The acronym ARIMA stands for “Auto-Regressive Integrated Moving Average.” Random-walk and random-trend models, autoregressive models, and exponential smoothing models (i.e., exponential weighted moving averages) are all special cases of ARIMA models. An ARIMA model is classified as an “ARIMA(p,d,q)” model, where the current value y is determined by:
* p — the number of lagged terms (AR),
* d — the number of integrations, and
* q — the number of moving average terms (MA).
Here is a good introduction to ARIMA modelling. Normally, models are either only-AR or only-MA terms, because including both kinds of terms in the same model can lead to overfitting of the data and non-uniqueness of the coefficients.
VS proposed an ARIMA (3,1,0) or AR model. B&V concluded an ARIMA((0,1,2) or MA model. I downloaded the GISSTEMP dataset (J-D average), and force fitted both models using a brute-force non-linear solver to minimize RSS in both instances. (I also tested and confirmed VSâ€™s rejection of a drift term in the AR model, not only for his three term solution but also for all combinations of a two-lag solution.)
The GISSTEMP dataset, 1880 to 1909 consists of 130 datapoints with a variance of 0.062388. The best-fit ARIMA(3,1,0) model looks like this(the lower plot being the residual error):
For comparison, the B&V model looks like this:
The RSS for the best-fit VS model (ARIMA(3,1,0) is 1.4784. This fit uses 3 degrees of freedom for coefficient estimation and 4 degrees of freedom for initial boundary conditions. The adjusted sd of the residual is then 0.1096 â€“ similar to that reported by VS on Bartâ€™s blog.
By comparison, the RSS for the best-fit B&V model (ARIMA(0,1,2)) is 1.3227. This fit uses 4 degrees of freedom for coefficients and 1 degree of freedom for initial boundary conditions, a total of 5. The adjusted sd of the residual is then 0.1029.
Normally comparisons within nested models are statistically straightforward and between models of different structure a much more complicated problem. However, in this instance, the comparison does look straightforward, and, on a purely statistical basis, I would have to conclude that the B&V model is superior.
Equally importantly, a visual inspection of the model matches suggests that the B&V moving average model captures the data character better. More speculatively, although both models produce residuals which do not reject stationarity, non-zero bias nor normality, the VS model retains a high degree of improbable excursion in its residual error function which is not apparent in the MA model proposed by B&V.
Overall, therefore, I am inclined to accept the B&V model (moving average) over the VS (AutoRegressive) model. As I mentioned in my previous post, if my finding is valid (and I openly acknowledge that it is neither complete nor rigorous) this has a significant effect on your a priori assumptions, David.
DS Note: My model was at the same integration order q=1 as yours are, and the conclusion is the same — no need for a non-zero bias or drift term (attributable to global warming). Perhaps I should work it through with those models, but it seems to have already been done.
It is worth reading wiki here on the difference between an AR model called an ‘infinite impulse’ and a MA model called a ‘finite impulse’ model. The finite vs infinite aspect might provide some intuition.