Additive outliers enter observations; innovation outliers enter process shocks. A fixed AR(1) example separates their paths and residual signatures.
Additive and innovation outlier
An additive outlier (AO) is a one-time disturbance added to an observed time-series value. An innovation outlier (IO), also called an innovational outlier, enters the process's shock term and propagates through its dynamics. The distinction matters when forecasting a financial spread: feeding a recording spike into the model as a genuine state change can carry the error into later forecasts. These are model locations for an outlier, rather than diagnoses of what happened in the market.
Put the disturbance in the equation
Let xt be a latent, centred series following an autoregressive model,
xt=ϕxt−1+εt,
and let yt be the recorded series. Here εt is the ordinary innovation: the new unpredictable component under the specified information set. Write It(T)=1 when t=T and zero otherwise. For a disturbance of size ω:
Mechanism
Where the disturbance enters
What is observed
AO
Observation equation
yt=xt+ωIt(T); the latent recursion is unchanged
IO
Process equation
xt=ϕxt−1+εt+ωIt(T) and yt=xt
This observation-versus-process distinction also appears in state-space models. The Fisch, Eckley and Fearnhead preprint, §§1–2 explains why the two locations can call for different updates to a hidden state. Neither location alone identifies an economic cause.
In the causal, covariance-stationary AR(1) baseline, ∣ϕ∣<1 and the ordinary innovations have mean zero, finite constant variance and no serial correlation. Calling them conditional forecast innovations additionally requires zero conditional mean given the past; uncorrelated noise alone is insufficient. The one-time disturbance is superimposed on that baseline, so we are not claiming that the contaminated series remains stationary.
The same +10 spike, a different tomorrow
Consider an original deterministic illustration with ϕ=0.5, known initial value x0=0 and all ordinary innovations set to zero. Time is measured in equally spaced observations; values are in arbitrary units. This is a shock-response comparison, not a random sample from the stationary distribution.
At T=2, add ω=10. In the AO scenario, the recorded spike leaves the latent path at zero. In the IO scenario, the shock enters the recursion: x2=10, x3=0.5(10)=5 and x4=0.5(5)=2.5.
Now apply the same fixed AR filter to each recorded series:
rt=yt−0.5yt−1.
Use the known presample observation y0=0, so the first value is r1=y1−0.5y0=0. An estimated model with a different initialisation can produce different first residuals.
Time t
AO recorded yt
IO recorded yt
AO filtered rt
IO filtered rt
1
0
0
0
0
2
10
10
10
10
3
0
5
−5
0
4
0
2.5
0
0
Synthetic AR(1) with coefficient 0.5 and known initial value zero. An additive outlier gives observed values 0, 10, 0, 0 and filtered residuals 0, 10, minus 5, 0. An innovation outlier gives observed values 0, 10, 5, 2.5 and filtered residuals 0, 10, 0, 0.
Original deterministic shock-response illustration. Columns separate the disturbance locations; rows separate observed values from residuals under the same fixed filter. Lines only connect observation times.
The AO's −5 residual is not a second negative disturbance. At time 3, the filter predicts 0.5(10)=5 from the contaminated lag, then subtracts that prediction from the actual zero. For the IO path, 5−0.5(10)=0: the persistence is exactly what the assumed model predicts.
More generally, an AO contributes ωIt(T)−ϕωIt(T+1) to this AR(1) residual filter. An IO contributes only ωIt(T), while its effect on observations at T+h is ωϕh, for h≥0. That sequence is the model's scaled impulse response. With negative ϕ, its signs alternate; with other time-series models, its shape changes.
A signature is evidence, not a verdict
The table assumes the correct coefficient is known and ordinary noise is absent. In data, parameters are estimated, ordinary shocks continue and multiple disturbances can overlap. A pair of residuals with opposite signs does not prove a recording error. At the final observation, these AO and IO scenarios are identical through the data available; there is no subsequent path to compare. Chen and Liu's forecasting paper discusses insufficient information near the forecast origin and the consequences of misclassifying an outlier.
Persistence alone also does not settle the label. A separately specified temporary-change intervention can reproduce this geometric decay. The IO label says that propagation follows the fitted process's dynamics. A permanent level shift is a different intervention in this stationary AR(1) setting. The IMSL description sets out these distinct dynamic patterns.
For a suspected bad financial observation, check the underlying record, units, timestamp and source before correction. For a genuine event, assess whether the baseline dynamics still describe the series. Preserve the original data and compare forecasts under stated disturbance assumptions; fitting an AO or IO is not permission to delete an inconvenient extreme.
For the forecasting foundation, AR, MA, and ARMA develops shock propagation, stationarity, invertibility and residual checks. The full lesson requires membership; its public preview is available.
IMSL, tsOutlierIdentification, Description: AO, IO, temporary-change and level-shift intervention patterns, and the endpoint identification limitation. The cited page documents the 2020 wrapper; no software installation or current-version recommendation is implied.
