acf and pacf suggests MA but auto.arima gave AR

Question

I have this data which is residual series obtained from predicted values and observations. original series was a random walk with a very small drift(mean=0.0025).

err <- ts(c(0.6100, 1.3500, 1.0300, 0.9600, 1.1100, 0.8350 , 0.8800 , 1.0600 , 1.3800 , 1.6200,  1.5800 , 1.2800 , 1.3000 , 1.4300 , 2.1500 , 1.9100 , 1.8300 , 1.9500  ,1.9999, 1.8500 , 1.5500 , 1.9800  ,1.7044  ,1.8593 , 1.9900 , 2.0400, 1.8950,  2.0100 , 1.6900 , 2.1800 ,2.2150,  2.1293 , 2.1000 , 2.1200 , 2.0500 , 1.9000,  1.8350, 1.9000 ,1.9500 , 1.7800 , 1.5950,  1.8500 , 1.8400,  1.5800, 1.6100 , 1.7200 , 1.8500 , 1.6700,  1.8050,  1.9400,  1.5000 , 1.3100 , 1.4864,  1.2400 , 0.9300 , 1.1400, -0.6100, -0.4300 ,-0.4700 ,-0.3450), frequency = 7, start = c(23, 1), end = c(31, 4))

and I know this residual series has some seriel correlations and can be modeled by ARIMA.

acf(err[1:length(err)]);pacf(err[1:length(err)])
# x axis starts with zero.
# showing only integer lags here, same plot as full seasonal periods. 
# shows it typically can be fitted by a MA model.

I have attempted following fittings:

library(forecast)

m1 <- auto.arima(err, stationary=T, allowmean=T)
#output
# ARIMA(2,0,0) with zero mean 

# Coefficients:

#         ar1     ar2
#      0.7495  0.2254
# s.e.  0.1301  0.1306

# sigma^2 estimated as 0.104:  log likelihood=-17.65
# AIC=41.29   AICc=41.72   BIC=47.58

m2 <- auto.arima(err, allowmean=T)
# output
# ARIMA(0,2,2) 

# Coefficients:
#          ma1     ma2
#      -1.3053  0.3850
# s.e.   0.1456  0.1526

# sigma^2 estimated as 0.1043:  log likelihood=-16.97
# AIC=39.94   AICc=40.38   BIC=46.12

From the acf and pacf of err we can see that it is to be fitted by an MA model rather than AR, why does auto.arima give me an AR fit?

Answer 1

I am a bit unclear why you believe your (P)ACF plots suggest an MA model. Here are some indications:

The data may follow an ARIMA(p,d,0) model if the ACF and PACF plots of the differenced data show the following patterns:

• the ACF is exponentially decaying or sinusoidal;
• there is a significant spike at lag p in the PACF, but none beyond lag p.

The data may follow an ARIMA(0,d,q) model if the ACF and PACF plots of the differenced data show the following patterns:

• the PACF is exponentially decaying or sinusoidal;
• there is a significant spike at lag q in the ACF, but none beyond lag q.

Your plots fall squarely in the first case, with $p=1$.

If we fit an AR(1) model, we get an AIC of 65.08 and residual (P)ACF plots as follows:

Nothing is significant any more, so I would happily go with this.

Then again, auto.arima() gets a much lower AIC. And also, if you allow it to look for non-stationary models, it actually fits an ARIMA(0,2,2) model (note that AICs cannot be compared between models with different orders of differencing):

And as a matter of fact, that plot of the time series itself is revealing. The first thing you should do is not to fit an ARIMA model to your data. No, the first thing should be to investigate your data carefully and figure out where the sudden drop around time 30 comes from. Then model whatever happened there.