# Estimate weekly, yearly quantities from finite samples

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.

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You should consider restating your question because in its current state it seems unlikely to be understandable. –  SRKX Jul 27 '13 at 22:03
I just tried to improve the question... hope this is better now. –  shnauz Jul 27 '13 at 23:48

In increasing order of complexity:

• If you can assume that your process is lognormal then the way to go is doing stats on the highest frequency returns and then scaling the results to the desided horizon (see the two references here for this and the next point).
• For other transition probabilities (still assuming i.i.d. increments) the scaling can be less trivial but can still be performed at worst numerically (e.g. via PDF convolution)
• If you're interested in capturing features such as autocorrelation then the issue gets complicated, see e.g. Newey-West and this question.
• Time series models like GARCH capture evolving characteristics departing from i.i.d. increments. You fit the model to the highest frequency data and then using it you can recover the unconditional PDF at any given horizon (by resimulation or analytically).

The latter option seems to be what you're looking for. There are other techniques for dealing with correlated samples such as overlapping returns (without fitting a specific model), but their drawbacks are still significant in practice and may not lead to the hypotetical accuracy improvement.

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Thanks for the comment, I will have a look at this (note I'd like to avoid the iid hypothesis for the daily returns...) –  shnauz Jul 30 '13 at 3:50
I've read your additions on merging parallel nonoverlapping estimates. In the i.i.d. setting my gut feeling is that you're losing some information/efficiency since the boundary elementary returns participate less than the central ones, but that might still be worth it depending on your goal. Also averaging dependent values can be tricky. But what is your main goal? To capture some internal autocorrelation or the non-trivial horizon dependence even in the i.i.d. case first? Would you be fine with autocorrelation in a lognormal model? –  Quartz Jul 30 '13 at 11:47
My main goal is to estimate the more precisely/robustly possible and with the minimum assumptions the variance of returns over a "long" period - say you have one year of daily returns, how would you estimate the 1 month return variance. Depending on how I see it, I might have only 12 monthly returns to play with, or more if I can use overlapping data. There might be a standard practice that I am not aware of. –  shnauz Jul 30 '13 at 13:05
@shnauz check the edit in response to your comment. Sorry for delay. –  Quartz Feb 19 at 12:19
I believe you should use the "usual" weekly dates i.e. $r_t = \ln \frac{S_t}{S_{t-5}}$.