# Time series analysis (ACF, PACF)

I have this monthly time series with pronounced seasonality and a bit of trend:

The ACF and PACF for 4 years (48 months) are:

1. Can I suppose that the data don't need transformations like: log(data), diff(data)...?

ACF: A spike at lag 12 in the ACF but no other significant spikes.
PACF: ?

2. R proposes ARIMA(0,0,0)(1,0,0)[12]. Could it be a good model?

• 1. Depends on the model you plan to use. Traditional ARMA assumes stationary normality Commented Jan 7, 2016 at 1:39
• If your using an ARMA and the data is not stationary and/or not well approximated with a conditional normal then you would want to transform it. In your case it is hard to tell (about the normality part...it looks stationary. You could formally test stationarity if it would make you feel better though). The resulting ARMA would assume it is possible for the series to go below 0. If that is not posible with your data and the ARMA forecasts/fitted values go below 0 then maybe log transform. 2. Cannot be answered until you define what "good" means Commented Jan 7, 2016 at 1:49

It looks like a plausible model, though I'd possibly start with seasonal differencing and see if the residuals from that didn't look over-differenced.

If you do go with the suggested model you have, you'd still look to see if there was autocorrelation in the residuals.

• Regarding the first sentence of the answer: I guess the function auto.arima was used to arrive to ARIMA(0,0,0)(1,0,0)[12]. If I am not mistaken, it tests for a seasonal unit root on the way, so if it did not suggest one, the test must have indicated absense of seasonal unit root. Commented Jan 7, 2016 at 8:36
• @Richard (edit) You are right, it uses the Osborn-Chui-Smith-Birchenhall test. [The discussion of deterministic seasonality in that paper would mean -- if I understand the paper right -- that deterministic seasonality could still be present.] Commented Jan 7, 2016 at 10:52

Based on the text here, you will need to compute the p-value and thus identify if there your time series is stationary.