# Interaction terms and causal interpretation

I'm estimating the following model:

$$Y_{i,t} = \alpha_{0} + \alpha_{1}X^{1}_{i,t} + \alpha_{2}*T_{t} + \alpha_{3}X^{1}_{i,t}*T_{t}$$

In which $$T_{t}$$ is the year treated as a continuous variable, tend to measure the linear trend of $$Y_{i,t}$$.

To give you some background, this work targets a field where using hypothesizing plus regression (not those like DID, RD, IV, etc.) analysis is still comfortable for researchers to infer causality in certain ways. But they still care about the standard issues (e.g., multicollinearity, omitted variable; measurement error, etc.) in interpreting results from the causal view.

Anyway, in this particular project of ours, we have the following hypotheses:

1, $$X^{1}_{i,t}$$ affect $$Y_{i,t}$$ .

2, the effect of $$X^{1}_{i,t}$$ on $$Y_{i,t}$$ changes linearly with time (i.e., year).

The model above is to test hypothesis 2. Therefore, we expect $$\alpha_{3}$$ to be statistically significant. And that is actually what we got.

But to interpret $$\alpha_{3}$$ in a causal sense, someone raised the concern that the effect of the interaction term may come from $$T_{t}$$ if it is correlated with $$X^{1}_{i,t}*T_{t}$$.

Thanks to the answers, I realized that this is a multicollinearity issue.

First, there is multicollinearity between interaction term ($$X^{1}_{i,t}*T_{t}$$) and variables ($$X^{1}_{i,t}$$, $$T_{t}$$) constitute it.

Second, there may be multicollinearity between $$X^{1}_{i,t}*T_{t}$$ and $$T_{t}$$ arise from the data generating process.

The first one can be taken good care by centering before generating the interaction term. But what if there is still multicollinearity between $$X^{1}_{i,t}*T_{t}$$ and $$T_{t}$$ even after the centering procedure?

1, How can we test whether the second type of multicollinearity exists?

2, How to deal with it if it does emerge from the result of the test?

Your model is a regression with a (linear) time component and a time-treatment interaction. Otherwise you explain nothing about the problem or the data you are working with. In this very general setting, you cannot interpret regression coefficients (either main effects or interactions) causally without further assumptions about the data generating process.

The reason: correlation is not causation as you probably already know. Regression tells us something about the relationships between the outcome Y and the predictors X and T (ie, about correlations). Extra conditions are required to interpret those relationships as causal. In other words, it's true that if X causes Y and its causal effect increases/decreases with time, then the interaction term is non-zero. But the interaction term is non-zero under a variety of other situations where X doesn't cause Y.

Since there is no generic argument that regression reveals causal effects, you need to focus on your specific problem/data/question to demonstrate causality.

(Free) Resources to learn about causal inference:
[1] Causal Inference for The Brave and True causality according to a data scientist
[2] Causal Inference: The Mixtape causality according to an economist
[3] The Effect: An Introduction to Research Design and Causality causality according to an economist, v2
[4] Causal Inference: What If causality according to epidemiologists
[5] Introduction to Causal Inference from a Machine Learning Perspective causality according to an ML scientist

• +1 I didn't know about those sites. I have a copy of Judea Pearl's 2009 textbook. Any important differences between Pearl's work and these books? May 6 at 15:06
• @AgnesianOperator These books are easier (possible?) to understand. Also check out Which causal inference book you should read. Brady Neal also has a book (more theoretical than the these three) and a great set of lectures. Should add it to the list actually. May 6 at 15:22
• Thanks for your response @dipetkov, really appreciated it. My bad to make the question too vague. I edited the problem and the question to focus on multicollinearity issues in regression from which one attempts to draw some causal interpretations. May 7 at 0:36
• It's not about multicollinearity either. As long as you stay within a general framework you won't be able to make causal conclusions because there are (many) counter-examples to your interpretation. For example: There is another variable Z, not included in the model, that causes X, causes Y and varies with time. From causal point of view this is very different from the hypothesis that X causes Y but the regression will happily detect the relationship between X and Y (they are related because they are both caused by Z). May 7 at 0:52
• @dipetkov, your response is very insightful, I believe it helps both me and those that run into this question. I understand that there are possible alternative explanations from the regression results. That's why I added to the question that we had developed a hypothesis about the explanation I posted here. I know this is not enough but it's acceptable in certain fields. Therefore, the aim of this question, as I understand it, is how to deal with the multicollinearity issue in regression models that one intends to draw causal claims. while putting all other concerns aside, for the time being May 7 at 3:25

You should accept the answer from @dipetkov (+1), as the difficulties of causal interpretation are key.

In terms of statistical analysis per se, note that $$T_{t}$$ is necessarily correlated with $$X^{1}_{i,t}*T_{t}$$; consider the perfect correlation if $$X^{1}_{i,t}$$ is constant with time. That's inherent in an interaction term.

The statistical model addresses whether including the $$X^{1}_{i,t}*T_{t}$$ term improves the model with just $$X^{1}_{i,t}$$ and $$T_{t}$$ as separate predictors. A significant coefficient for the interaction doesn't just gauge whether "the effect of $$X^{1}_{i,t}$$ on $$Y_{i,t}$$ changes linearly with time"; it also indicates that the effect of $$T_{t}$$ on $$Y_{i,t}$$ depends on the values of $$X^{1}_{i,t}$$. Those might have different "causal" interpretations.

And furthermore, with this type of time-series data a spurious correlation can wreak havoc with causal interpretation.

• EdM @EdM, I appreciate your response and took your advice to accept dipetkov's answer as the accepted one for my question. But I still hope that, despite all the difficulties of interpreting results from regression analysis in a causal sense, there is something we can do to improve our confidence in doing so. For this particular question, I wish to hear how people in the causal inference field think, although it's a classical question in regression analysis. May 7 at 0:46
• @JasonGoal the classical answer in regression analysis is that interactions don't necessarily imply multicollinearity, or vice-versa. See this page for an introduction. Multicollinearity is perhaps overemphasized in many presentations; important in big-data situations but less so in smaller data/simpler models. It affects the reliability of individual coefficient estimates, but not the ability of the model to fit the data at hand or even the ability of the model to predict from new data in many situations.
– EdM
May 7 at 1:41
• I agree with you that interaction does not necessarily imply multicollinearity. What I'm saying in the question is the multicollinearity between variables (X1, X2) and the interaction they formed (X1*X2). I also agree with you that multicollinearity is kind of over-emphasized in predictive models. I think it's also necessary to point out that multicollinearity is a concern in models to establish causal relationships. statisticalhorizons.com/multicollinearity May 7 at 3:10