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I simulate around 16000 stock prices with using auto.arima and arima.sim and I have two questions.

  1. Do I need to use plain prices or log prices?

    I am not a big fan of using logs without proper reasoning. What is a good reasoning? The only reasoning that comes into my mind is testing for heteroscedasticity.

  2. Is it better to use returns?

    Some people use returns and convert them back intro prices. As far as I understand that does not make sense when using auto.arima as it will figure out the right differencing steps anyway. Why use returns then (I need to account for this later in the simulation again if I would)?

I know that the results with auto.arima and without manual inspection might not be perfect but in the end, I have 16k datasets and any manual inspection would be impossible. I just want to simulate price series and check for visual chart patterns.

[edit] Background information:

Chart analysts claim that chart patterns (e.g. Should-Head-Shoulder) appear because of the behavior of the market participants. The participants act similar in comparable situations, what results in chart patterns. If true, an original time series has an inherent systematic caused by the participant's behavior. By bootstrapping I destroyed this systematic but kept the overall properties. I compared the number of patterns and it was significantly different. Hence, there seems to be some systematic in the original data that bootstrapping did not preserve. But this systematic does not need to be caused by what chart analysts claim. It could be the result of eg. AR MA effects. If I could find the same amount of patterns in a simulated series, it would be a hint that this systematic is caused by the used model. E.g. ARMA, GARCH, etc. But if I do not have a model, I have no cause or explanation for the chart pattern's appearance. I need to fit the models, to have similar conditions for original data and simulated because I then compare the number of patterns.

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One would normally simulate log-returns as their distribution is closer to normal (even though still not normal, usually more heavy-tailed and somewhat skewed) than the distribution of prices. For log-returns, you could either use the normal distribution or (to reflect the stylized facts better) skewed Student-$t$ or Johnsons's $S_U$. To obtain prices from simulated log-returns, take the exponent of their cumulative sum. You will recover prices up to a level shift.

One would also normally simulate i.i.d. observations or (to reflect the stylized facts better) a GARCH process with a constant mean, not ARIMA. Daily stock returns have very weak autoregressive patterns, so a constant conditional mean is closer to what we observe in the markets than an ARIMA conditional mean is.

Edit: After a discussion in the comments, I would say, if you are sure you want to have ARMA, ignore my last sentence and go ahead. The rest of my points should still apply. I would however think twice before including ARMA. Patterns that you see may have other causes than ARMA. Even if they can be fit by ARMA in sample, I doubt ARMA would predict well out of sample.

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  • $\begingroup$ Thank you! I'll give a bit more background. I check the number of visual chart patterns in a time series and compare it to the result of a simulated series. I used bootstrapping and there was a significant difference in the number of occurring patterns. Now, I try to find an explanation (e.g. AR or MA effects). Hence, I wanted to est. a model for each of the 16k stocks, simulate a new series and compare the number there. If I simply simulate based on e.g. skewed Student-t distribution, I would not have an explanation what effect causes the same amount of patterns, or am I wrong? $\endgroup$
    – kn1g
    May 23, 2021 at 10:26
  • $\begingroup$ @kn1g, I am not sure why you want to simulate from an estimated model and what you intend to achieve by that. Could you explain in a bit more detail? $\endgroup$ May 23, 2021 at 12:20
  • $\begingroup$ Sure, thanks for asking. Chart analysts claim that chart patterns (e.g. Should-Head-Shoulder) appear because of the behavior of the market participants. The participants act similar in comparable situations, what results in chart patterns. If true, an original time series has an inherent systematic caused by the participant's behavior. By bootstrapping I destroyed this systematic but kept the overall properties. I compared the number of patterns and it was significantly different. Hence, there seems to be some systematic in the original data that bootstrapping did not preserve. $\endgroup$
    – kn1g
    May 23, 2021 at 12:27
  • $\begingroup$ But this systematic does not need to be caused by what chart analyst claim. It could be the result of eg. AR MA effects. If I could find the same amount of patterns in a simulated series, it would be a hint that this systematic is caused by the used model. E.g. ARMA, GARCH, etc. But if I do not have a model, I have no cause or explanation for the chart pattern's appearance. I need to fit the models, to have similar conditions for original data and simulated because I then compare the number of patterns. I hope this was somehow understandable. Else, I try to rephrase :) $\endgroup$
    – kn1g
    May 23, 2021 at 12:31
  • $\begingroup$ @kn1g, this was a very good explanation. Consider editing it into your original post. That will help the users who do not read the comments thoroughly. In any case, I think my answer still applies to a large degree. If you are sure you want to have ARMA, go ahead. The points that are not related to ARMA in my answer should still apply. I would just think twice before including ARMA, since patterns that you see may have other causes than ARMA. Even if they can be fit by ARMA in sample, I doubt ARMA would predict well out of sample. $\endgroup$ May 23, 2021 at 12:50

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