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Economics 310
Handout # VIa Truncated and Censored Regression Model
I. The Truncated Regression Model
A. Truncated distributions : Suppose that x is a random variable with pdf f(x).
B. Truncated normal distributions : Let x be a normally distributed random variable.
C. Truncated Regression Model
The truncated regression model occurs when part of the data is missing. For example, if
we had data on wages and years of schooling for a sample of employed persons. Some
persons are excluded from the sample because the wage that they would receive if they
were employed is less than the minimum wage. So the data would be missing for all
persons who were not working. If we are just interested in the relationship between
wages and education for the subpopulation of employed workers then OLS might be
appropriate. But if we are interested in this relationship for all workers, employed or not,
then OLS would give misleading results.
D. Estimation. The above model can be estimated by OLS or by maximum likelihood.
Which is more appropriate depends upon what use you intend to use the results for. If
you are interested in the subpopulation use the OLS results. If you are interested
in the whole population use the MLE estimates.
E. Example
. g x = 100uniform() . g y = 100+2x+50*invnorm(uniform()) . reg y x Source | SS df MS Number of obs = 753 -------------+------------------------------ F( 1, 751) = 1182. Model | 2662995.79 1 2662995.79 Prob > F = 0. Residual | 1690804.10 751 2251.40359 R-squared = 0. -------------+------------------------------ Adj R-squared = 0. Total | 4353799.89 752 5789.62752 Root MSE = 47.
y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | 2.056637 .0597997 34.39 0.000 1.939242 2. _cons | 98.90442 3.498202 28.27 0.000 92.037 105.
The above gives the results for the whole population. Now suppose the data is truncated (ie missing) for y < 130. Then the OLS estimation for the subsample would be
OLS Estimation
reg y x if y > Source | SS df MS Number of obs = 621 -------------+------------------------------ F( 1, 619) = 607. Model | 1125353.39 1 1125353.39 Prob > F = 0. Residual | 1145965.29 619 1851.31711 R-squared = 0. -------------+------------------------------ Adj R-squared = 0. Total | 2271318.69 620 3663.41724 Root MSE = 43.
y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | 1.617359 .0655998 24.65 0.000 1.488534 1. _cons | 133.5722 4.167651 32.05 0.000 125.3878 141.
Notice that the coefficient of x is estimated to be 1.617 significantly below the true value of 2.0.
MLE estimation
truncreg y x , ll(130) (note: 132 obs. truncated)
Fitting full model:
Iteration 0: log likelihood = -3174. Iteration 4: log likelihood = -3153.
. tobit tickets price , ul Tobit estimates Number of obs = 50 LR chi2(1) = 72. Prob > chi2 = 0. Log likelihood = -100.56717 Pseudo R2 = 0.
tickets | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- price | -34.97672 4.717039 -7.415 0.000 -44.45597 -25. _cons | 2217.171 197.6837 11.216 0.000 1819.911 2614. ---------+-------------------------------------------------------------------- _se | 103.9596 18.72889 (Ancillary parameter)
Obs. summary: 16 uncensored observations 34 right-censored observations at tickets>=
. reg insure income
Source | SS df MS Number of obs = 200 ---------+------------------------------ F( 1, 198) = 527. Model | 166757.822 1 166757.822 Prob > F = 0. Residual | 62636.8196 198 316.347574 R-squared = 0. ---------+------------------------------ Adj R-squared = 0. Total | 229394.642 199 1152.73689 Root MSE = 17.
insure | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- income | .9704973 .0422701 22.959 0.000 .8871398 1. _cons | -23.42143 2.490869 -9.403 0.000 -28.33346 -18.
. reg insure income if insure > 0
Source | SS df MS Number of obs = 107 ---------+------------------------------ F( 1, 105) = 902. Model | 100691.287 1 100691.287 Prob > F = 0. Residual | 11713.6334 105 111.558413 R-squared = 0. ---------+------------------------------ Adj R-squared = 0. Total | 112404.921 106 1060.42378 Root MSE = 10.
insure | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- income | 1.940817 .0646011 30.043 0.000 1.812725 2. _cons | -95.09566 4.887191 -19.458 0.000 -104.7861 -85.
. tobit insure income ,ll Tobit estimates Number of obs = 200 LR chi2(1) = 468. Prob > chi2 = 0. Log likelihood = -416.40595 Pseudo R2 = 0.
insure | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+-------------------------------------------------------------------- income | 2.0489 .0597651 34.283 0.000 1.931046 2. _cons | -104.2937 4.41718 -23.611 0.000 -113.0042 -95. ---------+-------------------------------------------------------------------- _se | 10.73973 .7259328 (Ancillary parameter)
Obs. summary: 93 left-censored observations at insure<= 107 uncensored observations
