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I would like to have a reliable measure of similarity between several time series. These represents swap rates of different currencies.

I would like to start simple but also make sure I am not missing anything obvious. Knowing some of the shortcomings of naively computing cross-correlations as similarity measure, I thought that at least I should compute it on related processes that are as (weak-sense) stationary iid as possible. I did the following:

  1. Based on ACF and PACF plots of the differenced data, I have fitted an AR(1) model of the conditional mean to my differences time series.

  2. Residual ACF and PACF looks as white noise, but squared residuals suggests heteroskedacity.

  3. I have fitted then a AR(1)-GARCH(1,1) using Gaussian distribution for the likelihood.

I have found the following:

  • The distribution of the standardized residuals seems to be respected according to histogram (maybe student-t would be better as a refinement).

  • ACF and PAC both for standardized residuals and their square don’t show serial dependencies.

I might assume that standardized residual are white noise and compute cross-correlations between standardized residuals of univariate GARCH models. I guess all of this tells me nothing about the stationarity of the residuals, as here we are modelling conditional mean and variance. My question therefore is: is this needed? Should I proceed differently?

Apologies if I wrote too much nonsense but I’m new in this kind of analyses.

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  • $\begingroup$ How do you define similarity? Why do you think your approach should work? What do you mean by standardized residuals? This term is used in GARCH models, but your steps 1.-3. do not involve GARCH. $\endgroup$ Commented Mar 1 at 20:34
  • $\begingroup$ Apologies Richard, I stupidly forgot that I have fitted an AR(1)-GARCH(1,1) to the differenced series. Many thanks for the fast comment. $\endgroup$ Commented Mar 1 at 21:21
  • $\begingroup$ For “Similarity” I’m thinking about correlation, and why this should work is part of the question: one reason is because I think a cross correlation between iid stationary (but this I’m not sure) might be meaningful. $\endgroup$ Commented Mar 2 at 6:41
  • $\begingroup$ You are asking whether something you do makes sense without giving a clear definition of what you want to achieve. I am not saying the latter is easy, but we cannot evaluate the former without knowing the latter. Perhaps this thought experiment will help: generate an i.i.d. sample and from it, generate two ARIMA-GARCH trajectories with different parameters. The similarity of the standardized errors is 100% (they are identical), while the visual similarity, correlation and other measures of similarity of the raw observed series can be anything. If that is what you are looking after, it is fine. $\endgroup$ Commented Mar 2 at 8:41
  • $\begingroup$ I'm trying to compute some meaningful cross-correlation so I was trying to remove anything that could give unreliable results, hence considering the residuals. Something like an "enhanced pre-whitening". The experiment you proposed though, makes me understand this approach is totally nonsense actually. I have to think of a better way to assess similarity. Thanks a lot Richard. $\endgroup$ Commented Mar 2 at 14:55

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Perhaps this thought experiment will help:

  1. Take two sets of parameters that define two rather different ARIMA-GARCH processes. E.g. one with positive mean and another with negative, one with positive autocorrelations at certain lags, another with negative, one with highly persistent volatility and another without.
  2. Generate an i.i.d. sample.
  3. Using the i.i.d. sample as standardized errors and the two sets of parameters, simulate two ARIMA-GARCH trajectories.

The two time series trajectories will look fairly different when plotted against time and will have rather different characteristics in terms of their level, autocorrelation function and conditional heteroskedasticity. However, the similarity of the standardized errors of both trajectories is 100%, as they are identical. (We have not rigorously defined similarity, but a logical thing to require from a definition is that an object is entirely similar to itself.)

If that is what you are looking after, you are fine by focusing on standardized residuals. If not, consider looking at some measures applied on the raw time series (mean squared difference, correlation, etc.) or some other transformations (e.g. mean-adjusted, standardized, normalized, etc.). Each one will have its pros and cons depending on the context and on what properties defines similarity for you.

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