Inferring Treatment Effects After Testing Instrument Strength in Linear Models.

A common practice in IV studies is to check for instrument strength, i.e. its association to the treatment, with an F-test from regression. If the F-statistic is above some threshold, usually 10, the instrument is deemed to satisfy one of the three core IV assumptions and used to test for the treatment effect. However, in many cases, the inference on the treatment effect does not take into account the strength test done a priori. In this paper, we show that not accounting for this pretest can severely distort the distribution of the test statistic and propose a method to correct this distortion, producing valid inference. A key insight in our method is to frame the F-test as a randomized convex optimization problem and to leverage recent methods in selective inference. We prove that our method provides conditional and marginal Type I error control. We also extend our method to weak instrument settings. We conclude with a reanalysis of studies concerning the effect of education on earning where we show that not accounting for pre-testing can dramatically alter the original conclusion about education's effects.

[1]  Dennis L. Sun,et al.  Exact post-selection inference, with application to the lasso , 2013, 1311.6238.

[2]  J. MacKinnon,et al.  Estimation and inference in econometrics , 1994 .

[3]  Stephen G. Donald,et al.  Testing Identifiability and Specification in Instrumental Variable Models , 1993, Econometric Theory.

[4]  Timothy B. Armstrong,et al.  Unbiased Instrumental Variables Estimation Under Known First-Stage Sign , 2015, 1501.06630.

[5]  Anna Mikusheva,et al.  Robust Confidence Sets in the Presence of Weak Instruments , 2005 .

[6]  A. Belloni,et al.  SPARSE MODELS AND METHODS FOR OPTIMAL INSTRUMENTS WITH AN APPLICATION TO EMINENT DOMAIN , 2012 .

[7]  Jeffrey M. Woodbridge.  Econometric Analysis of Cross Section and Panel Data , 2002 .

[8]  Jonathan E. Taylor,et al.  Exact Post Model Selection Inference for Marginal Screening , 2014, NIPS.

[9]  J. Stock,et al.  Instrumental Variables Regression with Weak Instruments , 1994 .

[10]  Miguel A Hernán,et al.  Commentary: how to report instrumental variable analyses (suggestions welcome). , 2013, Epidemiology.

[11]  Snigdha Panigrahi,et al.  Selective sampling after solving a convex problem , 2016, 1609.05609.

[12]  Jean-Marie Dufour,et al.  Identification, Weak Instruments, and Statistical Inference in Econometrics , 2003 .

[13]  Jonathan E. Taylor,et al.  Inference After Selecting Plausibly Valid Instruments with Application to Mendelian Randomization , 2019, 1911.03985.

[14]  J. Stock,et al.  Inference with Weak Instruments , 2005 .

[15]  Dylan S. Small,et al.  Instrumental Variables Estimation With Some Invalid Instruments and its Application to Mendelian Randomization , 2014, 1401.5755.

[16]  Dennis L. Sun,et al.  Optimal Inference After Model Selection , 2014, 1410.2597.

[17]  Jinyong Hahn,et al.  A New Specification Test for the Validity of Instrumental Variables , 2000 .

[18]  David A. Jaeger,et al.  Problems with Instrumental Variables Estimation when the Correlation between the Instruments and the Endogenous Explanatory Variable is Weak , 1995 .

[19]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[20]  Jonathan E. Taylor,et al.  Bootstrap inference after using multiple queries for model selection , 2016, 1612.07811.

[21]  J. Angrist,et al.  Does Compulsory School Attendance Affect Schooling and Earnings? , 1990 .

[22]  Jonathan E. Taylor,et al.  Selective inference with a randomized response , 2015, 1507.06739.

[23]  J. Angrist,et al.  Journal of Economic Perspectives—Volume 15, Number 4—Fall 2001—Pages 69–85 Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments , 2022 .

[24]  David Card,et al.  Using Geographic Variation in College Proximity to Estimate the Return to Schooling , 1993 .

[25]  Marcelo J. Moreira.  Tests with correct size when instruments can be arbitrarily weak , 2009 .

[26]  Jonathan H. Wright,et al.  A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments , 2002 .

[27]  Marcelo J. Moreira.  A Conditional Likelihood Ratio Test for Structural Models , 2003 .

[28]  M. Baiocchi,et al.  Instrumental variable methods for causal inference , 2014, Statistics in medicine.

[29]  Joshua R. Loftus,et al.  A significance test for forward stepwise model selection , 2014, 1405.3920.

[30]  T. W. Anderson,et al.  Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations , 1949 .

[31]  J. Robins,et al.  Instruments for Causal Inference: An Epidemiologist's Dream? , 2006, Epidemiology.

[32]  Donald W. K. Andrews,et al.  Optimal Two‐Sided Invariant Similar Tests for Instrumental Variables Regression , 2006 .