# Interpretation of ACF and PACF

First, I am a French student, so forgive me for my English which can be not clear at all.

I have to analyze a financial series. I have some difficulties to make the second part of the work which focuses on ARMA model. I can't read (interpret?) my Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) correlogramm.

To my mind, here we have a MA(0) but I don't know if it's possible. In this case, what does it mean ?

And I don't know also with the PACF graph, if we are in a AR(3) or AR(6). And why the first stick is not over the "confidence level" ?

It is possible that my interpretations are false, please tell me.

• What software are you using? – Alecos Papadopoulos Dec 21 '15 at 15:48
• I am using Rstudio – Marc Dec 21 '15 at 21:19

## 1 Answer

In some statistical software (not all), in a correlogram the "zero lag" is also depicted - but the zero lag is just the correlation of a random variable with its own self, so it is by construction equal to unity. This appears to be happening in your first graph where the autocorrelation function is calculated.

The second graph is the partial autocorrelation function which calculates the correlation coefficients after the effect of all "previous" lags (i.e. of lower order) has been removed (by linear projection estimation).

First impression? The process is white noise. What lags appear to exceed the "statistical significance threshold", they nevertheless indicate very small values, never greater than $0.05$. This is truly economically negligible correlation even if it exists (irrespective of whether it is "statistically significant"), and most likely is due to "finite-sample variation", rather than indication of any actual relation.

And apart from that, do the lags appearing "significant" make sense? It always matters what the process you are examining is, and what is its "typical properties" based on past scientific work (but don't take this to mean that "past typical properties" always carry over to all the realizations of the process).

In any case, it would be instructive to "follow blindly" the Correlograms and see what you get, by trying various combinations based on the lags that appear statistically significant.

• Thank you so much for your answer :) ! I have to say to you that it is the first time I have to interpret an ACF and a PACF plot, and it's not easy for me because it seems to be not "typical" like in what we study, so I am a little lost. According to you, if I "follow blindly" the Correlograms and see what I get, are the following combinations true comparing the results given by correlograms ? Can I choose to ignore the lag number 6 in the PACF ? - ARMA(2,0,0) - ARMA(3,0,0) Or can you tell me what combination do you think is also appropriate ... ? Thank you in advance. – Marc Dec 21 '15 at 21:18
• @MaximeP. As I already wrote in my answer, to me this process looks like simple white noise. So I cannot really pick a combination of lags as more appropriate than another. You should try the various combinations and check the diagnostics for each model, to see which performs better. – Alecos Papadopoulos Dec 21 '15 at 22:08