ABSTRACT We investigate how school closures affect educational outcomes through parental responses. Using quasi‐experimental variation in school‐level closure days and linked administrative‐survey data, we find that 10 additional closure days raise private tutoring expenditure by 4.1% and within‐school test‐score dispersion by 25%; over the average 53‐day closure, low‐income students lose 0.09 SD. To quantify substitution between schooling and parental investment and evaluate compensatory policy, we estimate a tournament model in which schooling and tutoring jointly produce test scores. Tutoring substitutes for in‐person days, burdening low‐income households disproportionately. Counterfactuals show a means‐tested voucher fully offsets the resulting rise in inequality.
ABSTRACT I propose a new imputation estimator for missing covariate values in nonlinear models. The estimator provides efficiency gains relative to just using the complete cases, and is consistent for any model that falls under M‐estimation. This is unlike the commonly used dummy variable method and regression imputation, which I show to be generally inconsistent in nonlinear models. The proposed estimator is straightforward to implement, and relies only on the commonly used assumptions on missingness. To test these assumptions, I provide a novel and simple variable addition test. I show how the framework applies to nonlinear models for fractional and nonnegative responses.
ABSTRACT Wage markup shocks, an important driver of business cycles, are hard to pin down. We quantify the importance of exogenous variation in wage bargaining power, one source of those shocks, from German minimum wage introduction episodes and strikes. This disciplines the impulse responses of unemployment and output, and sharpens inference for other variables, which behave consistently with predictions from search‐and‐matching models. We find that wage bargaining shocks meaningfully contribute to aggregate fluctuations in unemployment and inflation, exhibit substantial pass‐through to prices, and imply plausible dynamics for the vacancy rate, firms' profits, and the labor share.
This paper proposes a simple smooth specification test for the propensity score in the spirit of Neyman (1937). We derive a particular restriction on the propensity score and construct the specification test based on it. Using the density-weighting method, we transform the original specification testing problem into a test of the joint significance of the generalized Fourier coefficients. We propose a novel projection method to eliminate the parameter estimation effect arising from estimating unknown nuisance parameters. The proposed test is shown to follow a standard distribution under the null hypothesis and to possess nontrivial asymptotic power against local alternatives that converge to the null at the parametric rate , where is the sample size. To enhance the practical applicability of the procedure, we further introduce a data-driven selection method for determining the testing order, enabling automatic selection based on the observed data. Compared to existing methods, our test does not suffer from the "curse of dimensionality" faced by Shaikh et al. (2009); at the same time, our test is both asymptotically pivotal and more computationally efficient than that of Sant'Anna and Song (2019). Extensive simulations demonstrate that the proposed test yields satisfactory empirical size and power. Compared with previous tests, our test exhibits comparable overall performance and delivers clear improvements in some cases, especially for high-frequency alternatives. Two empirical applications further illustrate the practical applicability of our proposal, examining propensity score specifications in the Soviet 156 Program and the US Job Corps Program.
We estimate average equivalised consumption measures across 370 local authority districts in Great Britain, using small-area estimation methods that combine information from a household budget survey, a much larger survey of local demographics and employment, and area-level information on bank account outflows and energy consumption. We show that including bank account data substantially improves our estimates, showing that these and other financial footprints data can play an important role in measuring local consumption and hence living standards. We also compare consumption measures that correspond to welfare under different assumptions about mobility and the capitalisation of local amenities into house prices, as well as traditional local income measures, and show that the rankings of local authorities are sensitive to the choice of measure.
