A Synthetic Instrumental Variable Method: Using the Dual Tendency Condition for Coplanar Instruments

Traditional instrumental variable (IV) methods often struggle with weak or invalid instruments and rely heavily on external data. We introduce a Synthetic Instrumental Variable (SIV) approach that constructs valid instruments using only existing data. Our method leverages a data-driven dual tendency (DT) condition to identify valid instruments without requiring external variables. SIV is robust to heteroscedasticity and can determine the true sign of the correlation between endogenous regressors and errors–an assumption typically imposed in empirical work. Through simulations and real-world applications, we show that SIV improves causal inference by mitigating common IV limitations and reducing dependence on scarce instruments. This approach has broad implications for economics, epidemiology, and policy evaluation.

💡 Research Summary

The paper tackles three persistent shortcomings of conventional instrumental variable (IV) techniques: (i) the difficulty of finding external variables that satisfy relevance and exogeneity, (ii) the bias and weak‑instrument problems that arise when the first‑stage relationship is modest, and (iii) the complications introduced by using multiple instruments, especially when some are weak. The authors propose a Synthetic Instrumental Variable (SIV) method that constructs valid instruments solely from the observed data, eliminating the need for any external shocks or auxiliary datasets.

The key insight is geometric. In the simple structural model y = βx + u, the outcome y, the endogenous regressor x, and the structural error u all lie in the two‑dimensional subspace W = span{x, y}. Consequently any valid instrument can be expressed as a linear combination of vectors inside this plane: z₀ = x + k δ₀ r, where r is a vector in W orthogonal to x, k ∈ {−1,+1} captures the sign of cov(x,u), and δ₀ is an unknown scalar. The identification problem therefore reduces to finding the correct δ₀ (and the sign k).

To locate δ₀ the authors introduce the Dual Tendency (DT) condition, a testable pair of moment restrictions. The first requirement is the usual exogeneity condition E