Granger Causality: A Review and Recent Advances

1 INTRODUCTION

• Granger causality asks whether the history of x_i reduces the prediction variance of x_j, framing causality through prediction rather than standard causal definitions.Source: §1 · p. 2
• The framework’s validity for causal analysis remains debated, though the review sees utility in analyzing time-series interactions defined by association.Source: §1 · p. 2
• Traditional Granger analysis relies on linear VAR coefficient tests in bivariate settings, which can confound inferences when systems involve many series (L¨utkepohl, 2005; L¨utkepohl, 1982).Source: §1 · p. 2
• Standard linear VAR analysis assumes real-valued series, linear dynamics, a known lag count, and fixed discrete sampling, while modern data may be nonlinear or irregularly sampled.Source: §1 · p. 2
• Recent advances loosen these restrictions and broaden the settings in which Granger causality can be applied.Source: §1 · p. 2

1.1 Outline of Review

• The review covers the original definition and early tests, network Granger causality, lag selection and non-stationary VARs, then advances for discrete-valued series, nonlinear dynamics, and differing sampling rates.Source: §1.1 · p. 2

2.1 Definition

• Granger (1969) defined y as causal for x when including y’s history reduces the variance of x’s optimal prediction error, not as proof of a causal effect.Under assumptions about temporal ordering, Granger argued that predictability could imply a causal mechanism.Source: §2.1 · p. 3
• In the linear model, the coefficient matrices are generally not identifiable unless A0 is diagonal; that special case is the simple VAR model, distinct from SVARs with instantaneous effects.SVARs may be identified under certain parameter restrictions.Source: §2.1 · p. 3
• The framework requires restrictive assumptions, including a complete system without unmeasured confounders, and its assumptions are unlikely to hold together in practice.Early bivariate approaches were especially vulnerable because they did not account for other variables.Source: §2.1 · p. 4

2.2 Early approaches and applications

• Early methods were largely bivariate, while Granger’s and Sims’s characterizations were equivalent and could be tested by comparing full and reduced models.The F-test compares a model with past values of both series against one using only the target series’ past values.Source: §2.2 · p. 5
• Ignoring x1 in a three-variable VAR can produce an apparent Granger-causal direction between x2 and x3 that differs across model settings.The figure illustrates why bivariate analyses that omit relevant variables can mislead.Source: Figure 1 · p. 6, §2.2 · p. 6
• Despite limitations, bivariate Granger-causality tests were widely used in economics, finance, neuroscience, and meteorology.Source: §2.2 · p. 6
• Recent developments aimed to mitigate this limitation by analyzing a potentially large set of variables.Source: §2.2 · p. 6

3 NETWORK GRANGER CAUSALITY

• Network Granger analysis either controls for other relevant variables when studying a few endogenous series or models relationships among all variables as a graph.The latter setting arises in systems such as gene regulatory networks and brain connectivity.Source: §3 · p. 7

3.1 Granger causality based on VARs

• In a VAR, xi Granger-causes xj when the corresponding lag-matrix entries are nonzero; nonzero entries of Σ−1 represent instantaneous dependencies (Eichler, 2012).The compact graph combines interactions across past lags, while the expanded graph represents variables at multiple time points.Source: §3.1 · pp. 7–8, Figure 2 · p. 8
• Standard VARs have O(p^2) parameters and rarely use more than six to eight variables, limiting their ability to account for all relevant information (Bernanke et al., 2005).Scaling to many series applies whether the additional series are exogenous or all series are endogenous.Source: §3.1 · p. 8
• Factor-augmented VARs use unobserved factors to represent many exogenous variables, estimated either by a principal-components two-step procedure or maximum likelihood (Bernanke et al., 2005).This approach targets settings with a small number of endogenous variables and many exogenous series.Source: §3.1 · pp. 8–9
• In VARs with many endogenous variables, recent work directly selects nonzero entries of the coefficient matrices using sparsity-inducing penalties added to the VAR loss; a lasso penalty is one example.Source: §3.1 · p. 9
• Group lasso recovers the true sparsity pattern only when grouped coefficients have similar magnitudes and may provide only directional consistency; thresholded group lasso was proposed for consistent pattern learning.Its resulting estimates can facilitate interpretation of Granger effects in the settings illustrated in Figure 4.Source: §3.1 · p. 10, Figure 4 · p. 10
• High-dimensional VAR work links the sample size needed for lasso consistency to the spectral density matrix’s eigen-structure and develops asymptotically valid parameter inference (Basu and Michailidis, 2015; Neykov et al., 2018; Zheng et al., 2019; Zhu and Liu, 2020).The cited consistency results address settings where p ≫ T.Source: §3.1 · pp. 10–11
• In large VARs, Ba´nbura et al. (2010) report better performance when prior tightness increases with model size; later work considers sparsity-inducing and coefficient-clustering priors.Source: §3.1 · p. 11

3.2 Lag Selection and Non-Stationary VAR Models

• An unspecified or misspecified maximum lag can change Granger-causality conclusions: short lags miss longer connections, while long lags risk overfitting, especially in high-dimensional VARs.Source: §3.2 · p. 11
• Shojaie and Michailidis (2010) use a truncating lasso with data-driven lag weights; block-coordinate descent converges to a local minimum and is consistent for network selection in high-dimensional panels.Source: §3.2 · pp. 11–12
• When the truncating lasso’s decay assumption may fail, Shojaie et al. (2012) propose an adaptive thresholded lasso that can zero entire lag matrices while retaining others.Source: §3.2 · p. 12
• Fox et al. (2011) relax stationarity with a switching VAR whose lag matrices depend on a latent state, and a switching state-space model allows noisy observations.Source: §3.2 · p. 12
• Nakajima and West (2013) induce continuously varying sparsity in a time-varying VAR using latent thresholds that zero elements of evolving lag matrices.Source: §3.2 · p. 12
• For high-dimensional piece-wise VARs with structural breaks, Safikhani and Shojaie (2020) combine lag-matrix changes, lasso estimation, and BIC to identify breaks and estimate regime-specific parameters.Source: §3.2 · pp. 12–13
• Bai et al. (2020) extend related ideas to non-stationary VARs with sparse and low-rank lag-matrix components, accounting for unobserved exogenous variables.Source: §3.2 · p. 13

4 MORE GENERAL NOTIONS OF GRANGER CAUSALITY

• For nonlinear series, a component-wise model x_ti = g_i(pasts) + e_t extends Granger causality: x_j is non-causal for x_i when g_i is invariant to x_<j.The model specifies how all series' pasts map to each series; linear dynamics are not required for this definition.Source: §4 · p. 13
• Classical Granger causality assumes real-valued Gaussian series, restricting analysis of applications involving count-valued observations.Source: §4 · p. 13
• Conditional-independence formulations extend non-causality to arbitrary nonlinear relationships: x_i's conditional distribution is unchanged by including x_<j.The function-based model still assumes additive noise, whereas this formulation compares distributions conditional on all histories with and without x_j's history.Source: §4 · pp. 13–14
• The section notes that subsampling can miss true interactions and add spurious ones, and reviews approaches to identifying Granger causality in subsampled and mixed-frequency time series (Zhou et al., 2014; Silvestrini and Veredas, 2008; Boot et al., 1967; Breitung and Swanson, 2002; Gong et al., 2015; Tank et al., 2019).Source: §4 · p. 14

4.1 Discrete-valued time series

• Multivariate discrete-valued series include count, binary, and categorical data, and continuous recordings may also be quantized for downstream analysis.The examples span voting records, patient health states, player actions, weather, wind, stock returns, and sales volume.Source: §4.1 · p. 14

4.1.1 Categorical time series

• For categorical series, Tank et al. (2021) reparameterize the MTD to make estimation convex and identify non-causality by equal columns of the corresponding Zj matrix.Equivalent MTD parameterizations yield the same Granger-causality conclusions, and the convex formulation supports algorithms that scale to higher dimensions.Source: §4.1.1 · pp. 16, 18

4.1.2 Alternative formulation for categorical time series

• Tank et al. (2021) propose an mLTD alternative; for the identifiable mLTD, xj is Granger non-causal for xi iff Zj = 0, mirroring the identifiable MTD in Proposition 3.Source: §4.1.2 · p. 18

4.1.3 Estimating networks of binary and count time series

• Hall et al. (2016) study high-dimensional GLAR models for Bernoulli and log-linear Poisson autoregressions, deriving sample-complexity and mean-squared-error bounds for sparsity-regularized maximum likelihood.Their analysis addresses correlations and potential heteroscedasticity in GLAR observations.Source: §4.1.3 · p. 19
• A multivariate Poisson INAR with diagonal thinning uses Dirichlet-process shrinkage on rate parameters to cluster series, making pairs in disjoint clusters strictly Granger-noncausal.In the multivariate case, margins are no longer Poisson unless the thinning matrix is diagonal.Source: §4.1.3 · p. 19
• INGARCH models use an AR-like conditional mean and can model overdispersed counts, but have mostly been used in low dimensions, leaving scaling them for Granger analysis open.The conditional mean is M_t = E[x_t | x_<t] = α_1 x_{t−1} + β_0; conditional distributions can include Poisson, binomial, or negative binomial specifications.Source: §4.1.3 · pp. 19–20

4.1.4 Granger causal interactions in point processes

• Granger analysis of irregularly and asynchronously observed series motivates continuous-time point-process models, but inference in general multivariate point processes is difficult because histories are intractable to represent.The standard framework assumes observations on a fixed discrete-time grid.Source: §4.1.4 · p. 20
• For multivariate Hawkes processes, Eichler et al. (2017) give link-function conditions for Granger non-causality and derive a nonparametric estimation procedure.In the stated criterion, N_j does not Granger cause N_i if and only if φ_ij(u) = 0 for all u ∈ R.Source: §4.1.4 · p. 20
• Hawkes-network work adds sparsity-based high-dimensional inference (Xu et al., 2016), inhibitory interactions and screening (Chen et al., 2017a, 2017c), and a high-dimensional inference framework (Wang et al., 2020).Source: §4.1.4 · p. 20

4.2 Methods for capturing interactions in non-linear time series

• Transfer entropy and directed information (Amblard and Michel, 2011) detect nonlinear dependencies with minimal predictive assumptions, but their estimators need large samples and struggle with high dimensionality.The review notes their high variance and curse of dimensionality.Source: §4.2 · p. 21
• ODE-based methods represent nonlinear dynamics through state-to-derivative functions, with additive formulations linking Granger non-causality to zero interaction functions; Chen et al. (2017b) establish network-GC consistency.These approaches are limited to additive interaction mechanisms.Source: §4.2 · p. 21
• Neural forecasting models can capture complex nonlinear interactions, but shared hidden layers make Granger relationships hard to interpret, and joint networks can require much data in high-dimensional settings.A joint network also assumes series depend on the same past lags, although their relevant lags may differ.Source: §4.2 · p. 22
• Tank et al. (2018) use component-wise neural transition functions and structured sparsity penalties to identify nonlinear Granger links and, in the MLP setting, their lags.Their framework is designed to address sample complexity and lag selection; the LSTM formulation avoids specifying lag order.Source: §4.2 · p. 22

4.2.1 MLP

• In a cMLP, zero first-layer weights for an input series imply Granger non-causality, while group, sparse-group, and hierarchical penalties select causal series and relevant lags.The hierarchical penalty also selects interaction lag order.Source: §4.2.1 · pp. 23–24

4.2.2 RNN

• Standard RNNs, especially LSTMs, make it difficult to disentangle how each series affects another, although Tank et al. (2018) use LSTMs to model complex temporal dependencies.Their LSTM-based formulation sidesteps lag selection by modeling long-range dependencies.Source: §4.2 · p. 22, §4.2.2 · p. 24

RNN

• The LSTM’s additive cell-state update can preserve long-range dependencies when forget gates remain close to one, allowing long-range nonlinear representations.Cell states from far in the past may still influence the state at time t.Source: §4.2.2 · p. 25
• A zero jth column of W1 is sufficient for series xj to be Granger non-causal for series xi in the cLSTM.The input matrices determine how past series affect the hidden representation and prediction of xi.Source: §4.2.2 · p. 25
• Tank et al. (2018) use group lasso across W1 columns to select Granger connections; larger λ values zero more columns.The cLSTM’s implicit recurrent structure defines gi, and Figure 10 shows shared column sparsity across its gate matrices.Source: §4.2.2 · pp. 25–26, Figure 10 · p. 25

4.3 Subsampled and mixed-frequency time series

• Sampling slower than the causal timescale can hide true interactions and add spurious ones, while ignoring subsampling can distort lagged and instantaneous effects (Zhou et al., 2014; Silvestrini and Veredas, 2008; Gong et al., 2015).In particular, A_ij = 0 does not imply (A^k)_ij = 0, or vice versa.Source: §4.3 · pp. 26, 28
• Subsampling yields a transition matrix A^k and a weighted sum of shocks from unobserved times, leaving multiple shocks per series and obscuring instantaneous-effect interpretation.Each shock is weighted by A raised to the power of its time lag.Source: §4.3 · pp. 27–28
• In the subsampling illustration, analysis of the observed data finds no lagged causal effect between x1 and x2 but a relatively large instantaneous interaction.This illustrates how subsampling can confound inferred lagged and instantaneous effects.Source: Figure 12 · p. 28, §4.3 · p. 28
• Mixed-frequency models use series-specific sampling rates and indicator matrices, and ignoring their structure can produce further mistakes beyond those caused by subsampling alone.Tank et al. (2019) derive a process analogous to the subsampled case.Source: §4.3 · p. 28
• With independent non-Gaussian shocks, Tank et al. (2019) show that C is identifiable up to column permutation and sign scaling, while A is identifiable under additional stated conditions.For mixed-frequency data, the theorem gives an additional condition for identifying A; in the pure subsampling case, further assumptions are required.Source: §4.3 · p. 29
• A one-time-step difference between when two series are sampled can identify A_ij and help resolve sign ambiguities in mixed-frequency settings (Tank et al., 2019).The result applies to the standard mixed-frequency setting and to the depicted case where one series is observed one time step before another.Source: §4.3 · p. 29
• When instantaneous causal effects form a directed acyclic graph, its structure is identifiable without prior specification of causal ordering.Corollary 1 states that the structural matrix is unique across representations under its assumptions.Source: §4.3 · pp. 29–30
• With independent, asymmetric shocks, the complete causal diagram of lagged and instantaneous effects is identifiable from subsampled data.This follows from Theorem 1 and Corollary 1.Source: §4.3 · p. 30
• Tank et al. (2019) model non-Gaussian SVAR errors as an m-component Gaussian mixture and use EM to jointly estimate parameters from observed subsampled and mixed-frequency data.The same method is used across scenarios.Source: §4.3 · p. 30

• Classical approaches’ shortcomings stemmed from restrictive, unattainable assumptions for inferring causal effects from time-series data and limitations of the simple methods historically used.Source: §5 · p. 30
• Recent methods address high-dimensional series, automatic lag selection, non-stationarity, non-Gaussian and non-continuous observations, and mismatches between causal and observed time scales, expanding Granger causality’s applications.Source: §5 · p. 30
• These developments offer opportunities to investigate interactions among components of complex systems from a systems perspective.Source: §5 · p. 30
• Further work is needed on flexible nonparametric methods that handle many observed series while accounting for unmeasured variables and non-stationarity.Source: §5 · p. 30
• Emerging intervention and perturbation data can support causal-effect discovery and, at minimum, help restrict the set of possible causal hypotheses as a first step toward causal inference.Source: §5 · p. 30