I'd like to estimate from a daily prices serie $P_t$ with $N$ observations a quantity such as the variance of the weekly returns. I will use $\ln\left(\frac{P_{T+5}}{P_T}\right)$ assuming 5 days in a week for a weekly return.
The good method (or so it seems) would be to use non-overlapping data, that is to say I will end up with a sample of $N/5$ data points. Let's say I use $\left(\ln\left(\frac{P_{5}}{P_0}\right), \ln\left(\frac{P_{10}}{P_5}\right), ..., \ln\left(\frac{P_{N}}{P_{N-5}}\right) \right)$ as sample to compute the variance (assuming $N$ is a multiple of 5). While this looks like the most robust method, the number of observations is greatly reduced.
Is there a way to do better? I am thinking about using overlapping data, but then the samples I end up with (for example $\left(\ln\left(\frac{P_{6}}{P_1}\right), \ln\left(\frac{P_{11}}{P_6}\right), ... \right)$) are not independent one from each other.
Is there a practical method to deal with this?
EDIT
Thank you all for your answers. Following the advice of SRKX, I am taking a real example. I have used the SPY returns from Jan 4th 2012 up to Dec 28th 2012. This gives an initial data set of 753 prices from which I consider the 3 days returns, to make things easier hopefully.
Out of this sample I can build 3 non-overlapping 3 days returns sets of 250 values each. By 3 days return I indeed mean $R_t = \ln \frac{S_t}{S_{t-3}}$. If $r_t = \ln \frac{S_t}{S_{t-1}}$ then obviously $R_t = r_{t-2}+r_{t-1}+r_t$ (apologizes for the sloppy notation).
If one assigns a number from 1 to 753 to the previous close prices, the first sample built with non-overlapping data is then: $$Sample_1 = (\ln \frac{S_4}{S_1}, \ln \frac{S_7}{S_4}, ..., \ln \frac{S_{751}}{S_{748}})$$ and the second and third are $Sample_2 = (\ln \frac{S_5}{S_2}, \ln \frac{S_8}{S_5}, ..., \ln \frac{S_{752}}{S_{749}})$, $Sample_3 = (\ln \frac{S_6}{S_3}, \ln \frac{S_9}{S_6}, ...\ln \frac{S_{753}}{S_{750}})$.
I don't want to choose a priori a specific distribution for the daily returns $r_t = \ln \frac{S_t}{S_{t-1}}$. All I will assume is that $E[r_i r_j] = 0$ for $i \neq j$, $E[r_i]=0$ (reasonable for the considered sample) and $E[r_i^2] = \sigma^2$ (this is not quite the case but let's keep this anyway).
Under these assumptions, I expect that the empirical correlation between the sample 1 and 2 will be $\frac{2}{3}$: $\ln \frac{S_4}{S_1} = r_2+r_3+r_4$ and $\ln \frac{S_5}{S_2} = r_3+r_4+r_5$, so the first elements of these samples have 2 daily returns in common, the remaining terms ($r_2$ and $r_5$) are not correlated (with each other and with $r_3+r_4$), same for all the terms of the sample, hence the result. Same goes for the correlation between the samples 2 and 3, and the correlation between sample 1 and 3 will be $\frac{1}{3}$ (under these assumptions the correlation for the $N$ days returns with $p$ days overlap is $\frac{p}{N}$ if I am not mistaken).
I am lucky, for the data I have chosen it works quite well, I measured the empirical correlations between sample 1 and 2 = $0.63,$ sample 2 and 3 = $0.6$, sample 1 and 3 = $0.3$.
So far so good, now I can compute the 3 resulting (annualized) volatilities out of the 3 samples: I found $\sigma_1 = 18.4\%$, $\sigma_2 = 17.1 \%$ and $\sigma_3 = 18.5\%$. The average of these values is $18\%$.
I can also compute the same thing by fusioning the 3 samples (so everything is mixed), which gives $18\%$.
As SRKX said all these results are finally almost the same... But my question really is: does it make sense to do this? Does computing the 3 sample variances add some information or not? It looks like for $N$ days returns, if the overlap $p$ is small the correlation between the samples can be reduced down to a "small enough" value (for large $N$, which is what I am interested in): in this case can it be used?
Or I should just use other techniques like bootstrapping the first sample and not try to play with the overlapping samples?
I hope I was not too long and clear enough... It might be completely trivial but for now something is not clear for me.