A Within Estimator for Three-Level Data: An Application to the WTO Effect on Trade Flows

This paper proposes a within estimator for three-level data, such as time-variant bilateral trade flows. The estimator helps to address the computational difficulties in estimating, for instance, the gravity model of bilateral trade that needs to control for unobserved country-pair and country-time heterogeneity using fixed effects. Unlike the traditional within transformation that removes cross-sectional heterogeneity, the proposed transformation method removes all three types of heterogeneity, each of which varies in two dimensions of three-level data. We demonstrate the properties of the estimator using empirical examples and Monte Carlo simulations. Simulation results show that the proposed estimators with the adjusted standard errors consistently estimate coefficient parameters and perform correct inference when no data are missing or the missing data are random. When missing data are not random, the proposed within transformation can reduce bias to various degrees, depending on the order of demeaning and sources of bias. As an empirical application, we investigate the WTO effect puzzle by applying the proposed estimator to show that WTO membership has a definite positive effect on bilateral trade flows.