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I would like to calculate some measure of volatility or noise for stationary time series data. This can be a measure for a single time series or a relative measure comparing multiple time series together. Let's assume a Dickey-Fuller test has already been conducted, and all the time series do not have a unit root.

What are some examples of such metrics to measure noise/volatility? I considered the simple "coefficient of variability" which is SD/mean. However, I am wondering if there are other ways to measure this. If it helps, I use R.

I know this is a vague request, and I apologize. I would really appreciate any suggestions or sources to learn about the topic.

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In finance the volatility measure is the standard deviation of the series. The means are often near zero, e.g. price returns, so it's not the coefficient of variation usually.

There are many ways to calculate the standard deviation though. For instance, even when the series are stationary they often have autocorrelations. In this case GARCH is a popular approach, which will give you the conditional variance. So, you can look at both long running and conditional variances. Sometimes the series exhibit stochastic variance behavior, in this case the models like Heston could be used.

Even with the simplest Gaussian, independent assumption there are multiple ways of estimating the variance. Take a look at this paper on how it's done in Bloomberg terminal.

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  • $\begingroup$ Thank you for your help! If I can just clarify, would the measures provided in the attached paper apply to sales data? It's mentioned that these are 'price returns' (with means often near 0 as you stated), so I am just wondering how applicable it might be to sales data which can have widely varying means. For example, there is a log(return) formula specified which is based on dividend and closing price. Would x(i) = log( sales(i)/sales(i-1) ) be a valid replacement? The data is weekly if it helps. I appreciate all your insight, especially on the GRACH models. $\endgroup$
    – user181973
    Oct 23, 2017 at 17:16
  • $\begingroup$ Anything in dollars (currency) has a potential to be nonstationary, both due to inflation and real growth. The firms in market economy are constantly expected to grow sales. That's why you may need to measure the volatility on growth rate terms. If you deal with short periods of time, this is not a concern usually $\endgroup$
    – Aksakal
    Oct 23, 2017 at 17:32
  • $\begingroup$ Here is a helpful link for implementing GARCH in python (article from Aug 2018) $\endgroup$
    – ascripter
    Aug 6, 2019 at 10:44
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As noted, typical statistical approaches based on L2 norms include the std dev as well as the coefficient of variation (which, for nonnegative metrics, produces a scale invariant measure) as well as the dispersion index (ratio of the variance to the mean). If the data is financial then it is also possible to calculate "upside" and/or "downside" measures of risk, aka above- or below-target semi-deviation, as described in these wiki articles (https://en.wikipedia.org/wiki/Downside_risk or https://en.wikipedia.org/wiki/Upside_risk).

L1 norm-based measures are possible, for instance, the MAD or mean absolute deviation and the MADM, the median absolute deviation from the median. Other nonparametric estimates include the interquartile range, the interdecile range, as well as metrics discussed by Rousseeuw and Croux in their paper, Alternatives to the Median Absolute Deviation (ungated copy here ... http://web.ipac.caltech.edu/staff/fmasci/home/astro_refs/BetterThanMAD.pdf).

Information-theoretic approaches include measures of entropy (https://en.wikipedia.org/wiki/Entropy_(information_theory)), such as Theil's U or the many variants of indexes of information diversity (e.g., https://en.wikipedia.org/wiki/Generalized_entropy_index).

Hyndman's contention is that his MASE metric is optimal for time series data. MASE is a normalized loss function. After creating train and test data, the test data residuals are normalized or divided by the average error in the training data. If MASE<1 then the proposed model is an improvement, on average, over a one-step ahead, random walk forecast.

See his paper, Hyndman and Koehler, Another look at measures of forecast accuracy, International Journal of Forecasting, 22(4):679-688, 2006, https://robjhyndman.com/papers/mase.pdf, p. 3

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