# What “more” does differencing (d>0) do in ARIMA than detrend?

I recently read a discussion about ARIMA models where someone said (referring to d as in ARIMA (p, d, q)):

Its true that d=1 takes out deterministic trends when they are present (they would appear only in the drift term.) But it does more than that.

I know that's not much context, but I seem to remember reading something similar in regards to detrending via differencing.

Two questions:

1. Does differencing (not just in an ARIMA context) do something more to your data than just detrend it? If so, what else does it do? (Add or remove?)

2. There are other detrending methods, such as fitting a curve (loess, linear regression) and using the residuals as detrended data. Would these other methods not do the "more than that" that differencing does? (Hence, might they be preferrable?)

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## 2 Answers

Differencing isn't actually the preferred way of removing a trend---detrending is. Detrending involves estimating the trend and calculating the deviation from the estimated trend in any particular period.

The main use of differencing is to remove the problem of unit roots. A unit root arises, for example, when $\rho=1$ in the simple AR(1) model $y_{t} = \rho y_{t-1} + \nu_t$. In this case, differencing yields a stationary white noise process $\nu_t$ that is appropriate for analysis.

Differencing a process without a unit root, but with a trend, can actually produce bad results (the new, differenced error term can have a strange distribution that contains autocorrelation, but of a tricky process). Similarly, detrending a process without a trend, but with a unit root can fail to eliminate the problem of non-stationarity (that is, it doesn't fix the unit root problem).

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To be fair, differencing can remove trend: polynomial depending on time. But yes this is more of a corner case, although mentioned in quite a few time series textbooks. –  mpiktas Sep 14 '11 at 7:23
@mpiktas, True, differencing can remove a trend. But, if there is a stationary noise component, differencing can produce a new random variable with a noise term that has a difficult structure to work with. So the trend may be gone, but the autocorrelation may be harder to deal with. –  Charlie Sep 14 '11 at 14:02
Thanks! Differencing was new to me and was introduced as a quick and non-parametric way to remove a trend. Sounds like it's a possible effect, but with a reasonably common side effect of introducing something I'd analogize to a "moire pattern" into the residuals. (Sort of like an unanswered question I have about log-transforming to get a multiplicative seasonality model: why not log-transform all the time, since it turns results into near-percentages which usually make more sense to me?) –  Wayne Sep 14 '11 at 14:38
:Wayne The end doesn't justify the means.Transformations(like drugs!) can be both good and bad for you.The idea is KISS.If the variance of the errors is non-constant AFTER you have removed outliers/level shifts/local time trends/seasonal pulses &removed any auto-correlative structure in the errors ( perhaps self-caused) &after fully g capturing/extracting the effect of leads and lags &after verifying that the parameters had not changed over time after fully verifying that GLS/WeightedLeastSquares was not applicable THEN one might investigate Power TRansforms e.g. reciprocals/sq root/logs. –  IrishStat Sep 15 '11 at 19:21

Unnecessary differencing or filtering can inject structure (see Slutsky Effect) . Sometimes a series can have a shift in the mean causing "non-statioanarity" ..the correct remedy is to neither difference or de-trend but to "de-mean" or use a Level Shift variable/filter to render the residual series stationary. Sometimes there is more than 1 trend requiring a number of trend variables/filters .... none of which have to start at the beginning if the series. Analysis will tell you which of these three approaches

1. differencing
2. de-meaning
3. de-trending

are suitable for your data.

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Very good. Part of my original question presented a false dichotomy, as it were. –  Wayne Sep 14 '11 at 14:40
Thanks also for the Slutsky Effect. Not listed in Wikipedia, but when I found a PDF of it, the explanation was pretty clear. (The direct application is to moving averages introducing autocorrelation, but I get the idea.) –  Wayne Sep 14 '11 at 18:49