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I wonder if every time series data should contain noise or not. For example I am taking the price of a ticker, say Yahoo, every hour and noting the values. Does this data contains noise or not?

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  • $\begingroup$ I'm no expert, but I'd say a ticker might contain two kinds of "noise". One might be rounding: the price might reported with less significant figures than the actual price. The other is time jitter: The price reported at a time instant may be the price at some indeterminate time in the past (say, anywhere between 0 to 10 seconds in the past). $\endgroup$ – MBaz Jan 20 '16 at 14:54
  • $\begingroup$ In addition to @Mbaz 's aspect of ticker data simply not being a digital signal, because it's not sure to be sampled at discrete points in time, I'd also like to mention quantization noise here, which, for things like stock exchange rates, might be highly correlated and impossible to model as additive noise $\endgroup$ – Marcus Müller Jan 20 '16 at 19:01
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Periodically taken ticker data is very likely not a set of samples of a band-limited function, thus can contain aliasing artifacts which can be considered noise. This is possibly a much larger contribution than quantization and time jitter noise.

Absence of noise requires the existence of well defined signal being measured. Proving that such a signal really exists is a separate and likely far more difficult problem for some types of time series.

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  • $\begingroup$ Interesting, I had never thought of that. However, I think it only becomes a problem when you try to interpolate or extrapolate. The samples themselves are not noisy (up to quantization and time jitter errors). $\endgroup$ – MBaz Jan 20 '16 at 22:30
  • $\begingroup$ Some might say that the last individual trades before any periodic price sampling times are just a lot of fat-tailed random noise (or can't be consistently differentiated from a random walk of such... especially with HFT systems perhaps using game theory to place some distribution of different bets every millisecond.) $\endgroup$ – hotpaw2 Jan 21 '16 at 2:18
  • $\begingroup$ I agree; that's why I believe you are correct to assume that the continuous ticker signal has a very large (if not infinite) bandwidth. Any naive interpolation or extrapolation based on the samples will likely be inaccurate because of this. On the other hand, Berlekamp made a lot of money applying information theory to the market; as far as I know he never explained exactly how he did it. $\endgroup$ – MBaz Jan 21 '16 at 15:48
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A huge yes. Perhaps stated in another way, due to the stochasticity, you can never be certain in practice that there is absolutely no noise in your data.

The timing at which you sample (almost periodic, every hour) might subject to small uncertainties (whether you do it by hand or by an automatic request). There is no infinite sampling precision.

Noting the values (do you note it by hand, actually?) is subject to perception and one can write figures noon accurately, forget a comma. Change $3.14$ as $31.4$ and you get a spurious peak on your time series.

Those are "copy or retrieval" noises.

As @hotpaw2 said, aliasing is a type of noise that occurs when converting a seemingly continuous data into a discrete one. It is worth pointing out (I have heard "I sample this day every 24 hour, so I have no noise". Actually, an irregular sampling might help in the case of a non band-limited signal.

Then you have almost the same issues with what Yahoo, considered here as a sensor, does. How long does it hold values? Sometimes free information is delivered at a slower pace than when it is paid? How does it round it? Does it smooth it over $3$ seconds? Moreover, does Yahoo always perform these actions in the same manner? In other words, is your signal seem stationary, or time invariant?

Those are system/sensor noises.

Finally, you take a price for a specific goal. It is a quantity that can be observed, that you may use to model a certain "real word" behavior. There might be events in the data that are "noises" for your purpose (slow trends, seasonality).

What is important is how those combined noises affect your modeling and processing.

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