# A smoothed series still exibits strong seasonality

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)[12]. 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.