Left-truncated discrete lifespans: The AFiD enterprise panel

Our model for the lifespan of an enterprise is the geometric distribution. We do not formulate a model for enterprise foundation, but assume that foundations and lifespans are independent. We aim to fit the model to information about foundation and closure of German enterprises in the AFiD panel. The lifespan for an enterprise that has been founded before the first wave of the panel is either left truncated, when the enterprise is contained in the panel, or missing, when it already closed down before the first wave. Marginalizing the likelihood to that part of the enterprise history after the first wave contributes to the aim of a closed-form estimate and standard error. Invariance under the foundation distribution is achived by conditioning on observability of the enterprises. The conditional marginal likelihood can be written as a function of a martingale. The later arises when calculating the compensator, with respect some filtration, of a process that counts the closures. The estimator itself can then also be written as a martingale transform and consistency as well as asymptotic normality are easily proven. The life expectancy of German enterprises, estimated from the demographic information about 1.4 million enterprises for the years 2018 and 2019, are ten years. The width of the confidence interval are two months. Closure after the last wave is taken into account as right censored.

💡 Research Summary

The paper investigates the lifespan of German enterprises using the AFiD panel, which records firm foundations and closures for the years 2018 and 2019. The authors model the lifetime of a firm as a discrete random variable X measured in years and assume a constant closure probability θ, implying that X follows a geometric distribution with probability mass function f_G(x)=θ(1−θ)^{x−1}. Because the panel starts in 2018, any firm founded before that year has an unobserved portion of its life prior to the observation window. Firms that closed before 2018 are completely absent from the data (left‑truncation), while those that survive past 2019 are right‑censored.

To avoid dependence on the distribution of foundation dates, the authors assume independence between foundation time and lifespan and condition on observability. They introduce a truncation age t (the number of years before 2018 a firm could have been founded) and consider only firms with X≥t+1, i.e., those still alive at the start of the panel. The geometric distribution’s memoryless property guarantees that the conditional distribution of the remaining lifetime remains geometric with the same θ.

The methodological core relies on counting processes. Define N(x)=1{X≤x} and its increment ΔN(x)=N(x)−N(x−1). To focus on the observable part, they construct tN(x)=N(x)−N(min{x,t}), which equals 1 only when the firm survives past the truncation point and then closes at or before x. The compensator of this process is A(x,θ)=θ·1{X≥x}, and the difference tN(x)−θ·x forms a martingale with respect to the filtration generated by the observable history. Using this martingale representation, the conditional likelihood of the observed data simplifies to a product of Bernoulli terms, leading to a closed‑form maximum‑likelihood estimator

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