chapter 4 outliers, influential observations and missing...
TRANSCRIPT
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Chapter 4
Outliers, Influential observations
and missing data.
Based on Peña et al. (2000), Chapter 6
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CHAPTER 4. CONTENTS.
4.1. Types of outliers in time series.
4.2. Procedures for outlier identification and estimation.
4.3. Influential observations.
4.4. Multiple outliers.
4.5. Missing-value estimation.
4.6. Forecasting with outliers.
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Two (complementary)approaches:
Diagnostic approach: outliers are indentified on the
residuals; a joint model is estimated; we obtain an
estimation of the outlier effects and robust parameter
estimates.
Robust approach: the estimation method is modified so
that the estimates are not contaminated by the presence
of outliers.
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Assumptions: the outliers happen on a time series which can be modelled by,
The model could be written in AR or MA forms:
* all the usual assumptions still apply.
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tt
d aByB )()(
tt ayB )( tt aBy )(
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.1. Types of outliers in time series.
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TYPES OF OUTLIERS IN TIME SERIES.
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- Additive outliers (AO). Correspong to an external error or
exogenous change of the time series ar a particular time
point
𝑧𝑡 = 𝑦𝑡 𝑡 ≠ 𝑇𝑦𝑡 + 𝜔𝐴 𝑡 = 𝑇
where 𝑇 is the time at which the outlier occurs and 𝜔𝐴 is the
magnitude of the outlier.
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TYPES OF OUTLIERS IN TIME SERIES.
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- An alternative representation is,
𝑧𝑡 = 𝜔𝐴𝐼𝑡(𝑇)
+ 𝜓(𝐵)𝑎𝑡
where 𝐼𝑡(𝑇)
is a dummy variable which is zero at all lags
except at time 𝑡 = 𝑇. Equivalently
𝑎𝑡 = 𝜋 𝐵 𝑧𝑡 − 𝜔𝐴𝐼𝑡 𝑇
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TYPES OF OUTLIERS IN TIME SERIES.
An AO can have serious effects on the properties of
the observed time series:
It will affect the estimated residuals.
It will affect the estimates of the parameter values.
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TYPES OF OUTLIERS IN TIME SERIES.
Estimated residuals. Assuming known parameters,
the residuals if the AO does not occur could be
obtained by
Whereas in the case of AO, they are obtained as
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tt yBa )(
))(()( )( tAttt IyBzBe
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TYPES OF OUTLIERS IN TIME SERIES.
The relation between them is,
So the effect of the AO on the residuals depend on
the weights. In a AR model, p residuals will be
affected.
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)()( tAtt IBae
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TYPES OF OUTLIERS IN TIME SERIES.
Estimated parameters: Consider a simple AR(1) . The
least-square estimate is
In case of an AO the estimate is
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2
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t
tt
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TYPES OF OUTLIERS IN TIME SERIES.
It can be shown that,
In general, a large AO will push all the autocorrelation coefficients towards zero.
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0ˆ A
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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- Innovative outliers (AO). Correspond to an internal error or
endogenous effect on the noise of the process
𝑧𝑡 = 𝑦𝑡 𝑡 < 𝑇
𝑦𝑡 + 𝜔𝐼𝜓 𝑗 𝑡 = 𝑇 + 𝑗, 𝑗 > 0
where 𝑇 is the time at which the outlier occurs and 𝜔𝐼 is the
magnitude of the outlier.
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TYPES OF OUTLIERS IN TIME SERIES.
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An alternative representation is
𝑧𝑡 = 𝜓(𝐵) ωIIt(T)
+ at
where It(T)
is an indicator variable which is zero at all lags
except at time 𝑡 = 𝑇. Equivalently,
𝜋 𝐵 𝑧𝑡 = ωIIt(T)
+ at
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TYPES OF OUTLIERS IN TIME SERIES.
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- Estimated residuals: for any ARIMA model
𝑒𝑇 = 𝑎𝑇 − 𝜔𝐼
𝑒𝑇+𝑗 = 𝑎𝑇+𝑗
for 𝑗 > 0
- Parameter estimates: In an AR(1)
𝑖𝑓 𝜔𝐼 → ∞ ⇒ 𝜙 → 𝜙 0
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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TYPES OF OUTLIERS IN TIME SERIES.
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- Level Shift outliers (LS). Correspond to a modification of the
local mean or level of the process starting from a specific point
and continuing until the end of the sample.
𝑧𝑡 = 𝑦𝑡 𝑡 < 𝑇𝑦𝑡 + 𝜔𝐿 𝑡 ≥ 𝑇
- It can be seen as a sequence of AO's of the same size.
Note: a level shift can transform a stationary into a nonstatonary
series.
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TYPES OF OUTLIERS IN TIME SERIES.
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The model for this type of outlier is
𝑧𝑡 = 𝜔𝐿𝑆𝑡(𝑇)
+ 𝜓(𝐵)𝑎𝑡
where 𝑆𝑡(𝑇)
is a step function that takes the value 0 before 𝑇
and 1 by 𝑡 ≥ 𝑇.
𝑆𝑡(𝑇)
=𝐼𝑡
(𝑇)
(1 − 𝐵)
The model can also be written as
𝜋 𝐵 𝑧𝑡 − 𝜔𝐿𝑆𝑡 𝑇
= 𝑎𝑡
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TYPES OF OUTLIERS IN TIME SERIES.
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- The effect of a LS on the residuals and on the parameter
estimates can be strong. Assuming known parameters
𝑒𝑡 = 𝑎𝑡 + 𝜋 𝐵 𝜔𝐿𝑆𝑡 𝑇 = 𝑎𝑡 +
𝜋 𝐵
(1 − 𝐵)𝜔𝐿𝐼𝑡
𝑇
- This means that all residuals after the LS can be affected
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TYPES OF OUTLIERS IN TIME SERIES.
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- The effect of a LS depends on (1) the model and (2) the
distance between the time at which the LS occurs and the end
of the observed sample.
- For 𝑛 − 𝑇 not too small
𝜔𝐿 → ∞ ⇒ 𝑟𝑧(1) → 1
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IF THE SHOCK OCCURS AT THE MIDDLE OF THE SAMPLE
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IF THE SHOCK OCCURS AT THE END OF THE SAMPLE
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TYPES OF OUTLIERS IN TIME SERIES.
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- Outliers and intervention analysis The types of outliers studied
can be considered as particular cases of interventions in a time
series (Box and Tiao, 1975)
𝑧𝑡 = 𝜔𝑉 𝐵 𝐼𝑡 𝑇 + 𝜓(𝐵)𝑎𝑡
in which 𝑉 𝐵 is the transfer function of the intervention.
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TYPES OF OUTLIERS IN TIME SERIES.
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Among many others:
- AO: 𝑉 𝐵 = 1
- IO: 𝑉 𝐵 = 1/𝜋(𝐵)
- LS: 𝑉 𝐵 = 1 1 − 𝐵
- TC: 𝑉 𝐵 = 1 1 − 𝛿𝐵
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.2. Procedures for outlier identification and
estimation.
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PROCEDURES FOR OUTLIER IDENTIFICATION.
In order to eliminate the effect of an outlier in a given
time series it is necessary to:
(1) detect the time at which the outlier happens
(2)identify the type of outlier
(3)remove its effect by estimating a model in which the
outlier is incorporated
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PROCEDURES FOR OUTLIER IDENTIFICATION.
Overview of the automatic procedure.
1. At each time point, we analyze what will be the most
lykely type of an outlier (likelihood ratio for the types
considered)
2. We choose as the candidate outlier timepoint the one
that has smallest p value and as outlier type the
corresponding outlier effect
3. Fit the appropiate intervention model to remove he
outlier effect.
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PROCEDURES FOR OUTLIER IDENTIFICATION.
One outlier-parameters known. Consider the
intervention model
The model could be rewritten in regression form,
Where is a (nx1) vector
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ttttt yIBVaBIBVz )()( )()()(
ttt yDz )(*
)(* )()( tt IBVD
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PROCEDURES FOR OUTLIER IDENTIFICATION.
in matrix notation
where
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YDZ *
)())(),...,((
)1(),...,(
)1(),...,(
**
1
*
1
1
nxnDDD
nxyyY
nxzzZ
n
n
n
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PROCEDURES FOR OUTLIER IDENTIFICATION.
This model is a multiple regression model with
autocorrelated residuals that can be estimated by
GLS. Once we obtain
where
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aDe
LLYVar
YLaZLeDLD
a ')( 2
11*1
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PROCEDURES FOR OUTLIER IDENTIFICATION.
it is possible to estimate the effect of the outlier as
Hint:
the matrix of weights (slide 49-lesson 1)
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211 )'()ˆvar(')'(ˆaDDeDDD
DDL 11
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PROCEDURES FOR OUTLIER IDENTIFICATION.
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To test the null hypothesis that the observation
at 𝑡 = 𝜏 is not an outlier we use the likelihood ratio
𝜆 =𝑤
𝑣𝑎𝑟(𝑤 )
which assuming known parameters follows a
N(0,1) distribution
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PROCEDURES FOR OUTLIER IDENTIFICATION.
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If we don’t know the outlier type, then the
statistic
𝜂 𝜏, 𝐼 = max𝑖
𝜆𝑖 𝑖 = 𝐴𝑂, 𝐼𝑂, 𝐿𝑆,𝑇𝐶
Finally, if the period is also unknown
𝜂 = max𝑡
𝜏𝑡 , 𝐼𝑡
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PROCEDURES FOR OUTLIER IDENTIFICATION.
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Citerion: an AO at time 𝑡 = 𝜏 will be detected
if
𝜂 = 𝜆𝐴.𝜏 > 𝐶
where C is a predetermined constant (3-4)
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PROCEDURES FOR OUTLIER IDENTIFICATION.
Multiple outliers-iterative procedure.
1. Model is fitted by ML assuming no outliers, obtain
residuals
2. Compute the LR for any timepoint and any outlier type.
3. If an outlier is detected, correct its effect on the
residuals are corrected.
4. Compute again the LR for the new residuals. Repite
until no new outliers can be identified.
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PROCEDURES FOR OUTLIER IDENTIFICATION.
5. Estimate jointly the sizes of the identified outliers and the
parameters, assuming k outliers
6. Eliminate the non-significative outliers and re-start the
process.
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k
jt
j
tjjt aBIBVz1
,
, )()(
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.3. Influential observations.
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INFLUENTIAL OBSERVATIONS.
We may have points that are no detected as outliers
and that still have a strong influence on the parameter
values of the model and hence on the forecasts.
The detection of influential observations is important
to understand the sensitivity of the parameter values
to a small fraction of the data.
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INFLUENTIAL OBSERVATIONS.
One possible way of measure the influence of an
observation is by making it a missing value(substitute
it by its conditional expectation given the rest of the
data).
Consider the AR(h) approximation of an ARIMA model
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t
h
iitit ayy
1
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INFLUENTIAL OBSERVATIONS.
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Then measure of the influence of observation
𝑡 = 𝜏 on the parameter values is
𝑃𝜋 𝜏 = 𝜋 − 𝜋 𝜏 ′Σ 𝜋
−1 𝜋 − 𝜋 𝜏
ℎ𝜎 𝑎2
𝜋 is the MLE (maximum likelihood estimation)
of the parameters
𝜋 𝜏 is the MLE assuming 𝑡 = 𝜏 is missing
Σ 𝜋𝜎 𝑎2 is the variance matrix of 𝜋 .
h is the number of parameters.
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INFLUENTIAL OBSERVATIONS.
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The influence measure can be written as
𝑃𝑍 𝑇 = 𝑍 − 𝑍 (𝑇)
′ 𝑍 − 𝑍 (𝑇)
ℎ𝜎𝑎2
where
𝑍 = 𝑋𝑧𝜋
𝑍 (𝑇) = 𝑋𝑧𝜋 (𝑇)
𝑋𝑧 =
𝑧ℎ … 𝑧1
⋮ ⋮𝑧𝑡−1 … 𝑧𝑡−ℎ
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INFLUENTIAL OBSERVATIONS.
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𝑍 = 𝑋𝑧𝜋
𝑍 (𝑇) = 𝑋𝑧𝜋 (𝑇)
𝑋𝑧 =
𝑧ℎ … 𝑧1
⋮ ⋮𝑧𝑡−1 … 𝑧𝑡−ℎ
𝑍 and 𝑍 (𝑇) are the estimated vectors of forecasts computed from
the two parameter vectors.
𝑍 − 𝑍 (𝑇) ′ 𝑍 − 𝑍 (𝑇) = 𝜋 − 𝜋 (𝑇)
′ 𝑋𝑧 ′𝑋𝑧 𝜋 − 𝜋 (𝑇)
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INFLUENTIAL OBSERVATIONS.
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The advantage of the last expression is that it can be computed
by the ARIMA representation of the model and the AR
approximation is not required.
ℎ = 𝑝 + 𝑞
Peña (1987) also proposed to measure the change on the
variance by
𝐷𝑣 𝑇 =𝜎 2 − 𝜎 (𝑇)
2
𝜎 (𝑇)2
where 𝜎 (𝑇)2 corresponds to a model in which observation 𝑇 is
assumed to be missing.
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.4. Multiple outliers.
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INFLUENTIAL OBSERVATIONS.
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The method just presented works well when the series has a
single outlier or a few isolated outliers.
However, sometimes the series is subject to patches of outliers
that may produce masking
The generalization of the outlier model for k outliers is
𝑧𝑡 = 𝜔𝑖𝑉𝑖(𝐵)𝐼𝑡(𝑇𝑖)
𝑘
𝑖=1
+ 𝑦𝑡
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INFLUENTIAL OBSERVATIONS.
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This model can be written as
𝑒𝑡 = 𝑥𝑡′𝛽 + 𝑎𝑡
where
𝛽 =
𝜔1
⋮𝜔𝑘
𝑥𝑡 =
𝑥1𝑡
⋮𝑥𝑘𝑡
with
𝑥𝑖𝑡 = 𝜋 𝐵 𝑉𝑖(𝐵)𝐼𝑡(𝑇𝑖)
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INFLUENTIAL OBSERVATIONS.
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Outlier identification methods are based on estimated the
effects of the outliers one by one.
These procedures are expected to work well when the
matrix 𝑥𝑡𝑥𝑡 ′𝑛𝑡=1 −1 is roughly diagonal.
but may lead to serious biases when the series have patches
of additive outliers and level shifts.
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INFLUENTIAL OBSERVATIONS.
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For an innovational outlier 𝑥𝑖𝑡 = 𝐼𝑡(𝑇𝑖), and therefore the
estimation of its effect is typically uncorrelated with other
effects
However, for additive outliers 𝑥𝑖𝑡 = 𝜋 𝐵 𝐼𝑡(𝑇𝑖) and the
correlation between the effects of consecutive additive outliers
can be very high.
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INFLUENTIAL OBSERVATIONS.
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For example, suppose that we have two consecutive additive
outliers of magnitudes 𝜔1 and 𝜔2 at times 𝑇 and 𝑇 + 1
The expected value of the estimator of 𝜔1 assuming that it is
the only outlier
𝐸 𝜔 1 𝑠 = 𝜔1 + 𝜔2
𝜋𝑖𝜋𝑖+1𝑛−𝑇−1𝑖=0
𝜋𝑖2𝑛−𝑇
𝑖=0
When the parameters are unknown, the problem is still more
series because a sequence of additive outliers can produce
important biases on the parameter estimates.
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.5. Missing value estimation.
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MISSING VALUE ESTIMATION
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The study of additive outliers and influential observations in
time series is closely related to missing-value analysis because
an outlier at T implies that the true value at this point is not
observed.
Suppose first that we have a stationary time series with a
missing observation at time T.
The estimation of the missing value is called the interpolation
problem and it is solved by computing the expectation of the
observed random variable given the rest of the data
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MISSING VALUE ESTIMATION
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Grenander and Rosenblatt (1957) showed that this
expectation is given by
𝐸 𝑧𝑇/𝑍(𝑇) = − 𝛿𝑖 𝑧𝑇+𝑖 + 𝑧𝑇−𝑖
∞
𝑖=1
where
𝛿𝑖 are the inverse autocorrelation coefficient,
𝑍(𝑇) includes all the data but the missing value.
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MISSING VALUE ESTIMATION
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A simple way to define the inverse autocorrelation function
can be found in Peña and Maravall (1991)
Define the dual process of an invertible ARIMA model as the
ARMA process
𝜃 𝐵 𝑧𝑡 = 𝜙(𝐵)∇dat
that is, the dual process is built by interchanging the role of the
AR and MA operators
Then the autocorrelation function of the dual process is the
inverse of the autocorrelation function of the original process.
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MISSING VALUE ESTIMATION
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Estimation of missing values
(1) Performed a first interpolation of the missing values,
identify the ARIMA model and estimate its parameters by
ML in the completed series
(2) obtain the inverse autocorrelation coefficient s and
compute the optimal interpolators of the missing values.
This procedure can be iterated
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CHAPTER 4. OUTLIERS, INFLUENTIAL AND
MISSING.
4.6. Forecasting with outliers.
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FORECASTING WITH OUTLIERS
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Usual measures of forecast uncertainty take into account two
sources of variability
1) Presence of noise in the model
2) parameters are unknown and must be estimated
However, no attention is given to model uncertainty: the
structure of the model is unknown. This is the most
important source of uncertainty in most cases.
Models with changes in regimes (Unit 6)…