested in studying Suppose we are interested in studying

how much an individual how much an individual desireddesired to give to to give to charity. charity.

•• For many people the amount we observe is For many people the amount we observe is zero, zero,

•• i.e. they give nothing to charity. i.e. they give nothing to charity.

•• For others, we observe the actual amount For others, we observe the actual amount they contributed. they contributed.


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Censored & Truncated Regression ModelCensored & Truncated Regression Model

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Truncated regression models Truncated regression models

are used for data, are used for data,

where whole observations are missing where whole observations are missing

so that the values for so that the values for

the dependent and the dependent and

the independent variable are unknown. the independent variable are unknown.


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Censored & Truncated Regression ModelCensored & Truncated Regression Model

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Censored regression models Censored regression models

are used for data, are used for data,

where only the value for the dependent where only the value for the dependent variable (hours of work for example) variable (hours of work for example)

is unknown is unknown

while the value of the independent variable while the value of the independent variable

(age, education, family status) (age, education, family status)

is still available is still available

TobitTobit ModelModel

Censored regression ? Censored regression ? oror

Truncated regression ?Truncated regression ?


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Original Original TobitTobit model suggested by model suggested by James Tobin (1918 James Tobin (1918 –– 2002)2002)


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Some examples in the empirical literatureSome examples in the empirical literature

Analyze a dependent variable that is zero for a Analyze a dependent variable that is zero for a significant fraction of the observations.significant fraction of the observations.


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•• The structural equation in the The structural equation in the TobitTobit model is:model is:

•• where where uuii ∼∼ N(0, N(0, σσ22))

y*y* is a latent variable that is observed for is a latent variable that is observed for

values greater than values greater than ττ and and

censored otherwise.censored otherwise.

ii*i uxy +β=


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•• The observed y is defined by the following The observed y is defined by the following measurement equationmeasurement equation

y*, if y* > y*, if y* > ττ

ττyy, if y* , if y* ≤≤ ττyyii = =

ii*i uxy +β= •• In the typical In the typical TobitTobit model, model,

•• we assume that we assume that ττ = 0 = 0 •• i.e. the data are censored at 0.i.e. the data are censored at 0.

•• Thus, we haveThus, we have


yyii = = y*, if y* > 0

0, if y* ≤ 0



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This model contains This model contains a a ProbitProbit model model

for for yyii being zero or positive being zero or positive

and a and a standard Regression model standard Regression model

for the positive values of for the positive values of yyii. .

yyii = = y*, if y* > 0

0, if y* ≤ 0

( )2i ,0N~u σ

ii*i uxy +β=


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The Probit model may, for example,

describe the influence of explanatory variables on the decision

whether or not to donate to charity,

while

the Regression model measures

the effect of the explanatory variables

on the size of the amount for

donating individuals.

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• Why Use the Tobit Model?

• Why not just use the observations for which y > 0 and estimate the model using OLS?

• The answer:

• if you do, your parameter estimates will be biased and inconsistent.

• The degree of bias will also increase as the number of observations that take on the value of zero increases.


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( )2i ,0N~u σ

yyii = =

yi*, if

0, if yi* ≤ 0

> 0

Neglecting the truncation can lead to biased estimates of α and β

=> ]x,0y|y[E iii

Why Use the Why Use the TobitTobit Model?Model?

]/)x[(

]/)x[(x

i

ii σβΦ

σβφσ+β

ii*i uxy +β=


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The last term on the RHS [The last term on the RHS [ σλσλ((αα)) ] :] :

the inverse Mills ratio / hazard function the inverse Mills ratio / hazard function

for the std N distribution.for the std N distribution.

φφ = = pdfpdf and and ΦΦ = = cdfcdf: p(: p(yyii > 0)> 0)


E[y | truncation] = µ + σλ(α)

]/)x[(

]/)x[(x

i

ii σβΦ

σβφσ+β=> ]x,0y|y[E iii


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Inverse Mills RatioNamed after John P. Mills, Named after John P. Mills,

The ratio of the Probability Density Function over theThe ratio of the Probability Density Function over the

Cumulative Distribution Function of a distribution.Cumulative Distribution Function of a distribution.

If If xx is a random variable distributed normallyis a random variable distributed normally

with mean with mean µµ and variance and variance σσ22, then, then

where where αα is a constant, is a constant,

ϕϕ denotes the standard normal denotes the standard normal pdfpdf, and , and

ΦΦ is the standard normal is the standard normal cdfcdf..

σµ−αΦ

σµ−αϕ

σ+µ=α> ]x|x[E

)z(

)z(

Φϕσ+µ=


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• Consider for example,

• the amount a person gives to charity.

• Suppose the true relationship between the amount a person wantswants to give to charity and that person’s income is

→



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The lower income people would actually like The lower income people would actually like to give negative amounts to give negative amounts (i.e. get money back!). (i.e. get money back!).

The red line indicates the true regression line for the relationship between income and donations


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In reality, we do not observe individuals In reality, we do not observe individuals making negative contributions. making negative contributions.

The observed data looks like this:The observed data looks like this:

What we observe is What we observe is they give nothing. they give nothing.



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If we simply estimated the model by OLS,

the parameter estimates would be biased downwards.True relationship

OLS regression line


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OLS tends to underestimate the magnitude of the slope.

the parameter estimates would be biased downwards.

True relationship

OLS regression line

A bit more complex than interpreting estimated coefficients from the OLS model.

In particular, the estimated coefficients represent the marginal effect of x on y*.

That is :

marginal effect of x on the latent variable y* not on the observed variable y.


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Interpreting Interpreting TobitTobit EstimatesEstimates

kk

*i

x

]x|y[Eβ=

∂∂


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What we want to explain is the observed amount of charitable observed amount of charitable contributions contributions

not the desired amount of charitable contributions. Thus,

what we want is the expected value of y

conditional on y being greater than zeroy being greater than zero: E[y | y > 0, x].


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What we want is the expected value of y conditional on y being greater than zero: E[y | y > 0, x]. That is:


=> ]x,0y|y[E iii

The desired marginal effects are the derivative of this function with respect to x.

=σβΦσβφ

σ+β]/)x[(

]/)x[(x

i

ii

)z(

)z(xi Φ

φσ+β

We have

E(y) = E(y | y > 0) P(y > 0)


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=> ]x,0y|y[E iii

)z(

)z(xi Φ

φσ+β

)z()z(

)z(X)y(E i Φ

Φφσ+β=

)z()z(x)y(E i σφ+Φβ=

)z(x

)y(Ek

k

Φβ=∂

∂k

k

*i

x

]x|y[Eβ=

∂∂

)ax(obPr

)x(f)ax|x(f

>=>


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Method of maximum likelihoodMethod of maximum likelihood

Olsen’s (1978) Olsen’s (1978) reparameterizationreparameterization simplifies ML simplifies ML estimation.estimation.

James Heckman has proposed a simple alternative to James Heckman has proposed a simple alternative to the ML method:the ML method:

J. J. Heckman, “Sample Selection Bias as a J. J. Heckman, “Sample Selection Bias as a Specification Error,” Specification Error,” EconometricaEconometrica, vol. 47, pp. , vol. 47, pp. 153153––161.161.

EstimationEstimation


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Heckman AlternativeHeckman Alternative

Consists of a twoConsists of a two--step estimating procedure:step estimating procedure:

Step 1: estimate the probability of, say, Step 1: estimate the probability of, say,

a consumer owning a house, a consumer owning a house,

on the basis of the on the basis of the probitprobit model. model.


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Step 2: estimate the model

by adding to it the inverse Mills ratio or the hazard rate that is derived from the probit estimate.

yyii = = yi*, if

0, if yi* ≤ 0

> 0

IMRIMR

=> ]x,0y|y[E iii]/)x[(

]/)x[(x

i

ii σβΦ

σβφσ+β

ii*i uxy +β=


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The Heckman procedure yieldsThe Heckman procedure yields

consistent estimates of the parameters, but consistent estimates of the parameters, but

they are not as efficientthey are not as efficient

as the ML estimates.as the ML estimates.

An Example of An Example of TobitTobitmodel: model:

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05

1015

Ext

ram

arita

l

10 15 20Education

451 observations lying along the horizontal axis. 451 observations lying along the horizontal axis. ⇒⇒ a censored sample, a censored sample,

⇒⇒ a a tobittobit model may be appropriate.model may be appropriate.

Of the 601 Of the 601

responses, 451 responses, 451

individuals had individuals had

no extramarital no extramarital

affairs, and 150affairs, and 150

individuals had individuals had

one or more one or more

affairs.affairs.


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Ray Fair Model: OLSRay Fair Model: OLS


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Ray Fair Model: Ray Fair Model: TobitTobit


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Compare OLS & Compare OLS & TobitTobit estimatesestimates

OLSOLS TobitTobit

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TobitTobit: Coefficient Estimates & Marginal Effects: Coefficient Estimates & Marginal Effects

Compare…..Compare…..


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TobitTobit in in StataStata

StatisticsLinear models & relatedCensored regressionTobit regression

Also

StatisticsPostestimation(1) Tests: Test parameters(2) Marginal effects or elasticities

cds m phil econometrics limited dependent variable models ... · 10/05/2011 · limited dependent...

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