# how to prove that $\hat \sigma^2$ is a consistent for $\sigma^2$?

Consider a regression model $Y_n=X_n\beta +\varepsilon$, where $X_n$ is a $n \times p_n$ matrix, and $\varepsilon=(\varepsilon_1,...,\varepsilon_n)'$ consists of independent and identically distributed variables with $E(\varepsilon_1)=0$ and $Var(\varepsilon_1)=\sigma^2$. Suppose that $\varepsilon_1 \sim N(0, \sigma^2)$ and suppose that $\hat \sigma ^2$ is the estimator of $\sigma^2$. Let

$$\hat \sigma ^2=\frac{\left\|Y_n-X_n\hat\beta\right\|^2}{n-p_n}.$$

Then, $\hat \sigma^2$ consistent for $\sigma^2$.

I can prove that $\hat \sigma^2$ is unbiased estimator for $\sigma^2$.

How can I continue to prove that $\hat \sigma^2$ consistent for $\sigma^2$?

• Thanks a lot. For this method, $X_i$ is random variable, but for my case, $X_i$ is fixed. My supervisor asked me not to use this method. He told me to prove unbiased estimator and then continue to prove the lim of variance. – Vivian Zhang Dec 1 '16 at 19:10