The Harmonic Synthetic Control Method

Abstract

Synthetic control methods can produce misleading counterfactual predictions when outcome series contain unit-specific stochastic trends, a common feature of nonstationary macroeconomic data. Existing remedies, such as pre-filtering or differencing, reduce spurious matching but may discard shared nonstationary variation that helps estimate donor weights. We propose Harmonic Synthetic Control (HSC), which replaces this binary choice with a soft allocation mechanism. HSC jointly estimates donor weights and a treated-unit-specific smooth residual component, then extrapolates this component into post-treatment periods using a time-series forecaster. A tuning parameter, selected by rolling-origin cross-validation, governs the division between donor matching and forecasting. As it varies, HSC continuously interpolates between synthetic control applied to differenced outcomes and synthetic control applied to raw outcomes with an intercept or trend. We provide a spectral interpretation showing how HSC downweights low-frequency residual components in donor matching and assigns them to the forecasting branch. A prediction-error decomposition separates weight-estimation distortion from residual-forecasting error. Monte Carlo exercises show that HSC adapts across regimes, performing well when stochastic trends are predominantly common or idiosyncratic, while estimators fixed to one regime can fail in the other.