The QLIM (qualitative and limited dependent variable model) procedure analyzes univariate and multivariate limited dependent variable models in which dependent variables take discrete values or in which dependent variables are observed only in a limited range of values. These models include logit, probit, tobit, selection, and multivariate models. The multivariate model can contain discrete choice and limited endogenous variables in addition to continuous endogenous variables.
The QLIM procedure supports the following models:
linear regression model with heteroscedasticity
Box-Cox regression with heteroscedasticity
probit with heteroscedasticity
logit with heteroscedasticity
tobit (censored and truncated) with heteroscedasticity
bivariate probit
bivariate tobit
sample selection and switching regression models
multivariate limited dependent variables
stochastic frontier production and cost models
In the linear regression models with heteroscedasticity, the assumption that error variance is constant across observations is relaxed. The QLIM procedure allows for a number of different linear and nonlinear variance specifications. Another way to make the linear model more appropriate to fit the data and reduce skewness is to apply Box-Cox transformation. If the nature of the data is such that the dependent variable is discrete and it takes only two possible values, ordinary least squares (OLS) estimates are inconsistent. The QLIM procedure offers probit and logit models to overcome these estimation problems. Assumptions about the error variance can also be relaxed in order to estimate probit or logit with heteroscedasticity.
The QLIM procedure also offers a class of models in which the dependent variable is censored or truncated from below or above or both. When a continuous dependent variable is observed only within a certain range and values outside this range are not available, the QLIM procedure offers a class of models that adjust for truncation. In some cases, the dependent variable is continuous only in a certain range and all values outside this range are reported as being on its boundary. For example, if it is not possible to observe negative values, the value of the dependent variable is reported as equal to 0. Because the data are censored, OLS results are inconsistent, and it cannot be guaranteed that the predicted values from the model fall in the appropriate region.
Most of the models in the QLIM procedure can be extended to accommodate bivariate and multivariate scenarios. The assumption that one variable is observed only if another variable takes on certain values lead to the introduction of sample selection models. If the dependent variables are mutually exclusive and observed only for certain ranges of the selection variable, the sample selection can be extended to include cases of switching regression. Stochastic frontier production and cost models allow for random shocks of the production or cost. They include a systematic positive component in the error term that adjusts for technological or cost inefficiency.
The QLIM procedure can use the maximum likelihood method for both univariate and multivariate models. PROC QLIM can use the Bayesian method for univariate models and for multivariate models in which the likelihood function is available in closed form. Initial starting values for the nonlinear optimizations are typically calculated by OLS.