Is a time series which is a deterministic linear trend + white noise considered an ARIMA model?

Question

This is, I guess, just a question of standard vocabulary.

All definitions I have seen for an ARIMA series just state that it is a series which becomes ARMA after differencing some number of times.

If you difference a time series model with linear trend

$$X_t = a + bt + w_t,$$

where $w_t$ is a white noise series, you get a stationary model

$$Y_t = b + (w_t - w_{t-1})$$

This would not be an ARMA model because of the nonzero mean, I guess, so $X_t$ doesn't fit the definition of ARIMA(0,1,1) I have seen. If you difference again, you get

$$Z_t = w_t -2 w_{t-1} + w_{t-2}$$

Thus, the series $X_t$ fits the definition I have seen of an ARIMA(0,2,2) model. Yet I see lots of websites and books informally suggest an ARIMA model should have only stochastic trends.

So my question is, do practitioners call a series like $X_t$ with a deterministic trend a type of ARIMA model, as would comport with the definitions I have seen? Or do the definitions I have seen need some revision to remove such series from consideration?

Also, would practitioners view the extra difference I applied to be a case of overdifferencing? I would think so, since a constant mean should be easy to deal with after that first difference.

Answer 1

No, $X_t$ is not considered an ARIMA model.

Walter Enders writes here in his textbook (3rd edition, p. 191)

We have shown that differencing can sometimes be used to transform a nonstationary model into a stationary model with an ARMA representation. This does not mean that all nonstationary models can be transformed into well-behaved ARMA models by appropriate differencing. Consider, for example, the a model that is the sum of of a deterministic trend and a pure noise component

$y_t = y_0 + a_1 t + \epsilon_t$

The first difference of $y_t$ is not well-behaved because

$\Delta y_t = a_1 + \epsilon_t - \epsilon_{t-1}$

Here $\Delta y_t$ is not invertible in the sense that $\Delta y_t$ cannot be expressed in the form of an autoregressive process. Recall that invertibility of a stationary process requires that the MA component does not have a unit root.

Answer 2

I personally consider Rob Hyndman's forecast package for R the gold standard in ARIMA modeling and forecasting. And this package will quite happily deal with time series of the form of your differenced series and call them "ARIMA with non-zero mean".

> set.seed(4); forecast::auto.arima(rnorm(100,5,1))
Series: rnorm(100, 5, 1) 
ARIMA(5,1,0)
... snip ...

Similarly, a deterministic trend plus white noise is modeled as "ARIMA with drift":

> set.seed(4); forecast::auto.arima(1:100+rnorm(100,5,1))
Series: 1:100 + rnorm(100, 5, 1) 
ARIMA(5,1,0) with drift

So yes, I would consider deterministic trends ARIMA processes.

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In addition, Brockwell & Davis' Introduction to Time Series and Forecasting (3rd ed., 2016) also consider "ARMA(p,q) processes with mean" on p. 74. I couldn't find an explicit discussion of trends that upon differencing turn into such ARMA(p,q) processes with (nonzero) mean, but I would say this extension is obvious enough to be accepted.

And I agree that this is a question of convention.