non stochastic regressors

Question

In the multiple linear regression analysis if regressors are non-stochastic the causal interpretation of parameters is automatically permitted? I think so, because it seems me that the model can be interpreted as "true causal model" but I'm not sure. Expecially I'm not sure about the role of something like that control variables.

Answer 1

In the multiple linear regression analysis if regressors are non-stochastic the causal interpretation of parameters is automatically permitted?

No, the fact that regressors are non-stochastic makes no difference for whether the estimates identify a causal effect. For causal interpretation of parameters you need a causal model.

For simplicity, assume everything has mean zero and unit variance. Let $X$ and $Z$ be two fixed vectors of size $n$ with $(n-1)^{-1}\sum_{i} X_{i}Z_{i} = \sigma_{xz} \neq 0$. Assume you do not observe $Z$.

Now let the structural equation for $Y$ be:

$$ Y = \gamma Z + U_{y} $$

Where $U_{y}$ is a zero mean, normally distributed random variable. Thus, there is no causal effect of $X$ on $Y$.

However, the regression of $X$ on $Y$ is given by,

$$ \begin{align} \hat{\beta} &= (n-1)^{-1}\sum_{i=1}^{n} X_{i}Y_{i} \\ &=(n-1)^{-1}\left(\gamma \sum_{i=1}^{n} X_{i}Z_{i} + \sum_{i=1}^{n} X_{i}U_{yi}\right)\\ &= \gamma\sigma_{xz} + (n-1)^{-1}\sum_{i=1}^{n} X_{i}U_{yi} \end{align} $$

Where the only "random" part is $U_{y}$, Thus, taking the expectation gives us:

$$ \begin{align} E[\hat{\beta}] &= \gamma\sigma_{xz} + (n-1)^{-1}\sum_{i=1}^{n} X_{i}E[U_{yi}]\\ &= \gamma\sigma_{xz} \end{align} $$

Which is different from zero and clearly does not have a causal meaning. We can also do asymptotics by letting the size of $X$ and $Z$ grow and keeping $\sigma_{xz}$ fixed. The very book you mention in the comments and in your other question, Brooks (2014), mentions how omitted variables can bias the estimate.

Bear in mind this is just an example of omitted variable bias, there are several other things that can go wrong, and they have nothing to do with whether you treat $X$ as stochastic or not. The bottom line here is that confounding, missing data, selection bias --- and many other problems---can still be present, whether you treat the regressors as random or "non-stochastic". You need to make assumptions about the presence or absence of those things, which are causal concepts, and for that you need a causal --- not a regression --- model.