The mathematical formulation of Granger causality could be represented as follows. Let’s suppose that we have two time series \( X_t \) and \( Y_t \), if we are trying to predict \( Y_{t+1} \), we can do it only using the past terms of \( Y_t \) so its future depends only on its own past, or we can use both time series \( X_t \) and \( Y_t \) to make the prediction. If the later prediction considering a relationship between the two time series is significantly better, then it is possible to argue that the past of \( X_t \) contains valuable information to predict \( Y_{t+1} \); therefore, there is a one-way relationship from \( X_t \) to \( Y_t \). In this case, the conclusion is \( X_t \) “Granger causes” \( Y_t \). This conclusion can be made because the analysis rests on the proposition of two funding axioms made by Granger (1980):

1. The cause happens before its effect.
2. The cause has unique information about the future values of its effect.

It is important to point out that the term causality can be a bit misleading. Granger causality shows that the change in the temporal evolution of one variable precedes the change of values on another. Since time proceeds only in one direction (as far as humans can perceive), the time delay establishes directionality or better time-bound predictability. Still, it is mute on the more profound question of the mechanisms of causality. People solve this debate by using the term “Granger causality” to distinguish between this practical definition and causality.

Analysis

To make an analysis using Granger causality, we need to be familiar with linear autoregressive models, which is a fancy name for a linear method used to predict future behavior based on past information of the same variable. A typical mathematical representation of an autoregressive model with one time-lag would be:

\[ Y(t) = \alpha_1 Y(t-1) + error_1 (t) \tag{1} \] 

Where \( \alpha \) represents the variable’s coefficient with one time-lag and the \( error_1 \) has a distribution with mean \( 0 \) and variance \( \sigma \).

As I mentioned earlier, the application of Granger causality compares two different models, one containing information from only one variable and the other from two. The model in (1) includes information on \( Y \) only, so we need a model with information about \( Y \) and another variable, which I will call \( X \), to proceed with the Granger causality analysis.

\[ Y(t) = \alpha_1 Y(t-1) + \beta_1 X(t-1) + error_2 (t) \tag{2} \]

Where \( \alpha \) represents the coefficient of the variable \( Y \) with one time-lag, \( \beta \) is the coefficient of the variable \( X \) with one time-lag, and the \( error_2 \) has a distribution with mean \( 0 \) and variance .

Finally, once we have the models described in (1) and (2), a F-test is performed with the null hypothesis of \( Y(t) \) equals to model (1) against the alternative hypothesis of \( Y(t) \) equals to model (2). The final result allows us to evaluate if \( X \) Granger causes \( Y \) or not. We say that \( X \) Granger causes \( Y \) if we reject the null hypothesis.

The previous formulation presents the case when we include only one time-lag of the variables under study in the models. You might be asking what is the formulation for the general case. Using the sum notation, we can extend models (1) and (2) as follows:

 \[ Y(t) = \sum_{j=1}^p \alpha_j Y(t-j) + error_1 (t) \tag{3} \]

\[ Y(t) = \sum_{j=1}^p \alpha_j Y(t-j) + \sum_{j=1}^p \beta_j X(t-j) + error_2 (t) \tag{4} \]

The same as the previous model in (3) and (4) \( \alpha_j \) and \( \beta_j \) are the coefficients of the model, \( error_1 \) and \( error_2 \) have a distribution with mean \( 0 \) and variance , but there is a new element \( p \). Here \( p \) represents the maximum number of lagged observations included in the model. The selection of an appropriate number of lags \( p \) is a crucial decision to make before performing the analysis; we can base this choice on statistical criteria for information such as FPE, AIC, HQIC, and BIC. These measures compare the inherent trade-off between model complexity and the model’s predictive power, following Occam’s insight that models should only be as complex as necessary. However, theoretical considerations are also important in selecting the final number of lags to be included in the model. Once the decision of the maximum lagged observations is made, we proceed in the same fashion previously described in the case of \( p=1 \) to conclude if \( Y \) Granger causes \( X \).

Limitations

The study of Granger causality was initially designed for linear equations. There are extensions to nonlinear cases nowadays, but those extensions can be more challenging to understand and use. Moreover, their statistical meaning is less comprehended. Among the extensions, we can find approaches that divide the global nonlinear data into smaller locally linear neighborhoods (Freiwald et al., 1999) or the use of a radial basis function method to perform global nonlinear regression (Ancona et al., 2004)

Another limitation is that one of the assumptions of Granger causality requires that the time series be stationary, which means that their mean and variance should remain constant over time. Non-stationary time series can be transformed into stationary ones using the differentiation method. Additionally, assuming that short windows of non-stationary data produce locally stationary signals, it is possible to perform the analysis in such windows (Hesse et al., 2003).

A Granger Causality Application

As a student in the field of Communication, one of my interests is the study of the information dynamics that people create on social media. In this section, I want to share an application of Granger causality I have worked on.

For this example, the objective is to analyze if there is a relationship between online communicated emotions in the aftermath of a natural disaster. The approach seeks to understand if the emotions expressed after a tragic event follow a pattern in which it is possible to predict the appearance of some emotions based on the previous expression of other emotions. For example, after an earthquake, people express fear initially, but once time passes, the expression of emotion changes to sadness, so we would like to know if the initial information about fear helps to predict the later appearance of sadness. In this case, we collected data from Twitter in the aftermath of the earthquake in Southern California on July 5, 2019. To assess the emotion, each tweet was processed using the Natural Language Understanding tool provided by IBM Watson. This tool analyzes the text to assign a value from 0 to 1 for the presence of five emotions: sadness, anger, fear, disgust, and joy.

The procedure has the following steps:

1. Check stationarity: We upload the data and check the stationarity of the time series.
2. Stationarity correction: If required, we transform the time series to make them stationary.
3. Lag selection: We selected the maximum lag for the analysis based on statistical and theoretical criteria.
4. VAR analysis: We conduct an initial multivariate analysis to identify predictive relationships between the five emotions studied.
5. Granger causality: Once we identify the predictive relationship between pairs of emotions, we conduct the analysis to determine if those relationships are Granger causal.