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Causal Identification in Time Series Models

Erik Jahn, Karthik Karnik, Leonard J. Schulman

TL;DR

The paper tackles identifiability of causal effects in time-series graphs with latent confounding, where graphs extend over infinite time. It introduces a finite-window bound $C$ that depends only on the number of variables per time step $w$ and the maximum latency $L$, proving that identifiability can be decided by running the Causal ID algorithm on a constant-size segment ${\mathcal{G}}[t_{\min}-C, t_{\max}]$ and that a unidentifiable effect implies a nearby time-shifted unidentifiability as well, with $C$ bounded by $C = L\cdot 2^{L w}\cdot (L w+1)^{2L w+2}$. The results establish an exponential upper bound and a linear lower bound through explicit constructions, and provide an algorithm to test identifiability for all future lags by evaluating a finite set of segments. The work advances practical causal analysis in long-horizon time-series with latent confounding, outlining directions for tightening bounds and exploring structure-induced improvements.

Abstract

In this paper, we analyze the applicability of the Causal Identification algorithm to causal time series graphs with latent confounders. Since these graphs extend over infinitely many time steps, deciding whether causal effects across arbitrary time intervals are identifiable appears to require computation on graph segments of unbounded size. Even for deciding the identifiability of intervention effects on variables that are close in time, no bound is known on how many time steps in the past need to be considered. We give a first bound of this kind that only depends on the number of variables per time step and the maximum time lag of any direct or latent causal effect. More generally, we show that applying the Causal Identification algorithm to a constant-size segment of the time series graph is sufficient to decide identifiability of causal effects, even across unbounded time intervals.

Causal Identification in Time Series Models

TL;DR

The paper tackles identifiability of causal effects in time-series graphs with latent confounding, where graphs extend over infinite time. It introduces a finite-window bound that depends only on the number of variables per time step and the maximum latency , proving that identifiability can be decided by running the Causal ID algorithm on a constant-size segment and that a unidentifiable effect implies a nearby time-shifted unidentifiability as well, with bounded by . The results establish an exponential upper bound and a linear lower bound through explicit constructions, and provide an algorithm to test identifiability for all future lags by evaluating a finite set of segments. The work advances practical causal analysis in long-horizon time-series with latent confounding, outlining directions for tightening bounds and exploring structure-induced improvements.

Abstract

In this paper, we analyze the applicability of the Causal Identification algorithm to causal time series graphs with latent confounders. Since these graphs extend over infinitely many time steps, deciding whether causal effects across arbitrary time intervals are identifiable appears to require computation on graph segments of unbounded size. Even for deciding the identifiability of intervention effects on variables that are close in time, no bound is known on how many time steps in the past need to be considered. We give a first bound of this kind that only depends on the number of variables per time step and the maximum time lag of any direct or latent causal effect. More generally, we show that applying the Causal Identification algorithm to a constant-size segment of the time series graph is sufficient to decide identifiability of causal effects, even across unbounded time intervals.
Paper Structure (7 sections, 7 theorems, 7 equations, 8 figures, 2 algorithms)

This paper contains 7 sections, 7 theorems, 7 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Proofs
  5. Cutting hedges to show Propositions \ref{['prop:past']}, \ref{['prop:future']}
  6. Constructing examples to show Theorem \ref{['thm:lowerbound']}
  7. Discussion

Key Result

Theorem 1

Let ${\mathcal{G}}$ be a periodic causal graph and let $\mathbf{X}, \mathbf{Y} \subseteq \mathbf{V}({\mathcal{G}})$ be disjoint subsets of variables. Let $t_{\min}$ be the smallest time index of a variable in $\mathbf{X}$ and $t_{\max}$ the largest time index of a variable in $\mathbf{Y}$. Then, the

Figures (8)

  • Figure 1: A priori, deciding if the causal effect of $X$ on $Y$ is identifiable requires running the causal ID algorithm on the entire part of the time series graph shown above - from its initialization up to the layer containing $Y$. Our results show that instead computation on a constant-size section around $X$ (i.e. just the middle part) suffices, using a shifted version $Y_{-\Delta}$ of $Y$.
  • Figure 2: Periodic graph of width $3$ and latency $1$. Directed edges are solid, bidirected dashed.
  • Figure 3: Causal ID algorithm (adapted from shpitser08)
  • Figure 4: The effect $P(X_{2,2} | \mathop{\mathrm{do}}\nolimits X_{1,1})$ is unidentifiable because of the unique hedge $({\mathcal{F}}, {\mathcal{F}}')$ with $\mathbf{V}({\mathcal{F}}) = \{X_{0,0}, X_{1,1}, X_{2,1}, X_{2,2}\}$ and $\mathbf{V}({\mathcal{F}}') = \{X_{2,2}\}.$ Note that this unidentifiability cannot be detected when only looking at the layers from $X_{1,1}$ onward.
  • Figure 5: Decides identifiability of $P(\mathbf{Y}_{+\Delta} | \mathop{\mathrm{do}}\nolimits \mathbf{X})$ for all $\Delta \geq 0$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Definition 1: periodic causal graph
  • Definition 2: segments
  • Definition 3: latency
  • Definition 4: distance between sets
  • Definition 5: time shifts
  • Definition 6: C-component
  • Definition 7: forest
  • Definition 8: C-forest
  • Definition 9: hedge
  • ...and 9 more