From Wikipedia,
"Let hat $\hat{L}$ be the maximum value of the likelihood function for the model."
To me, $\hat{L}$ should have two values, $\hat{\beta_0}$ and $\hat{\beta_1}$ in the simple linear case,
$$\hat{Y} =\hat{\beta_0} + \hat{\beta_1}X $$
Would someone be able to explain precisely what is meant by $\hat{L}$ in AIC?
If $y_i|x_i \sim \mathcal{N}(\beta_0 + \beta_1 x_i, \sigma^2)$, then its likelihood is given by $$ \prod_{i=1}^n f(y_i ;x_i) = \prod \frac{1}{\sqrt { 2\pi \sigma ^2} }\exp\left\{-\frac{(y_i - \beta_0-\beta_1x_i)^2}{2\sigma^2 }\right\}, $$ plug in the OLS (MLE) of $\beta_0$ and $\beta_1$ and $\sigma^2$ - and you get the maximum value of the likelihood function, i.e., $$ \hat{L} = \left( 2\hat{\sigma}^2 \pi \right)^{n/2}\exp\left\{ -\sum(y_i - \hat{y}_i )^2/(2\hat{\sigma}^2)\right\}, $$ where $\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1x_i$ and $\hat{\sigma}^2 = \frac{1}{n}\sum (y_i - \hat{y}_i)^2$,hence $$ \hat{L} = \left( 2\hat{\sigma}^2 \pi \right)^{n/2}\exp\left\{ -n/2\right\}. $$