Granger Causality—Statistical Analysis Under a Configural Perspective

Abstract

The concept of Granger causality can be used to examine putative causal relations between two series of scores. Based on regression models, it is asked whether one series can be considered the cause for the second series. In this article, we propose extending the pool of methods available for testing hypotheses that are compatible with Granger causation by adopting a configural perspective. This perspective allows researchers to assume that effects exist for specific categories only or for specific sectors of the data space, but not for other categories or sectors. Configural Frequency Analysis (CFA) is proposed as the method of analysis from a configural perspective. CFA base models are derived for the exploratory analysis of Granger causation. These models are specified so that they parallel the regression models used for variable-oriented analysis of hypotheses of Granger causation. An example from the development of aggression in adolescence is used. The example shows that only one pattern of change in aggressive impulses over time Granger-causes change in physical aggression against peers.

Appendix A: The Comparison Matrices in the Configural Approach to Examining Granger Causality

In the configural approach to the analysis of Granger causation, two base models are estimated. The first can be based on the series of Y observations only, the second uses both the X and the Y series of observations. Alternatively, both models can be based on the cross-classification that is spanned by both the X and the Y series of observations. In certain applications, it may be interesting to compare the type/antitype patterns from the first and the second model. This comparison allows researchers to identify the location of associations that are explained without and with consideration of the X scores. In the text of this article, we discuss the larger table, that is, the one that includes both the X and the Y series of observations. Here, we discuss the comparison of the smaller with the larger tables. The following outcomes of the comparison of type/antitype patterns are possible.

• 1. The emerging pattern of type and antitypes is the same for the two base models. In this case, there is no support for configural, Granger-type causality of the series of Y-scores as it is (not) explained by the series of X-scores. It should be noted that this outcome is very unlikely to occur. The reason for this is that also including the X observations results in a much larger cross-classification than using only the Y observations. For a type from the smaller cross-classification to also emerge in the larger table, the pattern of Y categories that constitutes this type must, in the larger table, constitute a type for every cell that includes the Y pattern that constituted the type. Specifically, consider a series of T observations of Y. The cross-classification of these observations has c _Y ^T cells, where c _Y is the number of categories of variable Y. Now, also including the T observations of X increases the number of cells by the factor c _X ^T, where c _X is the number of categories of variable X. By implication, each pattern of categories of Y appears c _X ^T times in the larger table. Thus, for a type to completely re-emerge in the larger cross-classification, it will have to emerge c _X ^T times—a rather unlikely event. This applies accordingly to antitypes. The following outcome is more likely.

• 2. A type or antitype from the smaller cross-classification re-surfaces only in some of the c _X ^T candidate cells of the larger cross-classification. This result is most interesting from a configural perspective. It suggests that, in a Granger causality sense, the series of X scores explains this type or antitype only for those patterns of X–Y observations for which types or antitypes emerge, but not for the remaining patterns. The reason for this interpretation is that the second CFA base model proposes independence between the X and the Y series of scores. Therefore, types and antitypes suggest where, in the cross-classification of the X and Y observations, there are local associations among the X and the Y observations.

• 3. Types or antitypes that emerged for the cross-classification of only the Y observations disappear entirely when the X observations are taken into account. In other words, none of the c _X ^T candidate cells emerges as a type or antitype in the cross-classification that is spanned by the Y and the X observations. This result indicates that a local relation between the X and the Y series does not exist, for this pattern. Including the information from the X series makes this type or antitype disappear. Because of the way the base model was specified, neither main effects of the X and the Y observations nor the interactions among the X observations or the Y observations can be the reason why a type or antitype emerges. This suggests that the assumption of no relation between the Y and the X series of observations cannot be rejected, for this configuration. This interpretation is parallel to the interpretation of changes in the appearance of configural mediation types or antitypes (von Eye and Mun 2013; von Eye et al. 2009).

• 4. Configurations that did not constitute types or antitypes in the cross-classification of the Y observations constitute types or antitypes for the cross-classification of the Y with the X observations. Here, the distinction between situations in which, for a configuration of Y categories, all c _X ^T candidate cells or only a subset of candidate cells constitute types or antitypes is not important. All we ask is whether types or antitypes emerge. If a configuration of Y and X categories constitutes a type or antitype under the second base model, the null hypothesis of no local relation between the Y and the X series of observations can be rejected. This suggests that a local relation exists that can be interpreted in the sense of Granger causation. In this article, we, therefore, focus on types and antitypes from the second base model.