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When are time series predictions causal? The potential system and dynamic causal effects

Jacob Carlson, Neil Shephard

Abstract

The potential system is a nonparametric time series model for assessing the causal impact of moving an assignment at time $t$ on an outcome at future time $t+h$, accounting for the presence of features. The potential system provides nonparametric content for, e.g., time series experiments, time series regression, local projection, impulse response functions and SVARs. It closes a gap between time series causality and nonparametric cross-sectional causal methods, and provides a foundation for many new methods which have causal content.

When are time series predictions causal? The potential system and dynamic causal effects

Abstract

The potential system is a nonparametric time series model for assessing the causal impact of moving an assignment at time on an outcome at future time , accounting for the presence of features. The potential system provides nonparametric content for, e.g., time series experiments, time series regression, local projection, impulse response functions and SVARs. It closes a gap between time series causality and nonparametric cross-sectional causal methods, and provides a foundation for many new methods which have causal content.
Paper Structure (24 sections, 10 theorems, 163 equations, 6 figures)

This paper contains 24 sections, 10 theorems, 163 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Defining dynamic causality
  3. Defining the potential system
  4. Defining causality with respect to a PS
  5. Examples of the potential system
  6. Structural equation model potential systems
  7. Linear potential systems
  8. Potential systems without features
  9. $m$-order potential systems
  10. PS-exogeneity
  11. From predictions to causality
  12. Assumptions on the sequential assignment mechanism
  13. Identification of causal effects through predictions
  14. Linear projection and covariance stationarity
  15. Causal summaries and strict stationarity
  16. ...and 9 more sections

Key Result

Theorem 1

Assume the SEM PS from Example ex:npsem.

Figures (6)

  • Figure 1: The left figure shows all the potential outcome paths for $T=3$. The right figure shows the observed outcome path $Z_{1:3}(A_{1:3})$ where $A_{1:3}=(1,1,0)^{\mathtt{T}}$, indicated by the thick blue line. The gray arrows indicate the missing data.
  • Figure 2: Time-$t$ system counterfactual paths. The left figure shows all the potential branches $D_{t,h}(a_{t})$ paths for horizon $h=0,1,2$ and $a_{t}\in \{0,1\}$. The right figure shows the observed outcome path $D_{t:t+2}=D_{t,0:2}(A_{t})$ where $A_{t}=1$, indicated by the thick blue line. The gray arrows indicate the missing data.
  • Figure 3: The PS drawn as a "Single World Intervention Template" (SWIT), under branch-sequential unconfoundedness --- the SAM.BSU- condition.
  • Figure 4: The PS drawn as a "Single World Intervention Template" (SWIT) under sequential randomization --- the SAM.BSR- case.
  • Figure 5: The PS drawn as a "Single World Intervention Template" (SWIT). Left-hand side shows the branch unconfoundedness --- the SAM.BU- case. The right-hand side shows the the branch randomization --- the SAM.BR- case.
  • ...and 1 more figures

Theorems & Definitions (55)

  • Definition 1: PS
  • Remark 1: Feature interpretation
  • Remark 2: Feature indexing
  • Remark 3: Non-interference
  • Remark 4: Branch potential outcomes
  • Remark 5: "Consistency"
  • Remark 6: Multi-period assignments
  • Definition 2: Dynamic causal effects
  • Definition 3: Describing dynamic causal effects
  • Remark 7: Total versus direct dynamic causal effects
  • ...and 45 more