I want to forecast prices $S(t)$ of some asset based on historical daily values. I want to use the geometric Brownian motion given by an SDE: $$dS=\mu S t + \sigma S dB,$$ where $B$ is a Brownian motion, for modeling. The historical prices are $$\{S_i\}_{i=1}^N,$$ from which I calculate the log-returns ($N-1$ in total) $$Z_i=\ln\frac{S_i}{S_{i-1}},$$ and the 1-day historical volatility as the standard deviation of the returns: $$\hat{\sigma} = \sqrt{Var\{Z_i\}}.$$
Q1: Let's say I want to forecast the prices $S(t)$ for 180 days. Should I take $\sigma$ in the SDE as the 1-day volatility $\hat{\sigma}$ or as $\sqrt{180}\hat{\sigma}$? I'd say that $\sigma=\hat{\sigma}$ as I'm modeling day by day, but is it correct?
Q2: How do I compute the drift $\hat{\mu}$ from the historical prices? Is it simply $$\hat{\mu}=\frac{1}{N-1}\sum\limits_{i=1}^{N-1}Z_i,$$ and is it (the above formula) a 1-day drift?