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Causal mediation analysis for difference-in-difference design and panel data

  • Pei-Hsuan Hsia , An-Shun Tai , Chu-Lan Michael Kao , Yu-Hsuan Lin und Sheng-Hsuan Lin ORCID logo EMAIL logo
Veröffentlicht/Copyright: 14. Juli 2025
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Epidemiologic Methods
Aus der Zeitschrift Epidemiologic Methods Band 14 Heft 1

Abstract

Objectives

Advantages of panel data, i.e., difference in difference (DID) design data, are a large sample size and easy availability. Therefore, panel data are widely used in epidemiology and in all social science fields. The literature on causal inference in panel data settings or DID designs has been expanding, although existing studies often focus on the exposed group (or treated group), limiting the development of mediation analysis methods.

Methods

In this study, we present a methodology for conducting causal mediation analysis in a DID design and panel data setting that encompasses the entire population including both exposed and unexposed groups, by proposing a general common trend assumption which has a form of exchangeability. We provide formal counterfactual definitions for controlled direct effect and natural direct and indirect effect in panel data setting and DID design, including the identification and required assumptions. We also demonstrate that, under the assumptions of linearity and additivity, controlled direct effects can be estimated by contrasting marginal and conditional DID estimators whereas natural indirect effects can be estimated by calculating the product of the exposure-mediator DID estimator and the mediator-outcome DID estimator. A panel regression-based approach is also proposed.

Results

The proposed method is then used to investigate mechanisms of the effects of the Covid-19 pandemic on the mental health status of the population. The results revealed that mobility restrictions mediated approximately 45 % of the causal effect of Covid-19 on mental health status.

Conclusions

The proposed approach offers a framework for mediation analysis in DID and panel data settings, addressing limitations of existing studies by including both exposed and unexposed groups.

Keywords: causal inference; mediation analysis; difference-in-difference; panel data
Corresponding author: Sheng-Hsuan Lin, Institute of Statistics, National Yang Ming Chiao Tung University, 1001 University Road, Hsinchu, 30010, Taiwan, E-mail:

Funding source: Ministry of Science and Technology in Taiwan

Award Identifier / Grant number: No. 109-2636-B-009 -001

Funding source: National Health Research Institutes

Award Identifier / Grant number: 14A1-PHGP10-052

  1. Research ethics: All data are open-access and country-wide statistics without containing any individual information. No ethical approval is required.

  2. Informed consent: Not applicable.

  3. Author contributions: SHL and KCL came up the original idea. HPH wrote the first version of manuscript. All authors edited the manuscript. SHL and KCL provided suggestions to refine the model and example data. YHL and HPH conducted data analysis.

  4. Use of Large Language Models, AI and Machine Learning Tools: An AI language model was used to improve the language and grammar of the manuscript.

  5. Conflict of interest: The authors have no conflicts of interest relevant to this article.

  6. Research funding: This study is supported by the grant from Ministry of Science and Technology in Taiwan (No. 109-2636-B-009-001) and National Health Research Institutes (14A1-PHGP10-052).

  7. Data availability: The data source analyzed in this study includes (1) Google search volumes of “insomnia”, (2) cell phone usage at the residence, (2) anonymized cell phone location data, and (3) Covid-19 confirmed case and death number statistics, from March 2020, to February 2021 in 40 countries. All data are open-access and country-wide statistics without containing any individual information. No ethical approval is required.

Appendix A: Counter examples for generalized common trend assumption and exchangeability assumption

Condition C1 (exchangeability assumption): Y it (a) ⊥ A i |Y i0 ∀a

Condition C2 (generalized common trend assumption): Y it (a) − Y i0(a) ⊥ A i ∀a

1 C1 does not imply C2

Consider the following example (as in Figure S1. (A)): let ϵ i be i.i.d. standard normal random variables. Let

Y i 0 = ε 1   Y it ( a ) = a − Y i 0 + ε 2   A i = Y i 0 + ε 3  

Figure S1: DAGs of the examples, showing that generalized common trend assumption and exchangeability assumption do not imply each other. In the sequential exchangeability assumption, Y it is directly influenced by Y i0. However, under the common trend assumption, there is no direct causal relationship between these two variables. Instead, they are both affected by a common cause, denoted as ε 1, which represents an underlying individual-level baseline factor. This factor influences not only Y it and Y i0 but also A i , thereby aligning with the core nature of the common trend assumption, which emphasizes homogeneity across exposed and unexposed group rather than individual-level exchangeability. Moreover, the common cause of Y it and A i violates exchangeability assumption because of residual confounding.
Figure S1:

DAGs of the examples, showing that generalized common trend assumption and exchangeability assumption do not imply each other. In the sequential exchangeability assumption, Y it is directly influenced by Y i0. However, under the common trend assumption, there is no direct causal relationship between these two variables. Instead, they are both affected by a common cause, denoted as ε 1, which represents an underlying individual-level baseline factor. This factor influences not only Y it and Y i0 but also A i , thereby aligning with the core nature of the common trend assumption, which emphasizes homogeneity across exposed and unexposed group rather than individual-level exchangeability. Moreover, the common cause of Y it and A i violates exchangeability assumption because of residual confounding.

Then,

  1. Given Y i0=y, (Y it , A i ) is a bivariate normal with mean vector (−y, y) and covariance matrix I 2, the 2 × 2 identity matrix. Hence, Y it ⊥ A i |Y g0.

  2. Since Y it − Y i0=a − 2Y g0 + ϵ 2, we have Cov (Y it − Y i0, A i )=Cov (−2Y i0, Y i0) = −2. In other words, Y it − Y i0is not independent to A i

2 C2 does not imply C1

Consider the following example (as in Figure S1. (B)): let ϵ i be i.i.d. standard normal random variables. Let

Y i 0 = ε 1 + ε 2   Y it ( a ) = a + ε 1 + ε 3   A i = ε 1 + ε 4  

Then,

  1. Since Y it − Y i0=ϵ 3 − ϵ 2 does not involve ϵ 1 or ϵ 4, it is independent to A i .

  2. Note that we can write Y it =Y i0 − ϵ 2 + ϵ 3 and A i =Y i0 − ϵ 2 + ϵ 4. This means that

Cov Y i t , A i | Y i 0 = Cov − ε 2 + ε 3 , − ε 2 + ε 4 = − 1

So Y it and A i are not conditionally independent.

Appendix B: Identification of total effect

Assumptions

NEPT

  • (A1) Y i 0 = Y i 0 a , for all a

Mean Exchangeability

  • (CT-0) E Y i t a − Y i 0 a = E Y i t a − Y i 0 a | A i = a ′ ∀ a ′

Detailed Process

T E : = E Y i t 1 − E Y i t 0 = E Y i t 1 − E Y i 0 − E Y i t 0 − E Y i 0 by A 1

With assumptions (A1) and (CT-0), it could be identified as follows.

Let ϕ TE(a) be a causal parameter for TE.

ϕ TE a : = Y i t a − E Y i 0 = E Y i t a − E Y i 0 a by A 1 = E Y i t a − Y i 0 a = E Y i t a − Y i 0 a | A i = a by CT  -  0 = E Y i t − Y i 0 | A i = a by consistency = E Y i t | A i = a − E Y i 0 | A i = a

T E = ϕ T E a 1 − ϕ T E a 0 = E Y i t − Y i 0 | A i = a 1 − E Y i t − Y i 0 | A i = a 0 = E Y i t | A i = a 1 − E Y i 0 | A i = a 1 − E Y i t | A i = a 0 − E Y i 0 | A i = a 0

Appendix C: Identification of controlled direct effect

Assumptions

NEPT

  • (A2) Y i 0 = Y i 0 a , m , for all a and m

Mean Exchangeability

  • (CT-1) E Y i t a , m − Y i 0 a , m = E Y i t a , m − Y i 0 a , m | A i = a ′ ∀ a ′

  • (CT-2) E Y i t a , m − Y i 0 a , m | A i = a ′ = E Y i t a , m − Y i 0 a , m | M i t = m , A i = a ′ ∀ a ′ , m

Detailed Process

C D E m = E Y i t a 1 , m − E Y i t a 0 , m = E Y i t a 1 , m − E Y i 0 − E Y i t a 0 , m − E Y i 0 by A 2

With assumptions (A2), (E1) and (E2), it could be identified as follows.

Let ϕ CDE(a) be a causal parameter for CDE.

ϕ CDE a : = E Y i t a , m − E Y i 0 = E Y i t a , m − E Y i 0 a , m by A 2 = E Y i t a , m − Y i 0 a , m = E Y i t a , m − Y i 0 a , m | A i = a by CT 1 = E Y i t a , m − Y i 0 a , m | A i = a , M i t = m by CT 2 = E Y i t − Y i 0 | A i = a , M i t = m by consistency

C D E m = ϕ CDE a 1 − ϕ CDE a 0 = E Y i t − Y i 0 | A i = a 1 , M i t = m − E Y i t − Y i 0 | A i = a 0 , M i t = m = E Y i t | A i = a 1 , M i t = m − E Y i 0 | A i = a 1 , M i t = m − E Y i t | A i = a 0 , M i t = m − E Y i 0 | A i = a 0 , M i t = m

Appendix D: Identification of natural direct effect and natural indirect effect

Assumptions

NEPT

  • (A1) Y i 0 = Y i 0 a , for all a

  • (A2) Y i 0 = Y i 0 a , m , for all a and m

  • (A3) Y i 0 = Y i 0 a , M i t a * , for all a and a *

  • (A4) M i 0 = M i 0 a , f o r a l l a

Mean Exchangeability

  • (CT-1) E Y i t a , m − Y i 0 a , m = E Y i t a , m − Y i 0 a , m | A i = a ′ ∀ a ′

  • (CT-2) E Y i t a , m − Y i 0 a , m | A i = a ′ = E Y i t a , m − Y i 0 a , m | M i t = m , A i = a ′ ∀ a ′ , m

  • (CT-3) E M i t a − M i 0 a = E M i t a − M i 0 a | A i = a ′ ∀ a ′

  • (CT-4) E Y i t a , m − Y i 0 a , m = E Y g i a , m − Y i 0 a , m | M i t a * = m ∀ a , a * , m

Detailed Process

With assumptions (A1)–(A4) and (CT-1)–(CT-4).

Let ϕ NE,Y(a, a *) be a causal parameter for the effect of M it on Y it .

ϕ NE , Y a , a * : = E Y i t a , M i t a * − E Y i 0 = E Y i t a , M i t a * − E Y i 0 a , M i t a * by A 3 = ∫ m E Y i t a , m − Y i 0 a , m | M i t a * = m f M i t a * m d m = ∫ m E Y i t a , m − Y i 0 a , m f M i t a * m d m by CT 4 = ∫ m E Y i t a , m − Y i 0 a , m | A i = a f M i t a * m d m by CT 1 = ∫ m E Y i t a , m − Y i 0 a , m | A i = a , M i t = m f M i t a * m i t d m by CT 2 = ∫ m E Y i t − Y i 0 | A i = a , M i t = m i t f M i t a * m d m by consistency

And with (A4) and (CT-3), let ϕ NE,M(a i ) be a causal parameter for the effect of A i on M it , we can also identify ϕ NE,M(a i ):

ϕ NE , M a i : = E M i t a − E M i 0 = E M i t a − E M i 0 a by A 4 = E M i t a − M i 0 a = E M i t a − M i 0 a | A i = a by CT 3 = E M i t − M i 0 | A i = a by consistency

The rest of the identification is related to the variable type of M it , so it will be discussed in the section of estimation.

Since M it is binary, the integration of m could be reduced to a summation of M it =1 and M it =0.

ϕ NE , Y a , a * = ∑ m = 0 1 E Y i t − Y i 0 | A i = a , M i t = m Pr M i t = m | A i = a * = Pr M i t = 1 | A i = a * E Y i t − Y i 0 | A i = a , M i t = 1 − E Y i t − Y i 0 | A i = a , M i t = 0 + E Y i t − Y i 0 | A i = a , M i t = 0

Then the process of identification is finished, it contains no more counterfactual terms. The original definition of NDE is E[Y it (a 1, M it (a 0)) − Y it (a 0, M it (a 0))] and NIE is E[Y it (a 1, M it (a 1)) − Y it (a 1, M it (a 0))], but by assumption (A3), it can be extended to ϕ NE,Y(a 1, a 0) − ϕ NE,Y(a 0, a 0) and ϕ NE,Y(a 1, a 1) − ϕ NE,Y(a 1, a 0). Then we can use the result of identified ϕ NE,Y(a, a *), and also estimate NDE and NIE by an DID estimator of Y.

N I E = E Y i t a 1 , M i t a 1 − Y i t a 1 , M i t a 0 = ϕ NE , Y a 1 , a 1 − ϕ NE , Y a 1 , a 0 = E M i t | A i = a 1 − E M i 0 | A i = a 1 − E M i t | A i = 0 − E M i 0 | A i = 0 E Y i t | A i = a 1 , M i t = 1   − E Y i 0 | A i = a 1 , M i t = 1 − E Y i t | A i = a 1 , M i t = 0 − E Y i 0 | A i = a 1 , M i t = 0

Appendix E: Equivalence of difference and product method

E Y i t | A i t = a i t = γ A a i t + γ t + γ i Model 1 E Y i t | A i t = a i t , M i t = m i t = θ A a i t + θ M m i t + θ t + θ i Model 2 E M i t | A i t = a i t = β A a i t + β t + β i Model 3

The equivalence of γ A ̂ − θ A ̂ and θ M ̂ β A ̂ :

E Y i t | A i t = a i t = ∑ m E Y i t | A i t = a i t , M i t = m i t × Pr M i t | A i t = a i t = ∑ m θ A a i t + θ M m i t + θ t + θ i × Pr M i t | A i t = a i t = θ A a i t + θ t + θ i + θ M ∑ m m i t × Pr M i t | A i t = a i t = θ A a i t + θ t + θ i + θ M E M i t | A i t = a i t = θ A a i t + θ t + θ i + θ M β A a i t + β t + β i = θ A a i t + θ M β A a i t + θ t + θ M β t + θ i + θ M β i γ A a i t + γ t + γ i = θ A a i t + θ M β A a i t + θ t + θ M β t + θ i + θ M β i ⇒ γ A − θ A a i t = θ M β A a i t

Appendix F: Country list for illustration data

Fourty countries: Argentina, Australia, Austria, Bolivia, Brazil, Canada, Chile, Colombia Croatia, Denmark, Egypt, Finland, France, Germany, Indonesia, Iraq, Ireland, Israel, Italy, Japan, Malaysia, Mexico, Netherlands, New Zealand, Peru, Philippines, Poland, Portugal, Saudi Arabia, Singapore, South Korea, Spain, Sweden, Taiwan, Thailand, Turkey, United Arab Emirates, United Kingdom, United States, and Venezuela.

Appendix G: Summary statistics for variables used in illustration

See Table S1.

Table S1:

The summary statistics for all variables across 40 countries and 12 months.

Mean Minimum Maximum SD
Covid-19 outbreak 128 0 3,137 311.5
Extent of mobility restriction 10.83 −2.133 39.57 7.363
Search volume 45.85 0 91 21.1

Appendix H: Built models for estimation of illustration

The built models for the search volumes of ‘insomnia’ and the extent of mobility restriction.

E [(search volume of keyword) it | (the extent of national covid-19 outbreak) it] = γ A  × (the extent of national covid-19 outbreak) it + γ t + γ i (Model 4)

E [(search volume of keyword) it | (extent of mobility restriction) it, (the extent of national covid-19 outbreak) it] = θ M  × (extent of mobility restriction) it + θ A  × (the extent of national covid-19 outbreak) it + θ t + θ i (Model 5)

E [(extent of mobility restriction) it | (the extent of national covid-19 outbreak) it] = β A  × (the extent of national covid-19 outbreak) it + β t + β i (Model 6)

Appendix I: Coefficient estimations in Model 2–4 of regression-based estimation.

See Table S2.

Table S2:

Coefficient estimations in Model 2–4.

Estimate, s SE 95 % CI p-Value
γ A 1.41 0.72 (−0.004, 2.82) 0.051
β A 0.75 0.28 (0.19, 1.31) 0.008
θ A 1 0.71 (−0.40, 2.39) 0.16
θ M 0.55 0.12 (0.32, 0.77) <0.001

Appendix J: Comparison of our work and previous studies

See Table S3.

  1. Exchangeability assumptions for time varying variables:

    1. Y a m ̄ ⊥ A t ∣ A ̄ t − 1 , M ̄ t − 1 , L ̄ t − 1 , C ∀ t

    2. Y a m ̄ ⊥ M t ∣ A ̄ t , M ̄ t − 1 , L ̄ t − 1 , C ∀ t

    3. M a ̄ t ⊥ A t ∣ A ̄ t − 1 , M ̄ t − 1 , L ̄ t − 1 , C ∀ t

  2. Consistency assumption: Y(a)=Y when A=a, M(a)=M when A=a, and Y(a, m)=Y when A=a and M=m

  3. Randomization assumption for instrumental variable T:

    1. (Y(t), A(t)) ⊥ T|C, ∀t ∈ {0, 1}

    2. (Y b , A(t)) ⊥ T|C, where Y b denote the baseline value of Y.

  4. Treatment randomization of the exposure: (a) Y(a, m) ⊥ A∣C, (b) M(a) ⊥ A∣C, given the confounders denoted by C

  5. Common trend assumptions in certain conditions including:

    1. Common trends for compliers and never takers under m=0 and a=0

    2. Common trends for compliers and always takers under m=0 and a=0 and homogeneous mean effects of A and M jointly across compliers and always takers/Zero direct effect on always takers

    3. Common trends for compliers and never takers under z=1 and d=0

    4. Common trends for compliers and always takers under a=0 and m=0 and homogeneous mean effect of D across compliers and always takers

  6. Extended common trend assumption:

    1. E Y i t a , m − Y i 0 a , m = E Y i t a , m − Y i 0 a , m | A i = a ∀ a

    2. E Y i t a , m − Y i 0 a , m | A i = a = E Y i t a , m − Y i 0 a , m | M i t = m , A i = a ∀ a , m

    3. E M i t ( a ) − M i 0 ( a ) = E M i t ( a ) − M i 0 ( a ) | A i = a ∀ a

    4. E Y i t a , m − Y i 0 a , m | A i = a = E Y i t a , m − Y i 0 a , m | M i t a * = m , A i = a ∀ a , a * , m

Table S3:

Comparison of our work and previous studies.

VanderWeele et al. (2017) Sawada (2019) Blackwell et al. (2022) Deuchert et al. (2019) Huber et al. (2022) This work
Difference in difference (DID) design No Yes Yes Yes Yes Yes
Estimated with DID estimates No No No No No Yes
Exchangeability assumption Yes, exchangeability assumptions for time varying variables1 are needed. No, but randomize assumptions for instrumental variable3 is assumed. Yes, treatment randomization of the exposure4 is assumed, and mediator parallel trends. Yes, treatment randomization of the exposure4 is assumed. Yes, treatment randomization of the exposure4 is assumed. No, extended common trend assumptions6 are used in identification.
Target population Entire population Entire population Entire population Defiers are excluded due to weak monotonicity assumption of the effect of A on M Defiers are excluded due to weak monotonicity assumption of the effect of A on M Entire population
Other assumptions Consistency assumptions2/random draw from the distribution of the mediator Weakened SUTVA (Stable Unit Treatment Value Assumption) Consistency assumptions2 Consistency assumptions2/Weak monotonicity assumption of A on M/Common trend assumptions under certain conditions5/No anticipation effect of M and A in the baseline period/Zero direct effect on always takers Consistency assumptions2/Weak monotonicity assumption of A on M/Strict monotonicity of Y(a,m)/No anticipation effect of M and A in the baseline period Consistency assumptions2/No effect at baseline period/No anticipation effect of M and A in the baseline period
Total effect Yes, average total effect over time could be estimated Yes, estimated as quantile difference. Not included. Yes Yes, estimated with quantile method Yes, estimated with DID estimate
Controlled direct effect Yes, average controlled direct effect over time could be estimated Not included. Yes, estimated with inverse probability weighting. Not included. Not included. Yes, estimated with DID estimate
Natural direct and indirect effect Yes, average natural direct and indirect effect over time could be estimated. Not included. Not included. Yes Yes, estimated with quantile method. Yes, estimated with DID estimate

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Received: 2024-09-30
Accepted: 2025-06-02
Published Online: 2025-07-14

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Causal mediation analysis for difference-in-difference design and panel data
  2. Research Articles
  3. Sensitivity analysis for unmeasured confounding for a joint effect with an application to survey data
  4. Investigating the association between school substance programs and student substance use: accounting for informative cluster size
  5. The quantiles of extreme differences matrix for evaluating discriminant validity
  6. Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions
  7. What if dependent causes of death were independent?
  8. Bot invasion: protecting the integrity of online surveys against spamming
  9. A study of a stochastic model and extinction phenomenon of meningitis epidemic
  10. Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis
  11. Time-varying reproductive number estimation for practical application in structured populations
  12. Perspective
  13. Should we still use pointwise confidence intervals for the Kaplan–Meier estimator?
https://doi.org/10.1515/em-2024-0025
Schlagwörter für diesen Artikel
causal inference; mediation analysis; difference-in-difference; panel data

Artikel in diesem Heft

  1. Causal mediation analysis for difference-in-difference design and panel data
  2. Research Articles
  3. Sensitivity analysis for unmeasured confounding for a joint effect with an application to survey data
  4. Investigating the association between school substance programs and student substance use: accounting for informative cluster size
  5. The quantiles of extreme differences matrix for evaluating discriminant validity
  6. Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions
  7. What if dependent causes of death were independent?
  8. Bot invasion: protecting the integrity of online surveys against spamming
  9. A study of a stochastic model and extinction phenomenon of meningitis epidemic
  10. Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis
  11. Time-varying reproductive number estimation for practical application in structured populations
  12. Perspective
  13. Should we still use pointwise confidence intervals for the Kaplan–Meier estimator?
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