# Modelling an undifferenced series as an ARMA process

I have series that shows that it requires differencing as the lag-1 value of ACF is almost equal to 1 and there is no significant spike in PACF.

However, if I do not difference it for some reason and model it as ARMA (1, 1) process. I get the following result.

Series: ret
ARIMA(1,0,1) with non-zero mean

Coefficients:
ar1     ma1    mean
-0.7310  0.5806  -2e-04
s.e.   0.2542  0.2814   3e-04

sigma^2 estimated as 6.064e-06:  log likelihood=345.57
AIC=-683.13   AICc=-682.56   BIC=-673.86


I have studied in these notes that an under differenced process will ask for AR parameter and an over differenced process will ask for MA parameter. It also shows that redundant AR and MA parameters may merely cancel each other even though they are statistically significant. Hence, I have the following questions.

1. How will the autocorrelation at lag-1 get reflected in AR and MA parameters?
2. What will be the signs of AR and MA parameters?
• Were the ACF and PACF that you plotted computed on the original, undifferenced series? If so, the spike you see in the ACF appears to be at lag zero, not 1, which is always exactly 1, and what you have there is consistent with white noise. Also, are the horizontal axes correct? Did you really compute lags up to 5000 but only plot a few of them? Commented Jun 5, 2017 at 12:13
• Yes. It is computed on original undifferenced data. I have used the acf and pacf functions in R on my data. And I am also wondering why it shows lags up to 5000. Commented Jun 5, 2017 at 12:29

There is no autocorrelation visible at lag 1 in your ACF or PACF. The spike that you see in the ACF is at lag zero; it's the correlation of the series with itself at the same moment in time, and is always 1, which isn't really useful to plot (stats::acf plots it but forecast::Acf does not, for this reason).

Evidence for differencing would present itself as slow decay in the ACF and a large spike at lag 1 in the PACF; there is no such evidence here. In fact, there is no evidence in either plot that there is any autocorrelation structure at all: it looks like you have white noise (i.e. ARMA(0,0)).

It's quite likely that the AR and MA parameters you estimate in your ARMA(1,1) model are therefore indeed "redundant". The function stats::arima uses the parametrization (assuming zero mean to make it simpler):

$$Y_t = \phi Y_{t-1} + \theta \varepsilon_{t-1} + \varepsilon_t$$

If $\phi = -\theta$ (which is approximately what you've estimated), then

$$(1-\phi L)Y_t = (1-\phi L)\varepsilon_t$$

This will hold if the true process is ARMA(0,0), regardless of the value of $\phi$ (which is therefore meaningless). You should probably go with the ARMA(0,0) model.

Lastly, stats::acf sets the horizontal axis based on the frequency of the input series, so most likely you have it set to some very small amount (maybe 1/365 where it should just be 365): the correlation at 1000 refers to 1000 years, not 1000 observations.

• Thank you for the answer. I provided the example to merely ask the questions given below. So, please assume that the series requires differencing, what will be the answers to the questions? Commented Jun 5, 2017 at 13:56