When we calculate the Null Deviance, we need calculate the log-likelihood of the Saturated Model first. For the linear regression, the log-likelihood function would be


for the saturated model, as $y=\hat{y}$, we will have


So how can we estimate the $\sigma$ here? Usually, in a non-saturated model,


If we do similar thing to saturated model, $\hat{\sigma}$ would be 0 and the log-likelihood of saturated model would not exist.


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