I am trying to compute the MSE of $\hat{\sigma}^2$, where $\hat{\sigma}^2= \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2$ for i.i.d samples $\{x_i\}_{i=1}^N$ normally distributed. I get the following calculation:
$$ \begin{align} \text{MSE} \, (\hat{\sigma}^2) &= \text{Var}(\hat{\sigma}^2) + \text{Bias}^2(\hat{\sigma}^2)\\ &= \mathbb{E}[\hat{\sigma}^4] + \mathbb{E} [\hat{\sigma}^2]^2 + \mathbb{E}[\hat{\sigma}^2] - \sigma^2 \end{align} $$
I am stuck here on how to calculate $\mathbb{E}[\hat{\sigma}^4]$ ? Any help or direction is appreciated a lot. Thanks!