# How to answer this multivariate time-series causal problem?

I seem to have a multivariate time-series causal problem. I am trying to forecast revenue based on variables that may have affected this revenue. I have a dataset containing marketing spend and marketing data through time for various campaigns and social media sites, for example, on the first week of April, the company spent 1M on Twitter marketing, spent \$ 3M on Facebook, received 300M of Facebook impressions, and so forth. The dataset also contains any other causal variables such as whether a given day was a bank holiday or whether the day was a holiday season. The business question is what will be the effect on revenue if we increased Facebook spend on this given week by 10%.

What tools can I employ to help solve such a task? It seems to be a causal question, so linear regression may not be of help?

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Jun 2, 2022 at 23:05

You could do both, (linear) regression or causal analysis. Here is the difference:

If you do regression, this will help you to approximately predict revenue for additional campaign data, provided your model is sufficiently correct. However, regression results, especially results of linear regression, are sometimes used in a way that might occasionally lead to wrong results. E.g., if some feature $$f$$ is given a large positive weight $$w_f$$, this is often (wrongly) interpreted in an interventional sense as follows: If this feature is increased by one unit, the revenue is increased by $$w_f$$. There are counterexamples showing that this is not always true. However, this doesn't happen very often, and this interpretation is quite common practice.

Nevertheless, this interventional interpretation is not correct, and regression results should only be used for prediction.

So, if you are interested in the question of how you should intervene in the marketing campaign to increase revenue, you should consider causal analysis, even though it is more complex.

• I thought Simpson's paradox was a reversal in association at the marginal vs. conditional level, i.e., due to confounding. I don't follow how this is related to regression coefficient interpretation? Regression coefficients give average, conditional increases. The interpretation is: a change in x of 1 unit will, on average, produce a change in y of β, assuming all other x variables stay the same. Linear regression assumes normally distributed errors, which is why we specify that the average change β is only seen in y when we average over many increases in x increasing by 1. Jun 3, 2022 at 12:22
• There's some really good discussion here about causation, including in the comments Jun 3, 2022 at 12:29
• @jdcrossval My point is that regression effects should not be interpreted interventionally, which, however, is often done. Jun 3, 2022 at 14:36