The dynamic generalized covariance measure for conditional independence testing with nonstationary time series

The paper addresses the problem of conditional independence testing for nonstationary nonlinear time series. The authors state: "Identifying relationships among stochastic processes is a core objective in many fields, such as economics. While the standard toolkit for multivariate time series analysis has many advantages, it can be difficult to capture nonlinear dynamics using linear vector autoregressive models. This difficulty has motivated the development of methods for causal discovery and variable selection for nonlinear time series, which routinely employ tests for conditional independence."

The authors note that "most conditional independence tests lack Type I error control guarantees outside the iid setting. At most, some methods provide guarantees for stationary mixing processes. To the best of our knowledge, only one conditional independence test has been proposed for the setting with one realization of a nonstationary process. They reference Malinsky and Spirtes [MS19], who introduce a conditional independence test for nonstationary linear vector autoregressions with iid Gaussian errors, focusing on processes with stochastic trends. In contrast, the authors state: we allow linear and nonlinear processes with general nonstationarity, time-varying regression functions, and non-iid non-Gaussian errors."

The paper's contributions are summarized as follows:

1. First conditional independence test for a single realization of a nonstationary nonlinear process: "We propose the first conditional independence test that can be used with a single realization of a nonstationary nonlinear process. In Theorem 3.1, we show that our test has asymptotic Type I error control, uniformly over a large family of null distributions."

2. Two types of test statistics: We provide l∞-type and l2-type test statistics to achieve high power against sparse and dense alternatives, respectively.

3. Double robustness property: "Our test statistics are based on the products of the residuals from nonlinear regressions of future or past values of the processes X and Y on the covariates Z at time t. Hence, our test is doubly robust, in the sense that slower convergence rates of one regression estimator can be compensated for by faster convergence rates of the other estimator."

4. Flexible error assumptions: "We allow the errors to be nonstationary and temporally dependent, in contrast with other residual-based conditional independence tests that require iid errors. Moreover, we allow the errors to depend on the covariates, whereas many other residual-based tests require covariate-independent errors, thereby ruling out heteroskedasticity."

5. Compatibility with any regression estimator: "Our test can be used with any regression estimator for nonstationary time series. However, we provide a guarantee for an instantiation of our test based on the sieve regression estimator from Ding and Zhou [DZ21] in Section A of the supplement."

6. Distribution-uniform strong Gaussian approximation: We introduce a distribution-uniform strong Gaussian approximation in Section C of the supplement, which we use to prove uniformly asymptotic Type I error control.

7. Unconditional independence test: "We introduce an unconditional independence test in Section D of the supplement. Thus, our work is the first to supply the two components used in constraint-based causal discovery algorithms for nonstationary nonlinear time series."

The paper works in a triangular array framework for high-dimensional nonstationary nonlinear time series. Let (Xt,n, Yt,n, Zt,n)t∈[n] be the observed sequence of length n ∈ N, where [n] = 1,..., n. The dimensions dX = dX,n, dY = dY,n, dZ = dZ,n can grow with n.

Time-offsets are introduced: Negative time-offsets are called lags, and positive time-offsets are called leads. Sets Ai, Bj, Ck contain the time-offsets of Xt,n,i, Yt,n,j, Zt,n,k under consideration, with the time-offsets Ck to be non-positive so that the covariates are known at time t.

The null hypothesis tested is: Xt,n ⊥⊥ Yt,n Zt,n for all times t ∈ Tn, which includes all leads and lags of interest.

For a fixed sample size n, distribution P, time t, and dimension/time-offset tuple m = (i, j, a, b) ∈ Dn, the paper decomposes:

Xt,n,i,a = fP,t,n,i,a(Zt,n) + εP,t,n,i,a, Yt,n,j,b = gP,t,n,j,b(Zt,n) + ξP,t,n,j,b,

where fP,t,n,i,a(z) = EP(Xt,n,i,aZt,n = z) and gP,t,n,j,b(z) = EP(Yt,n,j,bZt,n = z) are the time-varying regression functions, and RP,t,n,m = εP,t,n,i,aξP,t,n,j,b denotes the product of these errors.

The test statistic is defined as:

Sn,p(R̂n) = max s∈Tn,L (1/Tn,L) Σ t≤s R̂t,n p,

which is the max lp-norm, p ∈ [2, ∞], achieved by the scaled partial sum process of residual products.

The main idea: "under the null hypothesis (3), the products of the residuals will be small, as long as the regression functions are estimated well. Our test statistic (4) is designed to have power against alternatives in which the covariances of the errors from (1) are non-zero for some times."

The paper emphasizes: "As with the GCM test from Shah and Peters [SP20], our test only has power against alternatives with non-zero expected conditional covariance; see Section D of the supplement for the details. Hence, our test is not, in general, an omnibus test of conditional independence."

Algorithm 1 (the dGCM test) proceeds as follows:

1. Select hyperparameters for regression method via cross-validation

2. For each time t and dimension/time-offset tuple m, obtain estimates f̂t,n,i,a and ĝt,n,j,b, calculate residuals, and calculate the product of residuals R̂t,n,m

3. Select the lag-window size Ln using the minimum volatility method

4. Calculate covariances Σ̂R t,n using a local window

5. Simulate independent Gaussian random vectors R̆(r) t,n N(0, Σ̂R t,n)

6. Calculate test statistics for simulated data

7. Calculate the 1−α empirical quantile of simulated test statistics

8. Reject if the test statistic exceeds the quantile

The paper states: Suppose Assumptions 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 related to the temporal dependence and nonstationarity hold for the sequence of collections of distributions (P∗0,n)n∈N, where P∗0,n ⊂ PCI0,n for each n ∈ N.

The regression estimation errors must satisfy specific convergence rates, including:

• sup max EP(ŵfP,t,n,i,a2)(1/2) · max EP(ŵgP,t,n,j,b2)(1/2) = o(Tn(−1/2)τn(7)Dn(−3))

• sup max EP(ŵfP,t,n,i,a2)(1/2) = o(τn 7Dn(−5))

• sup max EP(ŵgP,t,n,j,b2)(1/2) = o(τn 7Dn(−5))

Then: "lim sup n→∞ sup P∈P∗0,n PP(Sn,p(R̂n) > q̂1−α+νn + τn) ≤ α."

The paper notes: "The dGCM test possesses a property known as rate double robustness, which means that we place stronger convergence rate requirements on the products of the L2(P) norms of the estimation errors than on each one individually."

Assumption 3.1 (Causal representations of observed processes): Each dimension of the observed sequence can be represented as a measurable function of iid inputs: Xt,n,i = GXt,n,i(HtX), Yt,n,j = GYt,n,j(HtY), Zt,n,k = GZt,n,k(HtZ).

Assumption 3.4 (Causal representations of error processes): εP,t,n,i,a = GεP,t,n,i,a(Hεt,a), ξP,t,n,j,b = GξP,t,n,j,b(Hξt,b), with EP(εP,t,n,i,aHĝt) = 0 and EP(ξP,t,n,j,bHf̂t) = 0.

Assumption 3.5 (Distribution-uniform decay of temporal dependence): Uses the functional dependence measure of Wu [Wu05], requiring polynomial decay: θe,∞P,t,n,l,d(h) ≤ Θ̄∞ · (h ∨ 1)(−β̄∞) with β̄∞ > 1, and for error products θRP,t,n,m(h, q̄R) ≤ Θ̄R · (h ∨ 1)(−β̄R) with β̄R > 3, q̄R > 4.

Assumption 3.6 (Distribution-uniform total variation condition for nonstationarity): Requires a bound on the total variation of the causal mechanism of error products over time.

The paper provides a guarantee for an instantiation of our test based on the sieve regression estimator from Ding and Zhou [DZ21] in the setting of locally stationary time series. The sieve estimator uses basis functions (Legendre polynomials in the simulations) for both time and covariate values, with the numbers of basis functions for time and for the covariate values, denoted by c̃n and d̃n, respectively, are chosen to increase with the sample size n at some rate.

Theorem A.1 states: "Suppose that Assumptions A.1, A.2, A.3, A.4, A.5, A.6, A.7 all hold... Further, suppose that we use the sieve time-varying regression estimator from Section A.3 with the basis functions ϕl1(u), φl2(z) chosen to be mapped Legendre polynomials, where the numbers of basis functions are chosen to satisfy c̃n = O(log(Tn)), d̃n = O(log(Tn)). Then Assumptions 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 hold for (P∗0,n)n∈N, and the sieve estimators will achieve the convergence rates required by Theorem 3.1."

The paper studies the setting with dX = 1, dY = 1, dZ = 1 and no time-offsets, so A = 0, B = 0, C = 0, and Tn = [n]. We test the null hypothesis Xt,n ⊥⊥ Yt,n Zt,n for all times t ∈ Tn.

The data generating process uses:

• A tvAR(1) covariate process: Zt,n = θZ(t/n)Zt−1,n + ηtZ with θZ(u) = 0.35 + 0.2 cos(2πu)

• Regression functions: fK(z, u) = (0.5 + 0.25 cos(2πu)) exp(−z2) sin(Kz) and gK(z, u) = (0.3 + 0.15 sin(πu)) exp(−z2) cos(Kz) with K ∈ 1, 2, 3, 4

• Correlated shocks with correlation ρ ∈ 0, 0.3, 0.6, 0.9

The paper compares Sieve-dGCM with the GCM test [SP20] using a generalized additive model, and the residual prediction test (RPT) [SB18; HPM18]. Results show: Our test holds the level even with fairly small sample sizes, and gains power as we increase the correlation and sample size. The other tests fail to hold the level.

The paper investigates how stock markets in the United States, United Kingdom, Hong Kong, and Japan are linked using "daily log returns based on the adjusted closing prices of the S&P 500, FTSE 100, Hang Seng, and Nikkei 225 from January 2022 to March 2025," yielding n = 845 observations.

Key findings: "We retain the null hypotheses that S&P(t) is independent of Nikkei(t) (BH-adjusted p-value 0.074) and that FTSE(t) is independent of Nikkei(t) (BH-adjusted p-value 0.055). All other null hypotheses of independence are rejected at the significance level α = 0.05."

For conditional independence: "We retain the null hypothesis that S&P(t) is independent of HangSeng(t) given FTSE(t) (BH-adjusted p-value 0.797). However, we reject the null that S&P(t) is independent of FTSE(t) given HangSeng(t) (BH-adjusted p-value 0.005)."

The paper concludes: "The dGCM test shows promise for detecting conditional dependencies among nonstationary nonlinear time series while controlling the Type I error in finite samples. Specifically, we find that the Sieve-dGCM test can hold the level if the sample size is large enough to reliably estimate the time-varying regression functions."

Based on the paper, I can improve AI systems in the following specific ways:

Improvement: Implement the Dynamic Generalized Covariance Measure (dGCM) test as a new statistical tool for AI systems that analyze time series data.

What the improved AI system can do:

• Test whether two time series variables are conditionally independent given a third variable, even when the underlying processes are nonstationary (e.g., financial markets, climate data, sensor readings)

• Detect nonlinear relationships that traditional linear VAR models miss

• Handle a single realization of a nonstationary process (no need for multiple trials)

• Control Type I error rates uniformly across a large family of null distributions

• Work with temporally dependent and heteroskedastic errors, unlike many existing tests

These improvements enable AI systems to handle the complex, nonstationary, nonlinear time series data that are common in economics, finance, climate science, and engineering, while providing statistically rigorous guarantees that are essential for high-stakes applications.