# Interpreting ACF/PACF of return series

Researching a return series on some currency pairs I grabbed 2 years worth of daily data and got to work trying to fit an ARIMA/GARCH model to it.

Fitting the (log) return series:

r = tick2ret(midPrice)

and then calculating the ACF and PACF

autocorr(r)

parcorr(r)

I get plots that look like:

Clearly the return series is mean reverting with it's mean hovering comfortably around zero.

About this time I usually say "a ha!" and spot the p and q I need to fit an ARIMA model from the ACF and PACF. However, the only lag here that is significant (considering ~5% of the lags touching will be by chance) is lag 0. This occurs on both the ACF and PACF. This means my return series is discrete white noise! That can't be right at all.

Going further and performing the ljung-box test on the return series:

[h,p] = lbqtest(r,'Lags', 8);

Shows h = 0 and p = 0.7746 indicating we almost certainly have no autocorrelation up to lag 8.

I feel like something is going wrong here. My intuition would tell me if the return series is mean reverting you would certainly have autocorrelation up to some lag.

What could be going wrong here? I'm still new to MATLAB (coming from R) so it's possible I'm doing something wrong...but I don't think so...

• "Clearly the return series is mean reverting..." Care to share your reasoning? The series doesn't look autocorrelated and none of your lags look particularly significant. Nov 25, 2017 at 4:02
• @Matthew Gunn My reasoning is simply that the series appears to always return to a mean of 0. I've never seen a log return series that didn't exhibit some form of autocorrelation, so I was taken back. Gap in my knowledge I suppose. Why would a series look this way and not exhibit autocorrelation?
– user20664
Nov 25, 2017 at 4:04
• This is what returns in an efficient market look like, very little or no autocorrelation. Nov 25, 2017 at 4:07
• @AlexC Interesting, that is what I had figured but it just seemed so strange to me. If you'd post that I'll mark this complete.
– user20664
Nov 25, 2017 at 4:08