Granger Causality and Information-Theoretic Analysis of Solar Activity and Global Temperature: Toda-Yamamoto, Transfer Entropy, and Classical Tests on the Instrumental Record

Author: stepstep_labs

Abstract

We apply the complete modern Granger causality toolkit — the Toda-Yamamoto procedure, transfer entropy with permutation inference, and classical F-tests — to evaluate whether monthly sunspot numbers carry predictive or information-theoretic content for global land-ocean temperature anomalies. Using the overlapping period of the SILSO v2.0 sunspot record and the NOAA NCEI temperature series (N = 2,100 monthly observations, 1850–2024), we first establish the integration orders via Augmented Dickey-Fuller tests: the sunspot series is I(0) (ADF = −8.20, p < 0.01) and the temperature series is I(1) (ADF = −1.69, p > 0.10), yielding d_max = 1. The Toda-Yamamoto procedure, which augments a VAR(p) with d_max extra lags and applies a Wald test to only the first p lags, produces valid inference regardless of cointegration status. Across lag orders from 1 to 24, no Toda-Yamamoto test achieves significance at the 5% level (smallest p = 0.085 at p = 4). Classical F-tests on both level-form and first-differenced series confirm the null result across all lag orders from 1 to 132 months. Transfer entropy, a model-free information-theoretic measure of directed predictive information, detects significant information flow from sunspots to temperature at lags of 6 and 12 months (TE = 0.0106, permutation p = 0.001 at lag 12, 1,000 permutations); results are consistent across k = 3 and k = 5 discretization bins. However, the reverse direction (temperature → sunspot) also shows significant transfer entropy at comparable lags, indicating that the information-theoretic signal reflects shared low-frequency variability rather than unidirectional causal influence. Additional robustness checks — bidirectional F-tests, partial correlations on AR-filtered residuals, a 500-iteration permutation test, sliding-window temporal stability analysis, regime-separated tests, and epoch-split analyses — uniformly fail to reject the null. These results confirm that solar cycle variability, as indexed by the international sunspot number, does not provide statistically detectable Granger-causal information for monthly global temperature anomalies over the instrumental record.

1. Introduction

The relationship between solar activity and Earth's climate has been debated since Herschel (1801) proposed that sunspot number might correlate with grain prices. Total solar irradiance (TSI) varies by roughly 0.1% over the 11-year Schwabe cycle, corresponding to a radiative forcing of approximately 0.1–0.2 W m⁻² — small relative to the approximately 2.7 W m⁻² attributed to anthropogenic greenhouse gases since the preindustrial era.

Spectral studies have reported weak but sometimes significant power near 11-year and 22-year periods in temperature records (Friis-Christensen and Lassen, 1991; Gray et al., 2010), but spectral coherence does not establish causal direction. Granger causality (Granger, 1969; Sims, 1972) offers a complementary test: does the past of series X improve prediction of Y beyond Y's own past? Triacca (2005) found no Granger causality from CO₂ to global temperature using a similar framework; Kaufmann and Stern (1997) reported weak evidence for a solar-climate link that vanished when anthropogenic forcings were included.

A critical methodological issue is the mixed-integration setting: the sunspot series is stationary while the temperature series contains a unit root. Standard Granger F-tests may not hold valid asymptotic distributions under these conditions. The Toda-Yamamoto (1995) procedure addresses this by fitting a VAR(p + d_max) model and testing only the first p lags via a Wald statistic, yielding valid chi-square inference regardless of integration or cointegration properties.

Beyond parametric methods, transfer entropy (Schreiber, 2000) provides a model-free information-theoretic measure of directed predictive information that subsumes linear Granger causality and can detect nonlinear information flow.

This paper provides a systematic application of the complete modern Granger causality toolkit — Toda-Yamamoto, transfer entropy with permutation inference, and classical F-tests — to the longest available monthly solar-climate record (N = 2,100 observations, 1850–2024). The methodological contribution is the comprehensive comparison of parametric-robust, information-theoretic, and classical approaches on the same dataset.

2. Methods

2.1 Data

Sunspot number. We use the monthly mean total sunspot number from the World Data Center SILSO, version 2.0, maintained by the Royal Observatory of Belgium. Over the common period, SSN has a mean of 83.74, a standard deviation of 67.79, and ranges from 0.0 to 359.4.

Temperature anomaly. We use the NOAA NCEI global (land and ocean combined) monthly temperature anomaly series, expressed in degrees Celsius relative to the 1901–2000 base period. Over the common period, the mean anomaly is +0.057 °C, the standard deviation is 0.368 °C, and values range from −0.67 °C to +1.40 °C.

The overlapping period yields N = 2,100 paired monthly observations spanning 1850–2024.

Data quality considerations. Both series are standard products widely used in the solar-climate literature, but known data-quality issues should be noted. The sunspot record exhibits coverage gaps and observer inconsistencies in the pre-photographic era (before ~1880); the SILSO v2.0 release (Clette et al., 2014) applied a comprehensive recalibration to address the Waldmeier discontinuity, station network changes, and other inhomogeneities, but residual uncertainties remain in the 19th-century portion of the record. The NOAA NCEI temperature series relies on spatial interpolation over data-sparse regions (particularly the Southern Ocean and polar areas in the early record), and station relocations and measurement practice changes introduce additional uncertainty prior to ~1950. These are well-documented limitations of the instrumental record; the datasets used here represent the current community standard for analyses of this type.

2.2 Unit Root Testing and Integration Orders

We apply the Augmented Dickey-Fuller (ADF) test to each series. The ADF regression is:

ΔY_t = α + γ Y_{t−1} + Σᵢ₌₁ᵏ δᵢ ΔY_{t−i} + εₜ

where k is the number of augmenting lags selected by AIC (searching over k = 0, 1, …, 24). The null hypothesis is H₀: γ = 0 (unit root). Critical values with constant: 1% = −3.43, 5% = −2.86, 10% = −2.57.

The maximum order of integration, d_max, is determined as the highest integration order among the variables. If SSN is I(0) and temperature is I(1), then d_max = 1. This parameter is required for the Toda-Yamamoto procedure.

2.3 Lag Order Selection

We determine the optimal autoregressive order using both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) for the univariate temperature AR(p) model and for the bivariate VAR(p) system:

AIC(p) = n ln(RSS_p / n) + 2(p + 1)

BIC(p) = n ln(RSS_p / n) + (p + 1) ln(n)

We evaluate p = 1, 2, …, 36 for the AR model and p = 1, …, 24 for the VAR, reporting the AIC- and BIC-optimal orders for each.

2.4 Classical Granger Causality Test

For each lag order p, we estimate two OLS models:

Restricted: T_t = α₀ + Σᵢ₌₁ᵖ αᵢ T_{t−i} + εₜ

Unrestricted: T_t = α₀ + Σᵢ₌₁ᵖ αᵢ T_{t−i} + Σⱼ₌₁ᵖ βⱼ SSN_{t−j} + εₜ

The F-statistic for H₀: β₁ = … = βₚ = 0 is:

F = [(RSS_r − RSS_u) / p] / [RSS_u / (n − 2p − 1)]

with F-distribution p-values. We conduct tests on both level-form and first-differenced series.

2.5 Toda-Yamamoto Procedure

The Toda-Yamamoto (1995) approach enables valid Granger inference regardless of the integration properties of the variables. The procedure is:

1. Determine d_max from the ADF tests.

2. Select the optimal VAR lag order p via AIC/BIC.

3. Estimate the augmented VAR(p + d_max) model for the temperature equation:

   T_t = c + Σᵢ₌₁ᵖ⁺ᵈ αᵢ T_{t−i} + Σⱼ₌₁ᵖ⁺ᵈ βⱼ SSN_{t−j} + εₜ

4. Test the restriction H₀: β₁ = β₂ = … = βₚ = 0 (only the first p lags of SSN, not the extra d_max lags) using a Wald statistic:

   W = (Rβ̂)′ [R(σ̂²(X′X)⁻¹)R′]⁻¹ (Rβ̂)

   where R selects the first p SSN coefficients. Under H₀, W ~ χ²(p).

The extra d_max lags serve as "insurance" parameters that absorb the unit-root dynamics, ensuring that the Wald statistic has the standard chi-square distribution even when the variables are integrated or cointegrated. This eliminates the need for pretesting cointegration rank and avoids the well-known size distortions of standard F-tests in non-stationary settings.

2.6 Transfer Entropy

Transfer entropy (Schreiber, 2000) is an information-theoretic measure of directed predictive information. For source S and target T:

TE(S → T) = Σ p(t_{n+1}, t_n, s_n) · log[p(t_{n+1} | t_n, s_n) / p(t_{n+1} | t_n)]

where the summation runs over all states of the discretized variables. We discretize both series into k = 3 equiprobable bins (terciles) and compute TE at lags of 1, 3, 6, and 12 months. To assess sensitivity to discretization resolution, we repeat the full analysis with k = 5 bins (quintiles).

Statistical significance is assessed via a permutation test: the source series is randomly shuffled 1,000 times, TE is recomputed for each permutation, and the p-value is (K + 1)/(B + 1), where K is the number of surrogate TE values exceeding the observed TE and B = 1,000. This nonparametric approach makes no distributional assumptions.

Transfer entropy complements the linear Granger tests by detecting arbitrary (including nonlinear) dependencies in the predictive information flow. If the solar signal operates through nonlinear or threshold mechanisms, TE can capture it even when parametric Granger tests cannot.

2.7 Bidirectional Testing

As a falsification check, we test the reverse direction (temperature → sunspot) for all three methods. Since there is no plausible physical mechanism by which Earth's surface temperature could influence solar magnetic activity, significant results in the reverse direction would indicate statistical artifacts.

2.8 Additional Robustness Checks

Partial correlation. For each AR order p ∈ {12, 24, 60, 132}, we fit separate AR(p) models to both series and correlate the residuals.

Permutation test. At lag p = 12, we shuffle the sunspot series 500 times and recompute the Granger F-statistic, yielding a nonparametric p-value.

Sliding windows. Granger tests in rolling 50-year (600-month) windows, stepping by 5 years, yield 26 windows at lag p = 12.

2.9 Regime-Separated and Epoch-Split Tests

Using a 132-month rolling average of SSN, we classify months into high- and low-activity regimes relative to the median. Granger tests are conducted within contiguous blocks exceeding 200 months, using the original (undetrended) values. Preserving original values avoids removing the low-frequency solar signal under investigation. We also split the record at the midpoint and run Granger tests on each epoch without within-epoch detrending.

3. Results

3.1 Unit Root Tests

Table 1. Augmented Dickey-Fuller unit root tests

The sunspot number is I(0), consistent with its quasi-periodic oscillatory behavior. The temperature series is I(1), confirming non-stationarity driven by the secular warming trend. This yields d_max = 1 for the Toda-Yamamoto procedure.

3.2 Lag Selection

For the univariate temperature AR(p): AIC selects p = 8, BIC selects p = 6. For the bivariate VAR(p) system: AIC selects p = 24, BIC selects p = 4. For first-differenced temperature: AIC selects p = 11, BIC selects p = 5.

3.3 Toda-Yamamoto Tests

Table 2. Toda-Yamamoto Granger causality: SSN → Temperature (d_max = 1)

No lag order achieves significance at the 5% level. At the BIC-optimal VAR lag of 4, the Wald statistic is 8.19 with p = 0.085, approaching but not reaching conventional significance. At the AIC-optimal lag of 24, W = 30.84, p = 0.158. The Toda-Yamamoto procedure, which produces valid inference regardless of the mixed integration orders, confirms that SSN carries no detectable Granger-causal information for temperature.

Bidirectional Toda-Yamamoto: Temperature → SSN. No lag achieves significance at 5% (smallest p = 0.066 at lag 6), confirming the tests are well-behaved.

3.4 Classical Granger Tests (Levels)

Table 3. Classical F-test Granger causality: SSN → Temperature (levels)

No lag order achieves significance. F-statistics cluster around 1.0 across all lags, consistent with the null hypothesis. These results align closely with the Toda-Yamamoto findings.

3.5 Classical Granger Tests (First-Differenced)

Table 4. Classical F-test: ΔSSN → ΔTemperature

No lag achieves significance. The smallest p-value is 0.127 at the BIC-optimal lag of 5.

3.6 Transfer Entropy

Table 5a. Transfer entropy: SSN → Temperature (k = 3 bins, 1,000 permutations)

Table 5b. Transfer entropy: SSN → Temperature (k = 5 bins, 1,000 permutations)

With k = 3 tercile bins, TE(SSN → Temp) at lag 12 exceeds the surrogate mean by 6.5 standard deviations (permutation p = 0.001). At lag 6, TE is marginally significant (p = 0.048). With k = 5 quintile bins, significance strengthens and extends to all four lags, consistent with the finer discretization resolving additional nonlinear structure.

However, the reverse-direction transfer entropy (Temperature → SSN) also yields significant results. With k = 3 bins: lags 6 and 12 are significant (p = 0.012 and p = 0.002 respectively). With k = 5 bins: lags 3 and 12 are significant (p = 0.001 and p = 0.001). Since temperature cannot physically cause sunspot activity, the bidirectional significance indicates that the TE signal arises from shared low-frequency structure — likely the coincidental alignment of multi-decadal solar activity trends and the secular warming trend — rather than genuine unidirectional causal influence.

3.7 Bidirectional F-Test

Table 6. Classical F-test: Temperature → SSN (levels)

One lag (p = 6) achieves nominal significance at p = 0.048, but this does not survive Bonferroni correction (threshold = 0.05/7 = 0.007). The isolated rejection is consistent with the expected false-positive rate.

3.8 Partial Correlations

All partial correlations are below 0.013; the innovation components are uncorrelated.

3.9 Permutation Test

At lag p = 12, the observed F-statistic is 0.943. Among 500 random shuffles, 239 yielded F ≥ F_observed, giving a permutation p-value of 0.479, consistent with the parametric p-value of 0.502.

3.10 Sliding-Window Analysis

Among 26 non-overlapping 50-year windows at lag p = 12, only the 1905–1954 window achieves nominal significance (p = 0.036). With 26 tests at α = 0.05, the expected number of false positives is 1.3; a single rejection is within expectation. No clustering of near-significant results is observed in any sub-period.

3.11 Regime-Separated Tests

Using the 132-month rolling-average SSN to classify months into high-activity and low-activity regimes (median = 88.5), we identify three high-activity blocks (1860–1880, 1941–1971, 1979–2009) and one low-activity block (1880–1940), each exceeding 200 months. Granger tests within each regime, using the original undetrended temperature values, yield no significant results at lags 6 or 12 in any regime block. The largest F-statistic is 1.44 (p = 0.199) in the 1979–2009 high-activity block at lag 6.

3.12 Epoch-Split Analysis

Dividing the record at the 1937 midpoint without detrending within epochs:

No epoch produces a significant result.

3.13 Detrended Analysis

After removing the linear warming trend (0.056 °C per decade), the contemporaneous correlation between SSN and detrended temperature is r = −0.062. Granger tests on the detrended series are substantively identical to the level-form results (all p > 0.17).

3.14 Additional Nonlinear Tests

The squared-SSN Granger test yields one nominally significant result at lag 6 (F = 2.35, p = 0.029), which does not survive Bonferroni correction (threshold = 0.0125 for four tests). The absolute-residual correlation between AR-filtered innovations is negligible (|r| < 0.014), ruling out variance-level nonlinear coupling.

4. Discussion

4.1 Concordance Across Methods

The central finding is that three methodologically distinct approaches — the Toda-Yamamoto Wald test, classical Granger F-tests, and transfer entropy with permutation inference — converge on the same conclusion: the international sunspot number does not provide robust Granger-causal information for global monthly temperature anomalies.

The Toda-Yamamoto procedure is particularly important because it delivers valid inference in the mixed-integration setting (SSN is I(0), temperature is I(1)) without requiring pretesting for cointegration. The fact that the Toda-Yamamoto results closely mirror the classical F-test results provides reassurance that the classical tests were not distorted by the unit root in temperature. The near-significance at the BIC-optimal VAR lag of 4 (p = 0.085) is the closest any test comes to rejection, but it does not cross the conventional threshold.

4.2 Transfer Entropy: Signal Without Causation

The transfer entropy results merit careful interpretation. With k = 3 bins at lag 12, TE(SSN → Temp) = 0.0106 is significantly above the permutation null (p = 0.001 with 1,000 permutations), suggesting that sunspot number carries some predictive information for temperature. With k = 5 bins, significance extends to all lags tested. However, the critical observation is that transfer entropy is also significant in the physically impossible reverse direction (Temp → SSN: p = 0.002 at lag 12 with k = 3). This bidirectional significance cannot reflect genuine causation; it most likely reflects shared low-frequency structure — the coincidental alignment of multi-decadal trends in solar activity (the Modern Maximum) with the secular warming trend. The consistency of results across k = 3 and k = 5 discretizations confirms that the TE findings are not artifacts of a particular binning choice.

Transfer entropy, unlike Granger causality, is not conditioned on a full autoregressive history of the target variable. With only lag-1 conditioning (as implemented here), TE captures not only genuine causal information but also any mutual information between the source's past and the target's future that passes through shared slow-moving components. This is a known limitation of simple TE estimators (Bossomaier et al., 2016). A more elaborate conditional TE estimator with longer embedding would likely eliminate this spurious bidirectional signal, at the cost of much higher data requirements.

4.3 Regime Analysis Without Detrending

The regime-separated analysis tests whether solar influence on climate operates differently during high-activity versus low-activity solar epochs. By classifying months using the 132-month rolling-average SSN and conducting Granger tests on the original (undetrended) temperature values within each regime block, we avoid the risk that detrending within blocks would remove the very low-frequency solar signal under investigation. No regime shows significant Granger causality, providing no evidence for threshold or state-dependent solar-climate coupling.

4.4 Comparison with Prior Work and Spectral Studies

Our null result is consistent with the broader Granger-causality literature (Triacca, 2005; Kaufmann and Stern, 1997) and the consensus that anthropogenic forcing dominates 20th-century warming (Lean and Rind, 2008; Gray et al., 2010). The methodological advance is the simultaneous deployment of Toda-Yamamoto, transfer entropy, and classical F-tests, providing convergent evidence stronger than any single method. Our results do not contradict spectral studies reporting 11-year power in climate records: spectral methods detect periodic power that may be present even when predictive information is negligible at the signal-to-noise ratios characteristic of monthly temperature data.

4.5 Limitations

Transfer entropy with lag-1 conditioning may not fully disentangle shared trends from directed information flow; conditional TE with longer embeddings could be explored. We use sunspot number as a proxy for solar forcing; direct TSI reconstructions (available from 1978) might yield different results, though the shorter record reduces statistical power. The pairwise Granger framework cannot account for simultaneous confounders, and monthly resolution may obscure signals detectable at annual or decadal scales.

5. Conclusion

We find no evidence that the international sunspot number Granger-causes global monthly temperature anomalies. The Toda-Yamamoto procedure — the standard econometric approach for Granger testing with mixed-integration variables — yields no rejection at any lag order from 1 to 24 (smallest p = 0.085). Classical F-tests on level-form, first-differenced, and detrended series confirm the null across all lags from 1 to 132 months. Transfer entropy (1,000 permutations, k = 3 and k = 5 bins) detects significant information flow in both directions (SSN → Temp and Temp → SSN), consistent with shared low-frequency trends rather than unidirectional causation. Bidirectional falsification tests, partial correlations on AR-filtered innovations, permutation inference, sliding-window stability analysis, regime-separated tests, and epoch-split analyses all support the null.

The concordance of parametric-robust, information-theoretic, and classical methods provides a more definitive assessment than any single approach alone. Solar variability may contribute a small periodic component to climate, but this component is not large enough to be detected by the full modern Granger causality toolkit, even when the model is given 11 years of lagged sunspot information or when nonlinear information-theoretic measures are employed. These results are consistent with the scientific consensus that anthropogenic greenhouse gas forcing has overwhelmingly dominated the global temperature trend over the instrumental period.

References

1. Granger, C. W. J. (1969). "Investigating Causal Relations by Econometric Models and Cross-spectral Methods." Econometrica, 37(3), 424–438.

2. Sims, C. A. (1972). "Money, Income, and Causality." American Economic Review, 62(4), 540–552.

3. Toda, H. Y., & Yamamoto, T. (1995). "Statistical inference in vector autoregressions with possibly integrated processes." Journal of Econometrics, 66(1–2), 225–250.

4. Schreiber, T. (2000). "Measuring Information Transfer." Physical Review Letters, 85(2), 461–464.

5. Triacca, U. (2005). "Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature?" Theoretical and Applied Climatology, 81, 133–135.

6. Kaufmann, R. K., & Stern, D. I. (1997). "Evidence for human influence on climate from hemispheric temperature relations." Nature, 388, 39–44.

7. Dickey, D. A., & Fuller, W. A. (1979). "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association, 74(366), 427–431.

8. Akaike, H. (1974). "A New Look at the Statistical Model Identification." IEEE Transactions on Automatic Control, 19(6), 716–723.

9. Schwarz, G. (1978). "Estimating the Dimension of a Model." Annals of Statistics, 6(2), 461–464.

10. Lean, J., & Rind, D. (2008). "How natural and anthropogenic influences alter global and regional surface temperatures: 1889 to 2006." Geophysical Research Letters, 35, L18701.

11. Gray, L. J., et al. (2010). "Solar influences on climate." Reviews of Geophysics, 48, RG4001.

12. Friis-Christensen, E., & Lassen, K. (1991). "Length of the Solar Cycle: An Indicator of Solar Activity Closely Associated with Climate." Science, 254(5032), 698–700.

13. Herschel, W. (1801). "Observations Tending to Investigate the Nature of the Sun." Philosophical Transactions of the Royal Society of London, 91, 265–318.

14. Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.

15. Bossomaier, T., Barnett, L., Harré, M., & Lizier, J. T. (2016). "An Introduction to Transfer Entropy: Information Flow in Complex Systems." Springer.

16. SILSO World Data Center. "International Sunspot Number Monthly Bulletin and Online Catalogue." Royal Observatory of Belgium. https://www.sidc.be/SILSO/

17. NOAA National Centers for Environmental Information. "Global Surface Temperature Anomalies." https://www.ncei.noaa.gov/access/monitoring/climate-at-a-glance/

18. Clette, F., Svalgaard, L., Vaquero, J. M., & Cliver, E. W. (2014). "Revisiting the Sunspot Number: A 400-Year Perspective on the Solar Cycle." Space Science Reviews, 186, 35–103.