# ARIMA (0,1,1) or (0,1,0) - or something else?

I've just started learning time series so please excuse me if it's painfully obvious; I haven't managed to find the answer elsewhere.

I have a data series showing a pretty obvious trend although it's quite noisy. I can take pretty much any division of the data and run classical tests to show a highly significant difference in means.

I decided to have a look at time series analysis to see if it could help describe the trend. An ARIMA(0,1,1) model comes out with AIC,BIC=34.3,37.3 (Stata), whilst an ARIMA(0,1,0) model comes out with AIC,BIC=55.1,58.1 - so I understand I'm supposed to prefer the (0,1,1) model.

However, the coefficient for the MA(1) is displaying as -0.9999997 (and not showing any p-values). If I try the same in SPSS I get an MA(1) coefficient of 1.000 (I assume SPSS uses opposite signs) with a p-value of 0.990 - does this mean it suggests I drop the term?

My understanding is that the effect of a MA(1) coefficient of -1 is basically to remove the old error term and convert the whole series to a linear trend. Does this mean ARIMA is totally unsuitable for my needs? On the plus side it gives me a sensible value for the trend. If I use the (0,1,0) model then I still get a reasonable value for the trend but it's not significant any more.

EDIT: Thanks for looking in. The trend looks like a fairly linear decrease; the data points seen to fairly noisily rattle around above and below a trend line. The ARIMA (0,1,1) model produces something that's not far off a straight line decrease which seems sensible - the (0,1,1) produces what is essentially a lagged version of the data, translated down by one month of trend. The data aren't stationary (due to the trend) - though the first differences seem to be. I don't think the (0,1,1) is a bad model - I'm just a little confused by the p-value seeming to suggest I should drop the MA term - or wondering if it means I should bin ARIMA entirely!

EDIT2 @vinux - thanks for the suggestion; that makes a lot of sense (and seems to be what the -1 MA term is trying to create?). I've uploaded as many graphs as I could think of as people had requested.

I've also put the monthly data up in CSV format at pastebin

• Can you be more specific about the 'trend'? – Glen_b Oct 14 '13 at 6:11
• Perhaps you could post a graph of the time series, its ACF, & its PACF. How long is the time series anyway? I'm sure @vinux is right about over-differencing. Did the error variance increase when you differenced? – Scortchi Oct 14 '13 at 13:57
• The data would be useful as the ACF is merely a descriptive summary statistic whose form can arise from a number of possible "causes". Please post your data so an informed analysis can proceed. – IrishStat Oct 14 '13 at 14:29
• @thigger, you don't need to difference a series for a deterministic trend. I guess in your case stationary model with trend ($X_t=a + bt + Z_t$, where $Z_t$ is stationary series) would fit your data. – vinux Oct 14 '13 at 15:28
• @vinux: Good point, & I think that might, if the error variance is small compared to the trend, explain the apparent need for a big MA term after differencing - you'd be introducing negative auto-correlation. – Scortchi Oct 14 '13 at 16:03

It is difficult to give the right answer without looking at the data. Here are some points that may help you in your modelling.

The coefficient of MA(1) very close to 1 indicates the sign of overdifferencing. This means unit root in Moving averages.

My suggestion would be: Check the original series is stationary (visually) or check the presence of unit root. If you observe deterministic trend (eg: linear), add the trend part with time series model. If the original series is stationary build the time series without differencing.

• (+1) For the OP's benefit, stationarity of the original series can be checked visually by examining whether the sample auto-correlation function falls off to noise quickly (exponentially). – Scortchi Oct 14 '13 at 13:30
• @Scortchi Whereas the acf can suggest non-stationarity, non-stationarity is a symptom and can have multiple causes. – IrishStat Oct 14 '13 at 14:32
• @vinux - just thought I'd add the other component to your answer here - It works well as (Xt=a+bt+Zt, where Zt is stationary series) - and the stationary series appears to be pretty much white noise. Thanks! My underlying error was a failure to appreciate the difference between trend removal by differencing and trend removal by subtraction of a linear trend - the -1 MA(1) term seems to have been trying to convert a random walk back into white noise. – thigger Oct 15 '13 at 10:18

As you say, the data is not stationary, we can find the stationary transformed data by differencing, and checked by the unit root test (e.g Augmented Dickey-Fuller test, Elliott-Rothenberg-Stock test, KPSS test, Phillips-Perron test, Schmidt-Phillips test, Zivot-Andrews test...) We can talk about ARMA model only after confirming the stationarity.

It is a classical way to identify the ARMA(p, q) by the ACF plot and PACF plot. ARMA(0,1) and ARMA(0,0) can be told here. Another method to identify p, q is about the EACF, but it is not widely used for univariate time series.

Empirical studies show that AIC usually tends to overfitting. The advantage of using AIC is for automatic algorithm to find the best model, but it is not usually recommended in traditional time series textbook.

• If the series has one or more shifts in the mean , I believe some if not most of your recommended tests will falsely conclude about the need for differencing as a remedy . Differencing is one from of remedy for non-stationarity but ny no means (play on words) is not the only form. See the following article on the flaws of differencing autobox.com/makridakis.pdf. Other possible causes for apparent non-stationarity that do not require differencing are 1:) time-varying parameters and 2) time varying error variance – IrishStat Oct 14 '13 at 15:27