Abstract
When a regression model is estimated using a censored subset of observations, coefficient estimates may be biased. Censored regression models have a long history in biometrics, engineering and other areas of applied statistics. The interest of economists in these models was stimulated by Tobin’s work on durable goods consumption in the late 1950s. It was Heckman’s publication of a simple two-step procedure for estimating censored regression models, however, that led to their widespread usage in applied econometric studies. Although this is not necessarily the best method for estimating all censored regression models, it has certain attractive properties. An understanding of this method is vital to the proper interpretation of the wealth of applied studies based on this approach. Also, valuable insights into the basic nature of sample selection problems can be gained from the formulation of the censored regression model popularized by Heckman.
We begin by exploring the estimation problems resulting from censoring and from certain properties of Heckman’s two-step estimation method. Procedures are developed for assessing the nature and extent of problems resulting from censoring; these procedures are then applied in an empirical analysis of the wage rates and hours of work of individuals in 10 different demographic groups using data from the Panel Study of Income Dynamics. One of our findings is that estimation using censored data can lead to bias and other related problems even when the degree of censoring is slight.
The authors express their appreciation to John Ham, Baldev Raj, Thanasis Stengos and two anonymous referees for their helpful comments on earlier versions of this paper.
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Notes
See, for instance, Buckley and James (1979), Dempster, Laird and Rubin (1977), Hartley (1958), Johnson and Kotz (1970, 1972), and Kalbfleisch and Prentice (1980) for an introduction to some of this history, including further references.
The empirical applications we are referring to are for models which can be classified as generalized Tobit models (Type 2 Tobit models in Amemiya, 1984, 1985), where the explanatory variables entering the regression model (to be estimated with a censored subset of observations) may differ from the explanatory variables entering the selection rule. Many problems in demography, education, marketing, finance, labor economics and other fields can be expressed using this sort of model. See Amemiya (1985, p. 365, Table 10.1) for a listing of applications covering a wide range of topics. Under various circumstances, estimation methods such as the EM algorithm, Powell’s least absolute deviations estimator (LAD), and various maximum likelihood procedures may yield better parameter estimates in terms of efficiency or mean squared error. See Amemiya (1973), Arabmazar and Schmidt (1981, 1982), Bera, Jarque and Lee (1984), Dempster, Laird and Rubin (1977), Dudley and Montmarquette (1976), Goldberger (1981, 1983), Huang (1964), Hurd (1979), Lee (1982), Nelson (1984), Olsen (1978, 1980), Paarsch (1984), Powell (1981, 1983a, 1983b, 1984), Robinson (1982), Wales and Woodland (1980) and Wu (1965). See Mroz (1987), for instance, concerning some of the substantive advantages of Heckman’s two-step estimation method.
This result is also given in Nelson (1984, p. 185, eq. 12). If predetermined variables are included in Z, then the asymptotic OLS bias vector may be defined as (σ1 ρ13 plimγ̂) provided that plimγ̂ exists.
For instance, in his discussion of Tobit-type models Kmenta (1986, p. 561) writes: “ The restriction on the observable range of the dependent variable matters if the probability of falling below the cut-off point is not negligible. In terms of our examples, if only a very small proportion of the households in the population did not purchase a durable good, or if only a very small proportion of all women were not in the labor force, the limited nature of the dependent variable could be ignored. Thus there is no problem in dealing with household expenditure on food or clothing, or with recorded wages of adult males.”.
This is because theoretically u 3i = v i-u 1i , and hence u 3i and u 1i will be correlated even if there is no correlation between the structural disturbance terms u 1i and v i.
Actually λ is a nonlinear function of Z, so \( R_{\lambda \cdot Z}^2 \) cannot equal 1 exactly. In actual applications of the Heckit procedure, we have often found that the R 2 for the OLS regression of λ on the other explanatory variables in the equation of interest is extremely close to 1.
See Kmenta (1986, pp. 437-438) for the usual OLS variance formula rewritten in a form so that the impacts of multicollinearity on the magnitudes of the variances of the coefficient estimates are explicit.
Multicollinearity problems can be considered in a more comprehensive way in the context of a principal components formulation of the model. See, for instance, Judge et al. (1985, Ch. 22) and Judge et al. (1988, Ch. 21). The approach adopted in this paper, however, may allow applied readers more readily to understand our main points.
See, for example, Heckman (1979, 1980) and Amemiya (1984, 1985) for the form of the variance-covariance matrix. Heckit estimation packages such as Greene (undated, p. 49) typically include some arbitrary procedure for avoiding negative estimates of the coefficient standard errors. Alternative estimates of the coefficient standard errors may be obtained using the procedure described in White (1980), as suggested by Lee (1982) and Amemiya (1984, p. 13). The latter estimates are used in this paper.
For a full description of the data bases and of the probit estimation results used in calculating the values of \( {\hat \lambda _i} \), see Nakamura and Nakamura (1985b, Section 2.7 and Chapter 3).
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Nakamura, A., Nakamura, M. (1989). Selection Bias: More than a Female Phenomenon. In: Raj, B. (eds) Advances in Econometrics and Modelling. Advanced Studies in Theoretical and Applied Econometrics, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7819-6_10
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