Source-linked AI summary
Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments
Victor Chernozhukov, Christian Hansen, Martin Spindler
TL;DR
The paper addresses causal and structural estimation with many instruments and controls, where selecting variables and conducting valid inference after selection are difficult. It combines sparse high-dimensional selection with orthogonal moment functions and double selection, obtaining theoretically valid inference and favorable simulation and empirical performance.
Problem
Estimating causal or structural parameters with many instruments and controls requires selecting variables while preserving valid post-selection inference.
Method
The approach selects instruments and controls using Lasso or Post-Lasso and combines this selection with orthogonal moment functions and a double-selection IV estimator.
Results
The estimator is asymptotically normal with a chi-squared score statistic and uniformly valid confidence intervals, while performing well relative to the Oracle estimator in simulation.
Takeaways & Limitations
The approach provides a practical way to estimate IV models with many instruments and controls using principled selection procedures coupled with selection-robust moments.
Takeaways & Limitations
The analysis relies on sparsity and regularity conditions, and the main discussion assumes exact sparsity while approximate sparsity is treated in supplementary material.
Abstract
from arXiv · showhide
In this note, we offer an approach to estimating causal/structural parameters in the presence of many instruments and controls based on methods for estimating sparse high-dimensional models. We use these high-dimensional methods to select both which instruments and which control variables to use. The approach we take extends BCCH2012, which covers selection of instruments for IV models with a small number of controls, and extends BCH2014, which covers selection of controls in models where the variable of interest is exogenous conditional on observables, to accommodate both a large number of controls and a large number of instruments. We illustrate the approach with a simulation and an empirical example. Technical supporting material is available in a supplementary online appendix.
1. Model and Estimation Approach
The paper develops an IV approach for estimating a causal coefficient with many instruments and controls by combining sparse-model selection with moment conditions robust to selection mistakes.
- Model: The linear IV model allows many exogenous controls and instruments, including cases where the number of controls exceeds the sample size.Instruments may be correlated with controls and become valid after conditioning on them.
- Model: Approximate or exact sparsity reduces estimation of α0 to selecting the instruments and controls with nonzero or substantively relevant coefficients.The paper assumes sparsity because informative estimation and inference are otherwise impossible when nuisance parameters outnumber observations.
- Selection problem: Variable selection can spuriously include irrelevant variables or omit relevant ones, and either mistake can distort post-selection estimation and inference for α.Omitted small predictors can materially affect inference, motivating procedures robust to selection errors.
- Estimation approach: Lasso or Post-Lasso provides principled selection that guards against false discovery and overfitting, while orthogonal moment functions reduce sensitivity to omitted small predictors.The paper uses tuning parameters following Belloni et al. (2012) and bases inference on moments locally insensitive to nuisance-estimation errors.
- Estimation approach: Algorithm 1 combines Lasso or Post-Lasso regressions for the endogenous variable, outcome, and controls, then estimates α0 by IV regression and uses robust standard errors.The proposed implementation is a double-selection strategy, with alternative algorithms having similar asymptotic properties.
- Inference: Under sparsity and regularity conditions, the estimator is asymptotically normal, its score statistic is chi-squared with one degree of freedom, and confidence intervals are uniformly valid over a large model class.These results apply whether Lasso or Post-Lasso is used.
2. Simulation Example
The simulation compares an infeasible Oracle, two naive procedures, and Double-Selection in a high-dimensional IV setting. Double-Selection performs close to the Oracle, while the naive procedures exhibit selection-related problems.
- The experiment compares Oracle, two naive estimators, and Double-Selection across 1000 replications.The estimators differ in whether they use known nuisance parameters, stepwise regression, non-orthogonal moments, or Double-Selection.
- Naive 2 has Bias .035, MAD .103, and Size .095, with size distortion despite relatively good MAD.Its moment condition lacks orthogonality and is formally valid when perfect model selection is possible.
- Oracle achieves Bias .006, MAD .095, and Size .043.
- Double-Selection achieves Bias .021, MAD .099, and Size .054.
- Principled selection combined with selection-robust moments produces estimation and inference performance close to the infeasible Oracle.
3. Empirical Example
The empirical example applies post-model-selection IV methods to automobile demand, selecting controls and instruments from baseline and larger variable sets. The selected specifications imply more elastic and theoretically more plausible demand estimates than the baseline.
- The automobile-demand model treats price as endogenous and uses product characteristics as controls and instruments.The specification is based on Berry et al. (1995), with market share modeled relative to an outside option.
- The baseline strategy uses five controls and ten instruments chosen from product characteristics using economic intuition and exchangeability.
- High-dimensional selection complements the economic intuition used to choose controls and instruments and addresses the ad hoc choice of utility attributes.
- Using only the original variables yields a price coefficient of -.185 with standard error .014, with four instruments selected.The paper attributes the difference from baseline largely to the changed instrument set and notes consistency with many-instrument bias toward OLS when irrelevant instruments are included.
- Using the larger variable set yields a price coefficient of -.221 with standard error .015 and selects additional controls and nonlinear terms.The selected variables suggest important nonlinearity was missed by the baseline specification.
- Inelastic-demand estimates fall from 670 products at baseline to 139 with selection from baseline variables and 12 with selection from the larger set.The authors describe the larger-variable-set results as the most reasonable estimates of own-price elasticities, while noting the simple logit specification’s usual drawbacks.
4. Conclusion
The paper proposes a high-dimensional IV approach that selects instruments and controls under approximate sparsity and combines selection with moment functions designed for valid post-selection inference.
- The approach targets causal or structural estimation when empirical models contain many controls and instruments.
- It relies on approximate sparsity, high-quality variable selection, and appropriate moment functions.
- The method extends earlier work on instrument selection and control selection to settings with large numbers of both instruments and controls.