Left-truncated discrete lifespans: The AFiD enterprise panel

1. Problem Statement

The authors aim to estimate the annual probability ($\theta$) that a German enterprise will close down (due to insolvency or other reasons) within the next year. They utilize data from the AFiD panel (Official Enterprise Data for Germany) provided by DESTATIS for the years 2018 and 2019.

The study faces two primary statistical challenges:

1. Left-Truncation: The panel data only captures enterprises that were still active at the start of the observation period (2018). Enterprises founded before 2018 but closed before 2018 are unobserved (missing). Enterprises founded before 2018 and still active in 2018 have their early survival history "truncated."
2. Variable Observation Windows & Random Foundation: Unlike a simple random sample where all subjects start at $t=0$, enterprises in the panel have different foundation years ($T$). Some are observed for only one year (founded in 2017), while others are observed for two years (founded earlier). The foundation year $T$ is a random variable, not a fixed covariate.
3. Right-Censoring: Enterprises that survive beyond the end of the observation period (2019) are right-censored.

The goal is to derive a consistent, asymptotically normal estimator for $\theta$ and its standard error without assuming a specific parametric distribution for the foundation year $T$ (semi-parametric approach).

2. Methodology

2.1 Statistical Model

• Lifespan Distribution: The lifespan $X$ (time from foundation to closure) is modeled using a Geometric Distribution with parameter $\theta$ (probability of closure).
  $$P(X=x) = \theta(1-\theta)^{x-1}, \quad x \in \mathbb{N}$$
• Independence Assumption: The model assumes that the foundation year $T$ and the lifespan $X$ are stochastically independent. This allows the analysis to remain semi-parametric regarding the distribution of $T$.
• Discrete Time: Time is treated as discrete (annual epochs) rather than continuous, aligning with the annual reporting nature of the official statistics.

2.2 Handling Left-Truncation and Filtering

The authors employ a rigorous martingale-based approach to handle the truncation and censoring:

• Filtration: They define a filtration (information set) $\mathcal{F}_x$ representing the history of an enterprise up to age $x$.
• Marginalization: To address left-truncation, they "forget" the survival history prior to the observation start. They define a truncated counting process $^tN$ (or $^TN$ for random truncation) which counts closures only after the truncation time.
• Compensator: They derive the compensator (the predictable part of the process) for the counting process of closures. For a geometric distribution, the compensator is linear in the "time at risk."
• Martingale Transform: The core innovation is showing that the difference between the observed counting process and its compensator forms a martingale with respect to the observed filtration.
  $$M(x) = N(x) - A(x, \theta)$$
  where $A(x, \theta)$ is the compensator.

2.3 Likelihood Construction

Instead of a standard likelihood, the authors construct a conditional marginal likelihood:

• They condition on the observability of the enterprise (i.e., $X \ge T + 1$).
• The likelihood contribution for an observed enterprise is derived as a product of Bernoulli probabilities based on the increments of the martingale.
• For enterprises closed before the observation window, the likelihood contribution is 1 (they do not affect the likelihood ratio), effectively removing them from the numerator while adjusting the denominator (time at risk) for those that remain.

2.4 Estimation and Asymptotics

• Estimator: The Maximum Likelihood Estimator (MLE) $\hat{\theta}$ is derived by maximizing the conditional likelihood. It takes the form of a ratio:
  $$\hat{\theta} = \frac{\text{Total number of observed closures}}{\text{Total time at risk (sum of observed years)}}$$
• Asymptotic Normality: Using the Rebolledo martingale central limit theorem (a weaker condition than Lindeberg-Lévy, suitable for dependent variances in panel data), the authors prove that $\sqrt{n}(\hat{\theta}_n - \theta_0)$ converges in distribution to a normal distribution.
• Standard Error: A closed-form expression for the asymptotic variance is derived, allowing for the calculation of confidence intervals without bootstrapping.

3. Key Contributions

1. Martingale Framework for Discrete Truncation: The paper successfully adapts continuous-time survival analysis tools (Doob-Meyer decomposition, compensators) to a discrete-time setting with random left-truncation. This avoids the need for complex product integrals required in continuous models.
2. Semi-Parametric Robustness: By conditioning on observability, the estimator for $\theta$ is invariant to the distribution of the foundation year $T$. This is crucial for business demography where the distribution of founding years is often unknown or non-stationary.
3. Closed-Form Standard Error: The derivation provides a direct formula for the standard error and confidence intervals, which is computationally efficient compared to simulation-based methods.
4. Handling of Right-Censoring: The methodology seamlessly integrates right-censoring (enterprises surviving past 2019) into the discrete counting process framework.

4. Results

• Data Application: The model was applied to approximately 1.4 million German enterprises founded between 2013 and 2017, observed in 2018 and 2019.
• Point Estimate: The estimated annual closure probability is $\hat{\theta} \approx 0.1009$.
• Life Expectancy: Based on the geometric distribution, the estimated mean lifespan (life expectancy) of a German enterprise is 10 years.
• Precision: The 95% confidence interval for the life expectancy is [9.88, 9.95] years, with a standard error of 0.036 years (approx. 2 months).
• Comparisons:
  • The results are consistent with previous German studies (e.g., Brüderl et al., 1992) regarding quantiles (e.g., 2-year and 5-year survival rates).
  • Compared to other countries, German enterprises show lower closure intensity than those in Portugal or Pakistan but higher intensity than older Japanese cohorts (though recent Japanese data shows increased closure rates).

5. Significance

• Methodological Rigor: The paper bridges the gap between business demography and advanced stochastic process theory. It demonstrates how martingale limit theorems can solve inference problems in complex panel data with truncation and censoring.
• Policy Relevance: Providing a precise, statistically sound estimate of enterprise life expectancy (10 years) with a narrow confidence interval offers reliable data for policymakers and economists analyzing business stability and economic resilience.
• Generalizability: The framework is applicable to any discrete-time event history data with left-truncation and random entry times, extending beyond business demography to fields like health (e.g., patient survival studies) or social sciences.
• Addressing "Immortal Time Bias": The paper explicitly addresses the bias introduced by ignoring the pre-observation survival history, ensuring that the estimated closure probability is not artificially deflated by the selection of only "surviving" firms.

In conclusion, Scholz and Weißbach provide a robust, semi-parametric solution for estimating enterprise survival probabilities in the presence of left-truncation and right-censoring, leveraging martingale theory to achieve asymptotic normality and precise standard errors for large-scale panel data.