Truncated regression models arise in many applications of statistics, for example in econometrics, 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.
Estimation of truncated regression models are usually done via parametric, semi-parametric and non-parametric maximum likelihood methods.
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.