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Generalized Synthetic Control Method for Causal Inference in Time-Series Cross-Sectional... Yiqing Xu Department of Political Science, Massachusetts Institute of Technology The Problem

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Generalized Synthetic Control Method for Causal Inference in Time-Series Cross-Sectional... Yiqing Xu Department of Political Science, Massachusetts Institute of Technology The Problem
Generalized Synthetic Control Method for Causal Inference in Time-Series Cross-Sectional Data
Yiqing Xu
Department of Political Science, Massachusetts Institute of Technology
Empirical Applications
The Problem
Objective: to identify causal effects with
time-series cross-sectional data
Problems of existing methods:
(1) DID: “parallel trends" may not be true
(2) Synthetic control: one treated unit
treatment
effect
control
T0
−2
Unifying the synthetic control method and
fixed-effect models in a simple framework
X Multiple treated units
X Variable treatment timing
X Conventional standard errors
X Robust DID estimates
X More efficient than the synthetic control method
X Fast and easy to implement
Quantities of Interest
• Individual
treatment effect: δit, ∀i ∈ T
1 P
• SATTt = N
δ
D
i∈T
it
it
tr
• PATTt = E[δit|Dit = 1]
Applying ideas of statistical learning to a
problem of causal inference
Identifying Assumptions
5
10
15
−3
Finding: On average, a high-level official’s visit
almost tripled a firm’s total outstanding loans in a
three-year period.
80
120
160
Case 2. One treated unit: Abadie, Diamond,
Hainmueller (2010): Proposition 99 on tobacco
consumption
40
Treated
Controls
Generalized synthetic control
1970
•I
use a resampling scheme to produce standard
errors for individual treatment effect and SATT .
→ The key is to correctly calculate the out-ofsample prediction error of the counterfaucal for
each treated unit
• When
the treatment group is relatively large and
Assumption 4 holds, inference of PATT can be
made using nonparametric bootstrap
show that in both cases I obtain the
correct coverage rates
1975
1980
1985
1990
1995
2000
1985
1990
1995
2000
Estimated treatment effect (GSynth)
Estimated treatment effect (Synth)
95% CI
1970
1975
1980
Finding: The estimated effect is similar in size to
that in Abadie et. al. though estimation uncertainty
increases.
Extensions
1
When It Doesn’t Work
Conditional ignorability
{Yi(1), Yi(0)}⊥
⊥Di|X, Λ, F
• Predict
the counterfactual for each treated unit in
the post-treatment period
• Take advantage of cross-sectional correlations
• Use pre-treatment periods to select models
Inference
• Simulations
Intuition
0
40
Main Contributions
4 Obtain treatment effect for each treated unit
−5
20
15
3 Predict counterfactuals for the treated units
−10
0
−0.5
−1.0
Important Special Cases
Ft,1 = 1: individual fixed effect
λi,1 = 1, Ft,1 = ηt: time fixed effect
Ft,1 = t: individual-specific linear time trend
Ft,1 = t, Ft,2 = t2: quadratic time trend
1−αt
t
Ft,1 = α , Ft,2 = 1−α : lagged dependent (AR1)
λi,1Ft,1 = Ziγt: effect of pre-treatment covariates
1 Estimate a factor model using the control
group units (Bai 2009)
2 Estimate factor loadings for the treated
units using least squares
Repeat 1-2 to minimize MSPE
−15
−40
+ it
+ it + δitDit
Loop starts: given the number of factors,
Estimates
95% CIs
−80
2.0
1.5
Coefficient
0.0
0.5
1.0
0
λiFt
0
λiFt
Algorithm
unobserved time-varying factors
• λi: individual-specific factor loadings
10
T0
Setup
Yit(0) = βXit +
Yit(1) = βXit +
5
control
T0
• Ft :
0
Coefficient
−1
0
1
control
Diff−in−diffs
−5
2
treated
λ0iFt = λi,1Ft,1 + λi,2Ft,2 + · · · λi,r Ft,r
−10
Generalized Synthetic Control
treated
I investigate the the effect of government officials’
visits to firms in China on firms’ access to loans
during the 2008 global crisis. DID fails because the
“parallel trends" assumption appears implausible.
−15
Case 1. Multiple treated units, variable
treatment timing: officials’ visits on firms’ access
to bank loans
3
treated
A Motivating Case
Estimates
95% CIs
treatment
effect
treatment
effect
∀i
Weak serial dependence of the error terms
3 Regularity conditions
4 Random sampling (if PATT is of interest)
2
• When
the number of pre-treatment periods or the
number of control group units is small
• When time-varying counfounders cannot be
decomposed (e.g., Wit 6= λi × Ft)
• When the SUTVA is violated
Yit(0) = Ziθt + αi + ηt + βXit +
0
λiFt
+ it
- αi and ηt: unit and time fixed effects
- Zi: pre-treatment time-invariant covariates
• The choice of identifying assumptions is the key
• The algorithm can also help choose the model
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