Causal Identification in Time Series Models (2504.20172v1)

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.

Analysis of Causal Identification in Time Series Models

The paper, "Causal Identification in Time Series Models," investigates the applicability of the Causal Identification algorithm to causal time series graphs, particularly focusing on models with latent confounders. Within this domain, causal graphs can span infinitely across time, raising complex challenges in determining whether causal effects from interventions are identifiable. Understanding these effects is crucial in fields such as economics, climate science, and other areas where large observational datasets are more readily available than experimental ones.

Key Contributions

The authors present novel findings in bounding the number of time steps required for assessing the identifiability of causal effects in these graphs. Specifically, they establish that the identifiability of causal effects does not necessitate computations over unbounded graph segments. Instead, mobilizing the Causal Identification algorithm over a constant-size segment of the graph suffices, regardless of the potentially infinite intervals involved. This result is encapsulated in Theorem 1, which introduces a constant $C$, related to the number of variables per time step and the latency of causal effects, that bounds the graph segment needed for identification.

Methodological Advances

The paper addresses two significant problems:

1. Identifiability Across Large Intervals: For graphs modeling complex systems like the Earth's climate, determining causal effects between two proximate variables could previously require extensive computation scaling with Earth's history. The established bounds, which function independently of such extensive time intervals, alleviate computational impracticality by curtailing necessary graph segments.
2. Uniform Identifiability Over Time: Often, researchers seek to determine if a causal effect maintains identifiability as time progresses infinitely. The paper advances a mechanism to decide such identifiability using a shifted variant of the target variable over a fixed segment $C$. Proposition 2 elucidates this shift and the conversion of identification challenges into a manageable computational problem.

The proof strategy employs a series of graph-theoretical tools, leveraging periodic graph structures and segment transformations. By demonstrating that for certain periodic time series graphs, a constant-size section encases all potential unidentifiability factors, the authors make impressive strides in reducing computational overhead.

Numerical Bounds and Theoretical Implications

The constants provided exhibit exponential scaling relative to latency and graph width, offering a first efficient bound applicable across varied periodic structures. However, the paper acknowledges this as a generic upper bound, noting that specific graphs might demonstrate significantly reduced complexity. This insight is exemplified in Theorem 4, showcasing constructions of graphs with lower bounds on the number of layers required for identification, specifically revealing linear scaling as plausible in specific cases.

Consequences and Future Work

These results imply substantial computational savings and methodological advancements for researchers utilizing causal inference in infinite time series models. This work facilitates scalable causal identification, expediting analysis across massive observational datasets without burdening computational resources.

For future research, improvements in the dependency of the constant $C$ on latent factors and graph width could enhance algorithm efficiency further. Additionally, exploring the condition number of the Causal ID mapping in periodic graphs remains an open domain, where tighter constraint discovery could bolster sample size efficiency. Empirical validation of these theoretical claims remains crucial, particularly in applications where environmental and societal factors intertwine, demanding nuanced causal insights.

The paper is pivotal in advancing causal inference methodologies within time-series data, offering researchers potent tools to navigate complex dynamic systems while maintaining rigorous computational feasibility.