We study the identification through instruments of a nonseparable function that relates a continuous outcome to a continuous endogenous variable. Using group and dynamical systems theories, we show that full identification can be achieved under strong exogeneity of the instrument and a dual monotonicity condition, even if the instrument is discrete. When identified, the model is also testable. Our results therefore highlight the identifying power of strong exogeneity when combined with monotonicity restrictions.

We study treatment-effect estimation using panel data. The treatment may be nonbinary, nonabsorbing, and the outcome may be affected by treatment lags. We make a parallel-trends assumption and propose event-study estimators of the effect of being exposed to a weakly higher treatment dose for ℓ periods. We also propose normalized estimators that estimate a weighted average of the effects of the current treatment and its lags. We also analyze commonly used two-way, fixed-effects regressions. Unlike our estimators, they can be biased in the presence of heterogeneous treatment effects. A local-projection version of those regressions is biased even with homogeneous effects.

Linear regressions with period and group fixed effects are widely used to estimate treatment effects. We show that they estimate weighted sums of the average treatment effects (ATE ) in each group and period, with weights that may be negative. Due to the negative weights, the linear regression coefficient may for instance be negative while all the ATEs are positive. We propose another estimator that solves this issue. In the two applications we revisit, it is significantly different from the linear regression estimator.

In markets where sellers are able to price discriminate, individuals pay different prices that may be unobserved by the econometrician. This article considers the structural estimation of a demand and supply model of differentiated products with such price discrimination and limited information on prices taking the form of, e.g., observing list prices from catalogues or average prices. Within this framework, identification is achieved not only with usual moment conditions on the demand side, but also through supply-side restrictions. The model can be estimated by GMM using a nested fixed point algorithm that extends the usual contraction mapping algorithm to our setting. We apply our methodology to estimate the demand and supply in the French new automobile market. Our results suggest that discounting arising from price discrimination is important. The average discount is estimated to be 9.6%, with large variation depending on buyers’ characteristics and cars’ specifications. Our results are consistent with other evidence on transaction prices in France.