Suppose $X_1,\ldots,X_n$ are iid normal random variables, $X_i \sim \mathcal{N}(\theta, \sigma^2)$. The method of moments gives estimations for $\theta$ and $\sigma^2$ by solving for them in $\bar{X}=\theta, \qquad \frac{1}{n}\sum X_i^2 = \theta^2+\sigma^2$ Giving $\theta = \bar{X}, \qquad \sigma^2=\frac{1}{n}\sum(X_i-\bar{X})^2$
Why does this give a biased estimation of $\sigma^2$? Since it seems to have $1/n$ instead of $1/(n-1)$ in front?