For my dissertation, I am looking at a time series of retail sales and want to identify an ARMA model to fit the data.
I have differenced the original data series as it was non-stationary and now when I look at the correlogram of the differenced series, the ACF and PACF are both significant at the first lag but then truncate afterwards into white noise. They also follow similar patterns.
I was wondering what kind of model this could indicate as it doesn't fit the usual mould of at least one of the ACF/PACF decaying slowly.
Graph of differenced series and correlogram attached.
Thanks.
From the correlogram graph:
I can say that ACF cuts off after 2 lags, and PACF decays, so MA(2) is the initial model and then you can use overfitting and underfitting to find the best model.
Hi Donald,
A distinctive feature of MA(q) models is that the ACF of order k>q cuts off.
A similar property for AR(p) holds true, but instead of ACF you need to look at the PACF.
For ARMA(p,q) models rules are more complex.
Something you do not mention in your question is that your data seem to have a seasonal pattern: there is a small cluster of non negligible PACFs of order 11 and 15. You can also see something like that in the ACF although the pattern is more subtle. Retail data should be seasonal, so the whole thing makes sens.
My recommendation is:
1. Fit a seasonal AR(p) model, recod the Akaike.
2. Fit a seasonal Ma(q) model, record the Akaike.
3. Fit a seasonal ARMA(p,q) and record the Akaike.
You could pick the model with the lowest Akaike.
Being more precise. Let us call Y(t) the first difference of retail sales. Try these models:
1. Y(t)=a +bY(t-1)+cY(t-2)+dY(t-12) + error (Seasonal AR(p))
2. Y(t)=a+error(t)+b*error(t-1)+c*error(t-12) (Seasonal MA(q))
3. Y(t)=a +bY(t-1)+cY(t-2)+dY(t-12) + error(t)+g*(error(t-1)) (Seasonal ARMA(p,q))
Good luck!
An ARIMA(1,1,1) will not capture the potential seasonal effects that seem to be present according to the PACF.
Thanks for the answers guys. Marwan Abdul hameed Ashour, I was also thinking this could be an ARIMA (1,1,1) but was hesitant because of the lack of decay in the ACF/PACF. Pablo Pincheira I will try to remove some of the seasonality that seems to be present (although the data *should* be seasonally adjusted) and then try fit the most appropriate model using IC and other statistics. Appreciate all your suggestions!
You may also test ARIMA (1,1,0), ARIMA(0,1,1) and ARIMA(1,1,1), then please check the significance all parameters, and you will see which one is the best for your case.
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