A Case Study of US GDP Growth

The Influence of Attention Score on Macroeconomic Indicators

The previous section explored the visual relationship between the attention metric and media coverage received by each clusterized group of topics. In this section, we conduct a quantitative analysis to measure the relationship between US GDP growth and the attention score.

The main hypothesis of this study is that attention score can provide valuable contextual insights that may impact the forecast of the macroeconomic indicators. To validate this hypothesis, we focus on the US GDP growth as the target variable. Note that at this stage, we are presenting preliminary results of the correlation and causality analysis between these variables.

For our analysis, we adopted a time-windowed approach, where each window spans a period of 12 quarters (3 years) with a 1-quarter step. To ensure consistency in the data frequency, the cluster's attention metric has been aggregated to a quarterly frequency. The correlation analysis considers both the output of the distance correlation test (Székely, G. J. et al., 2007) and the Spearman correlation (Dodge, Y., 2008). These approaches are particularly useful when we cannot assume a linear correlation between the variables. Furthermore, we conducted a Granger causality analysis (Granger, C. W., 1969) employing the same time-windowed methodology.

For this analysis, each window encompassed a duration of 32 quarters (8 years) with a 1-quarter step. Throughout all the conducted tests, we established a significance level of 0.1.

Distance correlation

The distance correlation test is a measure of independence between two random variables X and Y. Unlike Pearson correlation, which can only detect linear relationship patterns, the distance correlation test captures both linear and non-linear dependencies between variables.

The hypotheses used in the test are as follows:

H₀: dCor²(X, Y) = 0 (The random variables X and Y are independent.)
H₁: dCor²(X, Y) <> 0 (The random variables X and Y are not independent.),

where dCor²(X, Y) is the distance correlation between the two random variables. The calculation of dCor²(X, Y) requires additional statistics such as distance variance, distance standard deviation, and distance covariance.

The p-value associated with the test statistic is calculated through a permutation test. Initially, the distance correlation coefficient is calculated between the two variables, and then this value is compared to the distance correlation coefficient obtained from multiple shuffles of the data.

Table 1 (next page) presents the top 5 attention clusters that exhibit the highest correlation with the target variable. Notably, Cluster 46 which solely contains information about Topic 7 “Payments”, shows a correlation in 30 out of the 81 analysed periods, representing 37% of the windowed periods.

Figure 5 provides a further visual representation of the analysis. On the left-hand side, it shows the distribution of the distance correlation statistic over time and highlights the period where the null hypothesis is rejected. On the right-hand side, it presents the distribution of the p-values associated with each windowed period, and the red line represents the significance level of 0.1.

Spearman’s rank correlation

Spearman’s rank correlation is a non-parametric test that allows for measuring the relationship between two variables when the assumption of normality does not hold or in scenarios where one cannot assume a linear relationship between two variables. This test assesses whether the rank order of one variable (X) is independent of the rank order of the other variable (Y).

The hypothesis for this test can be formulated as follows:

H0: X and Y are mutually independent.
H1: X and Y are not mutually independent.

Upon initial examination, we have observed strong associations between the attention metric of certain clusters and US GDP growth over time. Table 2 (next page) presents the top 5 attention clusters (the ordering has been performed considering the number of correlated periods) that exhibit the highest correlation with our target variable. For instance, Cluster 31, which is associated with financial compliance, shows a correlation with the US GDP growth in 26 out of the 81 considered time windows, representing approximately 32.1% of the analysed periods.

Figure 6 illustrates the distribution of Spearman’s rank correlation statistic, on the left-hand side, highlighting the windows where the null hypothesis of independence is rejected with a 90% confidence level. On the right-hand side, the distribution of p-values is displayed, with a threshold line indicating the significance level of 0.1 used in this analysis.

Granger causality

The Granger Causality test provides insights into whether a time series X can be used to forecast another time series Y. If signal X "Granger-causes" Y, it suggests that past values of X, up to a certain lag, contain information that aids in predicting Y. The test should be applied bidirectionally to assess the feedback relationship between the two series. In our analysis, we evaluate whether the attention of a particular cluster Granger-causes US GDP growth and vice versa.

The hypotheses for the test are defined as follows:

H0: Variable X is not the cause of variable Y with a certain delay.
H1: Variable X is the cause of variable Y with a certain delay.

In contrast to the time-windowed approach used for the correlation analysis, the Granger Causality test employs a window
size of 32 quarters (8 years) with a 1-quarter step, allowing for a maximum lag of 9 quarters. Using this configuration, a total of 61 time-windowed periods are analysed.

Table 3 presents the results associated with clusters that exhibit a higher number of periods where cluster attention Granger-causes US GDP growth.

The table also includes information on the reverse relationship, where US GDP growth Granger-causes attention in the cluster. Observing the table, it is evident that attention values are more likely to have an impact on the target variable than the reverse relationship.

Figures 7 and 8, on the left-hand side, illustrate the distribution of the Granger test statistic over time, highlighting the periods where the null hypothesis is rejected and indicating the lags at which the causal relationship occurs.

Both charts display the distribution of p-values on the right-hand side, with the red line representing the significance level used in this analysis. The red dotted line in both plots denotes periods where the test could not be executed with the maximum allowed lag of 9. For Cluster 24, in particular, there are 49 periods where attention appears to have predictive power on the target variable, accounting for approximately 80.33% of the analysed periods.

References:

Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent Dirichlet allocation. Journal of Machine Learning Research, 3(Jan), 993-1022.

Dodge, Y. (2008). Spearman rank correlation coefficient. The concise encyclopedia of statistics, 502-505.

Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, 424-438.

Joyce, J. M. (2011). Kullback-Leibler divergence. In International Encyclopedia of statistical science (pp. 720-722). Springer, Berlin, Heidelberg.

Székely, G. J., Rizzo, M. L., & Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances.

Appendix A:

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Appendix A cont'd:

Appendix B:

Dendrogram obtained after applying the hierarchical recursive
agglomerative algorithm described in Section 1. (Click image to expand)