Journal of Econometrics 32 (1986) 5-34. North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP A SEMI-PARAMETRIC CENSORED REGRESSION ESTIMATOR Gregory M. DUNCAN * Washington State University, Pullman, WA 99164-4860, USA This paper introduces a semi-parametric method for estimating regression coefficients when the underlying parent population of errors in censored. The method is an example of the method of sieves: and it provides simultaneous estimates of the regression coefficients and the density of the underlying parent population. In the very simplest terms, the underlying unknown density is approximated by a spline with mesh size approaching zero with the sample size. The values of the density at the knots are then added to the list of the usual unknown parameters in a censored regression model, e.g., the regression coefficients and scale parameter. A quasi-likelihood function using the approximate spline density is then maximized over all the parameters mentioned above. The method is shown to result in strongly consistent parameter estimates. 1. Introduction This paper introduces a semi-parametric method for estimating regression coefficients when the underlying parent population of errors is censored. The method is an example of the method of sieves; and it provides simultaneous estimates of the regression coefficients and the density of the underlying parent population. In the very simplest terms, the underlying unknown density is approximated by a spline with mesh size approaching zero with the sample size. The values of the density at the knots are then added to the list of the usual unknown parameters in a censored regression model, e.g. the regression coefficients and scale parameter. A quasi-likelihood function using the approximate spline density is then maximized over all the parameters mentioned above. The method is shown to result in strongly consistent parameter estimates. 2. Background In a left (right) censored regression model, defined more formally below, the values of the dependent variable falling below (above) a given value are *I am indebted to Jim Heckman, Hal White, Alan Marcus, Chris Sims, Steve Cosslett, David Spencer, Rob Engle, and Dale Poirier for helpful comments. Ib Hansen, Cathleen Leue-Ronev. and Mark Thorna-are thanked for research and programming assistance. This research was funded under NSF grant SES-8109274. Participants at seminars at Minnesota, Wisconsin, Chicago, Northwestern, Bell Labs, Princeton, Columbia, UC San Diego, UC Berkeley, UC Santa Barbara and UC Riverside, are also thanked for their comments. 0304~4076/86/$3.500 1986, Elsevier Science Publishers B.V. (North-Holland)