Causal Lag Structure Discovery in Confounded Time Series via Orthogonalized Adaptive Estimation

Hong Kiat Tan; Isaac-Neil Zanoria; James Chen; Haoyang Lyu; Mihai Cucuringu

stat.MEOct 4, 2026

Causal Lag Structure Discovery in Confounded Time Series via Orthogonalized Adaptive Estimation

Authors: Hong Kiat Tan, Isaac-Neil Zanoria, James Chen, Haoyang Lyu, Mihai Cucuringu

Organizations: Department of Mathematics, University of California, Los Angeles · Department of Electrical and Computer Engineering, University of California, Los Angeles · Oxford-Man Institute of Quantitative Finance, University of Oxford · Department of Statistics, University of Oxford

Abstract

Finding which variables cause which others in multivariate time series, and at what lags, is central to science and policy, yet existing methods force a choice between flexible confounder adjustment, data-driven lag selection, and inference that controls the false discovery rate (FDR). ORACLE-VARX does all three in one pipeline. First, double/debiased machine learning (DML) removes nonlinear confounder effects from the outcomes and the lagged series. Second, adaptive causal lag estimation (ACLE) picks the lag order at each time step by sequential significance tests, tracking regime changes. Third, entry-wise $z$ -tests with Benjamini--Hochberg correction select directed edges at a target FDR. We prove that in each rolling window, the debiased coefficients are asymptotically normal around a window-averaged target, so their $z$ -tests are asymptotically valid. On a synthetic benchmark with time-varying structure and nonlinear confounding, ORACLE-VARX (LightGBM) tracks the true lag order best (RMSE $0.96$ vs $1.1$ -- $1.5$ ), has edge FDR $0.047$ , close to PCMCI ( $0.045$ ) and below VAR ( $0.129$ ) and VAR-LiNGAM ( $0.187$ ), and forecasts better than all three. On nine U.S. sector ETFs with macroeconomic confounders, it yields interpretable causal graphs whose lag order rises in high-volatility regimes.

Figures & tables

Panel of Figure 1 — Figure 1: An example of a time-unrolled causal graph of a 3-variable system. Solid blue arrows show the lag-1 cycle (coefficient $0.60$ ). Dashed red and dotted green arrows show lag-2 and lag-3 self-effects $a(t)$ and $b(t)$ that decay to zero (Section 5.1 ). Gray dashed arrows indicate nonlinear confounder effects.

Panel of Figure 2 — Figure 2: Sequential lag testing in ACLE. Left: the model with lags $1,\ldots,p{-}1$ . Right: testing $H_{0}\colon\mathbf{A}_{p}=0$ on the lag- $p$ block. If rejected at level $\alpha$ , lag $p$ is retained and we proceed to $p{+}1$ ; otherwise we stop and set $p_{\alpha}=p{-}1$ .

Method	Lag Sel.	Conf. Adj.	DML
VAR	RMSE	None	–
VARX	RMSE	Linear	–
ACLE-VAR	ACLE	None	–
ACLE-VARX	ACLE	Linear	–
OR-VARX	RMSE	Nonlinear	✓
ORACLE-VARX	ACLE	Nonlinear	✓

Table 1: Hierarchy of methods. Each row adds one capability relative to the row above. “RMSE” lag selection picks the lag order minimizing rolling validation RMSE ( 9 ) over

\{1,\ldots,p_{\max}\}

; “ACLE” uses significance-based sequential testing with validation-based

\alpha

-tuning (Section 3.3 ).

No confounders
Method	Lag	Coeff.	Fcst.	FDR	Power
	RMSE	MAE	MAE		$\uparrow$
VAR	1.475	0.060	0.110	0.129	0.655
ACLE-VAR	1.006	0.066	0.109	0.147	0.661
All confounders, no DML
VARX	1.239	0.063	0.109	0.076	0.650

Table 2: Synthetic benchmark results (All confounders observed unless noted). ET: Extra Trees; LGBM: LightGBM; TP: TabPFN. FDR: BH at

q=0.05

. Bold: best per column; lower is better except for Power.

Figure 3: Estimated vs. true edge coefficients over time for ORACLE-VARX (Extra Trees, all confounders). Each panel shows one edge; the black line is the true coefficient $\mathbf{A}_{k}^{*}[i,j](t)$ and the colored line is the rolling estimate $\hat{\mathbf{A}}_{k}[i,j](t)$ . Vertical dashed lines mark regime boundaries ( $t=1000$ , $t=2000$ ).

Panel of Figure 4 — Figure 4: Lag tracking: estimated $\hat{p}_{t}$ (blue) vs. true $p^{*}(t)$ (black step function) over time. Left : ACLE-VAR (no confounders). Right : ORACLE-VARX (Extra Trees, all confounders). The lag RMSE of every method is in Table 4 .

Figure 5: ETF application (ORACLE-VARX, Extra Trees, all10 confounders). Blue step function: adaptively selected lag order $\hat{p}_{t}$ over time (2004–2020). Orange dashed line: 21-day realized volatility of the S&P 500 (right axis). The lag order spikes during the 2008 financial crisis and COVID-2020, rising with the volatility.

Method	vix	macro5	all10
VAR	1.08
ACLE-VAR	1.32
VARX	1.11	0.43	0.75
ACLE-VARX	1.07	0.81	0.86
OR-VARX, ET	0.71	0.71	0.67
ORACLE-VARX, ET	0.83	0.85	0.68

Table 3: ETF Sharpe ratios (weighted strategy, 2004–2020) by confounder set; VAR and ACLE-VAR use none.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Three scenarios illustrating the role of causal sufficiency in Proposition A.13 . (a) When all confounders are observed ( $W_{t}$ ), DML correctly controls for them and the estimated coefficient $\hat{\mathbf{A}}_{k}[i,j]$ reflects the true causal relationship. (b) An unobserved common cause $U_{t}$ (dashed circle) of $Y^{(j)}$ and $Y^{(i)}$ creates a spurious association: DML estimates $\hat{\mathbf{A}}_{k}[i,j]\neq 0$ even though no direct causal link exists. (c) An unobserved mediator $U_{t}$ on the sole causal pathway from $Y^{(j)}$ to $Y^{(i)}$ : the indirect effect $Y^{(j)}\to U\to Y^{(i)}$ is not lost. It appears as a nonzero $\hat{\mathbf{A}}_{k}[i,j]$ at the lag where the two steps add up, so the edge is real but indirect. Causal sufficiency (Definition 5 ) rules out scenario (b) by requiring that all common causes are observed; scenario (c) does not violate it, because a mediator is not a common cause.

Non-DML methods
Method	Obs.	Nonzero MAE $\downarrow$	Fcst. MAE $\downarrow$	Lag RMSE $\downarrow$	FDR $\downarrow$	Power $\uparrow$	F1 $\uparrow$
VAR	none	0.0605	0.1100	1.475	0.129	0.655	0.734
ACLE-VAR	none	0.0655	0.1094	1.006	0.147	0.661	0.730
VARX	all	0.0627	0.1092	1.239	0.076	0.650	0.755
ACLE-VARX	all	0.0691	0.1081	1.099	0.080	0.651	0.754
VARX	partial-2	0.0604	0.1098	1.292	0.096	0.647	0.745

Appendix

Table 4: Complete synthetic benchmark results. All metrics are averaged over the rolling evaluation window. Obs.: observability level. FDR, Power and F1 use BH at level

q=0.05

. VARX treats the observed confounders as extra series; its lag

\hat{p}_{t}

is chosen on the equations of the

n

target series only, and only their edges are scored. Bold indicates the best value in each column.

Method	Obs.	Nonzero MAE $\downarrow$	Fcst. MAE $\downarrow$	Lag RMSE $\downarrow$	FDR $\downarrow$	Power $\uparrow$	F1 $\uparrow$
PCMCI + OLS refit	all	0.0646	0.1081	1.214	0.045	0.685	0.792
PCMCI + OLS refit	partial-2	0.0664	0.1089	1.223	0.052	0.672	0.780
PCMCI + OLS refit	partial-1	0.0664	0.1083	1.211	0.052	0.672	0.780
PCMCI + OLS refit	none	0.0657	0.1091	1.136	0.066	0.673	0.775
VAR-LiNGAM	all	0.0704	0.1074	1.132	0.187	0.641	0.703
VAR-LiNGAM	partial-2	0.0710	0.1082	1.132	0.196	0.641	0.697

Appendix

Table 5: Causal-discovery baselines on the synthetic benchmark, with ORACLE-VARX at full observability for reference. Same days and metrics as Table 4 .

Method	First-Stage Learner	Ann. Return (%)	Sharpe Ratio
VAR	—	5.93	1.08
ACLE-VAR	—	7.34	1.32
VARX	—	4.22	0.75
ACLE-VARX	—	4.80	0.86
OR-VARX	Extra Trees	3.51	0.67
ORACLE-VARX	Extra Trees	3.55	0.68

Appendix

Table 6: ETF portfolio performance (weighted strategy, market-adjusted). All DML methods use the all10 confounder preset (10 macroeconomic variables). Annualized return and Sharpe ratio are reported over the common test period (2004–2020).

Method	Learner	Naive	Weighted	Top 50%	Top 25%	Top 75%
VAR	—	4.32 / 1.04	5.93 / 1.08	6.16 / 0.95	5.12 / 1.01	8.41 / 0.86
ACLE-VAR	—	4.64 / 1.15	7.34 / 1.32	8.54 / 1.31	6.73 / 1.31	10.14 / 1.04
VARX	OLS	3.35 / 0.83	6.00 / 1.11	7.16 / 1.10	5.59 / 1.09	8.98 / 0.92
ACLE-VARX	OLS	3.09 / 0.78	5.83 / 1.07	6.47 / 1.00	5.85 / 1.14	8.76 / 0.90
OR-VARX	LightGBM	1.94 / 0.51	2.55 / 0.50	2.63 / 0.45	1.99 / 0.43	2.04 / 0.26
OR-VARX	XGBoost	0.98 / 0.27	1.43 / 0.29	2.07 / 0.36	1.79 / 0.38	3.21 / 0.38

Appendix

Table 7: Full strategy performance: vix confounder preset. Each cell shows Ann. Return (%) / Sharpe Ratio (market-adjusted, 2004–2020). VAR and ACLE-VAR use no confounders and are included as baselines. Best Sharpe per column in bold .

Method	Learner	Naive	Weighted	Top 50%	Top 25%	Top 75%
VAR	—	4.32 / 1.04	5.93 / 1.08	6.16 / 0.95	5.12 / 1.01	8.41 / 0.86
ACLE-VAR	—	4.64 / 1.15	7.34 / 1.32	8.54 / 1.31	6.73 / 1.31	10.14 / 1.04
VARX	OLS	0.87 / 0.23	2.34 / 0.43	2.51 / 0.40	1.51 / 0.32	2.99 / 0.35
ACLE-VARX	OLS	1.99 / 0.48	4.66 / 0.81	5.25 / 0.79	3.56 / 0.69	7.87 / 0.82
OR-VARX	LightGBM	0.11 / 0.05	0.47 / 0.11	$-$ 0.94 / $-$ 0.12	$-$ 0.36 / $-$ 0.05	0.52 / 0.10
OR-VARX	XGBoost	$-$ 0.21 / $-$ 0.03	$-$ 0.83 / $-$ 0.12	$-$ 0.95 / $-$ 0.12	$-$ 0.16 / $-$ 0.01	$-$ 2.96 / $-$ 0.27

Appendix

Table 8: Full strategy performance: macro5 confounder preset. Each cell shows Ann. Return (%) / Sharpe Ratio (market-adjusted, 2004–2020). Best Sharpe per column in bold .

Method	Learner	Naive	Weighted	Top 50%	Top 25%	Top 75%
VAR	—	4.32 / 1.04	5.93 / 1.08	6.16 / 0.95	5.12 / 1.01	8.41 / 0.86
ACLE-VAR	—	4.64 / 1.15	7.34 / 1.32	8.54 / 1.31	6.73 / 1.31	10.14 / 1.04
VARX	OLS	1.87 / 0.46	4.22 / 0.75	5.03 / 0.77	3.45 / 0.66	7.27 / 0.77
ACLE-VARX	OLS	1.89 / 0.47	4.80 / 0.86	5.68 / 0.86	3.38 / 0.65	5.98 / 0.66
OR-VARX	LightGBM	$-$ 0.44 / $-$ 0.09	0.13 / 0.05	$-$ 0.33 / $-$ 0.02	$-$ 0.46 / $-$ 0.07	1.44 / 0.19
OR-VARX	XGBoost	0.79 / 0.21	0.29 / 0.08	0.64 / 0.13	0.15 / 0.06	$-$ 2.95 / $-$ 0.27

Appendix

Table 9: Full strategy performance: all10 confounder preset. Each cell shows Ann. Return (%) / Sharpe Ratio (market-adjusted, 2004–2020). Best Sharpe per column in bold .

Panel of Figure 7 — Figure 7: Cumulative market-adjusted returns for three representative configurations. (a) The ACLE-VAR baseline achieves the highest Sharpe ratios overall (up to 1.32). (b) ORACLE-VARX with Extra Trees and the all10 preset—the configuration used for the lag-volatility analysis in Figure 5 —produces stable, positive returns across all strategies. (c) The macro5 preset with Extra Trees yields the highest DML Sharpe ratio (1.15, Top 50% strategy) among all DML experiments.

Panel A: Experiment Configuration
Parameter	Synthetic	ETF
$p_{\max}$	5	10
$\alpha$ -grid	$\{0.01,0.05,0.10,0.15,0.20,0.25,0.30\}$
OLS window	200 days	504 days
Tree training window	200 days	504 days
Validation days	20	21

Appendix

Table 10: Experiment configuration and first-stage learner hyperparameters.

	Synthetic	ETF
Windows	2,595	4,261
Blocks $F$	140	228
Fits per block $S$	60	585
Ours ( $F\cdot S$ )	8,400	133,380
(A) fresh split per window	155,700 (18.5 $\times$ )	2,492,685 (18.7 $\times$ )
(B) same estimator per window	1,704,960 (203 $\times$ )	62,198,370 (466 $\times$ )

Appendix

Table 11: First-stage fits per run (one learner, one observation level). Ratios are relative to our implementation.

Learner	Ours	Per window	(A)	(B)
Extra Trees	6.1 min	0.14 s	1.9 h	20.8 h
Random Forest	9.4 min	0.22 s	2.4 h	31.8 h
LightGBM	13.6 min	0.31 s	3.7 h	45.9 h
XGBoost	17.1 min	0.40 s	6.1 h	57.9 h

Appendix

Table 12: Wall-clock time of one synthetic run (Apple M5 laptop, CPU).

	Synthetic		ETF ( all10 )
Method	Window	Run	Window	Run
PCMCI (ParCorr)	0.16 s	7 min	4.4–9.1 s	5–11 h
PCMCI (CMIknn)	6.2–6.6 min	268–284 h	$>$ 1 h	$>$ 4,000 h
VAR-LiNGAM	0.02 s	1 min	9–15 s	11–17 h
Ours, Extra Trees	0.14 s	6.1 min	–	$\approx$ 1.6 h ∗
Ours, TabPFN (A100)	–	48–57 s	–	88 min

Appendix

Table 13: Time of the causal-discovery baselines on the Apple M5 laptop CPU (TabPFN: one A100 GPU). Baselines: one window at a time in one process, scaled to a full run (2,595 synthetic and 4,261 ETF windows); windows are independent, so these runs split evenly over cores. Ours uses 9 threads. CMIknn is PCMCI with the nonparametric conditional-mutual-information test.

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