# Causal Inference using Linear Regression

I have been reading recently on fitting linear regression to evaluate causal effect of some treatment. Let's call the variable in the model representing treatment as Xj.

From what I have read, we need to make sure to include in the model other variables that affect both the responsible variable 'y' and the treatment variable Xj. I understand that only variables that affect both will impact the coefficient of Xj.

However, if a variable Xi impacts the response variable 'y' but is independent of Xj , isn't it still important to include it in the model since it can reduce the error ? It won't change the coefficient of Xj but it will affect its standard error which is important when trying to establish whether treatment effect is significant or not.

Is my logic incorrect ? Do we only need to worry about adding variables that affect both response and treatment variable ?

The following holds generally, but the exact relationships may differ depending on the data.

Including a variable that is a predictor of the outcome and the treatment will reduce the bias and variance of the effect estimate. The more related to the outcome and the less related to the treatment, the more variance will be reduced.

Including a variable that is a predictor of the outcome and unrelated to the treatment will reduce variance without affecting bias.

Including a variable that is a predictor of the treatment and unrelated to the outcome (i.e., an instrument) will increase variance without affecting bias, as long as there is no unmeasured confounding. If there is unmeasured confounding, including instruments will increase the bias.

There are many other phenomena that can occur, too. Omitting a variable whose coefficient is smaller than its standard error can reduce the mean squared error even though doing so induces bias (Rao, 1971). Including variables with various properties can increase or decrease bias depending on those properties (Steiner & Kim, 2016). In general, though, your reasoning is correct.

Rao, P. (1971). Some Notes on Misspecification in Multiple Regressions. The American Statistician, 25(5), 37–39. https://doi.org/10.2307/2686082

Steiner, P. M., & Kim, Y. (2016). The Mechanics of Omitted Variable Bias: Bias Amplification and Cancellation of Offsetting Biases. Journal of Causal Inference, 4(2). https://doi.org/10.1515/jci-2016-0009