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Given an estimation procedure and real data, how would one compute the mean squared error? What value represents the "true" realized volatility in the case of calculating the Mean Squared Error in estimation? I'm specifically interested in intraday estimation error (one minute trade data for example)?

So for example:

  1. I want to estimate the realized volatility statistic on 1 day's worth of real 1 minute data

  2. I'm going to use block bootstrapping as my estimation procedure

  3. I run the block bootstrapping estimation algorithm and get an estimated realized volatility value

  4. To compute the estimation error, I need a true realized volatility value for this 1 day of 1 minute data.

What can I use for this "true" realized volatility value for the specific data that I am estimating from?

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  • $\begingroup$ What are you trying to do? Assess how reliable a volatility estimate is? $\endgroup$
    – SRKX
    Commented Jan 8, 2014 at 14:16
  • $\begingroup$ Yes. I edited the question with a specific example. I want to test how accurate bootstrapping is at estimating the true volatility. I chose the realized volatility because it's simple to calculate, but I'm stuck on what to use for estimation versus the value to use for the "true" volatility. $\endgroup$
    – Amir Sani
    Commented Jan 9, 2014 at 14:49
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    $\begingroup$ Hi @AmirSani I already answered to the "microstructure" part of your question here: quant.stackexchange.com/questions/2360/… $\endgroup$
    – lehalle
    Commented Jan 12, 2014 at 9:05

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Unfortunately, financial markets are not like physical measures, where you know the "true" value of a physical variable but you just access to it thanks to noised sensors. We do not know the "true" volatility, just because there is not such one value...

In statistics you have two kinds of modelling procedures:

  • the ones dedicated to estimate the unknown values of parameters of a known structural model. Here you have the usual "confidence intervals" approach.

  • the ones for simultaneously inferring from data the shape (i.e. "class") of the model and its parameters. Here you have the usual issues coming from "overfitting", and the usual approaches are "cross validation", "regularization - penalization", "Vapnik-Chervonenkis dimensions", etc.

In finance you are very often in the second case, and especially for volatility: for instance its value is not the same if your underlying model includes jumps or not... what is the "true" one? Moreover (as I commented) at your time scale (1 min) you face the microstructure noise, see my answer here for a brief.

But come back to my generic answer on "knowledge discovery" via statistical modelling: what can you test if you believe you have a good new estimation procedure?

You can test your prediction capability of course, but you will have to face a lot of ugly features of intraday volatility:

  • it is not iid, even not ergodic, since it has a seasonality (see Market Microstructure in Practice for intraday seasonalities).

  • once you removed the seasonality, it is clustered (we cannot ignore it since Robert Engle's Nobel prize).

  • moreover, it is path dependent (I mean, even inside a volatility cluster as defined by Engle)...

Thus if you wan to challenge existing estimation procedure, you will have to remove the first two features and demonstrate on the deseasonalized data, cluster by cluster. Of course you could alternatively try to perform a change of state space to estimate something else than volatility. Like use it to estimate the seasonality or the switching probabilities themselves...

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  • $\begingroup$ Hi Charles, Thank you for your detailed answer. When you say "switching" probabilities, are you referring to the cluster switching probabilities? Also, what procedures currently exist for estimating the seasonality? I'd like to see if it fits an estimation through "bootstrapping" paradigm. $\endgroup$
    – Amir Sani
    Commented Jan 13, 2014 at 11:21
  • $\begingroup$ @AmirSani intraday seasonality may be a better candidate than volatility itself. You can try seasonality of (1) the volatility (but complex), (2) the traded volume, (3) the bid ask spread. Open a question on intraday seasonalities (we do not have it I think) and we will discuss there. $\endgroup$
    – lehalle
    Commented Jan 14, 2014 at 7:03

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