Dynamic Covariate Balancing: Estimating Treatment Effects ...

Dynamic Covariate Balancing: Estimating TreatmentEffects over Time

Davide Viviano Jelena BradicUC San Diego

July, 2021

Davide Viviano and Jelena Bradic (UCSD) Dynamic Covariate Balancing: Estimating Treatment Effects over TimeJuly, 2021 1 / 48

Content

1 Introduction

2 Method

3 Numerical Studies and Empirical Applications

4 Conclusions

Two Examples

Ex1 “What is the effect of negative advertisement on election outcome?”(Blackwell, 2013)

Does negative advertisement the previous two, three, four weeks beforeelections positively or negatively impact election outcomes?Politicians may make strategic choices based on past observations(polls, competitor’s strategy, etc.).

Ex2 “What is the effect of democracy on economic growth?” (Acemogluet al., 2019)

Hard question.Long and short term impacts may be different;Heterogeneity may occur over time;Past growth may determine the selection into treatment;It is necessary to control for many potential confounders.

Inference on Treatment Histories

Suppose we collect data from an observational study with T periods and nindividuals.

How do we conduct inference on the effect of a treatment history (e.g.,being h periods under treatment compared to h periods under control)?

Challenges:

Individuals select dynamically based on arbitrary past information;

(Time-varying) covariates and outcomes depend on past treatmentassignments;

Many potential confounders (high-dimensional covariates);

The propensity score is unknown.

Data and Notation

Ex-post evaluation: we first collect data from T periods, we thenconduct inference.

We collect data(Xi ,1,Xi ,2, · · · ,Yi ,1, · · · ,Yi ,T ,Di ,1, · · · ,Di ,T

)ni=1∼i .i .d . P

Xi ,t ∈ Xt - time-varying covariates;

Yi ,t ∈ Y - intermediate outcomes;

Di ,t ∈ {0, 1} - dynamic treatments;

⇒ Potential outcomes: Yi ,t(d1, · · · , dt).

Goal: Inference on the long-term effects of two histories d1:T , d′1:T on the

end-line outcome YT .

Illustration: Two Periods

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

Dynamic model: D1

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

Dynamic model: D1

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

Dynamic model: D1

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

Dynamic model: D1 Y1,X2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

Goal Inference on E[Y2(1, 0)− Y2(0, 0)

](or conditional on baselines).

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

| | |t = 1 t = 2 t = 3

(X1,D1) (Y1,X2,D2) Y2

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

(Y1,X2)(1) Y2(1, 1)

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

(Y1,X2)(1) Y2(1, 1)

(Y1,X2)(0) Y2(1, 0)

(Y1,X2)(0) Y2(0, 0)

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

(Y1,X2)(1) Y2(1, 1)

(Y1,X2)(0) Y2(1, 0)

(Y1,X2)(0) Y2(0, 0)

Estimate each counterfactual?

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

(Y1,X2)(1) Y2(1, 1)

(Y1,X2)(0) Y2(1, 0)

(Y1,X2)(0) Y2(0, 0)

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

(Y1,X2)(1) Y2(1, 1)

(Y1,X2)(0) Y2(1, 0)

(Y1,X2)(0) Y2(0, 0)

i1(Y1(0),X2(0),Y2(0, 0)

i2(Y1(1),X2(1),Y2(0, 0),Y2(1, 1)

i3 ...

Review: Three Intuitive Estimators

1. Regress end-line outcomes on covariates:

Yi ,T =T−1∑t=0

αtDi ,t + Xi ,tβt +T−1∑t=1

Yi ,tφt + εi ,T .

Di ,t−1 also affects Yi ,t ⇒ we main underestimate the overall effect.

D1 Y1,X2

1. Regress end-line outcomes on covariates:

Yi ,T =T−1∑t=0

Yi ,tφt + εi ,T .

D1 Y1,X2

1. Regress the end-line outcomes on covariates:

Yi ,T =T−1∑t=0

Yi ,tφt + εi ,T .

D1 Y1,X2

2. Regress end-line outcomes without including the outcomes:

Yi ,T =T−1∑t=0

αtDi ,t + Xi ,1β + εi ,T .

⇒ omitted variable bias.

D1 Y1,X2

2. Regress end-line outcomes without including the outcomes:

Yi ,T =T−1∑t=0

αtDi ,t + Xi ,1β + εi ,T .

⇒ omitted variable bias.

D1 Y1,X2

3. Run separate regressions for

Yi ,t = Xi ,t−1βt+Di ,t−1αt+εi ,t , Xi ,t = Xi ,t−1β̃t−1+Di ,t−1α̃t−1+εx ,i ,t .

Prone to large estimation error in high dimensions.

D1 Y1,X2

3. Run separate regressions for

Yi ,t = Xi ,t−1βt+Di ,t−1αt+εi ,t , Xi ,t = Xi ,t−1β̃t−1+Di ,t−1α̃t−1+εx ,i ,t .

Prone to large estimation error in high dimensions.

D1 Y1,X2

Dynamic Covariate Balancing

Method

1. Outcome model using potential local projections:

⇒ It does not require to estimate each counterfactual (“adapt” localprojections of Jorda (2005) to dynamic treatments);

2. Estimation balances covariates directly (dynamically)

⇒ It does not require to specify and estimate the propensity score;⇒ It guarantees an asymptotic vanishing bias in high dimensions;⇒ More stable than (A)IPW estimators in the presence of poor overlap;⇒ Generalizes Zubizarreta (2015) and Athey et al. (2018) to dynamics.

Main conditions:

1. Sequential ignorability:

⇒ Treatments are assigned sequentially based on past observations andexogenous unobservables.

2. Linearity:

⇒ Linearity of outcomes on past outcomes and covariates.

Method

Main conditions:

2. Linearity:

Method

Main conditions:

2. Linearity:

Related Literature (Brief Overview)

Dynamic Treatments’ literature and marginal structural models (econ,biostatistics, and PoliSci)

High-dimensional panel data;

Balancing and Dynamic Treatments:

⇒ Here we formally characterizes the high-dimensional bias;⇒ Directly generalizes approximate residual balancing in one-period;⇒ Key intuition: need to estimate different weights for different periods

sequentially.

Other references: covariate balancing in i .i .d . setting, Panel data andDiD, Macroeconomics, ...

Content

1 Introduction

2 Method

4 Conclusions

Basic Model: Example

Basic model that we generalize in the following slides:

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

We can write

Yi ,2(d1, d2) =[Yi ,1(d1),Xi ,1, d1, d2

]β2 + εi ,2

Yi ,1(d1) =[Xi ,1, d1

]β1 + εi ,1

⇒ Yi ,2(d1, d2) is linear in Xi ,1 unconditionally on Yi ,1(d1):

Yi ,2(d1, d2) =[1, 1,Xi ,1

]βd1,d2 + νd1,d2

i ,1 , E[νd1,d2

∣∣∣Xi ,1,Di ,1

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

We can write

Yi ,2(d1, d2) =[Yi ,1(d1),Xi ,1, d1, d2

]β2 + εi ,2

Yi ,1(d1) =[Xi ,1, d1

]β1 + εi ,1

Yi ,2(d1, d2) =[1, 1,Xi ,1

]βd1,d2 + νd1,d2

i ,1 , E[νd1,d2

∣∣∣Xi ,1,Di ,1

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

We can write

Yi ,2(d1, d2) =[Yi ,1(d1),Xi ,1, d1, d2

]β2 + εi ,2

Yi ,1(d1) =[Xi ,1, d1

]β1 + εi ,1

Yi ,2(d1, d2) =[1, 1,Xi ,1

]βd1,d2 + νd1,d2

i ,1 , E[νd1,d2

∣∣∣Xi ,1,Di ,1

Example: Illustration

1. Model on potential outcomes:

Linear dependencies?

D1 Y1,X2

2. Remaining components are left unspecified.

Recall the starting model:

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

⇒ Why not a linear model on observed outcomes?

E[Yi ,2

∣∣∣Xi ,1,Di ,1

]=[Xi ,1,Di ,1

]γ + β2E

[Di ,2|Xi ,1,Di ,1

]︸︷︷︸

problematic

⇒ Model on the observed outcomes also depend on (X1,D1)→ D2;

⇒ Model on potential outcomes is more flexible.

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

E[Yi ,2

∣∣∣Xi ,1,Di ,1

]=[Xi ,1,Di ,1

]γ + β2E

[Di ,2|Xi ,1,Di ,1

]︸︷︷︸

problematic

Yi ,2 =[Yi ,1,Xi ,1,Di ,1,Di ,2

]β2 + εi ,2, εi ,2 ⊥ Di ,2

∣∣∣Di ,1,Xi ,1,Yi ,1

Yi ,1 =[Xi ,1,Di ,1

]β1 + εi ,1, (εi ,1, εi ,2) ⊥ Di ,1

∣∣∣Xi ,1.

E[Yi ,2

∣∣∣Xi ,1,Di ,1

]=[Xi ,1,Di ,1

]γ + β2E

[Di ,2|Xi ,1,Di ,1

]︸︷︷︸

problematic

Model Specification: General Case

Define Hi ,2(d1) =[Xi ,1,Xi ,2(d1),Yi ,1(d1)

](also with intercepts).

E[Yi ,2(d1, d2)

∣∣∣Hi ,2,Di ,1 = d1

]= Hi ,2(d1)β2

d1,d2,

E[Yi ,2(d1, d2)

∣∣∣Xi ,1

]= Xi ,1β

1d1,d2

It holds if Hi ,2(d1) is linear in Xi ,1;

It does not impose structural assumptions on (Di ,1,Di ,2);

Sequential ignorability:

Yi ,2(d1, d2) ⊥ Di ,2

∣∣∣Hi ,2,Di ,1,(Yi ,2(d1, d2),Hi ,1(d1)

)⊥ Di ,1

∣∣∣Xi ,1.

E[Yi ,2(d1, d2)

∣∣∣Hi ,2,Di ,1 = d1

]= Hi ,2(d1)β2

d1,d2,

E[Yi ,2(d1, d2)

∣∣∣Xi ,1

]= Xi ,1β

1d1,d2

Yi ,2(d1, d2) ⊥ Di ,2

∣∣∣Hi ,2,Di ,1,(Yi ,2(d1, d2),Hi ,1(d1)

)⊥ Di ,1

∣∣∣Xi ,1.

E[Yi ,2(d1, d2)

∣∣∣Hi ,2,Di ,1 = d1

]= Hi ,2(d1)β2

d1,d2,

E[Yi ,2(d1, d2)

∣∣∣Xi ,1

]= Xi ,1β

1d1,d2

Yi ,2(d1, d2) ⊥ Di ,2

∣∣∣Hi ,2,Di ,1,(Yi ,2(d1, d2),Hi ,1(d1)

)⊥ Di ,1

∣∣∣Xi ,1.

Identification and Estimation of the Linear model

! E[Yi ,2

∣∣∣Xi ,1,Di ,1 = 1,Di ,2 = 1]6= E

[Yi ,2(1, 1)

∣∣∣Xi ,1

Identification

E[E[Yi ,2

∣∣∣Hi ,2,Di ,1 = d1,Di ,2 = d2

]︸︷︷︸

=E[Yi,2(d1,d2)|Hi,2,Di,1=d1]

∣∣∣Xi ,1,Di ,1 = d1

[Yi ,2(d1, d2)

∣∣∣Xi ,1

Coefficients’ Estimation (with Lasso):

Y2 →[X2,X1,D1 = d1,D2 = d2

]→ H2β̂

2d1,d2

H2β̂d1,d2 →[X1,D1 = d1

]→ X1β̂

1d1,d2

Problem: bias due to high-dimensionality;

Unknown propensity score: (a)IPW can be prone to misspecificationand poor overlap

! E[Yi ,2

∣∣∣Xi ,1,Di ,1 = 1,Di ,2 = 1]6= E

[Yi ,2(1, 1)

∣∣∣Xi ,1

]Identification

E[E[Yi ,2

∣∣∣Hi ,2,Di ,1 = d1,Di ,2 = d2

]︸︷︷︸

=E[Yi,2(d1,d2)|Hi,2,Di,1=d1]

∣∣∣Xi ,1,Di ,1 = d1

[Yi ,2(d1, d2)

∣∣∣Xi ,1

Y2 →[X2,X1,D1 = d1,D2 = d2

]→ H2β̂

2d1,d2

H2β̂d1,d2 →[X1,D1 = d1

]→ X1β̂

1d1,d2

! E[Yi ,2

∣∣∣Xi ,1,Di ,1 = 1,Di ,2 = 1]6= E

[Yi ,2(1, 1)

∣∣∣Xi ,1

]Identification

E[E[Yi ,2

∣∣∣Hi ,2,Di ,1 = d1,Di ,2 = d2

]︸︷︷︸

=E[Yi,2(d1,d2)|Hi,2,Di,1=d1]

∣∣∣Xi ,1,Di ,1 = d1

[Yi ,2(d1, d2)

∣∣∣Xi ,1

Y2 →[X2,X1,D1 = d1,D2 = d2

]→ H2β̂

2d1,d2

H2β̂d1,d2 →[X1,D1 = d1

]→ X1β̂

1d1,d2

! E[Yi ,2

∣∣∣Xi ,1,Di ,1 = 1,Di ,2 = 1]6= E

[Yi ,2(1, 1)

∣∣∣Xi ,1

]Identification

E[E[Yi ,2

∣∣∣Hi ,2,Di ,1 = d1,Di ,2 = d2

]︸︷︷︸

=E[Yi,2(d1,d2)|Hi,2,Di,1=d1]

∣∣∣Xi ,1,Di ,1 = d1

[Yi ,2(d1, d2)

∣∣∣Xi ,1

Y2 →[X2,X1,D1 = d1,D2 = d2

]→ H2β̂

2d1,d2

H2β̂d1,d2 →[X1,D1 = d1

]→ X1β̂

1d1,d2

Balancing: Illustration

Estimating E[Y2(1, 1)]:D2 = 1 D2 = 0

D1 = 1

D1 = 0

Balancing conditions:

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

Estimating E[Y2(1, 1)]:

γ̂1 = 0 γ̂1 = 0

D2 = 1 D2 = 0

D1 = 1

D1 = 0

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

γ̂1 = 0 γ̂1 = 0

γ̂1 γ̂1

D2 = 1 D2 = 0

D1 = 1

D1 = 0

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

γ̂1 = 0 γ̂1 = 0

γ̂1 γ̂1

D2 = 1 D2 = 0

D1 = 1

D1 = 0

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

γ̂2 = 0 γ̂2 = 0

γ̂2 = 0

D2 = 1 D2 = 0

D1 = 1

D1 = 0

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

γ̂2 = 0 γ̂2 = 0

γ̂2 γ̂2 = 0

D2 = 1 D2 = 0

D1 = 1

D1 = 0

γ̂1 :∣∣∣∣∣∣X̄1 − γ̂>1 X1

∣∣∣∣∣∣∞≤ δn, γ̂2 :

∣∣∣∣∣∣γ̂>2 H2 − γ̂>1 H2

∣∣∣∣∣∣∞≤ δn

Estimators

⇒ Balancing to guarantee neglegible (op(1/√n)) estimation error;

⇒ It does not require the specification (and estimation) of thepropensity score.

Two periods: µ̂1,1 = γ̂>2 (Y2 − H2β̂1,1)︸︷︷︸Y2−Pred2

+γ̂>1 (H2β̂1,1 − X1β̂1,1)︸︷︷︸Pred2−Pred1

+ X̄1β̂1,1︸︷︷︸Est1

Step t (Sequential)

γ̂t = argminγt ||γt ||2∣∣∣∣∣∣ n∑

γ̂i ,t−1Hi ,t −n∑

γi ,tHi ,t

∣∣∣∣∣∣∞≤ δ(n, p),

n∑i=1

γi ,t = 1

||γt ||∞ ≤ log(n)n−2/3, 0 ≤ γi ,t ≤t∏

1{Di ,s = ds}

Estimators

⇒ Balancing to guarantee neglegible (op(1/√n)) estimation error;

⇒ It does not require the specification (and estimation) of thepropensity score.

Two periods: µ̂1,1 = γ̂>2 (Y2 − H2β̂1,1)︸︷︷︸Y2−Pred2

+γ̂>1 (H2β̂1,1 − X1β̂1,1)︸︷︷︸Pred2−Pred1

+ X̄1β̂1,1︸︷︷︸Est1

Step t (Sequential)

γ̂t = argminγt ||γt ||2∣∣∣∣∣∣ n∑

γ̂i ,t−1Hi ,t −n∑

γi ,tHi ,t

∣∣∣∣∣∣∞≤ δ(n, p),

n∑i=1

γi ,t = 1

||γt ||∞ ≤ log(n)n−2/3, 0 ≤ γi ,t ≤t∏

1{Di ,s = ds}

Guarantees

Under regularity assumptions:

1. For n large enough, the optimization problem admits a feasiblesolution which includes stabilized inverse-probability weights;

2. For log(pn)/n1/4 → 0 (high-dimensions)

µ̂T (d1:T )− µT (d1:T ) = Op(n−1/2)

i.e., the bias is asymptotic vanishing;

3. Results for asymptotic inference are derived.

Content

1 Introduction

2 Method

4 Conclusions

Numerical Study

Designs: β(j) = 1/j , β(j) = 1/j2

Linear model, different level of overlap.

Advertisment and Elections

ATE DCB ATE AIPW

S1 −1.767 −0.57(0.70) (1.45)

S2 −0.493 −0.22(0.764) (1.33)

Democracy and Growth

⇒ Study the effect from one to twenty years of democracy.

Content

1 Introduction

2 Method

4 Conclusions

Conclusion

Summary:

1. We introduce a method for estimating dynamic causal effects;

2. We introduce an identification strategy and estimation procedurewhich uses novel covariate balancing conditions;

3. We derive asymptotic properties and show feasibility and theparametric rate of convergence.

Questions?

Dynamic Covariate Balancing: Estimating Treatment Effects ...

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