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What are the most typical graphs for an auto-correlation function?

Is it usually decreasing exponentially fast? Do some non-periodic time series have very thick tail in the autocorrelation function?

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In order to use an auto-correlation function the data series must be weakly stationary. This means that it have a very fast exponential decay. If there is not a quick decay then there is presence of long memory in the data. Typical a decay that goes to zero by no more than 10-15 lag. There will often be evidence of thickness in the tails which is know as leptokurtic. You can you use the excess kurtosis check to test for the thickness of the tails. Often in financial data there will be thiuckness in the tails because of how the data was measured. In order to have a weak form stationary data series the first two moments must be constant, the data must come from a normal distribution and there also must be serial correlation in the data. I recommend reading Analysis of Financial Time Series by Ruey S. Tsay.
As long as a process in not stationary, I think the ACF will look thick. Dying down within the first few lags are signs of stationarity. Also, to really check if the process is stationary or not, I think the IACF and the PACF graphs greatly help in finding the predominant memory lags that add to the tail. So, in the process, when we take differences of the respective lags, we should be able to end up with a rather fast decreasing exponential!

I have also seen instances of the ACF, being alternatively correlated (positive & negative). So it all depends on how the correlation is in the series, and how long the lag effects are in place.
Hi, you can use Exponential Smoothing Method for non-stationary time series. Because there is no need to convert the time series into stationary for pattern recognisation.

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