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I am working extensively with financial time series models, mostly AR(I)MA, and Kalman.

One issue I keep facing is the sampling frequency. Initially I was thinking if offered the possibility to sample more frequently from an underlying process I should be sampling as frequently as possible so that I will have a much bigger number of samples, hence my model parameters will have less variation.

In reality this idea didn't turn out to be good. What happened is that if the underlying process is not exhibiting enough variation, increasing the sampling frequency actually meant getting a lot of repeating (same) values. And building a model on such values results in models with very very small model coefficients which don't predict well into the future (of course the definition of "well" is subjective and increased frequency requires to predict much more sample steps into the future to achieve the same time-step in a lower frequency setting). The model learns what it encounters the most - a flat line.

I wanted to do an adaptive sampling approach, i.e. sample more frequently when there is variation, and less frequently when there is not. This is not easy, however. First of all it is not clear what kind of bias I am introducing by doing so (and will differ depending on how I trigger the sample/skip). Secondly, time series models like ARIMA are not well suited for uneven sample steps.

Is there a good way to deal with this problem? It also makes me wonder how one achieves a seamless transition between continuous time models and discrete time models if models are so dramatically affected by sampling frequency (especially when time steps get smaller and smaller)? Any pointers to external resources will also be appreciated.

Thanks

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    $\begingroup$ "sample more frequently when there is variation, and less frequently when there is not" could work in sample, but that would be difficult to use for out-of-sample predictions. Are you interested in the former or the latter? Also, if you encounter regimes with low variation (or no variation at all) followed by regimes of high variation, you would naturally need separate models for the two. Otherwise you would have one model for the whole process and sampling at uneven intervals/frequencies would intuitively seem suboptimal. Also, the last paragraph qualifies as a stand-alone question, IMHO. $\endgroup$ – Richard Hardy Jan 1 '16 at 14:45
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    $\begingroup$ Also, you may consider making your title more informative, something to indicate the idea of sampling more frequently at points of large movements. $\endgroup$ – Richard Hardy Jan 1 '16 at 15:09
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    $\begingroup$ @RichardHardy I thought about regime switching models. However they are notoriously difficult to train. Do you know how to identify and train regime switching models in a dynamic fashion (automatically discovered without specifying the point of regime switch in advance)? Can you show some pointers? $\endgroup$ – Cowboy Trader Jan 1 '16 at 19:49
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ARIMA's may not be well suited to your purpose, but state space models are: you might sample as often as you wish (and in principle, the more the better) and perform a temporal update at fixed intervals, as the dynamics of your assumed process may demand. One of the beauties of state-space models is that the observation process is separate from the model process, and separate time intervals may be used for each.

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  • $\begingroup$ That is not solving my problem. Even in a state space model, model coefficients are first to be determined. State space update methodology applies to the state vector itself not to the coefficient matrices. $\endgroup$ – Cowboy Trader Jan 1 '16 at 19:35
  • $\begingroup$ I do not quite understand your comment. If you cast your model in state-space form you can compute the likelihood (assuming normality) using the filter Kalman, irrespective of the sampling frequency. Maximizing that likelihood you can estimate the parameters in the system matrices. $\endgroup$ – F. Tusell Jan 1 '16 at 19:42
  • $\begingroup$ That is true if you know the model in advance. When all the state transition matrices and noise covariance matrices are known you can do the updates, and you can do with skipping time steps. When you are given only the data first you need to infer the transition matrices. And those matrices will differ between a high volatility period and a low volatility period. $\endgroup$ – Cowboy Trader Jan 1 '16 at 19:47
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I'd like to point you to the article

Ghysels, E, P. Santa-Clara and R. Valkanov (2006): "Predicting volatility: Getting the most of return data sampled at different frequencies", Journal of Econometrics, vol. 131, pp. 59-95.

The authors employ a technique called MIDAS (mixed data sampling) by themselves in order to compare estimates of volatility based on data sampled at different frequencies. Admittedly this is not exactly what you were looking for but the authors claim that their technique is suitable for comparing the results in a meaningful way. Maybe this gives you at least a second way of analyzing your data. It seems that in particular in the field of macroeconomics this approach has gained some interest.

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    $\begingroup$ Thank you. The issue is not financial time series specific. Take any experimental situation and sample with high frequency in time dimension. You are ending up with a long flat line and the models learns that, a flat line. Because repeating samples overcrowd the meaningful samples that actually reflect the variation in the underlying process. This is really problematic, and I cannot find much related to this topic. $\endgroup$ – Cowboy Trader Jan 7 '16 at 12:05
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sample more frequently when there is variation, and less frequently when there is not

That could work in sample but would be difficult to use for out-of-sample predictions, unless you figure out how to predict the variability itself (and that need not be impossible). Also, if you encounter regimes with low variation (or no variation at all) followed by regimes of high variation, you would naturally need separate models for the two; having one model for the whole process and sampling at uneven intervals/frequencies would intuitively seem suboptimal. You mentioned regime switching models (when answering my comment), and that is a good illustration what you might need here.

I should be sampling as frequently as possible so that I will have a much bigger number of samples, hence my model parameters will have less variation.

This is not entirely true. In a time series setting, it is often the time span rather than the number of observations that matters. For example, 120 monthly observations (spanning 10 years) is a more informative sample than 209 weekly observation (spanning 4 years) when testing for presence of a unit root; see this Dave Giles' blog post and the last reference in it. Or consider a limiting case where you sample so frequently that you essentially measure the same thing multiple times. That would increase your sample size but would not bring in new information, leading to a spurious impression of estimate precision. So perhaps you should not spend too much time on increasing the sampling frequency and building some corresponding models?

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  • $\begingroup$ Post doesn't really answer the question. Regime switching is probably the way to go. $\endgroup$ – Cowboy Trader Feb 10 '17 at 17:12

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