Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Power calculations were conducted through simulated data generation under several candidate data-generating processes (DGPs), modeling different shapes of the transition equation between pre- and post-treatment formality. The simulations account for two-stage randomization (at cluster and restaurant level), stratification by baseline formality and local income classification, heterogeneous compliance rates across restaurant types (filers, registered non-filers, unregistered), and noise in formality outcomes.
Four distinct statistical tests were employed to detect key features of the transition dynamics:
Fully Parametric Test: Assumes that the true transition equation follows a specific S-shaped form, where formality evolves according to a parametric function with parameters governing the location and shape of steady states. The test focuses on whether the estimated parameters satisfy the mathematical conditions necessary for multiple equilibria, including an unstable steady state. Simulations show that the study has approximately 100% power to detect deviations from the null hypothesis when the true transition is S-shaped.
Concavity Test (Komarova and Hidalgo, 2023 method): Non-parametrically tests whether the empirical relationship between baseline- and endline formality is globally concave. Since the existence of an informality trap would imply an S-shape (non-concavity), rejection of global concavity supports the existence of multiple steady states. The study achieves over 90% power to reject global concavity under plausible data-generating processes.
Local Polynomial Test for Unstable Steady State: Estimates a smoothed non-parametric relationship between baseline and endline formality using local polynomial regression. An unstable steady state is detected if the fitted curve crosses the 45-degree line with a slope greater than one. Power simulations show over 80% power to detect such unstable steady states when they exist.
Cubic Polynomial Regression Test: Fits a flexible cubic polynomial regression to model the relationship between baseline and endline formality. The key test is whether the cubic term is statistically different from zero, indicating the presence of non-linear dynamics such as an S-shape. The study achieves over 90% power to reject the null hypothesis of no cubic structure when the true transition is sufficiently non-linear.
Power was estimated using 1,000 simulation replications for each design and hypothesis, with tests evaluated at a 5% significance level. These results confirm that the study is well-powered to detect the presence of non-linearities and multiple steady states in formality dynamics.