## Endogenous sampling and matching method in duration modelsEndogenous sampling with matching (also called "Mixed Sampling") occurs when the statistician samples from the non-right-censored subset at a predetermined proportion and matches on one or more exogenous variables when sampling from the right-censored subset. This is widely applied in the duration analysis of firm failures, loan defaults, insurer insolvencies etc. due to the low frequency of observing non-right-censored sample (bankrupt, default and insolvent observations in respective examples). However, the common practice of using estimation procedures intended for random sampling or for the qualitative response model will yield either inconsistent or inefficient estimator. This paper proposes a consistent and efficient estimator and investigates its asymptotic properties. In addition, this paper evaluates the magnitude of asymptotic bias when the model is estimated as if it were a random sample or an endogenous sample without matching. This paper also compares the relative efficiency of other commonly used estimators and provides a general guideline for optimally choosing sample designs. The Monte Carlo study with a simple example shows that random sampling yields an estimator of poor finite sample properties when the population is extremely unbalanced in terms of default and non-default cases while endogenous sampling and mixed sampling are robust in this situation.--Author's abstract. |

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Amemiya assume assumption Asymptotic Bias asymptotic distribution asymptotic efficiency asymptotic properties Asymptotic Variance Balanced AV0BSMLE AV0WMLE Consistency of MSMLE default and non-default default samples density derive the asymptotic dp dp dP(D duration analysis Duration Models Econometric empirical distribution empirical variance endogenous sampling Equation ESMLE and MSMLE estimates with varying Euclidean space exogenous variables extremely unbalanced Finite Sample Properties Finite sample variance frequency of defaults insurer insolvencies Log Likelihood Function Maximum Likelihood Estimator measurable function mixed sampling scheme Monte Carlo study Monte Carlo variance non-default samples non-right-censored Order Condition population is extremely population is relatively probability of default Proportional Hazards model qualitative response model random sampling relative efficiency relatively balanced replication RSMLE sample designs sample sizes sample without matching Sampling and Matching Sampling Maximum Likelihood simple example small sample statistician subset Takeshi Amemiya three estimators true value Unbalanced population variance and asymptotic Variance Balanced population Weighted Maximum Likelihood