On the Use of Linear Fixed Effects Regression Models for ausal Inference Kosuke Imai Department of Politics Princeton University Joint work with In Song Kim Atlantic ausal Inference onference Johns Hopkins May 23, 22 Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference / 2
Motivation Fixed effects models are a primary workhorse for causal inference Used for stratified experimental and observational data Also used to adjust for unobservables in observational studies: Good instruments are hard to find..., so we d like to have other tools to deal with unobserved confounders. his chapter considers... strategies that use data with a time or cohort dimension to control for unobserved but fixed omitted variables (Angrist & Pischke, Mostly Harmless Econometrics) fixed effects regression can scarcely be faulted for being the bearer of bad tidings (Green et al., Dirty Pool) Question: What are the exact causal assumptions underlying fixed effects regression models? Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 2 / 2
Main Methodological Results Standard (one-way and two-way) FE estimators are equivalent to particular matching estimators 2 Identify the information used implicitly to estimate counterfactual outcomes under FE models 3 Identify potential sources of bias and inefficiency in FE estimators 4 Propose simple ways to improve FE estimators using weighted FE regression 5 Within-unit matching, first differencing, propensity score weighting, difference-in-differences are all equivalent to weighted FE model with different regression weights 6 Offer a specification test for the standard FE model 7 Develop fast computation and open-source software Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 3 / 2
Matching and Regression in ross-section Settings Units reatment status Outcome 2 3 4 5 Y Y 2 Y 3 Y 4 Y 5 Estimating the Average reatment Effect (AE) via matching: Y 2 (Y 3 + Y 4 ) Y 2 2 (Y 3 + Y 4 ) 3 (Y + Y 2 + Y 5 ) Y 3 3 (Y + Y 2 + Y 5 ) Y 4 Y 5 2 (Y 3 + Y 4 ) Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 4 / 2
Matching Representation of Simple Regression ross-section simple linear regression model: Y i = α + βx i + ɛ i Binary treatment: X i {, } Equivalent matching estimator: where Ŷ i () = Ŷ i () = ˆβ = N N (Ŷi Ŷi()) () i= { Y i if X i = N N i = X i i = X i Y i if X i = { N N i = ( X i ) i = ( X i )Y i if X i = Y i if X i = reated units matched with the average of non-treated units Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 5 / 2
One-Way Fixed Effects Regression Simple (one-way) FE model: Y it = α i + βx it + ɛ it ommonly used by applied researchers: Stratified randomized experiments (Duflo et al. 27) Stratification and matching in observational studies Panel data, both experimental and observational ˆβ FE may be biased for the AE even if X it is exogenous within each unit It converges to the weighted average of conditional AEs: where σ 2 i ˆβ FE = t= (X it X i ) 2 / p E{AE i σi 2 } E(σi 2 ) How are counterfactual outcomes estimated under the FE model? Unit fixed effects = within-unit comparison Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 6 / 2
Mismatches in One-Way Fixed Effects Model ime periods Units : treated observations : control observations ircles: Proper matches riangles: Mismatches = attenuation bias Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 7 / 2
Matching Representation of Fixed Effects Regression Proposition Ŷ it (x) = K = ˆβ FE = K { N { N Y it t t Y it N X it i= t= N i= t= if X it = x if X it = x ) } (Ŷit () Ŷit(), for x =, ( X it ) + ( X it ) t t t t X it. K : average proportion of proper matches across all observations More mismatches = larger adjustment Adjustment is required except very special cases Fixes attenuation bias but this adjustment is not sufficient Fixed effects estimator is a special case of matching estimators Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 8 / 2
Unadjusted Matching Estimator ime periods Units onsistent if the treatment is exogenous within each unit Only equal to fixed effects estimator if heterogeneity in either treatment assignment or treatment effect is non-existent Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 9 / 2
Unadjusted Matching = Weighted FE Estimator Proposition 2 he unadjusted matching estimator where Ŷ it () = ˆβ M = N N i= Y it if X it = t = X it Y it t = X it if X it = (Ŷit Ŷit()) () t= and Ŷ it () = is equivalent to the weighted fixed effects model (ˆα M, ˆβ M ) = arg min W it (α,β) N i= W it (Y it α i βx it ) 2 t= t = X it if X it =, t = ( X it ) if X it =. t = ( X it )Y it t = ( X it ) if X it = Y it if X it = Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference / 2
Equal Weights reatment 2 2 2 2 Weights Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference / 2
Different Weights reatment 2 3 3 3 4 4 Weights Any within-unit matching estimator leads to weighted fixed effects regression with particular weights We derive regression weights given any matching estimator (e.g., first differences) for various quantities (e.g., AE, A) Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 2 / 2
Fast omputation and Standard Error alculation Standard FE estimator: demean both Y and X regress demeaned Y on demeaned X Weighted FE estimator: weighted-demean both Y and X regress weighted-demeaned Y on weighted-demeaned X Model-based standard error calculation Various robust sandwich estimators Easy standard error calculation for matching estimators Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 3 / 2
Specification est Should we use standard or weighted FE models? Standard FE estimator is more efficient if its assumption is correct Weighted FE estimator is consistent under the same assumption Specification test (White 98): Null hypothesis: standard FE model is correct Does the difference between standard and weighted FE estimators arise by chance? Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 4 / 2
Mismatches in wo-way FE Model Y it = α i + γ t + βx it + ɛ it Units ime periods riangles: wo kinds of mismatches Same treatment status Neither same unit nor same time Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 5 / 2
Mismatches in Weighted wo-way FE Model Units ime periods Some mismatches can be eliminated You can NEVER eliminate them all Adjustment is always required Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 6 / 2
Difference-in-Differences (DiD) Average Outcome..2.4.6.8. control group treatment group counterfactual time t time t+ Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 7 / 2
General DiD = Weighted wo-way (Unit and ime) FE 2 2: standard two-way fixed effects General setting: Multiple time periods, repeated treatments reatment Weights 2 2 2 2 Weights can be negative = the method of moments estimator Fast computation via projection on complex plane Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 8 / 2
Preliminary Empirical Results (a) Dyad with Both GA Members vs. One GA Member (b) Dyad with One GA Members vs. No GA Member Estimated Effects (log of trade ).2..2.4 Year Fixed Effects Average reatment Effect (AE) Without Restriction Equal Weights Propensity Score Weights Average reatment Effect (AE) With Restriction Equal Weights Propensity Score Weights.2..2.4 Year Fixed Effects Equal Weights Propensity Score Weights Average reatment Effect (AE) Without Restriction Equal Weights Propensity Score Weights Average reatment Effect (AE) With Restriction Without Restriction With Restriction Without Restriction With Restriction Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 9 / 2
oncluding Remarks Standard one-way and two-way FE estimators are adjusted matching estimators FE models are not a magic bullet solution to endogeneity In many cases, adjustment is not sufficient for removing bias Key Question: Where are the counterfactuals coming from? Different causal assumptions yield different weighted FE models Weighted FE models encompass a large class of causal assumptions: stratification, first difference, propensity score weighting, difference-in-differences Model-based standard error, specification test Open source software, R package wfe, available Kosuke Imai (Princeton) Fixed Effects for ausal Inference ausal Inference onference 2 / 2