# Simple Linear Regression Question: How does correlation between X and Y affect MSE?

Suppose we know that the correlation between $$X_1$$ and $$Y$$ is $$\rho_{X_1Y}$$ and the correlation between $$X_2$$ and $$Y$$ is $$\rho_{X_2Y}$$. Furthermore, suppose $$0 < \rho_{X_2Y} < \rho_{X_1Y}$$. Now, we fit the simple linear regression models $$Y \sim X_1$$ and $$Y \sim X_2$$, which one will have a lower MSE? i.e. which one has a lower $$\parallel Y - \hat{Y} \parallel_2^2 \hspace{0.2cm} = \hspace{0.2cm} \parallel \hat{\epsilon} \parallel_2^2$$

I am quite clueless. Is it even possible to say anything about MSE from just the correlation?

In simple linear regression, the squared correlation between the $$X$$ variable and $$Y$$ variable is equal to the $$R^2$$.
Also, $$R^2$$ has an equivalent representation involving MSE:
$$R^2=1-\dfrac{MSE}{Var(Y)}$$
In both of your simple linear regression equations, the $$Y$$ is the same, so the $$Var(Y)$$ in the denominator is the same for each. Consequently, the greater correlation magnitude corresponds to the higher $$R^2$$ and thus the lower $$MSE$$, with the extreme being an $$MSE$$ of zero (correlation magnitude equal to $$1$$, and $$R^2=1$$).