Output Options Examples References
TOBIT obtains estimates of the linear Tobit model, where the dependent variable is either zero or positive. The method used is maximum likelihood under the assumption of homoskedastic normal disturbances. For non-normal censored regression, see LAD.
TOBIT (LOWER=<lower limit>,MILLS=<name for output inverse Mills ratio>, UPPER=<upper limit>,WEIGHT=<weighting series>,<nonlinear options>) <dependent variable> <list of independent variables> ;
Usage
The basic TOBIT statement is like the PROBIT or OLSQ statements: first list the dependent variable and then the independent variables. If you wish to have an intercept term in the regression (usually recommended), include the special variable C or CONSTANT in your list of independent variables. You may have as many independent variables as you like subject to the overall limits on the number of arguments per statement and the amount of working space, but of course the number is limited by the number of data observations you have available.
The observations over which the regression is computed are determined by the current sample. If any of the observations have missing values within the current sample, TOBIT will print a warning message and will drop those observations.
The list of independent variables on the TOBIT command may include variables with explicit lags and leads as well as PDL (Polynomial Distributed Lag) variables. These distributed lag variables are a way to reduce the number of free coefficients when you are entering a large number of lagged variables in a regression by imposing smoothness on the coefficients. See PDL for a description of how to specify such variables.
The dependent variable need not be a strictly zero/positive variable. Negative values are treated as zero. The standard Tobit model involves truncation of the dependent variable below zero. Models with upper and/or lower truncation can be estimated by using the UPPER and/or LOWER option(s). See the Examples for more details.
The output of TOBIT begins with an equation title and the name of the dependent variable. Then the starting values and diagnostic output from the iterations are printed, followed by the convergence status.
The results printed are the mean of the dependent variable, the number of lower censored, uncensored, and upper censored observations, and a table of right hand side variable names, estimated coefficients, standard errors and associated t-statistics. The estimated standard deviation of the residual, SIGMA, is listed last in this table.
TOBIT also stores some of these results in data storage for your later use. The table below lists the results available after a TOBIT command.
|
variable |
type |
length |
description |
|
@LHV |
list |
1 |
Name of dependent variable |
|
@YMEAN |
scalar |
1 |
Fraction of positive observations |
|
@NOB |
scalar |
1 |
Number of observations |
|
@NPOS |
scalar |
1 |
Number of positive observations |
|
@LOGL |
scalar |
1 |
Log of likelihood function |
|
@IFCONV |
scalar |
1 |
1 if convergence achieved, 0 otherwise |
|
@NCOEF |
scalar |
1 |
Number of parameters (#params) including SIGMA |
|
@NCID |
scalar |
1 |
Number of identified coefficients |
|
@RNMS |
list |
#params |
list of names of independent variables |
|
@COEF |
vector |
#params |
Coefficient estimates |
|
@SES |
vector |
#params |
Standard errors |
|
@T |
vector |
#params |
T-statistics |
|
%T |
vector |
#params |
p-values for T-statistics |
|
@GRAD |
vector |
#params |
Gradient of log likelihood at convergence |
|
@VCOV |
matrix |
#params*#params |
Variance-covariance of estimated coefficients |
|
@DPDX |
matrix |
#vars*2 |
Means of probability derivatives |
|
@RES |
series |
#obs |
Residuals for non-truncated observations |
|
@MILLS |
series |
#obs |
Inverse Mills' ratios |
If the regression includes a PDL variable, the following will also be stored:
|
@SLAG |
scalar |
1 |
Sum of the lag coefficients |
|
@MLAG |
scalar |
1 |
Mean lag coefficient (number of time periods) |
|
@LAGF |
vector |
#lags |
Estimated lag coefficients, after "unscrambling" |
Method
TOBIT uses analytic first and second derivatives to obtain maximum likelihood estimates via the Newton-Raphson algorithm. This algorithm usually converges fairly quickly. Starting values for the parameters are obtained from a regression on the observations with positive values of the dependent variable. See Greene (1981), p. 508, formula (13) and footnote 5 for the details. Alternative starting values may be supplied in @START (see NONLINEAR). A globally concave parameterization of the likelihood function is used for iterations. Multicollinearity of the independent variables is handled with generalized inverses, as in all TSP regression procedures.
The numerical implementation involves evaluating the normal density and cumulative normal distribution functions. The cumulative normal distribution function is computed from an asymptotic expansion, since it has no closed form. See the references under CDF for the actual method used to evaluate CNORM(). The ratio of the density to the distribution function is also known as the inverse Mills ratio. This is used in the derivatives and with the MILLS= option.
LOWER= the value below which the dependent variable is not observed. The default is zero.
MILLS= the name of a series used to store the inverse Mills ratio series evaluated at the estimated parameters. The default is @MILLS.
WEIGHT= the name of a weighting series. The weights are applied directly to the likelihood function, and no normalization is performed.
UPPER= the value above which the dependent variable is not observed. The default is no limit.
Nonlinear options - see NONLINEAR.
Standard Tobit model with truncation below zero:
TOBIT CAR C INCOME RURAL MSTAT;
Truncation below two:
TOBIT (LOWER=2) CAR C INCOME RURAL MSTAT;
Truncation above ten:
TOBIT (UPPER=10) CAR C INCOME RURAL MSTAT;
Amemiya, Takeshi, "Tobit Models: A Survey," Journal of Econometrics 24, December 1981, pp. 3-61.
Greene, William H., "On the Asymptotic Bias of the Ordinary Least Squares Estimator of the Tobit Model," Econometrica 49, March 1981, pp. 505-513.
Maddala, G. S., Limited-dependent and Qualitative Variables in Econometrics, Cambridge University Press, New York, 1983, pp. 151-155.
Tobin, James, “Estimation of Relationships for Limited Dependent Variables,” Econometrica 31(1958), pp. 24 36.