Explaining the forecasts from an ARIMA model

Question

I am trying to explain to myself the forecasting result from applying an ARIMA model to a time-series dataset. The data is from the M1-Competition, the series is MNB65. I am try

Answer 1

1. No ARIMA(p,0,q) model will allow for a trend because the model is stationary. If you really want to include a trend, use ARIMA(p,1,q) with a drift term, or ARIMA(p,2,q). The fact that auto.arima() is suggesting 0 differences would usually indicate there is no clear trend.

2. The help file for arima() shows that the intercept is actually the mean. That is, the AR(1) model is (Y_t-c) = ϕ(Y_{t-1} - c) + e_t rather than Y_t = c + ϕY_{t-1} + e_t as you might expect.

3. auto.arima() uses a unit root test to determine the number of differences required. So check the results from the unit root test to see what's going on. You can always specify the required number of differences in auto.arima() if you think the unit root tests are not leading to a sensible model.

Here are the results from two tests for your data:

R> adf.test(x)

        Augmented Dickey-Fuller Test

data:  x 
Dickey-Fuller = -1.031, Lag order = 3, p-value = 0.9249
alternative hypothesis: stationary 

R> kpss.test(x)

        KPSS Test for Level Stationarity

data:  x 
KPSS Level = 0.3491, Truncation lag parameter = 1, p-value = 0.09909

So the ADF says strongly non-stationary (the null hypothesis in that case) while the KPSS doesn't quite reject stationarity (the null hypothesis for that test). auto.arima() uses the latter by default. You could use auto.arima(x,test="adf") if you wanted the first test. In that case, it suggests the model ARIMA(0,2,1) which does have a trend.