eqI am very confused about a very basic question. This is probably more statistics than quantitative finance, but still, should be useful for this stackexchange board as well.
Let's assume I have monthly returns $r_1, r_2, ...., r_T$.
I can compute average return and standard deviation as:
\begin{equation} \bar{r} = \frac{1}{T} \sum_{t=1}^T r_i \end{equation}
I can compute standard deviation of return as: \begin{equation} \sigma_r = \sqrt{\frac{1}{T-1} \sum_{t=1}^T(r_i - \bar{r})^2} \end{equation}
Thus if I want to test whether the mean is equal to zero I can build the t-statistic:
\begin{equation} t-stat = \frac{\bar{r}}{\sigma_r/\sqrt{T}} \end{equation}
Now assume I annualize the mean and the standard deviation by making:
\begin{equation} \bar{r}^{annual} = \bar{r} \times 12 \end{equation}
\begin{equation} \sigma_{r}^{annual} = \sigma_r \times \sqrt{12} \end{equation}
Now if I compute the same t-stat, as above I get:
\begin{equation} t-stat^{annual} = \frac{\bar{r}^{annual}}{\sigma^{annual}_r/\sqrt{T}} = \sqrt{12} (t-stat) \end{equation}
My question is what am I doing wroing? Why is the t-stat becoming multiplied by sqrt(12)? If instead I first annualize the returns series $r_i$ by making $r_i \times 12$ this doesn't happen, and I get the same t-stat, regardless of whether I annualize or not.