# Correlation coefficient between a stochastic and non-stochastic variable or, what is the difference between a non-stochastic variable and a constant?

I am asking in the context of simple linear regression.

If my regressor $$X_i$$ is non-stochastic, should $$\mathbb E(X_i) = \overline X$$ or $$X_i$$ i.e., treat it like a constant? I presume the latter because some of the derivations, the unbiased one for example treat it as a constant by taking $$X$$ terms out of the expectation.

Further, what is $$\mathrm{Corr}(Y_i,X_i)$$ where $$Y_i$$ is the regressand. If we treat $$X_i$$ as a constant like we did previously, then the correlation should be undefined since $$\mathrm{Cov}(Y_i,X_i)=0$$ and $${\sigma_x}^2 = 0 \quad \because X_i - \mathbb E(X_i) = 0$$. Intuitively, though I feel like it should be a number close to $$\pm 1$$ because they share a linear relationship.

If we go by undefined, then my question is what is correlation coeffecient? So far, I thought of it as a measure of how closely can we fit a straight line in a scatter plot but that definition seems to break down when we consider a non-stochastic variable.

What's more puzzling is that the book I am following, Basic Econometrics by Damodar Gujarati, states that the residuals $$\widehat u_i$$ are uncorrelated with $$X_i$$ because $$\sum_{i=1}^n \widehat u_i \, X_i = 0$$ (end of section 3.1, fifth edition).

This makes me think that correlation should not be undefined for a non-stochastic variable, thus my intuition serving right.

If we assume $$X_i$$ as stochastic i.e., $$\mathbb E(X_i) = \overline X$$, then $$\mathrm{Cov}(Y_i,X_i) = \mathbb E(u_i \, X_i)$$ which makes a lot more sense, but then some of the proofs that require it to be treated like a constant break down.

While writing this question, I realised that it may be that for unscripted terms i.e., constants, the expectation is itself but for scripted terms i.e., non-stochastic variable, there should be a bar, but that still doesn't solve the problems. With this interpretation, $$\mathbb E(u_i \, X_i) = \overline X \,E(u_i) = 0$$. Which again, doesn't make sense why two variables that share a linear relation should have zero covariance, and this still poses the problem of treating it like a constant in derivations. $$\mathbb E(X_i \, Y_i)$$ should $$= \overline X \, \mathbb E(Y_i)$$ not $$X_i \, \mathbb E(Y_i)$$.

Lastly, I know $$\mathrm{Corr}(Y_i,X_i) \neq 0$$ generally, since coeffecient of determination has a similar formula and that obviously won't always be 0.

• Let's say a car is on the road at 60km/h. It's position (as a function of time) is a non-stochastic variable that changes 60km every hour and therefore, not a constant. Its speed is a non-stochastic variable and also a constant. Jun 8 '21 at 17:39

This means that if $$X_i$$ is nonstochastic, then $$\mathbb{E}(X_i) = X_i$$. As for the $$Corr(Y_i, X_i)$$, you are correct, it is not defined. We basically say: "we have no information about the relationship", and it can be anything from $$-1$$ to $$1$$.
As for your question about stochastic regressors, $$\mathbb{E}(X_i)$$ can be very well approximated by $$\bar{X}$$ (by LLN in case of i.i.d. $$X_i$$), but it does not have to be equal to it. Moreover, it will most definitely be two different numbers, even with two different names: the expectation and the sample mean.
Covariance between $$Y_i$$ and $$X_i$$ can then be calculated as $$Cov(Y_i, X_i) = Cov(\beta_0 + \beta_1 X_i + u_i, X_i) \\ = Cov(\beta_0, X_i) + \beta_1 Cov( X_i, X_i) + Cov(u_i, X_i)\\ = \beta_1 Var(X_i)$$ because by the classical OLS assumptions $$Cov(u_i, X_i) = 0$$. I guess by plugging these values in the proofs that you were talking about would not break them down.