I have a monthly time series. It is basically a price level series (inflation data), and I converted it into monthly percentage changes (i.e. like the CPI measure). This time series exhibits extremely large seasonality on an annual period. I did a moving average filter of period 12, and my intuition was that this would smooth the seasonality. However, when I checked I found that the smoothed series ACF still had a large spike at 12 months, and no other statistically significant spikes, but the PACF has spikes at 12 and 24. The best fitting arima type was (0,1,0)(2,0,0). This is significantly better than any non-seasonal model. The SAR coefficients are (1) -0.5 and (2) -0.3.
If I do a seasonal decomposition, then I discover that the the MA has reduced the cyclical component (of period 12) to almost zero, as I expected.
What is the intuitive explanation of these results, does it perhaps mean that there is a longer cycle, of say 24 months? Decomposing on 24 months reveals a small but not entirely negligible seasonal component, yet the SAR coefficients are large.
In short, I am confused of the conditions under which a Moving average filter of appropriate period does not remove the need for seasonality from an ARIMA type model.