I'm working with two time series over the same time period, and by plotting the time series graphically there is an obvious relationship that exists between the two. When one's values peak, the other's instantaneously drop.
Is there a way I could calculate a significance test for such relationship? I know the Granger causality test models a series' current state as a function the previous states of itself and the other series. In my case the relationship is instantaneous, so it's not giving me anything tangible.
Best Answer
Granger causality is not causality. Granger causality is actually prediction of a time series based on distributed lags from that time series as well as other time series. Causality is the ability to infer a counterfactual difference in outcomes given you experimentally manipulate ("do") an exposure in a hypothetical research setting.
Instead, if you wish to measure how "instantaneously related" two time series are, calculate the cross-correlation of the two time series. This test can be non-specific, since it's possible that two ARMA processes simply follow the same seasonal trends. You can expand on the idea of cross correlation by fitting the following model:
$$E(Y_{(t)}, | Y_{(s<t)}, X_{(s<t)}, X_t) = \sum_{j=1}^s \beta_j Y_{(j)} + \sum_{j=1}^n\gamma_j X_{(j)} + \gamma_t X_{(t)}$$
Here the series are either discrete or pseudo-discrete and the linear combinations may be generalized to some form of GAM. The goal is performing semi-parametric inference on the term $\gamma_t$ which estimates a mean difference in the outcome, $Y_{t}$ controlling for all Granger predictive values (Granger causal is misleading in light of modern causal inference). If the $\gamma_t$ is statistically significant, we can infer that the apparent cross correlation of time series owes to an instantaneous value, rather than spurious synchronization, or a shared historical autocorrelation.