Estimation of the Two-Limit Probit Regression Model

Abstract

Some economic variables are restricted by an upper and lower limit but are continuous between the two limits. Measurements of such variables are sometimes available in their natural form and sometimes only in the form of three categories where information concerning the middle category is suppressed (unemployed, employed part time, employed full time, for example). Where such a variable is a continuous function of other variables between the two limits, the function can be estimated from data of either sort provided the function and the distribution of errors can be specified. WHEN THE LIMITED dependent variable technique developed by Tobin [3] is extended to provide for cases in which the dependent variable in a regression is subject to both an upper limit and a lower limit, a surprising property of the statistical model emerges.1 Estimates of the regression function can be obtained whether or not the exact values of the dependent variable are known for the nonlimit cases. Provided the functional form can be specified correctly, classification of the dependent variable into upper limit, lower limit, and non-limit observations provides enough information, along with observed values of the independent