Suppose we are using an Ordinary Least Squares (OLS) estimator of $\alpha_{0}$ and $\alpha_{1}$ for the simple linear regression below:
$$ H_{i} = \alpha_{0} + \alpha_{1}X_{i} + \epsilon_{i} $$
How does one show that the goodness-of-fit measure, $R^{2}$ will equal to the square of the correlation coefficient of $H_{i}$ and $\hat{H}_{i}$ which is the expression below:
$$ R^{2} = \left[ \frac{\sum_{i=1}^{N}(H_{i} - \bar{H})(\hat{H}_{i}-\bar{\hat{H}})}{\sqrt{\sum_{i=1}^{N}(H_{i}-\bar{H})^{2}}{\sqrt{\sum_{i=1}^{N}(\hat{H}_{i}-\bar{\hat{H}})^{2}}}} \right]^{2} $$
Help is greatly appreciated. Thank you.
