# How to extend Realized Volatilty to multiple periods

I'm trying to calculate 5-day realized volatility (as proxy for integrated volatility) using 5-min frequency data.

I'm working from the paper

CORRECTING THE ERRORS: VOLATILITY FORECAST EVALUATION USING HIGH-FREQUENCY DATA AND REALIZED VOLATILITIES
by Andersen et al (2005)

I'm able to use $$RV_t(h) \equiv \sum_{i=1}^{1/h} r^{(h)2}_{t-1+ih}$$ with $t = 1$ day and $h=81$ for 81 5-min samples per day
to get the daily realized variance.

In the paper, p.282, footnote 4

For notational simplicity, we focus our discussion on one-period return and volatility measures, but the general results and associated measurement error adjustment extend in a straightforward manner to the multiperiod case

I would be very grateful if someone could point out that straightforward extension to me.

Thanks

Edit
I'm going with the simple approach of adding the variances for 5 days, i.e. $h=81 * 5$
Annualized volatility calculated from this is in the same range as the Yang-Zang & Parkinsons estimators

Any input would still be appreciated.

I think your approach for adding the Realized Variances is correct. The focus on one period returns is simply because it eases notation. The only thing you would need to change is the units of time you are working with. This is $t=1$ (e.g. week) means five days and under that notation you would just need to change your $h$ to the new appropriate value $h/5$. This is equivalent to just adding the Realized Variances.
However, I believe the value of the time step $h$ for $5$ minutes is initially (considering 8 hours of trading) would be: $$h=\frac{5}{8*60}\approx0.0104$$ So that as your cited paper says:
[...] $1/h$ is assumed to be an integer [...]
For the $5$ days case: $$h=\frac{1}{8*60}\approx0.0021$$
Note that this extension to multiple periods does not take into account jumps that occur between closure and market opening (ignored by simple adding the daily $RV$'s) which have shown different behaviours compared to the jumps during trading hours. For a detail discussion on how to include close-open jumps see for example Yang&Zhang.