tobit

4. Tobit-Model

TobitTobit-Model

1University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

1. The Tobit - Model 2. An exampleWooldridge (2003), Introductory Econometrics, 2nd edition, Chap.17.2Other:Wooldridge (2002): Econometric Analysis of Cross Section and Panel Data, Chapter 16. Ruud (2000): An Introduction to Classical Econometric Theory, Chapter 28. Greene (2000): Econometric Analysis, 4th edition, Chapter 20.3


4. Tobit-Model

Problem: special attribute(s) of the dependent variable (DV)1. dependent variable constrained and 2. clustering of observations at the constraint

Examples: consumption (1. not 2.) wage changes (2. not 1.) Labor supply (1. and 2.)3University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

left- and right-censoring in the dataleft-censored, from below1

right-censored, top-coded3 Density 0 1 2

0

.2

.4

Density .6

.8

0

1 hourly benefits, $

2

3

2.5

3

3.5

4 logw_cens

4.5

5

Distribution of hourly benefits, Fringe.dta,4

command: hist hrbens

Distribution of log-wages in West-Germany, males, 1.1.1986, clerks, IABS01University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

2 different sorts of Models Data censoring Earnings variable (IABS) Demand for stadium tickets Duration in unemployment Corner solutions Labor Supply Household expenditures on holidaysUniversity of Freiburg WS 2007/2008 Alexander Spermann

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4. Tobit-Model

Censoring in a regression framework Ruud, Figure 28.2


4. Tobit-Model

If DV is constrained and if there is clustering

OLS on the complete sample biased and inconsistent, OLS on the unclustered part biased and inconsistent.


4. Tobit-Model

Solution possibility 1: Estimate a Probit Model

1 y! 0

if

y"0

Loses information on y.

R y y! 08

do not throw away information (Tobin 1958)University of Freiburg WS 2007/2008 Alexander Spermann

Solution: Tobit-regression

4. Tobit-Model

yi* Trick: introduce a latent variableAssume: linear conditional expectation for latent Var.

E(y | x) ! x FAssumption:

* i

' i

y ! x F Ii y yi ! 0 * i

* i

' i

I i ~ i.i.d. N(0, W 2 )y "0 yi* e 0University of Freiburg WS 2007/2008 Alexander Spermann

if if

* i

9

4. Tobit-Model

Random sample {(x i , yi ) : i ! 1, 2,..., N} Estimation of the parameters of the model: Non-linear LS estimation Maximum likelihood method


4. Tobit-Model

Maximum likelihood estimation: Likelihood-function consists in two parts 1. Probit-Part For censored observations we have:r(y i ! 0) ! r(y* e 0) i

Ii x i' F ! r(Ii e x i' F) ! r e W W x i' F x i' F ! * ! 1 * W W University of Freiburg WS 2007/2008 Alexander Spermann

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4. Tobit-Model

2. Linear part Can formulate a linear model for the part that is uncensored:lim Pr(yiQp 0

Yi ! y i Q | yi " 0, Q " 0) ! yi x 'iF x 'iF Q * ! W W

lim Qp 0

yi

1 yi x 'iF f (I i ) ! J W W 12University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

Likelihood- and Log-Likelihood-function: x i' F 1 yi x i' F ! 1 * J W yi " 0 W W yi ! 0 ln 1 yi x i' F x i' F ! ln 1 * ln J W yi " 0 W W yi ! 0

ln L is maximized wrt

and . and .

FOC yields estimator for13

and are asymptotically normal. Inference is standard.University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

Data: Dependent Variable: hours

working hours (yearly) of married women 753 Observations

14

428 women exchange work for money in the labor market (hours vary in the dataset between 12 and 4950) 325 women do not work.University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

explanatory variables: age educ exper nwifeinc kidslt6 kidsge6 age education in years of schooling experience in actual years of work family income (in 1000$) that is not generated by the woman number of kids age < 6 number of kids 6< age < 18University of Freiburg WS 2007/2008 Alexander Spermann

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4. Tobit-Model

Estimation of a Tobit-Model (in Stata):

Source: Wooldridge, Econometric Analysis of Cross Section and Panel Data (2002)

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estimated coefficients are to be interpreted as the effect of the regressors on the latent variable.University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

Direct comparison of OLS and Tobit output impossibleOLS nwifeinc educ exper exper2 age Kidslt 6 Kidsge 6 Constant Log- likelihood R2 -3.45 28.76 65.67 -0.700 -30.61 -442.09 -32.78 1330.48 ---0.266 750.18 Tobit -8.81 80.65 131.56 -1.86 -54.41 -894.02 -16.22 965.31 -3819.09 0.274 1122.02University of Freiburg WS 2007/2008 Alexander Spermann

Dependent variable: hours

17

W

4. Tobit-Model

1. Marginal effect on the latent variablexE y* x xxk

! F

k

Slope of dashed line: tobit

Slope of solid line: OLS18University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

2. Marginal effect on the actual variablexE y x xxk xF ! F k* W Probability that an observation is different from zero (if 1, then OLS=Tobit)

yGreen line!!

019

xUniversity of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

3.

Marginal effect on positive observations

xP c xE(y | x, y " 0) ! Fk Fk ! F k {1 P (c)[c P (c)]} F k xx k xc

Where (c) is called inverse Mills Ratio: xF J J(c) W P(c) ! ! * (c) xF * W

(c) captures the change in the population, we condition on (y>0), when changing x.20University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

4. Marginal effect on the probability, that an observation is uncensored. xF xF r(y " 0 x) ! 1 * ! * W W

It follows:

x r y " 0 x xxk

xF F k ! J W W

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NB: For coefficients 2-4 need choose an appropriate x-vector!University of Freiburg WS 2007/2008 Alexander Spermann

4. Tobit-Model

Comparison OLS - TOBIT on the basis of the marginal effect on actual DV (example educ, for an average individual): OLS TOBIT x'F Fk * W 80.65 0.60428.76 48,73

F k , OLS


4. Tobit-Model

Interpretation: On average, an additional year of education increases the labor supply by 48,7 hours (for an average individual). OLS underestimates the effect of education on the labor supply (in the average of the explanatory variables).


4. Tobit-Model

dtobit calculates the four different marginal effects (at the mean of the explanatory variables):


4. Tobit-Model


4. Tobit-Model


4. Tobit-Model

Specification Unobserved, independent heterogeneity not problematic, as OLS Endogeneity (left-out variables, simultaneity) standard-IV, similar to OLS Heteroskedasticity, nonormal errors inconsistency, different from OLS


4. Tobit-Model

alternatives for Tobit nonlinear estimation, eg. E(Y|x)=exp(xb) CLAD-estimator (for censoring problems) hurdle models, two-tiered models (for corner solution problems)


tobit

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Transcript of tobit