Looks like the rationale behind the accepted answer of this post is incorrect.

Under leave one out cross validation(LOOCV), the variance of its MSE estimator is $$var [\frac{\Sigma_i x_i}{n}] = \frac{var[\Sigma_i x_i]}{n^2}$$ where $x_i$ is an estimate of MSE from one particular iteration.

I agree that LOOCV has a higher enumerator (b/c of the covariance terms), but the denominator is larger as well because there are essentially n estimates (greater than k estimates as in the k-fold case).

Given this, why does LOOCV still have higher variance in estimating MSE and why does it have lower bias?

(This is against intuition b/c increasing sample should decrease variance and leaves bias unchanged for $\hat\theta$ and $\hat{y}$)