Specification testing with many auxiliary statistics

We develop a new test for the correct specification of a model which compares a large number of auxiliary statistics computed with observed data to their model-generated counterparts. We establish the asymptotic properties of our new specification test under general regularity assumptions. To further validate our method, we perform extensive Monte-Carlo simulations to empirically assess its finite sample properties across a range of models and parameter configurations. Our analysis reveals that the size of our test remains controlled consistently, with a strong ability to detect deviations from the true model. Further, our experiments convincingly show that allowing for a larger set of auxiliary statistics significantly enhances our test's ability to detect deviations from the true model. Finally, we illustrate the performance of our proposed test by evaluating the suitability of Champernowne (1953)'s parametric model to characterize wealth and income data from the U.S. In particular, we consider a broad set of auxiliary statistics, including percentiles, quantile ratios, and inequality measures.