Kaplan–Meier Estimation and Cox Proportional Hazards Modelling for Spanish Construction Firms

This repository contains the full survival analysis conducted on a sample of firms operating in the Spanish construction sector, using firm-level microdata obtained from the SABI database. The objective is to characterise time-to-failure dynamics and quantify the effects of financial and operational covariates on the hazard of market exit.

• Estimation of non-parametric survival functions using the Kaplan–Meier estimator.
• Comparison of group-specific survival distributions via Log-Rank, Breslow, and Tarone–Ware tests.
• Specification and estimation of a Cox Proportional Hazards (PH) model, including diagnostics for the PH assumption.
• Integration of time-dependent covariates where proportionality is violated.
• Identification of statistically significant determinants of the hazard rate of insolvency.

• Source: SABI (Bureau van Dijk)
• Initial sample: 500 firms in NACE Rev.2 code 410 – Construction.
• Final analytical sample: 194 firms with complete information.
• Event definition:
  • Event = 1 → firm dissolved / liquidated / in insolvency proceedings
  • Censored = 0 → firm still active or registry closure without confirmed failure
• Time variable: Firm age at censoring or event (years).

Key variables used in modelling:

• ROA (Return on Assets) — profitability metric.
• N_EMP — number of employees.
• BPE — profit per employee.
• State — event indicator.
• Time — duration until event/censoring.

The non-parametric estimator is computed to obtain the empirical survival function

• S_hat(t) = Π over all event times t_i ≤ t of (1 − d_i / n_i),

where d_i is the number of events at time t_i and n_i is the number of firms at risk just before t_i.

Survival curves are stratified by autonomous community to assess regional heterogeneity.

The equality of survival functions across groups is tested using:

• Log-Rank (equal weighting across time)
• Breslow / Generalized Wilcoxon (early-time weighting)
• Tarone–Ware (intermediate weighting)

All three tests reject equality (p < 0.01), indicating statistically significant differences in survival distributions across regions.

A semiparametric hazard specification is estimated as: [ h(t | X) = h_0(t)\exp(\beta' X), ]
with covariates representing profitability, size, and productivity.

Diagnostics for the PH assumption include:

• Schoenfeld residual-based tests
• Time-interaction terms (X × t)

The number of employees violates proportional hazards; hence a time-dependent effect is included.

The final hazard function includes:

• ROA (significant, positive effect on hazard)
• N_EMP (significant, initial positive effect)
• N_EMP × Time (significant, attenuating effect over time)
• BPE (non-significant)

Key interpretations:

• A one-unit increase in ROA increases the instantaneous hazard by approx. 0.9%.
• Higher employment levels increase early-stage failure risk, but the effect decreases by approx. 0.3% per year of firm age.