Regression discontinuity models are commonly used to nonparametrically identify and estimate a local average treatment effect (LATE).We show that the derivative of the treatment effect with respect to the running variable at the cutoff, referred to as the treatment effect derivative (TED), is nonparametrically identified, easily estimated, and has implications for testing external validity and extrapolating the estimated LATE away from the cutoff. Given a local policy invariance assumption, we further show this TED equals the change in the treatment effect that would result from a marginal change in the threshold, which we call the marginal threshold treatment effect (MTTE). We apply these results to Goodman (2008), who estimates the effect of a scholarship program on college choice. MTTE in this case identifies how this treatment effect would change if the test score threshold to qualify for a scholarship were changed, even though no such change in threshold is actually observed.

Shape invariance is a property of demand functions that is widely used for parametric and semiparametric modeling and is associated with a commonly employed class of equivalence scale models used for welfare calculations. This paper derives the set of all shape-invariant demand functions and associated preferences. All previously known shape-invariant demands were derived from utility functions that, up to monotonic transformation, are called IB/ESE (independent of base–equivalence scale exact) utility functions, because they yield IB/ESE equivalence scales. This paper shows that there exist exceptional shape-invariant demands that are not derived from a transform of IB/ESE utility and provides some simple tests for these exceptions. In particular, all the exceptions have rank 2, so any rank 3 or higher demand system is shape invariant if and only if it is derived from a transform of IB/ESE utility.

Relative prices are nonstationary and standard root- T inference is invalid for demand systems. But demand systems are nonlinear functions of relative prices, and standard methods for dealing with nonstationarity in linear models cannot be used. Demand system residuals are also frequently found to be highly persistent, further complicating estimation and inference. We propose a variant of the translog demand system, the NTLOG, and an associated estimator that can be applied in the presence of nonstationary prices with possibly nonstationary errors. The errors in the NTLOG can be interpreted as random utility parameters. The estimates have classical root- T limiting distributions. We also propose an explanation for the observed nonstationarity of aggregate demand errors, based on aggregation of consumers with heterogeneous preferences in a slowly changing population. Estimates using U.S. data are provided.

This paper presents a model of consumer demand that is consistent with the observed expenditure patterns of individual consumers in a long time series of expenditure surveys and is also able to provide a detailed welfare analysis of shifts in relative prices. A nonparametric analysis of consumer expenditure patterns suggests that Engel curves require quadratic terms in the logarithm of expenditure. While popular models of demand such as the Translog or the Almost Ideal Demand Systems do allow flexible price responses within a theoretically coherent structure, they have expenditure share Engel curves that are linear in the logarithm of total expenditure. We derive the complete class of integrable quadratic logarithmic expenditure share systems. A specification from this class is estimated on a large pooled data set of U.K. households. Models that fail to account for Engel curvature are found to generate important distortions in the patterns of welfare losses associated with a tax increase.