M.Sc. in EconomicsEconometrics Module I

Topic 7: Censored Regression Model

Carol Newman

Censored Regression Model The Tobit Model

Continuous dependent variable which is constrained or truncated at some value or values

Conventional regression models fail to account for the difference in the DGP for limit vs non-limit observations on the dependent variable

Two components to the DGP:

The probability that an observations will be at the limit

The distribution of the dependent variable for non-limit observations

OLS will yield biased and inconsistent estimates

Censored Regression Model Derivation of the standard tobit/censored regression model:

Probability of a limit observation assuming

The distribution of the dependent variable conditional on being positive

where the inverse mills ratio is defined as:

Model is composed of a discrete and a continuous part

i*i ey βix

otherwise 0

0 if

i

iii

y

*y*yy

βi

i

xx 10 |yP i

20 ,N~ei

βx

βxx iiiii y,|yE 0

..

.

Censored Regression Model Model is estimated using MLE

Limit observations:

Non-limit observations

Log-likelihood equation:

Note: in practice re-parameterised to improve convergence (see Olsen 1978)

00

1ii y

ii

y

ii,i

ylnlny|,Lln

βxβx

xβ

βx

xx iiiii |yP|f:y 1000

βxx ii

iiii

yy,|yf:y 00

Censored Regression Model Interpretation of the coefficients

McDonald and Moffitt Decomposition (1980)

Show that (in class):

iiiiiii |yP,y|yE|yE xxx 00

ki

iiiiiiii

ki

ii

ki

ii

x

,y|yE|yP,y|yE

x

|yP

x

|yE

xxx

xx 000

0

k

i

ki

ii

x

|yE

βxx

Censored Regression Model Properties:

Tobit model is inconsistent if assumptions of normality and homoscedasticity of the error term are violated

Lagrange multiplier tests can be used to test for violations of these assumptions

If errors are found to be non-normal or heteroscedastic likelihood equation must be adjusted