I've got two time series (parameters of a model for males and females) and aim to identify an appropriate ARIMA model in order to make forecasts. My time series looks like:
The plot and the ACF show non-stationary (the spikes of the ACF cut off very slowly). Thus, I use differencing and obtain:
This plot indicate that the series might now be stationary and the application of the kpss test and the adf test support this hypothesis.
Starting with the Male series, we make the following observations:
- The empirical autocorrelations at Lags 1,4,5,26 and 27 are significant different from zero.
- The ACF cuts off (?), but I'm concerned about the relatively big spikes at lag 26 and 27.
- Only the empirical partial autocorrelations at Lags 1 and 2 are significant different from zero.
On ground of these observations alone, if I had to choose a pure AR or MA model for the differenced time series, I would tend to choose either an AR(2) model by arguing that:
- We have no significant partial autocorrelations for lag greater than 2
- The ACF cuts off except for the region around lag 27. (Are these few outliers alone an indicator, that a mixed ARMA model would be appropriate?)
or an MA(1) model by arguing that:
- The PACF clearly cuts off
- We have for lags greater 1 only 4 spikes exceeding the critical value in magnitude. This is "only" one more than the 3 spikes (95% out of 60) which would be allowed to lie outside the dotted area.
There are no characteristica of an ARIMA(1,1,1) model and choosing orders of p and q of an ARIMA model on grounds of ACF and PACF for p+q > 2 gets difficult.
Using auto.arima() with the AIC criterion (Should I use AIC or AICC?) gives:
- ARIMA(2,1,1) with Drift; AIC=280.2783
- ARIMA(0,1,1) with Drift; AIC=280.2784
- ARIMA(2,1,0) with Drift; AIC=281.437
All three considered models show white noise residuals:
My summed up questions are:
- Can you still describe the ACF of the time series as cutting of despite the spikes around lag 26?
- Are these outliers an indicator that a mixed ARMA model might be more appropriate?
- Which Information Criterion should I choose? AIC? AICC?
- The residuals of the three models with the highest AIC do all show white noise behavior, but the difference in the AIC is only very small. Should I use the one with the fewest parameters, i.e. an ARIMA(0,1,1)?
- Is my argumentation in general plausible?
- Are their further possibilities to determine which model might be better or should I for example, the two with the highest AIC and perform backtests to test the plausibility of forecasts?
EDIT: Here is my data:
-5.9112948202 -5.3429985122 -4.7382340534 -3.1129015623 -3.0350910288 -2.3218904871 -1.7926701792 -1.1417358384 -0.6665592055 -0.2907748318 0.2899480865 0.4637205370 0.5826312749 0.3869227286 0.6268379174 0.7439125292 0.7641139207 0.7613140511 3.0143912244 -0.7339255839 2.0109976796 0.8282394650 -2.5668367983 5.9826406394 1.9569198553 2.3860893476 2.0883339390 1.9761894580 2.2601997245 2.2464027995 2.5131158613 3.4564765529 4.2307335557 4.0298688374 3.7626317439 3.1026407174 2.1690168737 1.5617407254 2.6790460788 0.4652054768 -0.0501046517 -1.0157683791 -0.5113698054 -0.0180401353 -1.9471272198 -0.2550365250 -1.1269988523 0.5152074134 0.2362626753 -2.9978337017 1.4924705528 -1.4907767844 -0.5492041416 -0.7313021018 -0.6531515868 -0.4094159299 -0.5525401626 -0.0611454515 -0.5256272882 -1.1235247363 -1.7299848758 -1.3807763611 -1.6999054476 -4.3155973110 -4.7843298990