When Should We Use Linear Fixed Effects Regression Models for Causal Inference with Longitudinal Data? Kosuke Imai Department of Politics Center for Statistics and Machine Learning Princeton University Joint work with In Song Kim (MIT) Seminar at The University of Tokyo July 7, 2016 Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 1 / 37
Fixed Effects Regressions in Causal Inference Linear fixed effects regression models are the primary workhorse for causal inference with longitudinal/panel data Researchers use them to adjust for unobserved time-invariant confounders (omitted variables, endogeneity, selection bias,...): Good instruments are hard to find..., so we d like to have other tools to deal with unobserved confounders. This 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) Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 2 / 37
Overview of the Talk Identify two under-appreciated causal assumptions of unit fixed effects regression estimators: 1 Past treatments do not directly affect current outcome 2 Past outcomes do not directly affect current treatments and time-varying confounders can be relaxed under a selection-on-observables approach New matching framework for causal inference with panel data: 1 propose within-unit matching estimators to relax linearity 2 incorporate various estimators, e.g., the before-and-after estimator 3 establish equivalence between matching estimators and weighted linear fixed effects regression estimators Extend the analysis to two-way fixed effects models, difference-in-differences design, and synthetic control method An empirical illustration: Effects of GATT on trade Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 3 / 37
Linear Regression with Unit Fixed Effects Balanced panel data with N units and T time periods Y it : outcome variable X it : causal or treatment variable of interest Assumption 1 (Linearity) Y it = α i + βx it + ɛ it U i : a vector of unobserved time-invariant confounders α i = h(u i ) for any function h( ) A flexible way to adjust for unobservables Average contemporaneous treatment effect: β = E(Y it (1) Y it (0)) Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 4 / 37
Strict Exogeneity and Least Squares Estimator Assumption 2 (Strict Exogeneity) ɛ it {X i, U i } Mean independence is sufficient: E(ɛ it X i, U i ) = E(ɛ it ) = 0 Least squares estimator based on de-meaning: ˆβ FE = arg min β N i=1 t=1 T {(Y it Y i ) β(x it X i )} 2 where X i and Y i are unit-specific sample means ATE among those units with variation in treatment: τ = E(Y it (1) Y it (0) C it = 1) where C it = 1{0 < T t=1 X it < T }. Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 5 / 37
Causal Directed Acyclic Graph (DAG) Y i1 Y i2 Y i3 X i1 X i2 X i3 U i arrow = direct causal effect absence of arrows causal assumptions Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 6 / 37
Nonparametric Structural Equation Model (NPSEM) One-to-one correspondence with a DAG: Y it = g 1 (X it, U i, ɛ it ) X it = g 2 (X i1,..., X i,t 1, U i, η it ) Nonparametric generalization of linear unit fixed effects model: Allows for nonlinear relationships, effect heterogeneity Strict exogeneity holds No arrows can be added without violating Assumptions 1 and 2 Causal assumptions: 1 No unobserved time-varying confounders 2 Past outcomes do not directly affect current outcome 3 Past outcomes do not directly affect current treatment 4 Past treatments do not directly affect current outcome Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 7 / 37
Potential Outcomes Framework DAG causal structure Potential outcomes treatment assignment mechanism Assumption 3 (No carryover effect) Past treatments do not directly affect current outcome Y it (X i1, X i2,..., X i,t 1, X it ) = Y it (X it ) What randomized experiment satisfies unit fixed effects model? 1 randomize X i1 given U i 2 randomize X i2 given X i1 and U i 3 randomize X i3 given X i2, X i1, and U i 4 and so on Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 8 / 37
Assumption 4 (Sequential Ignorability with Unobservables) {Y it (1), Y it (0)} T t=1 X i1 U i. {Y it (1), Y it (0)} T t=1 X it X i1,..., X i,t 1, U i. {Y it (1), Y it (0)} T t=1 X it X i1,..., X i,t 1, U i as-if random assumption without conditioning on past outcomes Past outcomes cannot directly affect current treatment Says nothing about whether past outcomes can directly affect current outcome Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 9 / 37
Past Outcomes Directly Affect Current Outcome Y i1 Y i2 Y i3 X i1 X i2 X i3 U i Strict exogeneity still holds Past outcomes do not confound X it Y it given U i No need to adjust for past outcomes Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 10 / 37
Past Treatments Directly Affect Current Outcome Y i1 Y i2 Y i3 X i1 X i2 X i3 U i Past treatments as confounders Need to adjust for past treatments Strict exogeneity holds given past treatments and U i Impossible to adjust for an entire treatment history and U i at the same time Adjust for a small number of past treatments often arbitrary Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 11 / 37
Past Outcomes Directly Affect Current Treatment Y i1 Y i2 Y i3 X i1 X i2 X i3 Correlation between error term and future treatments Violation of strict exogeneity No adjustment is sufficient U i Together with the previous assumption no feedback effect over time Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 12 / 37
Instrumental Variables Approach Y i1 Y i2 Y i3 X i1 X i2 X i3 U i Instruments: X i1, X i2, and Y i1 GMM: Arellano and Bond (1991) Exclusion restrictions Arbitrary choice of instruments Substantive justification rarely given Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 13 / 37
An Alternative Selection-on-Observables Approach Y i1 Y i2 Y i3 Absence of unobserved time-invariant confounders U i past treatments can directly affect current outcome X i1 X i2 X i3 past outcomes can directly affect current treatment Comparison across units within the same time rather than across different time periods within the same unit Marginal structural models can identify the average effect of an entire treatment sequence Trade-off no free lunch Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 14 / 37
Adjusting for Observed Time-varying Confounders Y i1 Y i2 Y i3 X i1 X i2 X i3 Z i1 Z i2 Z i3 U i past treatments cannot directly affect current outcome past outcomes cannot directly affect current treatment adjusting for Z it does not relax these assumptions past outcomes cannot indirectly affect current treatment through Z it Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 15 / 37
A New Matching Framework Even if these assumptions are satisfied, the the unit fixed effects estimator is inconsistent for the ATE: { ( T t=1 E C X ) } it Y T it t=1 i T (1 X it )Y it p t=1 ˆβ FE X T S 2 it t=1 1 X i it E(C i Si 2 τ ) where Si 2 = T t=1 (X it X i ) 2 /(T 1) is the unit-specific variance Key idea: comparison across time periods within the same unit The Within-unit matching estimator improves ˆβ FE by relaxing the linearity assumption: ˆτ match = 1 N i=1 C i N i=1 C i ( T t=1 X ity it T t=1 X it T t=1 (1 X it)y it T t=1 (1 X it) ) Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 16 / 37
Constructing a General Matching Estimator M it : matched set for observation (i, t) For the within-unit matching estimator, M match it = {(i, t ) : i = i, X i t = 1 X it} A general matching estimator: ˆτ match = N i=1 1 T t=1 D it N T D it (Ŷit(1) Ŷit(0)) i=1 t=1 where D it = 1{#M it > 0} and Ŷ it (x) = { Y it if X it = x 1 #M it (i,t ) M it Y i t if X it = 1 x Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 17 / 37
Before-and-After Design No time trend for the average potential outcomes: E(Y it (x) Y i,t 1 (x) X it X i,t 1 ) = 0 for x = 0, 1 with the quantity of interest E(Y it (1) Y it (0) X it X i,t 1 ) Or just the average potential outcome under the control condition E(Y it (0) Y i,t 1 (0) X it = 1, X i,t 1 = 0) = 0 This is a matching estimator with the following matched set: M BA it = {(i, t ) : i = i, t {t 1, t + 1}, X i t = 1 X it} Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 18 / 37
It is also the first differencing estimator: ˆβ FD = arg min β N i=1 t=2 T {(Y it Y i,t 1 ) β(x it X i,t 1 )} 2 We emphasize that the model and the interpretation of β are exactly as in [the linear fixed effects model]. What differs is our method for estimating β (Wooldridge; italics original). The identification assumptions is very different Slightly relaxing the assumption of no carryover effect But, still requires the assumption that past outcomes do not affect current treatment Regression toward the mean: suppose that the treatment is given when the previous outcome takes a value greater than its mean Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 19 / 37
Matching as a Weighted Unit Fixed Effects Estimator Any within-unit matching estimator can be written as a weighted unit fixed effects estimator with different regression weights The proposed within-matching estimator: ˆβ WFE = arg min β W it = N i=1 t=1 T D it W it {(Y it Y i ) β(x it X i )} 2 where X i and Y i are unit-specific weighted averages, and T T if X t =1 X it = 1, it T T t =1 (1 X it ) if X it = 0. Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 20 / 37
We show how to construct regression weights for different matching estimators (i.e., different matched sets) Idea: count the number of times each observation is used for matching Benefits: computational efficiency model-based standard errors robustness matching estimator is consistent even when linear unit fixed effects regression is the true model specification test (White 1980) null hypothesis: linear fixed effects regression is the true model Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 21 / 37
Linear Regression with Unit and Time Fixed Effects Model: Y it = α i + γ t + βx it + ɛ it where γ t flexibly adjusts for a vector of unobserved unit-invariant time effects V t, i.e., γ t = f (V t ) Estimator: ˆβ FE2 = arg min β N i=1 t=1 T {(Y it Y i Y t + Y ) β(x it X i X t + X)} 2 where Y t and X t are time-specific means, and Y and X are overall means Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 22 / 37
Understanding the Two-way Fixed Effects Estimator β FE : bias due to time effects β FEtime : bias due to unit effects β pool : bias due to both time and unit effects ˆβ FE2 = ω FE ˆβ FE + ω FEtime ˆβ FEtime ω pool ˆβ pool w FE + w FEtime w pool with sufficiently large N and T, the weights are given by, ω FE E(S 2 i ) = average unit-specific variance ω FEtime E(S 2 t ) = average time-specific variance ω pool S 2 = overall variance Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 23 / 37
Matching and Two-way Fixed Effects Estimators Problem: No other unit shares the same unit and time Units Time periods 4 3 2 C T C C T T C T T C C C T C C 1 Two kinds of mismatches 1 Same treatment status 2 Neither same unit nor same time T T T C T Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 24 / 37
We Can Never Eliminate Mismatches Units Time periods 4 3 2 1 C T C C T T C T T C C C T C C T T T C T To cancel time and unit effects, we must induce mismatches No weighted two-way fixed effects model eliminates mismatches Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 25 / 37
Difference-in-Differences Design Parallel trend assumption: E(Y it (0) Y i,t 1 (0) X it = 1, X i,t 1 = 0) = E(Y it (0) Y i,t 1 (0) X it = X i,t 1 = 0) Average Outcome 0.0 0.2 0.4 0.6 0.8 1.0 control group treatment group counterfactual time t time t+1 Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 26 / 37
General DiD = Weighted Two-Way FE Effects 2 2: equivalent to linear two-way fixed effects regression General setting: Multiple time periods, repeated treatments Units Time periods 4 3 2 1 C T C C T T C T T C C C T C C T T T C T Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 27 / 37
Units Time periods 4 3 2 1 0 0 0 0 0 0 0 1 2 0 1 1 2 0 1 1 2 1 2 0 0 0 0 0 Fast computation, standard error, specification test Still assumes that past outcomes don t affect current treatment Baseline outcome difference caused by unobserved time-invariant confounders It should not reflect causal effect of baseline outcome on treatment assignment Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 28 / 37
Synthetic Control Method (Abadie et al. 2010) One treated unit i receiving the treatment at time T Quantity of interest: Y i T Y i T (0) Create a synthetic control using past outcomes Weighted average: Yi T (0) = i i ŵ i Y it Estimate weights to balance past outcomes and past time-varying covariates A motivating autoregressive model: Y it (0) = ρ T Y i,t 1 (0) + δt Z it + ɛ it Z it = λ T 1 Y i,t 1 (0) + T Z i,t 1 + ν it Past outcomes can affect current treatment No unobserved time-invariant confounders Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 29 / 37
Causal Effect of ETA s Terrorism THE AMERICAN ECONOMIC REVIEW MARCH 2003 11- -5 / s / a / 210-5 0 9-, -Actual with terrorism - -Synthetic without terrorism 1955 1 1960 1965 1970 1975 1980 1985 1990 year 1995 2000 FIGURE1. PERCAPITA GDP FOR THE BASQUECOUNTRI Abadie and Gardeazabal (2003, AER) 4 Imai (Princeton) and Kim -- \ (MIT) Fixed Effects for Causal Inference GDt' gap Tokyo (July 7, 2016) 30 / 37
The main motivating model: Y it (0) = γ t + δ t Z it + ξ U i + ɛ it A generalization of the linear two-way fixed effects model How is it possible to adjust for unobserved time-invariant confounders by adjusting for past outcomes? The key assumption: there exist weights such that w i Z it = Z i t for all t T 1 and w i U i = U i i i i i In general, adjusting for observed confounders does not adjust for unobserved confounders The same tradeoff as before Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 31 / 37
Effects of GATT Membership on International Trade 1 Controversy Rose (2004): No effect of GATT membership on trade Tomz et al. (2007): Significant effect with non-member participants 2 The central role of fixed effects models: Rose (2004): one-way (year) fixed effects for dyadic data Tomz et al. (2007): two-way (year and dyad) fixed effects Rose (2005): I follow the profession in placing most confidence in the fixed effects estimators; I have no clear ranking between country-specific and country pair-specific effects. Tomz et al. (2007): We, too, prefer FE estimates over OLS on both theoretical and statistical ground Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 32 / 37
Data and Methods 1 Data Data set from Tomz et al. (2007) Effect of GATT: 1948 1994 162 countries, and 196,207 (dyad-year) observations 2 Year fixed effects model: ln Y it = α t + βx it + δ Z it + ɛ it Y it : trade volume X it : membership (formal/participants) Both vs. At most one Z it : 15 dyad-varying covariates (e.g., log product GDP) 3 Assumptions: past membership status doesn t directly affect current trade volume past trade volume doesn t affect current membership status Before-and-after increasing trend in trade volume Difference-in-differences after conditional on past outcome? Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 33 / 37
Empirical Results: Formal Membership Dyad with Both Members vs. One or None Member Estimated Effects (log of trade) 0.2 0.1 0.0 0.1 0.2 0.3 0.4 FE WFE FE WFE FD Year Fixed Effects Dyad Fixed Effects Year and Dyad Fixed Effects FE DID caliper DID Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 34 / 37
Empirical Results: Participants Included Dyad with Both Members vs. One or None Member Estimated Effects (log of trade) 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Formal Member Participants FE WFE FE WFE FE WFE FD FE WFE Year Fixed Effects Dyad Fixed Effects Year and Dyad Fixed Effects FD FE DID caliper DID FE DID caliper DID Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 35 / 37
Concluding Remarks When should we use linear fixed effects models? Key tradeoff: 1 unobserved time-invariant confounders fixed effects 2 causal dynamics between treatment and outcome selection-on-observables Two key (under-appreciated) causal assumptions of fixed effects: 1 past treatments do not directly affect current outcome 2 past outcomes do not directly affect current treatment A new matching estimator: 1 Within-unit matching estimator no linearity assumption 2 Various causal identification strategies can be incorporated including the before-and-after and difference-in-differences designs 3 Equivalent reqpresentation as a weighted linear fixed effects regression estimator R package wfe is available at CRAN Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 36 / 37
Send comments and suggestions to: kimai@princeton.edu More information about this and other research: http://imai.princeton.edu Imai (Princeton) and Kim (MIT) Fixed Effects for Causal Inference Tokyo (July 7, 2016) 37 / 37