Training error is optimistic

Let $y = X\beta + \varepsilon$ with $X$ a fixed $n \times p$ matrix of full column rank and $\varepsilon$ having mean zero and covariance $\sigma^2 I$. Fit OLS, $\hat\beta = (X^TX)^{-1}X^Ty$.

(a) Show $E\left[\|y - X\hat\beta\|^2\right] = \sigma^2(n - p)$.

(b) Let $y^* = X\beta + \varepsilon^*$ be fresh data at the same inputs, with $\varepsilon^*$ independent of $\varepsilon$ and distributed the same way. Show $E\left[\|y^* - X\hat\beta\|^2\right] = \sigma^2(n + p)$.

(c) With $n = 250$ daily observations and $p = 25$ predictors, by what factor does average training MSE understate average test MSE?