QuanTAlib/lib/statistics/granger/Granger.md

GRANGER: Granger Causality F-Statistic

"Correlation is not causation, but Granger causality is not causation either. It is prediction." -- Clive Granger

Introduction

The Granger Causality test asks a precise, falsifiable question: does knowing the history of series X improve your ability to predict series Y, beyond what Y's own history already provides? The answer arrives as an F-statistic from comparing two OLS regression models. Higher F means X contains predictive information about Y that Y itself does not. This implementation uses lag-1, runs in O(1) streaming mode via running sums, and handles bar corrections for live trading.

Historical Context

Clive Granger introduced this test in 1969, later refined in Granger (1980). The key insight: "causality" here means temporal predictive precedence, not physical causation. The test became a workhorse in econometrics for testing lead-lag relationships between economic variables, exchange rates, and commodity prices. In trading, it identifies which instruments lead others, informing pairs trading, cross-asset signals, and regime detection.

Standard implementations require batch matrix operations. This implementation maintains running statistics for O(1) per-bar updates, matching the batch result exactly while supporting streaming and bar correction.

Architecture and Physics

1. Dual-Input Streaming Design

The indicator takes two series: Y (dependent, the series you want to predict) and X (independent, the hypothesized cause). At each bar, it maintains three parallel ring buffers storing the lagged triplet (y_t, y_{t-1}, x_{t-1}) over a rolling window of size period.

2. Running Sum Statistics

Nine running sums track means, variances, and cross-covariances:

• sumY, sumYLag, sumXLag for means
• sumYY, sumYLagYLag, sumXLagXLag for variances
• sumYYLag, sumYXLag, sumYLagXLag for covariances

These enable O(1) updates: subtract the oldest triplet, add the newest. Periodic resync every 1000 bars corrects floating-point drift.

3. Bar Correction via isNew

When isNew=false, the indicator restores the previous state snapshot and replaces the newest triplet in all buffers and running sums. This handles tick updates within the same bar without re-processing the entire window.

Mathematical Foundation

Restricted Model (AR(1))

y_t = c_0 + c_1 \cdot y_{t-1} + \varepsilon_{1,t}

OLS coefficients:

c_1 = \frac{\text{Cov}(y_t, y_{t-1})}{\text{Var}(y_{t-1})} c_0 = \bar{y} - c_1 \cdot \bar{y}_{t-1}

Unrestricted Model (AR(1) + X lag)

y_t = d_0 + d_1 \cdot y_{t-1} + d_2 \cdot x_{t-1} + \varepsilon_{2,t}

Two-variable OLS via Cramer's rule:

D = \text{Var}(y_{t-1}) \cdot \text{Var}(x_{t-1}) - \text{Cov}(y_{t-1}, x_{t-1})^2 d_1 = \frac{\text{Cov}(y_t, y_{t-1}) \cdot \text{Var}(x_{t-1}) - \text{Cov}(y_t, x_{t-1}) \cdot \text{Cov}(y_{t-1}, x_{t-1})}{D} d_2 = \frac{\text{Cov}(y_t, x_{t-1}) \cdot \text{Var}(y_{t-1}) - \text{Cov}(y_t, y_{t-1}) \cdot \text{Cov}(y_{t-1}, x_{t-1})}{D}

F-Statistic

SSR_1 = \left(\text{Var}(y_t) - c_1^2 \cdot \text{Var}(y_{t-1})\right) \cdot N SSR_2 = \sum_{i=1}^{N} \left(y_i - d_0 - d_1 \cdot y_{i-1,\text{lag}} - d_2 \cdot x_{i-1,\text{lag}}\right)^2 F = \frac{(SSR_1 - SSR_2) / q}{SSR_2 / (N - k)}

where q = 1 (one restriction: d_2 = 0) and k = 3 (unrestricted model parameters). The F-statistic follows an F(1, N-3) distribution under the null hypothesis that X does not Granger-cause Y.

Performance Profile

Metric	Value
Update complexity	O(1) amortized, O(N) for SSR2 loop
Memory	3 ring buffers + 9 running sums
Allocations per Update	Zero
SIMD potential	Low (recursive lag dependency)
Warmup period	period + 1

Quality Metrics

Metric	Score (1-10)
Responsiveness	7
Smoothness	5
Lag	3 (inherent from windowed regression)
Noise rejection	6
Interpretability	8 (F-statistic, compare to critical values)

Validation

This indicator validates against statistical properties rather than external TA libraries, as Granger causality is not commonly found in standard TA packages.

Test	Description	Result
Causal relationship	Y = f(Y_lag, X_lag) + noise	F > 0, high
Independent series	Two independent GBMs	F finite, generally low
Asymmetric detection	X causes Y but Y does not cause X	F(Y,X) > F(X,Y)
Batch vs streaming	TSeries batch matches streaming	Exact match
Span vs streaming	Span API matches streaming	Exact match
Bar correction	isNew=false restores state	Values match

Common Pitfalls

1. Not true causation. Granger causality tests temporal precedence in prediction, not physical causation. A spurious correlation with a lagged third variable can produce high F.
2. Period too small. Period must exceed 3 for the F-statistic to have positive degrees of freedom. Small periods amplify noise. Use 20+ for meaningful results.
3. Constant or near-constant series. Zero variance in the lag produces NaN (division by zero in OLS). This is mathematically correct behavior.
4. Multicollinearity. If y_lag and x_lag are nearly perfectly correlated, the denominator D approaches zero, producing NaN. This indicates the two predictors carry redundant information.
5. Confusing direction. F(Y,X) tests whether X helps predict Y. F(X,Y) tests the reverse. Always verify which direction matters for your trading thesis.
6. Critical values depend on sample size. For F(1, N-3): at 5% significance, critical value is approximately 4.0 for N=20, declining toward 3.84 for large N.
7. Floating-point drift. Running sums accumulate rounding errors over thousands of bars. The built-in resync every 1000 bars limits this to negligible levels.

References

• Granger, C.W.J. (1969). "Investigating Causal Relations by Econometric Models and Cross-spectral Methods." Econometrica, 37(3), 424-438.
• Granger, C.W.J. (1980). "Testing for Causality: A Personal Viewpoint." Journal of Economic Dynamics and Control, 2, 329-352.
• Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press. Chapter 11.
• Sims, C.A. (1972). "Money, Income, and Causality." American Economic Review, 62(4), 540-552.