Causal Discovery

These algorithms use conditional independence tests to prune edges between variables. They systematically test whether two variables are independent given a conditioning set of other variables. If independence is found, the edge is removed.

• PC Algorithm: The foundational method; starts with a fully connected graph and iteratively removes edges.
• FCI Algorithm: An extension that handles latent confounders, outputting a Partial Ancestral Graph (PAG).
• Key Test: Often relies on Fisher's Z-test for partial correlation in Gaussian data.
• Limitation: Sensitive to the order of testing and errors in independence tests can cascade.

These methods search over the space of possible Directed Acyclic Graphs (DAGs) and assign a score to each, optimizing for the graph that best fits the data according to a specific criterion. They treat causal discovery as a combinatorial optimization problem.

• Greedy Equivalence Search (GES): A two-phase greedy search that adds and then removes edges to optimize a score like the Bayesian Information Criterion (BIC).
• Scoring Functions: Common scores include BIC, Akaike Information Criterion (AIC), and Bayesian Dirichlet equivalence score.
• Advantage: Directly optimizes for model fit, often more robust to small sample errors than constraint-based tests.
• Challenge: The search space of DAGs grows super-exponentially, making exhaustive search infeasible.

FCMs exploit asymmetries in the data-generating process to break Markov equivalence and identify the direction of causation. They assume the effect is a function of the cause plus independent noise.

• LiNGAM: Assumes linear relationships and non-Gaussian noise. The non-Gaussianity allows for identifying the full causal order, not just an equivalence class.
• Additive Noise Models (ANM): Assumes the noise is additive and independent of the cause. Tests for independence of the residual from the hypothesized cause to infer direction.
• Post-Nonlinear Models: A more flexible class where the cause is nonlinearly transformed before adding noise, applicable to complex financial relationships.
• Key Insight: If the model is correctly specified, these methods can distinguish X → Y from Y → X without intervention.

Hybrid approaches combine the computational efficiency of constraint-based methods with the statistical robustness of score-based methods to achieve a better balance of speed and accuracy.

• Max-Min Hill-Climbing (MMHC): First uses a constraint-based local discovery phase (Max-Min Parents and Children) to build a skeleton, then applies a score-based Bayesian hill-climbing search to orient edges.
• GFCI: A hybrid that combines the GES score-based approach with the FCI constraint-based algorithm to handle latent confounders efficiently.
• Benefit: Reduces the search space for the score-based phase, making the algorithm tractable for high-dimensional financial datasets with dozens of variables.
• Application: Well-suited for discovering causal relationships among a large set of macroeconomic indicators and asset returns.

A modern approach that formulates causal discovery as a continuous optimization problem over an adjacency matrix, enabling the use of powerful gradient descent techniques and GPU acceleration.

• NOTEARS: Introduces a smooth, differentiable acyclicity constraint, transforming the combinatorial DAG search into a continuous constrained optimization problem solvable with standard optimizers like L-BFGS.
• DAG-GNN: Uses a graph neural network to model the functional relationships and a variational autoencoder framework to learn the causal graph.
• Advantage: Scales efficiently to thousands of variables and can be integrated with deep learning architectures for nonlinear causal discovery.
• Financial Use Case: Discovering causal networks among hundreds of stock returns in a high-dimensional portfolio.

The fundamental graphical output of causal discovery algorithms. A DAG represents variables as nodes and direct causal effects as directed edges (arrows), with the strict constraint that no path loops back to its origin.

• Encodes conditional independence relationships
• Forms the basis for the Markov property: a node is independent of its non-descendants given its parents
• Common algorithms: PC algorithm, GES, LiNGAM
• In finance: maps causal channels between macroeconomic indicators and asset returns

A statistical hypothesis test determining whether one time series is useful in forecasting another. A variable X 'Granger-causes' Y if past values of X contain information that helps predict Y beyond the information in past values of Y alone.

• Key assumption: causes must precede effects in time
• Does not prove true structural causality—only predictive utility
• Critical for lead-lag analysis in high-frequency trading
• Limitation: fails under contemporaneous causation or unobserved confounding

A non-parametric, information-theoretic measure of directed information flow between two stochastic processes. It quantifies the reduction in uncertainty about the future of Y given the past of X, beyond Y's own past.

• Based on Shannon entropy and Kullback-Leibler divergence
• Captures non-linear dependencies that linear Granger tests miss
• No assumption about functional form or distribution
• Applied in finance to detect information flow between market venues during price discovery

A graphical rule for identifying which variables must be conditioned on to isolate a causal effect. It blocks all spurious, non-causal paths (backdoor paths) between a treatment and outcome while preserving genuine causal pathways.

• Requires a known or hypothesized DAG structure
• Determines the minimal sufficient adjustment set
• Prevents collider bias and M-bias from incorrect conditioning
• Essential for translating discovered graphs into valid causal estimates

An extraneous variable that influences both the treatment and the outcome, creating a spurious association that distorts the true causal effect. Unmeasured confounding is the primary threat to causal discovery from observational data.

• Creates non-causal correlation between variables
• Causal discovery algorithms attempt to detect latent confounders
• In markets: volatility regimes often confound the relationship between order flow and price impact
• FCI and RFCI algorithms can model unobserved confounders

A fundamental property where a stochastic process has constant statistical moments (mean, variance, autocorrelation) over time. Most causal discovery methods assume stationarity, making it a critical preprocessing check for financial time series.

• Non-stationary data produces spurious causal links
• Tests: Augmented Dickey-Fuller, KPSS, Phillips-Perron
• Solutions: differencing, cointegration modeling, regime-switching frameworks
• Violation is common in markets due to structural breaks and volatility clustering