1. Introduction
The establishment of cause and effect is a fundamental, if elusive, driver of climate science research. While causality is much sought after, it is challenging to establish, especially in observations—recall the adage “correlation does not equal causation.” Determining true causality requires not only the establishment of a relationship between two variables, but also the far more difficult task of determining a direction of causality. Although they do not provide information regarding directionality, correlation-based methods, such as lagged linear regression, remain popular and useful tools for identifying lagged relationships between climate variables.
A lagged regression model can provide a straightforward assessment of spatial and temporal variability. Lagged regression analysis has been a popular technique in climate science for nearly 100 years (e.g., Walker 1923, 1924). Since 1988, the phrases “lagged regression,” “lag regression,” “lagged correlation,” and “lag correlation” have appeared in a combined total of over 800 manuscripts in the Journal of Climate alone. Lagged linear regression analysis has been used in a wide variety of climate science applications, including, but not limited to, stratosphere–troposphere interactions (e.g., Polvani and Waugh 2004); tropical variability patterns, such as the Madden–Julian oscillation and El Niño–Southern Oscillation (e.g., Klein et al. 1999; Hendon et al. 2007); Arctic sea ice extent (e.g., Blanchard-Wrigglesworth et al. 2011); and sea surface temperature variability (e.g., Yu et al. 2010). This is just a small sampling of the hundreds of studies across atmospheric and climate science that utilize linear lagged regression analysis.
While lagged regression can be a straightforward and effective tool for identifying covarying patterns in space and time, lagged regression also has its drawbacks. First, while lagged regression can show the existence of instantaneous and lagged relationships between variables, lagged regression alone cannot indicate the direction of causality. Lagged regression may indicate that two variables are related to each other when in actuality they are linked or driven by a third variable (e.g., Fig. 3 in Kretschmer et al. 2016). Finally, lagged regression can be interpreted to suggest that one variable causes a response in the other when in fact it does not. This can occur when one variable has high memory, or autocorrelation (e.g., Runge et al. 2014; Kretschmer et al. 2016), and this is the scenario that will be explored here.
As an example, consider the relationship between tropical Pacific sea surface temperatures [i.e., El Niño–Southern Oscillation (ENSO)], and surface temperature over North and South America. ENSO is considered to be a primary driver of surface temperature anomalies in these regions (e.g., Ropelewski and Halpert 1986; Gu and Adler 2011). However, on monthly time scales, SST anomalies are quite persistent—the 1-month lag autocorrelation of the Niño-3.4 SST index (anomaly form, with the 1951–2000 mean removed; Rayner et al. 2003) is 0.91, meaning that over 80% of the variability in tropical Pacific SST in the Niño-3.4 region is determined by the previous month. The Niño-3.4 index takes over 6 months to decorrelate (defined using its e-folding time). This memory in ENSO can lead to ambiguity when applying lagged linear regression. For example, Fig. 1 shows the lagged relationship between ENSO and land surface temperature 
Using lagged regression to test the hypothesis that (a) ENSO drives 
Citation: Journal of Climate 31, 8; 10.1175/JCLI-D-17-0334.1
Decades of research on ENSO and its impact on surface temperature over the Americas point to ENSO driving surface temperature, not the other way around (e.g., Ropelewski and Halpert 1986; Gu and Adler 2011). However, that conclusion is not clear from Fig. 1—the lagged regression results are ambiguous. One potential cause of this ambiguity could be the high autocorrelation in the Niño-3.4 index. Instead of asking, “Can we use 
In this paper, we aim to demonstrate the following:
Granger causality is typically superior to traditional lagged regression when one or more datasets has substantial memory;
Granger causality and lagged regression tend to yield similar results when there is a true causal relationship; and
Granger causality is only slightly more difficult to implement than traditional lagged regression.
2. Statistical model
Granger causality (Granger 1969) was first developed as a predictive model in economics. More recently, Granger causality has found applications in climate science, such as determining the influence of snow cover and vegetation on surface temperature (e.g., Kaufmann et al. 2003), the impact of sea surface temperature on the North Atlantic Oscillation (e.g., Mosedale et al. 2006) or on Atlantic hurricane strength (e.g., Elsner 2006, 2007), ENSO’s impact on the Indian monsoon (e.g., Mohkov et al. 2011), and attributing global temperature increases to increases in global atmospheric CO2 (see Attanasio et al. 2013 and references therein). However, the use of Granger causality remains far behind that of lagged regression. We use a Monte Carlo simulation to demonstrate that Granger causality is straightforward and, under specific circumstances, is less likely than lagged regression to lead to the inference of a causal relationship when there is not one.
We perform a Monte Carlo simulation in which we vary α, γ, and τ. First, we create a D time series with 550 steps following Eq. (1). After discarding the first 50 values of D, we create R following Eq. (2). We perform our regression analysis (discussed in the next section) and repeat this process 5000 times for each combination of α, γ, and τ. We test 20 values of α, ranging from 0 to 1; 20 values of γ, ranging from 0.005 to 15; and 15 values of τ, ranging from 1 to 15, to ensure that our results are robust.
At least one value of b must be significant according to a two-sided t test.
All values of b collectively must increase the variance explained by the regression according to an F test.
For both the standard lagged regression and the Granger causality analysis, we perform the regressions in both directions: the direction we know to be correct (D driving R) and the direction we know to be incorrect (R driving D). In this way, we can evaluate whether or not Granger causality outperforms standard lagged regression, as defined by a lower risk of false detection, given the same identification rate of correct relationships. It is also worth noting that selecting the maximum lag k is an important and potentially challenging part of Granger causality analysis. Typically, k is selected based on a common metric for model selection, such as the Akaike information criterion or the Bayesian, or Schwarz, information criterion (e.g., Mosedale et al. 2006). In both cases, the preferred model is the one with the k value that minimizes the selection criteria and thus limits the model from becoming overfitted. Finally, the approach that we detail here is a relatively straightforward approach to Granger causality that has been used in climate sciences in recent years to great success; it is worth noting, however, that there are alternative ways of calculating Granger causality, many of which have been developed in neuroscience (e.g., Barnett and Seth 2014; Stokes and Purdon 2017).
3. Monte Carlo results
First, we compare the performance of lagged regression and Granger causality by evaluating the ability of D to predict R. Recall that R was created using D, so our models should suggest a causal relationship. Figure 2 shows the percentage of significant results (e.g., the model reports a significant causal relationship for the hypothesis that D drives R at 95%) as a function of memory α (y axis) and noise γ in R (x axis) for the lagged regression model (Fig. 2a) and the Granger causality model (Fig. 2b). Darker colors imply that the model indicated a causal result (in this case, D causes R) more often. Both panels of Fig. 2 look similar—in this case, lagged regression and Granger causality yield comparable results. Both methods show a dependence on γ—that is, as R becomes noisier, both models are less able to predict R from D. Both methods also exhibit minimal dependence on α, demonstrating that in general, both models are quite capable of predicting R, even when D has a very high memory. Here, we note that this lack of dependence on α is specific to the AR-1 process modeled in Eq. (1), where the variance of the noise [the 
Testing the hypothesis that D drives R using (a) lagged regression and (b) Granger causality. Shading indicates the percentage of significant results at 95% confidence.
Citation: Journal of Climate 31, 8; 10.1175/JCLI-D-17-0334.1
While Fig. 2 demonstrates that lagged regression and Granger causality generally yield similar results in the case of D driving R, there is one notable exception: when memory is very high (
Next, we evaluate lagged regression and Granger causality by using R to predict D; we compare the outcomes of the two methods when we look for causality in the wrong direction (recall that R was created from D). In this case, we would hope that the models do not suggest a causal relationship between R and D. This hypothesis of R driving D is tested in Fig. 3. Figure 3 is laid out similarly to Fig. 2, with darker colors indicating that the model reported a causal relationship more frequently. In Fig. 3, the advantages of Granger causality become apparent. Figure 3a shows that the lagged regression model exhibits a strong dependence on α—as D’s memory increases, the lagged regression model is increasingly more likely to suggest that R drives D, which we know to be incorrect. Even at moderate values of α, the lagged regression model implies that there is a causal relationship in the wrong direction. While low values of α show a false positive rate between 5% and 10% (recall that significance is assessed at 95% confidence, meaning we would expect a significant result of 5% merely by random chance), at 
Testing the hypothesis that R drives D using (a) lagged regression and (b) Granger causality. Shading indicates the percentage of significant results at 95% confidence.
Citation: Journal of Climate 31, 8; 10.1175/JCLI-D-17-0334.1
There is no such dependence on memory for the Granger causality method, as seen in Fig. 3b. Indeed, Fig. 3b indicates that the results of the Granger causality method are simply noise, with Granger causality yielding a significant result about 5% of the time, consistent with our 95% significance testing. These results are not dependent on lag τ; memory α, or noise γ in R. In this case, Granger causality’s insensitivity to α, or memory in D, shows an improvement over a typical lagged regression model for variables with high memory.
Recall that the 1-month autocorrelation of Niño 3.4 is 0.91. Figure 3a demonstrates that a lagged regression analysis involving Niño 3.4 could be susceptible to reporting a causal relationship when there is none—the lagged regression analysis is simply picking up the memory α in Niño 3.4. Granger causality analysis, on the other hand (as seen in Fig. 3b), would likely not be susceptible to this problem, as the results of the Granger causality analysis do not depend on α, even when α is very high [see Runge et al. (2014) for a more in-depth discussion of this effect]. This will be explored in the following section.
4. Applications in climate variability
a. ENSO and surface temperature
We now apply the results of our statistical model to the apparent paradox of Fig. 1. We know that ENSO’s memory is large; do the benefits of Granger causality seen in the statistical model carry over to climate variability problems? This time, we perform lagged regression and Granger causality analysis in both directions—we use ENSO to predict 
Figure 4 compares lagged regression (Figs. 4a,b) and Granger causality (Figs. 4c,d) to test the hypothesis that ENSO drives 
Using (a),(b) lagged regression and (c),(d) Granger causality to test the hypothesis that (left) ENSO drives 
Citation: Journal of Climate 31, 8; 10.1175/JCLI-D-17-0334.1
Since ENSO dynamics and teleconnections have been well studied and largely understood for decades, climate scientists are unlikely to misinterpret Fig. 4b. The memories of the two variables are vastly different, and the ENSO–
b. Arctic–midlatitude connections: Another example
Finally, we use Granger causality analysis and lagged regression to investigate the relationship between Arctic temperature and low-level winds across the mid-to-high latitudes. The topic of the impact of Arctic warming on midlatitude weather and climate is one of much scientific discussion and debate (e.g., Walsh 2014; Barnes and Screen 2015 and references therein). However, the direction of the causality of this Arctic–midlatitude relationship is not clear—how much does the Arctic temperature drive midlatitude weather, and how much does midlatitude weather drive changes in Arctic temperature? We do not fully address these questions here; we simply seek to point out that Granger causality can provide information about the direction(s) of causal relationships that cannot be determined from traditional lagged regression.
To analyze the relationship between Arctic temperature and low-level winds, we define Arctic temperature 
Figure 5 displays the results of lagged (Figs. 5a,b) and Granger (Figs. 5c,d) regression analysis for 
Using (a),(b) lagged regression and (c),(d) Granger causality to test the hypothesis that (left) polar mean temperature drives 700-hPa zonal winds and (right) 700-hPa zonal winds drive polar mean temperature. Red indicates a significant lagged relationship identified at up to 20 days (
Citation: Journal of Climate 31, 8; 10.1175/JCLI-D-17-0334.1
The case of 700-hPa winds driving Arctic temperatures (Figs. 5b,d) presents a somewhat different picture. Again, the lagged regression (Fig. 5b) shows large-scale responses over much of the Northern Hemisphere—the Atlantic and Pacific storm tracks, much of continental North America, nearly the entire sub-Arctic (poleward of 60°N), most of Europe, and much of Siberia. Granger causality analysis (Fig. 5d) has a more limited large-scale response than that given by lagged regression; notably, Granger causality does not show a significant response over Siberia and shows a weaker, less spatially homogeneous response in the sub-Arctic region when compared to lagged regression. Previous work has linked changes in midlatitude circulation and sea surface temperatures to warmer Arctic temperatures (e.g., Graversen 2006; Screen et al. 2012; Wettstein and Deser 2014; Baggett and Lee 2015); however, as Fig. 5 demonstrates, the details of these circulation changes differ with different methodologies.
5. Discussion
In this manuscript, we have tried to present a clear, concise, and compelling argument for an increased use of Granger causality analysis in climate variability studies. We have emphasized Granger causality’s superior performance, as compared to lagged regression, in situations in which one or more variables has substantial memory. However, like any approach, Granger causality analysis has its own limitations. One obvious drawback is the possibility of a confounding variable—that is, an additional process or variable could be driving the modeled variables (e.g., in the bivariate case, a third process Z, could influence the independent X and dependent Y variables—
Moreover, Granger causality is simply one approach to causal analysis. Granger causality provides an opportunity for incremental improvement to the already-extant lagged regression analysis framework that has gained so much traction in climate variability studies. Multiple regression-based approaches, such as vector autoregressive (VAR) models, have built upon this Granger causality approach and have been applied to climate variability studies focused on the influence of sea ice on midlatitude circulation (e.g., Strong et al. 2009; Matthewman and Magnusdottir 2011), intraseasonal variability of sea ice (e.g., Wang et al. 2016), paleoclimate data (Davidson et al. 2016), and the relationship between the North Atlantic Oscillation and North Atlantic sea surface temperatures (e.g., Wang et al. 2004).
Even more recently, probabilistic graphical models based on Pearl causality have been introduced to climate science and represent the current state of the art in causal detection theory [see Ebert-Uphoff and Deng (2012) for a thorough introduction of graphical models in climate research]. This graphical approach to causality was first proposed in the 1980s (e.g., Rebane and Pearl 1987; Pearl 1988) and has since been refined and further developed. (e.g., Spirtes et al. 1991). Granger causality has, in fact, been incorporated into these graphical models, creating an approach known as graphical Granger models (e.g., Arnold et al. 2007). Ebert-Uphoff and Deng (2012) and Runge et al. (2014) have demonstrated the utility of these graphical approaches to causality in climate science, and we encourage readers to refer to these papers for more thorough discussions of these graphical models and their advantages in climate variability studies.
6. Conclusions
While lagged regression is a straightforward, popular, and often effective analysis technique in climate variability studies, it is vulnerable to overstating causal relationships in situations in which one or more datasets has significant memory (e.g., Runge et al. 2014). We use a Monte Carlo model to demonstrate the following:
Granger causality outperforms (i.e., lowers the risk of false detection) lagged linear regression when one or more variables has substantial memory;
Granger causality and lagged linear regression yield similar results when there is a true causal relationship between the variables (except in the case of very high autocorrelation); and
Granger causality analysis is only slightly more challenging to implement than traditional lagged linear regression analysis, as it simply consists of a lagged autoregression and a lagged multiple linear regression.
Acknowledgments
Many thanks to Greg Herman, Thomas Birner, Eric Maloney, David Thompson, and Lauren McGough for their suggestions and feedback regarding this work. Many thanks also to the three anonymous reviewers and the editor, whose thorough reviews have greatly strengthened this work. The MERRA-2 data used in this study have been provided by the Global Modeling and Assimilation Office (GMAO) at NASA Goddard Space Flight Center. MCM and EAB are supported by the National Science Foundation under Grant AGS-1545675.
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