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A Practical Approach to Causal Inference over Time

Martina Cinquini, Isacco Beretta, Salvatore Ruggieri, Isabel Valera

TL;DR

The paper develops a principled bridge between discrete-time stochastic processes and structural causal models by mapping stable VAR(p) dynamics to linear SCMs. It defines two time-aware intervention classes (additive and forcing) and proves that, under suitable conditions, the long-run behavior of a DSP can be captured by an SCM, enabling interventional and counterfactual reasoning from observational time series. A key construct is the ${\mathcal{T}}$-transformation, which relates time series equilibria to SCMs, preserving causal effects at infinity and enabling a commutative analysis of interventions. Empirical evaluation on synthetic and real data shows that causal VARs achieve strong observational forecasting while yielding accurate estimates of intervention effects, with practical implications for policy analysis and decision support. The approach opens pathways to extend causal inference over time to nonlinear DSPs and high-dimensional settings, broadening its applicability in fields like climate science and economics.

Abstract

In this paper, we focus on estimating the causal effect of an intervention over time on a dynamical system. To that end, we formally define causal interventions and their effects over time on discrete-time stochastic processes (DSPs). Then, we show under which conditions the equilibrium states of a DSP, both before and after a causal intervention, can be captured by a structural causal model (SCM). With such an equivalence at hand, we provide an explicit mapping from vector autoregressive models (VARs), broadly applied in econometrics, to linear, but potentially cyclic and/or affected by unmeasured confounders, SCMs. The resulting causal VAR framework allows us to perform causal inference over time from observational time series data. Our experiments on synthetic and real-world datasets show that the proposed framework achieves strong performance in terms of observational forecasting while enabling accurate estimation of the causal effect of interventions on dynamical systems. We demonstrate, through a case study, the potential practical questions that can be addressed using the proposed causal VAR framework.

A Practical Approach to Causal Inference over Time

TL;DR

The paper develops a principled bridge between discrete-time stochastic processes and structural causal models by mapping stable VAR(p) dynamics to linear SCMs. It defines two time-aware intervention classes (additive and forcing) and proves that, under suitable conditions, the long-run behavior of a DSP can be captured by an SCM, enabling interventional and counterfactual reasoning from observational time series. A key construct is the -transformation, which relates time series equilibria to SCMs, preserving causal effects at infinity and enabling a commutative analysis of interventions. Empirical evaluation on synthetic and real data shows that causal VARs achieve strong observational forecasting while yielding accurate estimates of intervention effects, with practical implications for policy analysis and decision support. The approach opens pathways to extend causal inference over time to nonlinear DSPs and high-dimensional settings, broadening its applicability in fields like climate science and economics.

Abstract

In this paper, we focus on estimating the causal effect of an intervention over time on a dynamical system. To that end, we formally define causal interventions and their effects over time on discrete-time stochastic processes (DSPs). Then, we show under which conditions the equilibrium states of a DSP, both before and after a causal intervention, can be captured by a structural causal model (SCM). With such an equivalence at hand, we provide an explicit mapping from vector autoregressive models (VARs), broadly applied in econometrics, to linear, but potentially cyclic and/or affected by unmeasured confounders, SCMs. The resulting causal VAR framework allows us to perform causal inference over time from observational time series data. Our experiments on synthetic and real-world datasets show that the proposed framework achieves strong performance in terms of observational forecasting while enabling accurate estimation of the causal effect of interventions on dynamical systems. We demonstrate, through a case study, the potential practical questions that can be addressed using the proposed causal VAR framework.

Paper Structure

This paper contains 79 sections, 3 theorems, 56 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries and background
  4. Structural Causal Models
  5. Equation
  6. Graph
  7. Intervention
  8. Discrete-time Stochastic Processes
  9. Equation
  10. Graph
  11. Vector Autoregressive models
  12. Equation
  13. Graph
  14. Causal perspective on Discrete-time Stochastic Processes
  15. Causal Interventions on SDEs
  16. ...and 64 more sections

Key Result

Theorem 1

Given a stable $\textsc{VAR}{}(p)$$\mathcal{D}\xspace$ defined by eq:VAR, there exists a linear SCM${\mathcal{M}}\xspace$ with structural equations For simplicity we set $\bm{\nu}=0$, i.e., we assume $\mathbb{E}[{{\bm{X}}_t}]=0$. The theorem applies in the general case up to a translation of both th such that, given the transformation $\bm{Z}_t = \frac{1}{\sqrt{t}}\sum_{i=1}^t{\bm{X}}_i$, the foll

Figures (11)

  • Figure 1: ${\mathcal{T}}$-transformation transfers causal information from the temporal to the cross-sectional dimension, and thus to the joint distribution $P(\bm{Z}_t)$. The diagram commutes, i.e., red and blue paths produce the same result.
  • Figure 2: Causal graphs. The causal graph for (a) and (b) is known, while for (c), it is assumed. In (b), nodes are labeled with the initials of each feature: Expertise, Responsibility, Loan Amount, Duration, Income, Savings, and Credit Score. In (c), $0-14$, $15-64$, and $65-99$ represent age groups.
  • Figure 3: Additive Intervention. (Left) Intevention on Expertise with $\bm{F} = 0.2$. (Right) Effect on Credit Score. Shaded regions in both plots denote $95\%$ confidence bounds.
  • Figure 4: Forcing Intervention. (Left) Intevention on Expertise with $\bm{F} = 1$ and target $\hat{E}=5$. (Right) Effect on Credit Score. Note that confidence bounds are narrower w.r.t. \ref{['fig:additive_int']}.
  • Figure 5: German. Effect of increasing Expertise on Credit Score. (Left) The time each loan applicant takes to cross or not the acceptance threshold. The histogram shows the distribution of crossing times. (Right) Comparison of two loan applicants, i.e., trajectories, with similar scores at intervention time. After the intervention, they diverge significantly, with only an applicant being accepted at the maximum time. Forecasts are dashed for observational and dotted for interventional.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 1: Causal Effect over time ($\mathtt{CE}\xspace{}_t$)
  • Theorem 1
  • Remark
  • Remark
  • Example 1: RDE
  • Remark 1
  • Example 2: RDE equilibration
  • Theorem 2
  • Remark
  • Theorem 3