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Currently I am trying to determine whether a sample of monthly returns can be seen as the outcome of a random sample. I plotted the ACF for 20 lags and got the following plot:

I am uncertain by the result. There is very little autocorrelation except for the 5th lag, which is barely significant. Is this single significant point evidence against the data being from a random sample?

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    $\begingroup$ Not really; given that you're testing at the 95% level of confidence (I suspect) you'd expect around 1 out of 20 sample autocorrelations to appear significant just by random chance, and there's also no a priori reason to expect monthly returns to be autocorrelated with just a five month lag. $\endgroup$
    – jbowman
    Commented Nov 30, 2017 at 15:10

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In this case, I would not conclude the ACF plot as evidence that the data is not from a random/white noise process.

Note: Blue lines in these plots often indicate 95% confidence intervals. As such, there's a 1/20 probability that one of the lags will be significant due to random chance.

In your scenario, this appears to be exactly what has happened. Only one of your 20 lags (excluding the first lag, which always equals 1) shows significance. And the significance is not strong. And it's at a lag, which does not make contextual sense with your data.

Conclusion: This appears to be an ACF plot for white noise.

Suggestion: Don't forget to check PACF plot as well. Other ways to check for seasonality would be with a Periodogram, and by investigating the time series plot in general.

Hope this helps!

Edit: As @IrishStat aludes to in his answer, understanding the underlying process of a sample can not be determined by the ACF alone. If you are unfamiliar with Time Series methods, seek out a professional that specializes in it (in person) and have them help formally answer your question. Best of luck!

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  • $\begingroup$ What you say is correct if and only if what I reflected on in my comment is true. $\endgroup$
    – IrishStat
    Commented Nov 30, 2017 at 17:37
  • $\begingroup$ In this scenario, it seemed the user was looking for a more straight-forward answer and not the nitty gritty details of what Time Series is. User wanted an interpretation at face-value. So based off what he has given us, this felt like the most straight-forward, understandable answer to his question. $\endgroup$
    – creutzml
    Commented Nov 30, 2017 at 17:45
  • $\begingroup$ Interpreting the ACF plot to answer the question whether or not a plot of the ACF can answer the question of randomness assumes no pulses , no step/level shifts , no local time trends , no seasonal pulses, constant error process and constant ARMA structure over time. Since the original data was not presented or analyzed these assumptions are unverified thus the question can not be answered in a straight-forward way. $\endgroup$
    – IrishStat
    Commented Nov 30, 2017 at 21:39
  • $\begingroup$ I think you're reading too far into the question. I saw this question as a simple request for "Interpretation of an ACF plot." So I responded with an interpretation of the ACF plot shown to me, without any other information given, and what that interpretation might imply. $\endgroup$
    – creutzml
    Commented Nov 30, 2017 at 21:47
  • $\begingroup$ that is precisely the problem, you responded without sufficient information. An analysis might have disclosed a pulse and an ar(1) model. If you don't treat the pulse the ACF is downwards biased. This is referred to as "the Alice in Wonderland problem" where the untreated pulse causes you to not see the underlying structure.Simulate an AR(1) process and inject an increasingly larger pulse at 1 time period >1 and see what the resultant acf is. You will find that the presence of an untreated outlier/pulse obfuscates model identification due to increased variance and covariance.. $\endgroup$
    – IrishStat
    Commented Nov 30, 2017 at 22:25
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Interpreting the ACF plot to identify ARIMA structure premises no pulses , no step/level shifts , no local time trends , no seasonal pulses, constant error process and constant ARMA structure over time. If you wish can post your data and I will attempt to verify these assumptions.

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