1. Tobit-Model Einordnung:
Gruppe der Limited dependent Variable models; zB 0-1 Variable Probit/Logit
Sonderfall beschränkter Variablen
Nicht mit KQ-Methode schätzbar; sondern Maximum Likelihood

2. The Tobit - Model An example
Wooldridge (2003), Introductory Econometrics, 2nd edition, Chap.17.2
Other:
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

3. Problem: special attribute(s) of the dependent variable (DV)
1. Tobit - Model
Problem: special attribute(s) of the dependent variable (DV)
dependent variable constrained and
clustering of observations at the constraint
Examples:
consumption (1. not 2.)
wage changes (2. not 1.)
Labor supply (1. and 2.)

4. left- and right-censoring in the data
1. Tobit - Model
left- and right-censoring in the data
left-censored, from below
right-censored, top-coded
Distribution of hourly benefits, Fringe.dta,
command: hist hrbens
Distribution of log-wages in West-Germany, males, , clerks, IABS01

5. 1. 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 holidays

6. Censoring in a regression framework
1. Tobit - Model
Censoring in a regression framework
Ruud, Figure 28.2

7. OLS on the complete sample biased and inconsistent,
1. 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.
Intuition (gesamtstichprobe): Effekt von x auf y nicht konstant für alle x
Intuition (Teilstichprobe):
Ignoriert den Übergang von Nullern zu positiven Werten
NB: OLS zur Null verzerrt, aber richtiges Vorzeichen
Nichtlinearität: Zeichne ein Schaubild, wenn Steigung der Gerade ungleich Null schneidet sie irgendwann die Achse
Konstante partielle Effekte unplausibel bspw. im AA Kontext. Bei Randlösung. Erhöhung des Stundelohnes hat keinen Effekt auf AA, aber im Positiven Bereich natürlich schon
Figure 28.2, Ruud

8. 1. Tobit - Model  do not throw away information (Tobin 1958)
Solution possibility 1: Estimate a Probit Model if
Loses information on y.
 do not throw away information (Tobin 1958)
Solution: Tobit-regression

9. 1. Tobit - Model latent variable Trick: introduce a latent variable
Assume: linear conditional expectation for latent Var.
Assumption: if

10. 1. Tobit - Model Estimation Random sample
Estimation of the parameters of the model:
Non-linear LS estimation
Maximum likelihood method

11. Maximum Likelihood estimation
1. Tobit - Model
Maximum Likelihood estimation
Maximum likelihood estimation:
Likelihood-function consists in two parts
Probit-Part
For censored observations we have:

12. Maximum Likelihood estimation
1. Tobit - Model
Maximum Likelihood estimation
2. Linear part
Can formulate a linear model for the part that is uncensored:

13. 1. Tobit - Model Likelihood function
Likelihood- and Log-Likelihood-function:
ln L is maximized wrt β and σ.
Intuition for ML:
Choose parameters such that the probability of having observed this sample is maximized
FOC yields estimator for β and σ.
β and σ are asymptotically normal. Inference is standard.

14. 2. An Example Data: Dependent Variable: 753 Observations
hours working hours (yearly) of married women
753 Observations
428 women exchange work for money in the labor market
(hours vary in the dataset between 12 and 4950)
325 women do not work.

15. 2. An Example explanatory variables: age age educ education
in years of schooling
exper experience
in actual years of work
nwifeinc family income (in 1000$) that is not generated by the woman
kidslt6 number of kids
age < 6
kidsge6 number of kids
6< age < 18

16. 2. An Example Estimation of a Tobit-Model (in Stata):
Source: Wooldridge, Econometric Analysis of Cross Section and Panel Data (2002)
 estimated coefficients are to be interpreted as the effect of the regressors on the latent variable.

17. 2. An example Comparison OLS - Tobit
Direct comparison of OLS and Tobit output impossible
OLS
Tobit
nwifeinc
-3.45
-8.81
educ
28.76
80.65
exper
65.67
131.56
exper2
-0.700
-1.86
age
-30.61
-54.41
Kidslt 6
Kidsge 6
-32.78
-16.22
Constant
965.31
Log- likelihood
---- R2
0.266
0.274
750.18
Dependent variable: hours
Grund:
OLS: beta ist der Effekt von x auf y
Tobit: beta ist der Effekt von x auf y*
Funny: MLE-Schätzer ugf. OLS geteilt durch anzahl der nichtzensierten Beobachtungen

18. Interpretation of coefficients
2. An example
Interpretation of coefficients
Marginal effect on the latent variable
Slope of dashed line: tobit
Slope of solid line: OLS
Formeln aus WO, p. 521ff.

19. Interpretation of coefficients
2. An example
Interpretation of coefficients
Marginal effect on the actual variable
Probability that an observation is different from zero (if 1, then OLS=Tobit) y
Green line!! x

20. Interpretation of coefficients
2. An example
Interpretation of coefficients
Marginal effect on positive observations
Where λ(c) is called inverse Mills Ratio:
Formeln aus WO, P. 521ff
Intuition für 3
Wenn sich x ändert, ändert sich die Zusammensetzung derjenigen, für die gilt y>0. Which means that the effect on the observations which are already positive, and where the effect is beta has to be adjusted by the effect on those individuals that didn‘t have positive y‘s before
λ(c) captures the change in the population, we condition on (y>0), when changing x.

21. Interpretation of coefficients
2. An example
Interpretation of coefficients
Marginal effect on the probability, that an observation is uncensored.
It follows:
Intuition für 4:
Prob(y=0|x)=Prob(y<=0|x)=Prob(u<=-xb|x)=Prob(u>=xb|x)=1-Phi(xb|x)
NB: For coefficients 2-4 need choose an appropriate x-vector!

22. Interpretation of coefficients
2. An example
Interpretation of coefficients
Comparison OLS - TOBIT on the basis of the marginal effect on actual DV (example educ, for an average individual):
OLS TOBIT
E(Y|x) einmal mit OLS und einmal mit Tobit at mean values
Bivariat gilt das immer: OLS verzerrt zur Null (bei gleichem Vorzeichen und ähnlicher Signifikanz..)
48,73
28.76

23. Interpretation of coefficients
2. An example
Interpretation of coefficients
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).

24. marginal effects: dtobit
2. An example
marginal effects: dtobit
dtobit calculates the four different marginal effects (at the mean of the explanatory variables):

25. marginal effects: dtobit
2. An example
marginal effects: dtobit

26. marginal effects: dtobit
2. An example
marginal effects: dtobit

27. 3. Extensions 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

28. 3. Extensions 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)