A better approach to this would probably be treating the data as panel data and either doing monthwise hierarchical Bayesian regressions, and testing if the coefficients pooled over time are significantly non-zero, or else using a sort of distributed lag model or instrumental variables model on the monthwise simple regressions. In these cases, it’s better to roll up lagged features of the exogenous time series to serve as constructed covariates in the lag model, than to rely on differencing and time series tests which are like a landmine of interpretation issues.
Checking causality in models like this is very tricky, so a lot of care has to be taken, and you should expect work to proceed slowly and require a lot of posterior checks.
(I am fully aware that I can google it but in my opinion it's usually better to ask someone who is already familiar with the topic because he/she may know good learning materials.)
- < http://www.stat.columbia.edu/~gelman/arm/ >
If you happen to use Python, pymc3 is a great place to look for hierarchical and Bayesian timeseries models with full examples.