Causal Inference Opening Workshop - A Bracketing Relationship between Difference-in-Differences and Lagged-Dependent-Variable Adjustment - Peng Ding, December 11, 2019
Difference-in-differences is a widely used evaluation strategy that draws causal inference from observational panel data. Its causal identification relies on the assumption of parallel trends, which is scale-dependent and may be questionable in some applications. A common alternative is a regression model that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes. In the context of linear models, Angrist and Pischke (2009) show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect. We show that the same bracketing relationship holds in general nonparametric (model-free) settings. We also extend the result to semiparametric estimation based on inverse probability weighting.
Hypothesis testings on individualized treatment rulesYoung-Geun Choi
Invited talk in Joint Statistical Meetings 2017, Baltimore, Maryland.
Individualized treatment rules (ITR) assign treatments according to different patient's characteristics. Despite recent advances on the estimation of ITRs, much less attention has been given to uncertainty assessments for the estimated rules. We propose a hypothesis testing procedure for the estimated ITRs from a general framework that directly optimizes overall treatment benefit. Specifically, we construct a local test for testing low dimensional components of high-dimensional linear decision rules. Our test extends the decorrelated score test proposed in Nang and Liu (2017) and is valid no matter whether model selection consistency for the true parameters holds or not. The proposed methodology is illustrated with numerical study and data examples.
2017-07, Research Seminar at Keio University, Metric Perspective of Stochasti...asahiushio1
In this talk, I explain several major stochastic optimizers from the perspective of the metric, that is the definition of the parameter space of the model.
Model Selection with Piecewise Regular GaugesGabriel Peyré
Talk given at Sampta 2013.
The corresponding paper is :
Model Selection with Piecewise Regular Gauges (S. Vaiter, M. Golbabaee, J. Fadili, G. Peyré), Technical report, Preprint hal-00842603, 2013.
http://hal.archives-ouvertes.fr/hal-00842603/
Asset Prices in Segmented and Integrated Marketsguasoni
This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.
Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospect...Jaehyun Song
社会科学方法論研究会 (2018年3月9日)報告資料
Title: Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia.” Political Analysis. 22 (4) pp. 434-456
Data: Mar. 9, 2018
Location: Social Science Methodology Workshop, Osaka University
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
I will discuss paradigmatic statistical models of inference and learning from high dimensional data, such as sparse PCA and the perceptron neural network, in the sub-linear sparsity regime. In this limit the underlying hidden signal, i.e., the low-rank matrix in PCA or the neural network weights, has a number of non-zero components that scales sub-linearly with the total dimension of the vector. I will provide explicit low-dimensional variational formulas for the asymptotic mutual information between the signal and the data in suitable sparse limits. In the setting of support recovery these formulas imply sharp 0-1 phase transitions for the asymptotic minimum mean-square-error (or generalization error in the neural network setting). A similar phase transition was analyzed recently in the context of sparse high-dimensional linear regression by Reeves et al.
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Hypothesis testings on individualized treatment rulesYoung-Geun Choi
Invited talk in Joint Statistical Meetings 2017, Baltimore, Maryland.
Individualized treatment rules (ITR) assign treatments according to different patient's characteristics. Despite recent advances on the estimation of ITRs, much less attention has been given to uncertainty assessments for the estimated rules. We propose a hypothesis testing procedure for the estimated ITRs from a general framework that directly optimizes overall treatment benefit. Specifically, we construct a local test for testing low dimensional components of high-dimensional linear decision rules. Our test extends the decorrelated score test proposed in Nang and Liu (2017) and is valid no matter whether model selection consistency for the true parameters holds or not. The proposed methodology is illustrated with numerical study and data examples.
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Talk given at Sampta 2013.
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Model Selection with Piecewise Regular Gauges (S. Vaiter, M. Golbabaee, J. Fadili, G. Peyré), Technical report, Preprint hal-00842603, 2013.
http://hal.archives-ouvertes.fr/hal-00842603/
Asset Prices in Segmented and Integrated Marketsguasoni
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Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospect...Jaehyun Song
社会科学方法論研究会 (2018年3月9日)報告資料
Title: Review: Cyrus, Samii, Laura Paler, and Sarah Zukerman Daly. 2016. “Retrospective Causal Inference with Machine Learning Ensembles: An Application to Anti-recidivism Policies in Colombia.” Political Analysis. 22 (4) pp. 434-456
Data: Mar. 9, 2018
Location: Social Science Methodology Workshop, Osaka University
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
Similar to Causal Inference Opening Workshop - A Bracketing Relationship between Difference-in-Differences and Lagged-Dependent-Variable Adjustment - Peng Ding, December 11, 2019 (20)
Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
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Causal Inference Opening Workshop - A Bracketing Relationship between Difference-in-Differences and Lagged-Dependent-Variable Adjustment - Peng Ding, December 11, 2019
1. A bracketing relationship between
difference-in-differences and
lagged-dependent-variable adjustment
Peng Ding
UC Berkeley, Statistics
December 10, 2019 at SAMSI
With Fan Li at Duke Statistics, published in Political Analysis
1 / 20
2. Classic Card and Krueger (1994)
▶ Minimum wage increased in New Jersey in 1992, not in Pennsylvania
▶ Observed employment in fast food restaurants before and after
▶ Figure from Angrist and Pischke (2008) book:
2 / 20
3. The basic two-period two-group panel design
▶ Units: i = 1, . . . , n
▶ Two periods—“before” and “after”: T = t, t + 1
▶ Two groups—control and treatment: Gi = 0, 1
▶ Treatment is only assigned to group Gi = 1 in the “after” period
▶ DiT : the observed treatment status at time T
▶ Dit ≡ 0 (all control in the “before” period)
▶ Di,t+1 = 1 for the units in group Gi = 1 =⇒ Gi = Di,t+1
▶ Outcome YiT : i = 1, . . . , n and T = t, t + 1
3 / 20
4. Potential outcomes and causal effects
▶ Potential outcomes {YiT (1), YiT (0)}
▶ Observed outcomes: YiT = YiT (DiT )
before T = t after T = t + 1
control group G = 0 Yit = Yit(0) Yi,t+1 = Yi,t+1(0)
treatment group G = 1 Yit = Yit(0) Yi,t+1 = Yi,t+1(1)
▶ Causal estimand — average effect on the treated:
τATT = E{Yi,t+1(1) − Yi,t+1(0) | Gi = 1} = µ1 − µ0
▶ µ1 = E{Yi,t+1(1) | Gi = 1} = E(Yi,t+1 | Gi = 1) identifiable
▶ µ0 = E{Yi,t+1(0) | Gi = 1}: counterfactual
▶ key: inferring µ0 based on observables 4 / 20
5. Difference-in-differences (DID)
Assumption (Parallel trends conditioning on covariates Xi )
E{Yi,t+1(0) − Yi,t(0) | Xi , Gi = 1} = E{Yi,t+1(0) − Yi,t(0) | Xi , Gi = 0}
▶ Nonparametric identification of µ0 = E{Yi,t+1(0) | Gi = 1}
µ0,DID = E [E{Yit (0) | Xi , Gi = 1} + E{Yi,t+1(0) − Yit (0) | Xi , Gi = 0} | Gi = 1]
= E(Yit | Gi = 1) + E{E(Yi,t+1 − Yit | Xi , Gi = 0) | Gi = 1}
▶ Without covariates — difference-in-difference
▶ nonparametric identification:
µ0 = E(Yit | Gi = 1) + E(Yi,t+1 | Gi = 0) − E(Yit | Gi = 0)
▶ moment estimator: ˆτDID = ( ¯Y1,t+1 − ¯Y1,t) − ( ¯Y0,t+1 − ¯Y0,t)
5 / 20
6. The scale dependent issue of DID
▶ Parallel trends may hold for the original Y but not for a nonlinear
monotone transformation of Y , for example, log Y
▶ This restricts the use of DID in general settings
▶ Athey and Imbens (2006): “parallel trends” on the CDF level
▶ Sofer et al. (2016): an negative outcome control approach
6 / 20
7. Lagged-dependent-variable adjustment (LDV)
Assumption (Ignorability conditional on lagged dependent variable)
Yi,t+1(0) ⊥⊥ Gi | (Yit, Xi )
▶ Nonparametric identification of µ0 (conditioning on X implicitly):
µ0,LDV = E{E(Yt+1 | G = 0, Yt) | G = 1}
=
∫
E(Yt+1 | G = 0, Yt = y)FYt (dy | G = 1)
▶ FYt (y | G = g) = pr(Yt ≤ y | G = g)
▶ The assumption is scale-free
7 / 20
8. A bracketing relationship based on linear model fitting
a little more general than Angrist and Pischke (2009)
▶ Ignore covariates X
▶ Two versions of LDV
▶ fit ˆE(Yt+1 | G = 0, Yt = y) = ˆα + ˆβYt under control:
ˆτLDV = ( ¯Y1,t+1 − ¯Y0,t+1) − ˆβ( ¯Y1,t − ¯Y0,t)
▶ fit ˆE(Yt+1 | G, Yt) = ˆα + ˆτ′
LDVG + ˆβ′
Yt using all units:
ˆτ′
LDV = ( ¯Y1,t+1 − ¯Y0,t+1) − ˆβ′
( ¯Y1,t − ¯Y0,t)
▶ Compared to DID
ˆτDID = ( ¯Y1,t+1 − ¯Y0,t+1) − ( ¯Y1,t − ¯Y0,t)
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9. Interpreting the bracketing relationship under linear models
▶ Consider the case with ˆβ or ˆβ′ smaller than 1
▶ The sign of ˆτDID − ˆτLDV or ˆτDID − ˆτ′
LDV depends on the sign of ¯Y1,t − ¯Y0,t
▶ Treatment group has smaller Yt on average =⇒ ˆτDID > ˆτLDV
▶ Treatment group has larger Yt on average =⇒ ˆτDID < ˆτLDV
▶ How much ˆβ or ˆβ′ deviates from 1
=⇒ how different the DID and LDV estimates are
▶ They are identical if ˆβ = 1 or ˆβ′ = 1
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10. A lemma: conditioning on X implicitly
Lemma
The difference between µ0,DID and µ0,LDV is
µ0,LDV − µ0,DID =
∫
∆(y)FYt (dy | G = 1) −
∫
∆(y)FYt (dy | G = 0)
▶ Depends on average changed outcome given Yt in control group:
∆(y) = E(Yt+1 | G = 0, Yt = y) − y
= E(Yt+1 − Yt | G = 0, Yt = y)
▶ Depends on the difference between the distribution of Yt in the
treated and control groups
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11. Two testable conditions
Condition (Stationarity)
∂E(Yt+1 | G = 0, Yt = y)/∂y < 1 for all y.
▶ Linear model: in the control group, the coefficient of the outcome
Yt+1 on Yt is smaller than 1 (Angrist and Pischke 2009)
▶ The time series of the outcomes would not evolve to infinity
Condition (Stochastic Monotonicity)
(1) FYt (y | G = 1) ≥ FYt (y | G = 0) for all y;
(2) FYt (y | G = 1) ≤ FYt (y | G = 0) for all y.
▶ (1) implies that the treated group has smaller lagged outcome
▶ (2) implies the opposite relationship
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12. A theorem
Theorem
If Stationarity and Stochastic Monotonicity(1) hold, then
µ0,DID ≤ µ0,LDV, τDID ≥ τLDV.
If Stationarity and Stochastic Monotonicity(2) hold, then
µ0,DID ≥ µ0,LDV, τDID ≤ τLDV.
▶ It does not require the parallel trends or the ignorability
▶ Simply a result on the relative magnitude between τDID and τLDV
▶ Extends Angrist and Pischke (2009) to the nonparametric setting
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13. Interpretations of the theorem
▶ Under Stationarity and Stochastic Monotonicity(1),
τDID ≥ τLDV
▶ Both of them can be biased for the true causal effect τATT
▶ τDID ≥ τLDV ≥ τATT =⇒ τDID over-estimates τATT more than τLDV
▶ τATT ≥ τDID ≥ τLDV =⇒ τLDV under-estimates τATT more than τDID
▶ τDID ≥ τATT ≥ τLDV =⇒ τDID and τLDV are the upper and lower bounds
▶ In the last case, [τLDV, τDID] bracket the true causal effect
▶ Analogous results under Stationarity and Stochastic Monotonicity(2)
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14. Example 1: Card and Krueger (1994) study
▶ Effect of a minimum wage increase on employment
▶ Employment information in New Jersey and Pennsylvania before and
after a minimum wage increase in New Jersey in 1992
▶ Outcome = # employees at each fast food restaurant
▶ Estimates:
ˆτDID = 2.446, ˆτLDV = 0.302, ˆτ′
LDV = 0.865
▶ Coefficients of the lag outcome ˆβ = 0.288 < 1 and ˆβ′ = 0.475 < 1
▶ The same conclusion under a quadratic model
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15. Example 1: graphical checks
10 20 30 40 50 60 70
5102030
G = 0
Yt
Yt+1
0 20 40 60
0.00.20.40.60.81.0
Yt
FYt
(y|G)
G=1
G=0
Left: linear and quadratic fitted lines of E(Yt+1 | G = 0, Yt).
Right: FYt (y | G = g) (g = 0, 1) satisfy Stochastic Monotonicity.
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16. Example 2: Bechtel and Hainmueller (2011) study
▶ Electoral returns to beneficial policy
▶ We focus on the short-term electoral returns by analyzing the causal
effect of a disaster relief aid due to the 2002 Elbe flooding in Germany
▶ Before period = 1998; After period = 2002
▶ The units of analysis are electoral districts
▶ Treatment = the indicator whether a district is affected by the flood
▶ Outcome = the vote share that the Social Democratic Party attains
▶ Estimates:
ˆτDID = 7.144, ˆτLDV = 7.160, ˆτ′
LDV = 7.121
▶ Coefficients of the lag outcome ˆβ = 1.002 > 1 and ˆβ′ = 0.997 < 1
▶ These estimates are almost identical 16 / 20
17. Example 2: graphical checks
30 40 50 60
2030405060
G = 0
Yt
Yt+1
30 40 50 60
0.00.20.40.60.81.0
Yt
FYt
(y|G)
G=1
G=0
Left: linear fitted lines of E(Yt+1 | G = 0, Yt).
Right: FYt (y | G = g) (g = 0, 1) satisfy Stochastic Monotonicity.
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18. Example 3: data
▶ Evaluating the effects of rumble strips on vehicle crashes
▶ Units: n = 1986 road segments in Pennsylvania
▶ Crash counts before (year 2008) and after (year 2012) the intervention
Table: Crash counts (3+ means 3 or more crashes).
(a) control group G = 0 (b) treated group G = 1
Yt+1
0 1 2 3+
Yt
0 789 238 57 18
1 235 95 40 15
2 61 37 11 6
3+ 26 21 4 2
Yt+1
0 1 2 3+
Yt
0 183 39 7 3
1 40 22 5 2
2 16 4 0 1
3+ 2 6 0 1
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19. Example 3: results
▶ Stationarity holds for all y = 0, 1, 2, 3+:
ˆE(Yt+1 | G = 0, Yt = y) = .374, .572, .670, .660
▶ Stochastic Monotonicity(1) holds for y = 0, 1, 2, pr(Yt ≤ y | G = g)
(700, .909, .973) for g = 1; (666, .898, .968) for g = 0
▶ Nonparametric estimate of µ0 under ignorability:
ˆµ0,LDV =
∑
y
ˆE(Yt+1 | G = 0, Yt = y)pr(Yt = y | G = 1) = .438
▶ Under the parallel trend: ˆµ0,DID = .395
▶ Matches the theoretical prediction
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20. Discussion
▶ Create a super-model that incorporates both DID and LDV
▶ requires multiple time periods: T = t + 1, . . . , t + K
E(Yi,T | Xi , Yi,T−1, Gi ) = αi + λT + βYi,T−1 + τGi + θ⊤
Xi
▶ Nickell (1981) and Hausman–Taylor (1981) identification and
estimation under this model require much stronger assumptions
▶ Practical suggestion
▶ assumptions for DID and LDV: not nested, cannot be validated by data
▶ report results from both approaches
▶ conduct sensitivity analyses allowing for violations of these assumptions
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