I've gone through all the steps to fit a good ARIMA model - I plotted the data, I looked at the ADF tests, I looked at the ACF plot with no AR and MA terms just a constants. I came up with an ARMA(0,1,1) model as the ACF cut off after two lags (the second was negative) and the PACF decayed exponentially from the first order being negative.

The problem is I cannot get rid of a large positive spike at the first order lag in the ACF plot of the residuals once my model is fitted. I've tried increasing the number of MA terms, but the lag doesn't go away and the second term isn't significant. Given it is an MA model, I don't think further orders of differencing are need.

I've coded my model in R

Any advice?

Edit: I've included some of the graphs below. I tested my model with and without a constant, as the constant was not significant, however the AIC was lower with a constant. Theoretically thought, I think no constant makes more sense as I don't think there is a constant average trend. However, excluding the constant (with the argument include.mean=FALSE) doesn't change the ACF or PACF of ther residuals. The code I used to fit the models is:

ArmaOhdifc01 = armaFit( ~arma(0, 1), data=OhDiff)
ArmaOhdif01 = armaFit( ~arma(0, 1), data=OhDiff, include.mean=FALSE)

Below are the graphs - I didn't include the ACF and PACF of the first model becuase there is no difference to that of the second.

The first graph is just a plot of the 1st differenced data

The second graph is a plot of the ACF of a model fitted with just a constant on the differenced data, and no AR or MA terms

The third graph is the PACF on the same model with just a constant

The 4th plot is the ACF on the residuals of an ARMA(0,1,1) with a constant

The 6th plot is the PACF on the residuals of an ARMA(0,1,1) with a constant

  • $\begingroup$ The fact that the residuals are not entirely uncorrelated does not mean that the model must be entirely rejected. You may use other measures, e.g. how well it predicts the data. $\endgroup$
    – emcor
    Dec 10, 2014 at 14:41
  • $\begingroup$ What is the input data? Can you provide the values or a plot of the acf or pacf of the data? Here you find a chapter of the online textbook which I post in this forum nearly every second day. $\endgroup$
    – Richi Wa
    Dec 11, 2014 at 14:23
  • $\begingroup$ You difference the data once ... mabye there is something with using a constant as described in the chapter above ... $\endgroup$
    – Richi Wa
    Dec 11, 2014 at 14:24
  • $\begingroup$ @Richard I actually looked through that e-book, as well as some others, and it was helpful (note: it would be great if I could order a pdf copy to read on my ipad). I've added some more details above. I had tested the model without a constant as well as with, althought it didn't change the ACF/PACF (might be a coding error?) $\endgroup$
    – Celeste
    Dec 15, 2014 at 22:33
  • $\begingroup$ I think you can order the book as pdf - it is definately worth the money. I bought it as paperback. As I see in the remark below your problem is solved ;) $\endgroup$
    – Richi Wa
    Dec 16, 2014 at 8:08

1 Answer 1


It is a classical misunderstanding, your model is right, you always have a acf equal to one at lag zero (and not one) since if there is no lag acf = covariance(x , x_lag 0) / variance x = variance x / variance x = 1.

So you need to pay attention to the x axis , some software displays ACF starting at lag zero and some others from 1 (which make better sense) .

  • $\begingroup$ That's great, and I also feel rather stupid not paying attention to that. Thanks @Malick $\endgroup$
    – Celeste
    Dec 16, 2014 at 2:33

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