Truncated regression model

Truncated regression models arise in many applications of statistics, for example in econometrics,^[1][2][3][4] in cases where observations with values in the outcome variable below or above certain thresholds are systematically excluded from the sample. Therefore, whole observations are missing, so that neither the dependent nor the independent variable is known.

Truncated regression models are often confused with censored regression models where only the value of the dependent variable is clustered at a lower threshold, an upper threshold, or both, while the value for independent variables is available.

Example

One example of truncated samples come from historical military height records. Many armies imposed a minimum height requirement (MHR) on soldiers. This implies that men shorter than the MHR are not included in the sample. This implies that samples drawn from such records are perforce deficient i.e., incomplete, inasmuch as a substantial portion of the underlying population's height distribution is unavailable for analysis. Consequently, without proper statistical correction, any results obtained from such deficient samples, such as means, correlations, or regression coefficients are wrong (biased). In such a case truncated regression has the considerable advantage of immediately providing consistent and unbiased estimates of the coefficients of the independent variables, as well as their standard errors, thereby allowing for further statistical inference, such as the calculation of the t-values of the estimates.

See also

• Sampling bias
• Truncation (statistics)
• Truncated dependent variable

Footnotes

1. Amemiya, T. (1973): “Regression Analysis When the Dependent Variable is Truncated Normal,” Econometrica, 41, 997–1016.
2. Heckman, J. J. (1976): “The Common Structure of Statistical Models of truncate, Sample Selection, and Limited Dependent Variables and a Simple Estimator for Such Models,” Annals of Economic and Social Measurement, 15, 475–492.
3. Lewbel, A. and O. Linton, (2002), Nonparametric censored and truncated regression, Econometrica, 70, 765–779.
4. Park, B.U., L. Simar, and V. Zelenyuk (2008). "Local likelihood estimation of truncated regression and its partial derivatives: Theory and application," Journal of Econometrics 146(1), pages 185-198.

References

• A'Hearn, Brian (2004). "A Restricted Maximum Likelihood Estimator for Truncated Height Samples". Economics and Human Biology. 2 (1): 5–20. doi:10.1016/j.ehb.2003.12.003.
• Komlos, John (2004). "How to (and How Not to) Analyze Deficient Height Samples: an Introduction". Historical Methods. 37 (4): 160–173. doi:10.3200/HMTS.37.4.160-173.
• Wolynetz, M. S. (1979). "Maximum Likelihood estimation in a Linear model from Confined and Censored Normal Data". Journal of the Royal Statistical Society. Series C (Applied Statistics). 28 (2): 195–206. doi:10.2307/2346748. JSTOR 2346749.