When are instruments generated from geographic characteristics in bilateral relationships invalid?

In their highly influential paper, “Does Trade Cause Growth?,” Frankel and Romer estimate a trade equation to predict bilateral trade shares, which are in turn aggregated to construct an instrument for trade openness in income regressions. The Frankel-Romer approach has gained widespread popularity as a method to generate instruments for trade and many other outcome variables from estimated bilateral relationships. In this paper, we show analytically and empirically that the Frankel–Romer instrument gives inconsistent estimates for the effect of trade on income when fitted values for zero and missing bilateral shares are omitted from the instrument. This leads to a violation of the exclusion restriction because the instrument captures the endogenous variation in the observed probability that countries engage in bilateral trade, as reflected in the number of observed trading partners. This violation disappears when predicted bilateral shares for all observed and unobserved bilateral trade relationships are included in the instrument for trade openness.