# Correlation in least square estimator k-fold cross validation

I have model

$$Y = \beta + \epsilon$$

where $Y$ is scalar and $\beta \in \mathbb{R}$. $\epsilon$ has mean $0$ and variance $\sigma^2$.

If I perform a k-fold cross validation, what is the correlation between $\hat{\beta^1}$ and $\hat{\beta^2}$?

$\hat{\beta^1}$ and $\hat{\beta^2}$ are the least square estimator for 1st and 2nd fold.

This is how I am approaching it:

$$Cor(\hat{\beta^1}, \hat{\beta^2}) = \dfrac{cov(\hat{\beta^1}, \hat{\beta^2})}{\sigma_{\hat{\beta^1}} \sigma_{\hat{\beta^2}}}$$.

The number of elements in training set for $\hat{\beta^1}$, $\hat{\beta^2}$ should be same but I do not know where go on from there.

Update: I have reached $$Cor(\hat{\beta^1}, \hat{\beta^2}) = \dfrac{k^3}{(k-1)^3} \cdot \dfrac{1}{n} \cdot \dfrac{cov(\hat{\beta^1}, \hat{\beta^2})}{\sigma^2}$$