SCHOOL OF STATISTICS
UNIVERSITY OF THE PHILIPPINES DILIMAN
WORKING PAPER SERIES
Robust Estimation of A Time Series Model
with Structural Change
by
Wendell Q. Campano
and
Erniel B. Barrios (correspondence)
UPSS Working Paper No. 2009-10
July 2009
School of Statistics
Ramon Magsaysay Avenue
U.P. Diliman, Quezon City
Telefax: 928-08-81
Email: updstat@yahoo.com
1
Robust Estimation of A Time Series Model with Structural Change
Wendell Q. Campano
University of the Philippines Diliman
Magsaysay Ave., Diliman, Quezon City Philiipines
E-mail: wqcampano@up.edu.ph
Erniel B. Barrios (correspondence)
University of the Philippines Diliman
Magsaysay Ave., Diliman, Quezon City Philiipines
E-mail: ebbarrios@up.edu.ph
Tel. No.:(63 2) 9280881
Abstract
A Procedure for estimating a time series models with structural change is proposed.
Nonparametric bootstrap (block bootstrap or AR Sieve) is applied to a series of estimates
obtained through a modified forward search algorithm. The forward search algorithm is
implemented with overlapping and independent blocks of time points.
The procedure can
mitigate the difficulty in estimation when there is a temporary structural change. The simulation
study indicated robustness of estimates from the estimation method when temporary structural
change is introduced into the model provided that the time series is fairly long.
We also
provided a procedure for detecting structural change and the subsequent adjustment of the
overall model if indeed, there is a structural change.
Keywords: nonparametric bootstrap, ARIMA model, structural change, forward search
AMS Classification Codes: 62G05, 62G09, 62G35
1. Introduction
Advances in computing facilities and methods have profound impact on the way we
analyze and investigate time series data. Implementation of complex, iterative, nonparametric
procedures and simulations are now simpler. Detection of outliers and structural breaks in time
series, mostly iterative in nature, has become an integral part of model diagnostics (Tsay, 2000).
2
The occurrence of unusual shocks can result to time series outliers or if prolonged a little longer
can create temporary structural break or sometimes permanent structural change. This paper
explores two such computing-intensive methods, the forward search and the nonparametric
bootstrap methods in time series analysis that is influenced by random disturbances that
temporarily alters the model behavior.
The forward search (FS) is a powerful algorithm introduced initially for detecting
atypical observations and their effects on models fitted to a data. Atkinson and Riani (2000)
noted that this method was originally developed for models that assumed independent
observations in various modeling frameworks, e.g., linear and nonlinear regression, generalized
linear models, multivariate analysis, among others. The FS has been effective in detecting
aberrant observations and hidden data structure even in the presence of masked effect due to the
contamination caused by unusual observations. The method is also successful in detecting
clustered observations for both continuous and categorical data (Cerioli et al, 2007).
The forward search starts by fitting a model from an initial subset of observations
considered outlier-free, progressing with the search by adding an observation or a set of
observations according to their ‘closeness’ to the fitted model measured by some similarity or
distance measure, e.g., residuals. Thus, the forward search is made up of three phases: choosing
an initial subset, progressing (forward) in the search, and diagnostic monitoring, (Riani, 2004).
Riani (2004) extended the forward search in time series data still focusing on outlier
detection. The initial subset is robustly chosen among k blocks of contiguous observations of
fixed dimension b. The idea of block sampling is to conserve the dependence structure of the
3
observations in the series. Then progressing with the search by moving to higher dimension, say
b+1, using the least squared standardized prediction residual, prediction continue until the
highest possible dimension is obtained. This paper modified further the forward search to
account the dependence structure in time series data. Instead of searching by moving to higher
dimension, the search is done using blocks of the same length. The main idea of maintaining the
length of the blocks is to isolate certain perturbations present in some segments in the series.
This is subsequently expected to reveal the underlying behavior of the time series. The forward
search algorithm will be employed to independent (non-overlapping) and overlapping blocks of
data. Independent blocks can address longer time series data while overlapping blocks can
address the problems usually associated with short time series data.
The focus of this study is to develop a procedure that will produce a robust estimate of
model parameters in the presence structural change. These structural perturbations can be
considered as contiguous outliers or persistent shocks because these observations exhibit some
structure of their own different from the bulk of the data. Thus, presence of these “irregular”
segments may result in an inadequate or biased time series model (Chen and Liu, 1993).
2. Estimation with Data Perturbation
Data perturbations such as structural changes and other outlying observations are
common in time series. These perturbations are sets of observations which are in some way
different from the bulk of the data and may have a structure of their own (Konis and Laurini,
2007). Tsay (1986) noted that these aberrant observations or structures could seriously affect the
statistics calculated from the data such as the autocorrelation functions: SACF, SPACF, and
4
ESACF in the ARMA model. Hence, it is necessary to correctly or at least robustly estimate a
model in the presence of these observations.
The general procedure to deal with these atypical segments in the series is to first identify
where occurrence of these segments are, then measure the effect of these observations to the
specified model. Tsay (1986) proposed an iterative procedure to model time series in the
presence of outliers using a linear regression technique. Two classes of single outlier models
were considered: innovational outlier (IO) and additive outlier (AO). Once the outlier is detected,
an iterative method to specify tentative models for the outlier-contaminated series will be used
using the ESACF, then remove the effect of the atypical observations from the model.
2.3
The Bootstrap Method
Nonparametric bootstrap is a computing-intensive method that involves repeated
“resampling” technique in order to generate an empirical estimate of the distribution of certain
statistics (Mooney and Duval, 1993). The generated empirical distribution function (EDF) F̂ is
then used to estimate the unknown cumulative distribution function (CDF) F and then use F̂
just as we would a parametric model (Davison and Hinkley, 1997). Mooney and Duval (1993)
noted that both bootstrap and parametric inference have the same underlying purpose: making an
inference about  using a statistic ˆ . The only difference is how they obtain the sampling
distribution of ˆ . Parametric inference makes distributional assumptions on the distribution of
ˆ while nonparametric bootstrapping involves finding first this distribution of ˆ before making
any inferences. The basic steps in the nonparametric bootstrap procedure, as described by
Mooney and Duval (1993), are as follows:
5
1. Construct an empirical probability distribution Fˆ ( x) , from the sample by placing a
probability of 1
n
at each point, x1, ..., x n . This is the EDF of x , which is a
nonparametric maximum likelihood estimate (MLE) of the population distribution
function, F ( X ) .
2. From the EDF, Fˆ ( x) , draw a simple random sample of size n with replacement. This
*
is a “resample”, xb .
3. Calculate the statistic of interest, ˆ , from this resample, yielding ˆb* .
4. Repeat steps 2 and 3 B times, where B is a large number.
5. Construct a probability distribution from B ˆb* ’s by placing a probability of 1/B at
each point, ˆ1* , ˆ2* ,...,ˆB* . This distribution is the bootstrapped estimate of the
 
sampling distribution of ˆ, Fˆ * ˆ * .
We used the percentile method of constructing the confidence interval since it does not
require parametric assumption that is otherwise needed in the normal approximation and BC
method.
 

The percentile method takes literally the notion that Fˆ * ˆ * approximates F ˆ . An  level confidence interval includes all the values of ˆ * between the  / 2 and 1   / 2 percentiles
 
of the Fˆ * ˆ * distribution (Efron, 1982 and Stine, 1990 as cited by Mooney and Duval, 1993).
DiCiccio and Romano (1988) noted that none of these bootstrap confidence interval methods
offers the best confidence intervals for the general case since the criteria for judging the quality
of their results vary widely.
6
2.4
Bootstraps for Time Series
Davison and Hinkley (1997) discussed two approaches to resampling in time series: the
model-based resampling and block resampling. In the model-based resampling, the idea is to fit a
suitable model to the data, followed by the computation of residuals from the fitted model, and
then to generate new series by incorporating random samples from the residuals into the fitted
model. Typically, the residuals are centered to have the same mean as the innovations of the
model. This type of resampling is based on applying model equation(s) of the series to
innovations resampled from the residuals. To illustrate, suppose an AR(1) model
Yt  Yt 1   t , t  Z ,   1
(1)
is fitted to the realizations Y1 ,..., Yn giving estimated AR coefficient ˆ and innovations
ˆt  Yt  ˆYt 1 , t  2,..., n.
(2)
Note that ˆ1 is unobtainable because Y0 is unknown. Model-based resampling then proceed by
random sampling with replacement from centered residuals ˆ2   ,..., ˆn   to obtain simulated
innovations  0* ,...,  n* , and then setting Y0*   0* and
Yt *  ˆYt *1   t* , t  1,..., n.
(3)
The major drawback with model-based resampling is that the parameters of a model and its
structure must be identified from the data. If the chosen structure is not appropriate, the
resampled series will be generated from a wrong model, and hence they will not have the same
statistical properties as the original data.
The second approach is the block resampling that involves resampling of blocks of
consecutive observations. The block bootstrap tries to mimic the behavior of an estimator by
independent and identically distributed resampling of blocks of consecutive observations. The
7
simplest version of this approach is to divide the data into b non-overlapping blocks of the same
length l (e.g., n  bl ). The procedure is to take a bootstrap sample from the blocks with equal
probabilities, and then paste the selected blocks end-to-end to form a new series. The idea is to
preserve the original time series structure within a block. If the blocks are long enough, enough
of the original dependence structure will be preserved in the resampled series. This method will
work best if the dependence is weak and the blocks are as long as possible. However, using the
block bootstrap, the dependence between the blocks is neglected and the bootstrap sample is not
(conditionally) stationary and exhibit artifacts which are caused by linking the randomly selected
blocks (Bühlmann, 1997). Bühlmann (1997) proposed using the sieve bootstrap instead to
address these issues.
Sieve bootstrap is done by fitting parametric models first, using for instance the Akaike
information criterion, and then resampling is performed over the residuals. An infinitedimensional nonparametric model is approximated by a sequence of finite-dimensional
parametric models, a strategy known as the method of sieves. Specifically, the true underlying
stationary process is approximated by an autoregressive model of order p [AR(p)], where
p  p(n) is a function of the sample size with p(n)   , p(n)  o(n) (n  ) . The sieve
bootstrap relies heavily on the crucial assumption that the data X 1 ,..., X n is a finite realization of
an AR() process


j
( X t  j   )   t , 0  1,
(4)
j 0

with

2
j
  . The AR() representation (4) includes the important class of ARMA(p,q)
j0
models
8
p
q
X t    j X t  j    k  t k   t , t  Z ,
j 1
(5)
k 1
q
with invertible generating MA-poynomial, i.e., ( z)  1   k z k , z  C has its roots outside
k 1
the unit disk z  C ; z  1(Bühlmann, 2002). The definition of the sieve bootstrap, provided in
Bühlmann (2002), is given below.
Let X t , t  Z a real-valued stationary process with E[ X t ]   . Represent X t  as a one-sided
infinite order AR process as in (4) and denote by X 1 ,..., X n a sample from the process X t ,
tZ .
1. Fit an autoregressive process, with increasing order p(n) as the sample size n
increases and estimate the coefficients ˆ1, n ,...,ˆp ,n corresponding to model (4). Note
that this procedure yields residuals
p(n)
ˆt , n 
 ˆ
j ,n
( X t  j  X ),  0, n  1 (t  p  1,..., n) .
(6)
j 0
2. Construct the resampling based on this autoregressive approximation. Center the
residuals
~t , n  ˆt ,n  (n  p) 1
n
 ˆ
t ,n
(t  p  1,...n)
(7)
t  p 1
n
and denote the empirical cumulative distribution function of ~t , n t  p 1 by
Fˆ ,n (.)  (n  p ) 1 1[~t ,n .].
(8)
3. Resample for any t
 t* i..i..d . ~ Fˆ , n
(9)
9
 
4. Define X t
*
t Z
by the recursion
p ( n)
 ˆ
j,n
( X t* j  X )   t* .
(10)
j 0
In practice, the sieve bootstrap sample X 1* ,..., X n* , as described in Bühlmann (2002), are
constructed in the following way:
1.
Choose starting values, e.g., equal to zero.
2. Generate an AR( p(n) ) process according to (10) until ‘stationarity’ is reached and
then discard the first p(n) generated values.
3. Consider any statistic Tn  Tn ( X 1 ,..., X n ) , where Tn is a measurable function of n
observations. The bootstrapped statistic Tn* is defined as Tn*  Tn ( X 1* ,..., X n* ) .
The sieve bootstrap yields a (conditionally) stationary bootstrap sample and does not
exhibit artifacts in the dependence structure like in the block bootstrap where the dependence
between blocks is neglected. The method does not require ‘pre-vectorizing’ the original
observations. Also, the sieve bootstrap sample is not a subset of the original sample. It has been
shown that this method outperforms the more general block bootstrap within the class of linear
invertible time series. Details in the comparison between sieve bootstrap and other time series
bootstrap methods are in Bühlmann (2002).
3. Statistical Inference in the Presence of Temporary Structural Change
Statistical inference for models given data containing structural change can be done in
two phases. In Phase I, a robust model estimate reflecting the underlying overall behavior of the
10
series will be obtained. Without loss of generality, assume that the time series y t  t  1,..., n. is
stationary. Suppose y t follow an autoregressive moving average [ARMA(p, q)] model,
 ( B) yt   ( B)at
(11)
Where  ( B )  1  1B  ...   p B p and  ( B )  1  1B  ...   q B q are polynomials in B , B is the
backshift operator, and a t  is a white noise process. Suppose further there are k contiguous
segments each of dimension b, denoted by s (i ) , time segments i  1,..., k , defined from the series.
Note
that
for
independent
segments,
s (1)  y1 ,..., y b , s ( 2)  y b 1 ,..., y 2b ,..., s ( k )  y ( k 1)b 1 ,..., y kb . For overlapping segments, for
example first b years then adding and deleting one time point at a time, the segments are formed
as: s (1)  y1 ,..., y b , s ( 2)  y 2 ,..., y b 1 ,..., s ( k )  y k ,..., y b k 1  .
3.1
Phase I: The Estimation Procedure
We describe a method of estimating the overall behavior of the time series contaminated
with temporary structural change. The procedure consists of implementing a modified forward
search algorithm and a nonparametric bootstrap method. We initially present the steps necessary
regardless of the nature of the blocks (overlapping or independent). Then later, specific steps
needed specifically for the nature of the blocks will be presented.
Modified Forward Search
Step 1: For each of the k segments, fit model equation (11) and obtain the estimates of each
parameter in the equation. The following matrices of estimates are obtained:
11
ˆ11 ˆ21
ˆ

ˆ22
ˆ
 FS   12
 ... ...

ˆ
ˆ
1k  2k
... ˆp1 

... ˆp 2 
... ... 

... ˆpk 
and
ˆFS
ˆ11
ˆ

  12
 ...

ˆ
 1k
ˆ21
ˆ22
...
ˆ
 2k
... ˆq1 

... ˆq 2 
... ... 

... ˆqk 
The estimates from Step 1 are called the series of “forward searched” parameter estimates
of equation (11), a total of (p + q) set of series. When the segments are not independent
(overlapping), each series of forward search estimates represents the estimates obtained using
AR-Sieve.
The goal for the modification of the forward search algorithm in this procedure is to
reveal the overall picture of the time series by partitioning the series into blocks of constant
length. Any temporary structural change is therefore localized in specific segments only and will
not necessarily contaminate the entire time series.
The purpose of the FS for time series
discussed by Riani (2004) is to order the observations according to their closeness with the fitted
model and hence focuses in outlier detection.
Nonparametric Bootstrap Estimation
The estimation routine is continued given the estimates from Step 1 above. This time, an
AR Sieve bootstrap is used to revise the estimates from the modified forward search algorithm.
Step 2: Using nonparametric bootstrap, estimate each parameters in equation (11) based on the
series of forward search estimates obtained in Step 1. That is, the bootstrap estimate of  i
12
in (11) denoted by ˆi BS using the series ˆi FS
ˆi1 
ˆ 

  i2  ,
 ... 
 
ˆik 
i  1,..., p and the bootstrap
estimate of  j in (11) denoted by ˆ j BS using the series ˆ j FS
ˆ j1 
ˆ 

  j 2  , j  1,..., q . Also,
 ... 
 
ˆ jk 
compute the Monte Carlo Variance and a (1   )100% bootstrap confidence interval
(BCI) for each parameter in equation (1).
When the segments are independent blocks,
block bootstrap is used. On the other hand, if the segments are overlapping, AR Sieve is
used.
When data perturbations or structural change are localized in certain blocks of
observations, the forward search part should be able to identify the underlying model structure
from the majority of the blocks. Unlike in independent data case where sample size increases
during the forward search algorithm, block size is maintained to be constant and the search is
applied on the blocks. If the block size is progressively increased considering the premise that
structural change occurred, erratic parameter estimates from an ARIMA model can be expected.
It is advantageous to maintain a constant block size so that specific segments where structural
change occurred can be isolated from the rest of the segments. Hence, robustness can be
achieved. The application of the nonparametric bootstrap will aid in filtering out the effect of
structural change in certain localized segments. Thus, a temporary structural change should not
prominently influence the “majority” of the blocks, and the bootstrap can come up with a more
stable version of the series of forward searched estimates.
13
Bootstrap Procedure for Independent Blocks
The bootstrap procedure for the independent blocks uses the series of estimates obtained
from the forward search. For each series of estimates obtained through FS, let  be the parameter
to be estimated.
(i)
Resample from the series of estimates obtained for each block.
(ii)
Compute the mean ˆ b  
(iii)
Repeat steps (i) and (ii) B (replication size) times, where B is large.
(iv)
Compute the bootstrap estimate ˆ BS 
ˆ BS 
1 n
ˆ k .
n k 1
1 B  j
 ˆ , Monte Carlo Variance
B j 1
1 B
(ˆ ( j )  ˆ BS ) 2 , and BCI.

B 1 j
Bootstrap Procedure for Overlapping Blocks (AR-Sieve)
For overlapping blocks, there is dependence between blocks and hence, forward search
estimates obtained from these blocks are also not independent. The nonparametric bootstrap
relies heavily on the assumption that the values being resampled are independent. Thus, in order
to apply nonparametric bootstrap to the series of forward search estimates from overlapping
blocks, the dependencies between the estimates should be removed. The AR-Sieve is
implemented to obtain a new series of forward search estimates that are indepedent. The ARSieve procedure follows:
For each block,
(i)
Estimate simultaneously the parameters in equation (11) and obtain the
residuals.
14
(ii)
Generate new residual series a t with mean 0 and variance equal to the
mean square error (MSE) of the residuals in (i).
(iii)
Choose starting values, e.g., equal to zero, and generate new series y t
based on estimates in (i) and the new residual series a t in (ii).
(iv)
Estimate the model equation using the generated series y t in (iii) and store
the parameter estimates.
(v)
Repeat steps (iii) and (iv) B times, where B is a large number.
(vi)
Compute the mean of each parameter from the B models estimated. This is
now the forward search estimate for the block.
After exhausting all the blocks, nonparametric bootstrap procedure used for
independent blocks will be applied to the forward search estimates obtained using ARSieve.
3.2
Phase II: A Procedure on Identifying and Modeling Structural Change
In this section, a procedure of how to identify and model the segments with structural
change is presented. The procedure uses a BCI for detecting segments with perturbation. Once
these segments are identified, the estimated model in Phase I is then adjusted so that the resulting
model represents the underlying structure as well as the structural changes in the series. The
basic steps of this procedure follow:
Step 1: Using the BCI for i obtained in Step 2 of Section 3.1, compare the individual
elements ˆil , l  1,..., k in the vector ˆi FS
ˆi1 
ˆ 

  i 2  with the BCI. Identify the segments
 ... 
 
ˆ
ik 
15
where the ˆil , l  1,..., k are outside the BCI and for each segment, compute the
difference im  BCIl  ˆim , where m  the segment where ˆim  BCI for ˆi and BCI l is
the BCI limit nearest to ˆim . The idea here is that a segment with estimate(s) outside the
BCI indicates presence of perturbation in that particular segment.
Thus, this step
suggests the use of BCI as a method for detecting structural changes in the time series.
~
Step 2: The adjusted estimate of ˆi , denoted by i is, ˆi BS   im I t (tm ) where I t (t m )  1 if t
m
belongs to time segment m and 0, otherwise.
Apply this procedure for all the parameters in equation (11) to generate adjusted parameter
estimates:
~
i  ˆi BS  im I t (tm ) for i , i  1,..., p
m
and
~
 j  ˆ j BS    jm I t (t m ) for  j , j  1,..., q
m
These parameter estimates yields the adjusted model estimates that represents the general
structure of the time series and the perturbations in the series.
4
Simulation Studies
The proposed procedure was illustrated and evaluated through a simulation study.
Simulated data were generated from AR(1), MA(1), and ARMA(1,1) processes with embedded
structural change at the following locations in the series: start; middle; end; start and middle;
start and end; middle and end; start, middle, and end.
Embedding of structural change was done by simply replacing the selected segments
(start, middle, or end) of the uncontaminated series with observations generated from a model of
the supposed structural change. For example, consider a simulated series X t , t  1,...,100 and
16
suppose a 5-timepoint structural change will be embedded at the start and end of the series. To
*
*
incorporate the perturbation, generate X 1* ,..., X 5*  and X 96
,..., X 100
 using the structural change
model, then replace
respectively.
X
*
1
X 1 ,..., X 5 
Thus,
the
and
new
X 96 ,..., X 100 
series
with
treated
X
*
1
,..., X 5*
as
the

*
*
and X 96
,..., X 100
,
realization
is

*
*
,..., X 5* , X 6 ,..., X 95 , X 96
,..., X 100
.
Short and long series were considered in the evaluation. For the long series, different
lengths of embedded perturbations were used to monitor the effect of these occasional
perturbations to the model parameter estimates. A comparison between bootstrap estimates using
moving blocks and independent blocks was made. The forward search was implemented to the
following segmentations:
(i) Long Series (130 years of monthly data)
a. 10-year independent data segments
b. first 10 years, then adding and deleting 5 years at a time
c. first 10 years, then adding and deleting one year at a time
(ii) Short Series (5 years of monthly data)
a. 1-year independent data segments
b. first year, then adding and deleting 1 month at a time
Thus, two cases of overlapping segments were compared. One is moving segment by
relatively short period and the other is moving by relatively long block. A span of 1 year, 5
years, and 10 years of structural perturbation were embedded in the long series while a span of 3
months were considered in the short series.
17
The evaluation of the proposed procedure involved examining the robustness of model
estimates vis-à-vis conditional least squares under following scenarios:
(i)
near nonstationarity and near noninvertibility
(ii)
location and length of perturbation
(iii)
length of the series
(iv)
bootstrap method
A model is considered robust if the model parameter estimates are sensitive to the overall
memory pattern of a time series but are insensitive to occasional outliers (Chen and Liu, 1993).
Robustness means that the parameter estimates should be close to the underlying overall model
parameters even in the presence of perturbations.
The long series consists of 1560 values (130 years of monthly data or less than 5 years of
daily data) and 60 values (5 years of monthly data or approximately 3 months of daily data) for
the short series. The models where data were simulated are summarized in Table 1.
[Table 1 Here]
All the processes are stationary and invertible. However, investigations were also done near the
boundaries of nonstationarity and noninvertibility.
Table 2 shows the structural change
embedded in the series generated from each model in Table 1. After embedding the said
perturbations, the (conditional) least squares estimates (CLS) already showed significant
deviations from the true model parameters even with just few contaminations, illustrating the
data-sensitive character of ARIMA models.
[Table 2 Here]
18
4.1
Parameters Estimates
Tables 3 summarizes
the
parameter
estimates
of
the
AR(1)
models
(1  0.5 B)(Yt  10)  a t (stationary) and (1  0.95B )(Yt  10)  a t (near nonstationary) each with
temporary structural change at the start, middle and end of the series. On the long stationary
series, the bootstrap estimates (BS1, BS2, and BS3) of  were much closer to the true value than
the CLS estimates. The estimates of the mean  , did not differ much for the two procedures.
However, for the near nonstationary long series, almost all the CLS estimates of mean and of 
were nearer the true values. This is so because slicing the time series into blocks can highlight
the near nonstationary behaviors or even result to nonstationarity within each block. For the case
of the short series, the estimates from the proposed procedure were evidently not reliable because
of high absolute percent differences (Table 4). The bootstrap estimate of  was off by large
amount as explained by the fact that the proposed algorithm involved segmenting the time series
into blocks. With short time series, the overall dependence structure is hardly passed on to the
individual blocks of even shorter length. Thus, with the introduction of perturbations, the
behavior of each block produced from a short time series can generally produce erratic picture of
the dependence structure of the time series.
[Table 3 Here]
[Table 4 Here]
Similar results were obtained from MA(1) models. On the invertible long series
[ (Yt  10)  (1  0.5B)a t ] , the estimates of  from the new procedure yield smaller (absolute)
percent differences compared to those obtained from CLS, even with many points of
contamination. The estimates using the proposed procedure and the CLS were similar for the
near noninvertible long series [ (Yt  10)  (1  0.95B)a t ], see Table 5 for details. The estimates
19
were relatively close to the true value for both procedures. Like in the case of AR(1), the
proposed procedure did not work well for short series (Table 6 )
[Table 5 Here]
[Table 6 Here]
The parameter estimates in Table 7 proved further the advantage of the proposed
procedure with stationary and invertible ARMA(1,1) [ (1  0.4 B)(Yt  10)  (1  0.5)a t ]. The
estimates of the mean,  and  were closer to the true values compared to the CLS estimates.
However,
in the case of near nonstationary and near
noninvertible long series
[ (1  0.95 B)(Yt  10)  (1  0.95)at ], the CLS method performed better, see Table 7 for details.
Like in the case of AR(1) and MA(1), the estimates for the short ARMA(1,1) series were also
not reliable (Table 8).
[Table 7 Here]
[Table 8 Here]
4.2
Effect of Near Nonstationarity and Near Noninvertibility
In the case of AR(1), being near nonstationarity caused the estimates of the mean to be
unstable using the proposed procedure. The estimates of  however, were still robust (Table 3).
Near nonstationarity has greater effect on the estimates of the mean than of  for AR(1) models,
both for long and short time series. Majority of the blocks of near nonstationary series can not
represent the overall picture of the time series and hence, the bootstrap estimates failed to capture
the underlying structure.
For the MA(1) models shown in Tables 5 and 6, the degree of invertibility does not
really affect the estimates produced using the proposed procedure. Even if the series is near
20
noninvertible, the estimates are quite robust. The independent block method is capable further of
filtering the perturbations localized in each segment, that could have blended into the
noninvertibility character of MA(1).
The results for the ARMA(1,1) presented in Table 7
showed that near nonstationary
and almost noninvertible series produced unstable estimates of the mean with the proposed
procedure. However, the estimates of  and  remain robust.
Near nonstationarity of the time series can greatly influence the optimality of ARIMA
modeling which relies heavily on the dependence structure of the data. Blocks defined from near
nonstationary series can highlight more the nonstationary behavior of the time series. Some, if
not majority of the blocks can even exhibit nonstationarity. Thus, bootstrapping estimates from
these blocks will not give optimal results. Stationarity implies that any time series segment of the
same length should exhibit similar dependence structure regardless of the section of the time
series it is extracted from. The proposed method benefits from this since the algorithm explicitly
partitions the time series into blocks of constant length and estimates of the model parameters are
generated from each block. Furthermore, any temporary structural change occurring in the time
series can be localized only in some blocks. Other blocks that are not affected by these
perturbations should be able to clearly characterize the dependence structure of the time series.
4.3
Effect of Location and Length of Structural Change
The location and length of the perturbation significantly affected estimates obtained from
CLS. Longer length of contamination can produce greater deviation of the estimated model from
the true model and occasional perturbations can introduce bias in model-fitting. This was not the
21
case for the estimates obtained using the proposed procedure, the parameter estimates were
relatively closer to the true model even with longer contamination length compared to the CLS
estimates. This confirms the robustness of the estimates obtained from the blended resamplingforward search estimates. This is true only for stationary and relatively longer time series.
Moreover, the location of the contamination did not really matter when the proposed estimation
procedure is used. The estimates obtained were robust whether the contaminations were at the
start, middle, end, or a combination.
4.4
Overlapping Blocks vs. Independent Blocks
The bootstrap procedure using overlapping and independent segments (BS1, BS2, and
BS3) produced similar estimates for the long series, in general. Results from overlapping
segments moving by short blocks or by relatively long blocks were also similar. However, note
that there seemed to be an advantage of using non-overlapping segments against overlapping
segments
when
the
series
is
near
nonstationary
[ (1  0.95B )(Yt  10)  a t
and
(1  0.95B)(Yt  10)  (1  0.95)a t ]. The estimates produced by bootstrapping independent blocks
(BS3), especially of the mean, were more robust. There are only few blocks that can be formed if
they are non-overlapping compared to overlapping one. Thus, logically, if the series is near
nonstationary, fewer blocks can emphasize the near nonstationarity (or even nonstationarity)
condition of the complete series.
4.5
Comparison between CLS and the Proposed Method
The estimates obtained using the proposed procedure were more robust compared to the
least square estimates that uses the entire time series at once when the series was long and
stationary/invertible. The new procedure was able to produce better estimates than CLS in terms
22
of sensitivity to the overall structure and insensitivity to occasional structural changes provided
that the series is long and stationary/invertible. Simulation results for the series from AR(1),
MA(1), and ARMA(1,1) showed that CLS estimates were more sensitive to structural changes
in time series compared to bootstrap estimates.
In the case of short time series data, the implementation of the proposed procedure did
not yield desirable results. The estimates obtained were neither robust nor stable, percent
difference of estimates from true values and the standard errors were relatively large. CLS
estimates were better than the bootstrap estimates for short series. The CLS took advantage of
using the whole time series in estimating an ARIMA model at once. The poor estimates obtained
using the proposed procedure can be attributed to the very small number of observations
involved in the estimation per block. The dependence structure in a block can be entirely
different from the global dependence structure estimated in an ARIMA model. Using few data
points yield poor estimates in terms of robustness and stability, in general. Another reason is that,
altering or contaminating a short series can completely destroy the general structure of the data
unlike in the case of the long series where the general structure is preserved.
Forecast accuracy measured by MAPE was also analyzed. Both estimation procedures
yield comparable results. Despite the robustness of the bootstrap estimates over the CLS
estimates, there was no significant gain in forecasting performance. In fact, there were many
cases wherein the MAPE of the CLS models were lower than those of the proposed procedure.
One reason for this is the insensitivity of the bootstrap estimates to the perturbations introduced
in the series. The bootstrap estimates can be highly penalized in the segments where structural
change occurred. Forecast based on CLS models outperformed those coming from the models
23
estimated using the proposed procedure in the segments with structural change. However, in the
segments without structural change, the proposed procedure outperformed the CLS, especially
for the stationary (and invertible) long time series.
4.6
Identifying and Modeling Structural Change
In this section, the Phase II procedure is illustrated using the simulated series with five
years perturbation. A 95% BCI for the mean is (9.96, 10.50) and a 95% BCI for  is (0.46,
0.68). The Forward Search estimates are presented in Table 9 below.
[Table 9 Here]
This leads to the following adjusted estimates of the model parameters:
For  : 10.20 + 1.121I (1,...,120 ) (t ) - 0.03 I ( 241,..., 360 ) (t ) - 0.24 I ( 361,..., 480 ) (t )
- 0.38 I ( 721,..., 840 ) (t ) - 0.13 I (841,..., 960 ) (t ) + 0.51 I (1441,...,1560 ) (t )
For  : 0.57 + 0.131I (1,...,120 ) (t ) - 0.11 I ( 241,..., 360 ) (t ) - 0.05 I ( 481,..., 600 ) (t ) - 0.08 I ( 601,..., 720 ) (t )
+ 0.28 I ( 721,..., 840 ) (t ) - 0.06 I (1201,...,1320 ) (t ) + 0.25 I (1441,...,1560 ) (t )
Note that the true structural change occurred at time points 1-120, 721-840, and 1441-1560. The
procedure did detect the occurrence of the perturbation. However, there were also points that
were misclassified. While the method is highly sensitive to detect structural change, it is not
specific because of the confounding effect of the temporary shocks on the general behavior of
the time series. Even after the implementation of the procedure, there was still a need to
subjectively classify whether there was an occurrence of structural change in the series or none.
For example, in Table 9, if only the top 3 highest distance from BCI were considered, then the
method had correctly identified the contaminations in the series.
24
Tables 10-12 present a closer examination of the sensitivity of the BCI in identifying
structural changes in the selected series from the simulation study. Recall that when the FS
estimate(s) for a particular segment is outside a BCI, it was proposed that structural change
should have occurred in that segment. The method of detecting structural change using a BCI is
highly sensitive to the structural change induced in the simulated data. With 95% coverage
probability, all the segments were considered as having structural change. Note that only a small
fraction of the series was embedded with perturbations but the method indicated otherwise.
Relaxing the coverage probability to 99%, BCI can reduce the number of misclassified segments;
however, the overall sensitivity is still very high. The extreme sensitivity of the method is a
consequence of the high level of accuracy exhibited by bootstrap confidence intervals, i.e., the
bootstrap confidence intervals have narrow width. Thus, even estimates from segments without
structural change are very vulnerable of falling outside a BCI.
[Table 10 Here]
[Table 11 Here]
[Table 12 Here]
5. Conclusions
The simulation study confirmed the bias in estimating parameters of a time series model
when different perturbations are introduced to the parameters. The perturbations resemble
temporary structural change in the model that will revert back to the original behavior after a
short period. These contaminations can conceal the underlying general structure of the time
series and may result to poor estimation. Bootstrapping of estimates obtained through forward
search algorithm is proven to reveal the underlying model.
25
The estimation procedure yields robust estimates of ARIMA models with temporary
structural change generated from AR(1), MA(1), and ARMA(1,1) processes. The estimates
obtained from the nonparametric bootstrap of a series of estimates from the modified forward
search algorithm are superior over the estimates obtained using the conditional least squares in
terms of robustness and capturing the overall structure of the data provided that the time series is
relatively long, stationary, and invertible. The length of the times series is an important
prerequisite for the optimality of the proposed method because of the segmentation into blocks of
equal length. The global pattern of the time series is easily passed on to the local behavior of the
blocks when the time series is longer. Stationarity is also important since it implies that the
overall state of dependence structure is preserved within each block. High-frequency data that is
vulnerable to temporary shocks and short-term structural changes can benefit from the proposed
estimation procedure. If the interest is to unveil the underlying model, free from any temporary
shocks, then a forward search-nonparametric bootstrap algorithm proposed in this paper should
be able to produce robust and stable estimates of parameters of such models.
The proposed procedure performed poorly when the series is relatively short. Segmenting
the short time series into blocks will tend to aggregate the bias brought about by the temporary
structural change. Because of the data-sensitivity of ARIMA modeling procedures, short time
series cut into even shorter blocks will become highly vulnerable to any data point that deviates
from the underlying model. Similar arguments can be made on the poor performance of the
procedure on near nonstationary models. If the time series (whether short or long) is cut into
shorter blocks, near nonstationarity can further be highlighted in each block. When there is
already near nonstationary behavior in the global scenario, the localized nonstationarity can be
easily observed. Thus, estimation of an ARIMA model will suffer at the block level.
26
The forecast accuracy of the new procedure is comparable to the CLS. The models
obtained using the new procedure can outperform the CLS models in the uncontaminated
segments. Thus, if the interest is forecasting future values that would represent the true
underlying model, then the proposed procedure can provide more accurate results.
References
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27
Tsay, R. S. (1986). Time Series Model Specification in the Presence of Outliers. J. of the
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American Stat. Assoc. 95:638-643.
Table 1
Models Considered in the Simulation.
Process
AR(1)
MA(1)
Model
Nature of the Process
(1  0.5 B)(Yt  10)  a t
Stationary
(1  0.95B )(Yt  10)  a t
Near Nonstationary
(Yt  10)  (1  0.5B)a t
Invertible
(Yt  10)  (1  0.95B)a t
Near Noninvertible
28
ARMA(1,1)
(1  0.4 B)(Yt  10)  (1  0.5 B)a t
(1  0.95B )(Yt  10)  (1  0.95B)a t
Stationary and Invertible
Near Nonstationary and Near
Noninvertible
Note: a t ~ N (0,1) for all the models.
Table 2
Structural Change Models
Process
Structural Change Model
AR(1)
(1  0.8 B)(Yt  13)  a t
MA(1)
(Yt  13)  (1  0.8B )a t
ARMA(1,1)
(1  0.8 B)(Yt  13)  (1  0.8 B)a t
Note: a t ~ N (0,1) for all the models.
Table 3
Parameter Estimates for the Long AR(1) Series With Temporary Structural Change (S.C) at the
Start, Middle, and End of the Series.
(1  0.5 B)(Yt  10)  a t
Method
mean
(s.e.)
%
diff
phi1
(s.e.)
% diff
(1  0.95B)(Yt  10)  a t
MAPE
MSE
20.59
8.01
1.01
0.44
8.05
1.03
3.16
8.05
1.02
mean
(s.e.)
% diff
phi1
(s.e.)
% diff
MAPE
MSE
0.03
9.84
0.99
2.84
10.21
1.24
3.26
9.98
1.11
(a) 1 year temporary structural change
CLS
BS1
BS2
10.16
(0.064)
10.10
(0.017)
10.14
(0.041)
1.59
0.98
1.37
0.60
(0.02)
0.50
(0.008)
0.52
(0.018)
10.15
(0.452)
4.08
(3.012)
6.21
(3.353)
1.48
59.19
37.88
0.95
(0.008)
0.92
(0.006)
0.92
(0.014)
29
BS3
10.18
(0.057)
1.77
0.55
(0.03)
9.90
8.05
1.01
10.42
(0.861)
4.22
0.94
(0.017)
1.11
9.95
0.99
57.38
8.38
1.13
4.78
9.07
1.11
8.61
1.32
4.71
9.11
1.11
7.13
8.60
1.32
4.90
9.14
1.11
13.48
8.48
1.28
0.93
(0.01)
0.91
(0.006)
0.90
(0.011)
0.91
(0.016)
2.34
6.68
10.48
(0.35)
10.06
(0.216)
10.21
(0.395)
9.87
(0.614)
4.21
9.07
1.11
61.42
8.38
1.21
16.90
9.30
0.98
8.92
1.47
2.67
9.56
0.99
3.14
9.11
1.51
3.92
9.75
1.00
4.83
9.08
1.49
0.95
(0.008)
0.92
(0.006)
0.91
(0.015)
0.92
(0.025)
0.29
7.75
11.69
(0.471)
11.49
(0.301)
11.85
(0.888)
10.59
(1.195)
3.37
9.53
1.00
(b) 5 years temporary structural change
CLS
BS1
BS2
BS3
10.50
(0.126)
10.36
(0.093)
10.34
(0.188)
10.20
(0.145)
5.00
3.62
3.38
2.02
0.79
(0.016)
0.53
(0.017)
0.54
(0.04)
0.57
(0.054)
0.57
2.11
1.27
(c) 10 years temporary structural change
CLS
BS1
BS2
BS3
10.76
(0.143)
10.51
(0.101)
10.64
(0.223)
10.67
(0.36)
7.61
5.06
6.42
6.68
0.81
(0.015)
0.54
(0.018)
0.52
(0.037)
0.52
(0.043)
14.87
18.53
5.94
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 year
BS2 – overlapping segments moving a length of 5 years, BS3 – non-overlapping 10-year block
Table 4
Parameter Estimates for the Short AR(1) Series Simulated with Temporary Structural Change
(S.C) at the Start, Middle, and End of the Series.
(1  0.5 B)(Yt  10)  at
(1  0.95B )(Yt  10)  a t
Method
̂
(s.e.)
CLS
BS1
BS2
10.26
(0.239)
9.92
(0.217)
10.32
(0.206)
%
diff
2.60
0.83
3.22
ˆ
% diff
MAPE
MSE
(s.e.)
0.33
(0.124)
0.09
(0.027)
0.12
(0.19)
34.93
9.10
1.54
82.64
9.11
1.72
75.68
9.64
1.61
̂
(s.e.)
9.74
(0.842)
16.23
(2.12)
10.25
(0.559)
% diff
ˆ
% diff
MAPE
MSE
7.12
8.53
1.22
52.74
37.16
13.43
36.32
10.81
1.58
(s.e.)
2.62
62.32
2.49
0.88
(0.065)
0.45
(0.031)
0.60
(0.127)
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 month, BS2 – nonoverlapping 1-year block
30
Table 5
Parameter Estimates for the Long MA(1) Series Simulated
with Temporary Structural Change (S.C) at the Start, Middle, and End of the Series.
(Yt  10)  (1  0.5B)a t
(Yt  10)  (1  .95 B)a t
Method
mean
(s.e.)
%
diff
theta1
(s.e.)
% diff
MAPE
MSE
1.31
8.60
1.22
7.37
8.57
1.22
5.74
8.58
1.22
2.46
8.59
1.22
29.82
9.33
1.47
10.67
9.23
1.51
11.67
9.26
1.50
13.74
9.36
1.49
38.44
9.86
1.68
17.24
9.79
1.74
17.91
9.85
1.73
14.81
9.99
1.73
mean
(s.e.)
%
diff
theta1
(s.e.)
10.07
(0.052)
10.01
(0.019)
10.03
(0.037)
10.08
(0.045)
0.67
10.35
(0.055)
10.17
(0.047)
10.23
(0.111)
10.35
(0.179)
3.45
10.75
(0.059)
10.57
(0.083)
10.65
(0.215)
10.73
(0.361)
7.52
% diff
MAPE
MSE
-0.91
(0.011)
-0.88
(0.005)
-0.88
(0.012)
-0.89
(0.02)
4.60
8.62
1.17
7.27
8.63
1.17
7.68
8.65
1.17
6.80
8.67
1.17
-0.93
(0.01)
-0.88
(0.003)
-0.88
(0.007)
-0.91
(0.015)
2.43
8.73
1.27
7.35
8.72
1.29
7.11
8.74
1.28
4.54
8.77
1.27
-0.90
(0.011)
-0.87
(0.005)
-0.88
(0.012)
-0.90
(0.02)
5.64
9.31
1.51
7.99
9.23
1.52
7.55
9.27
1.51
5.30
9.29
1.51
(a) 1 year temporary structural change
CLS
BS1
BS2
BS3
10.13
(0.042)
10.10
(0.015)
10.12
(0.038)
10.13
(0.058)
1.32
1.00
1.20
1.31
-0.51
(0.022)
-0.46
(0.008)
-0.47
(0.018)
-0.49
(0.027)
0.11
0.35
0.81
(b) 5 years temporary structural change
CLS
BS1
BS2
BS3
10.25
(0.051)
10.10
(0.044)
10.15
(0.106)
10.25
(0.185)
2.53
1.00
1.47
2.49
-0.65
(0.019)
-0.55
(0.011)
-0.56
(0.028)
-0.57
(0.048)
1.68
2.25
3.46
(c) 10 years temporary structural change
CLS
BS1
BS2
BS3
10.70
(0.056)
10.49
(0.079)
10.58
(0.197)
10.69
(0.341)
6.98
4.93
5.80
6.91
-0.69
(0.018)
-0.59
(0.011)
-0.59
(0.024)
-0.57
(0.036)
5.71
6.51
7.30
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 year
BS2 – overlapping segments moving a length of 5 years, BS3 – non-overlapping 10-year block
Table 6
Parameter Estimates for the Short MA(1) Series Simulated with Temporary Structural Change
(S.C) at the Start, Middle, and End of the Series.
(Yt  10)  (1  0.5B)a t
(Yt  10)  (1  .95 B)a t
Method
̂
(s.e.)
CLS
BS1
BS2
10.36
(0.289)
10.04
(0.071)
10.35
(0.272)
%
diff
3.63
0.44
3.47
ˆ
% diff
MAPE
MSE
(s.e.)
-0.41
(0.122)
-0.25
(0.039)
-0.42
(0.163)
17.66
12.36
2.46
49.86
12.34
2.62
16.72
12.33
2.46
̂
(s.e.)
10.54
(0.268)
10.43
(0.082)
10.72
(0.317)
%
diff
5.44
4.26
7.18
ˆ
% diff
MAPE
MSE
41.08
10.07
1.75
42.74
10.02
1.76
29.07
10.01
1.80
(s.e.)
-0.56
(0.109)
-0.54
(0.03)
-0.67
(0.137)
31
Table 7
Parameter Estimates for the Long ARMA(1,1) Series Simulated with Temporary Structural
Change (S.C) at the Start, Middle, and End of the Series.
(1  0.4 B)(Yt  10)  (1  0.5)a t
Method
mean
(s.e.)
phi1
(s.e.)
% diff
% diff
theta1
(s.e.)
% diff
MAPE1
MAPE2
-0.52
(0.028)
-0.53
(0.01)
-0.54
(0.023)
-0.55
(0.039)
4.88
13.53
8.04
6.27
14.03
8.01
7.59
13.85
8.03
9.55
13.63
8.04
-0.41
(0.028)
-0.50
(0.01)
-0.49
(0.02)
-0.49
(0.026)
17.78
8.07
8.44
0.91
10.03
8.21
1.52
10.07
8.18
2.76
9.74
8.25
-0.40
(0.026)
-0.52
(0.012)
-0.55
(0.029)
-0.56
(0.045)
20.04
7.24
8.66
4.05
9.13
8.47
9.89
8.58
8.84
12.49
7.83
9.45
(a) 1 year temporary structural change
CLS
BS1
BS2
BS3
10.01
(0.067)
9.96
(0.017)
9.99
(0.044)
10.02
(0.068)
0.06
0.42
(0.03)
0.35
(0.012)
0.37
(0.023)
0.39
(0.039)
0.42
0.08
0.22
5.79
11.45
8.19
3.27
(b) 5 years temporary structural change
CLS
BS1
BS2
BS3
10.37
(0.109)
10.21
(0.069)
10.16
(0.099)
10.28
(0.172)
3.70
0.65
(0.023)
0.42
(0.017)
0.43
(0.035)
0.44
(0.052)
2.13
1.56
2.83
63.14
5.81
7.30
11.02
(c) 10 years temporary structural change
CLS
BS1
BS2
BS3
10.85
(0.15)
10.55
(0.13)
10.87
(0.405)
11.35
(0.73)
8.47
0.74
(0.02)
0.45
(0.02)
0.45
(0.049)
0.47
(0.066)
5.52
8.69
13.47
84.55
12.13
11.85
18.59
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 year
BS2 – overlapping segments moving a length of 5 years, BS3 – non-overlapping 10-year block
Table 7 (Cont.)
Parameter Estimates for the Long ARMA(1,1) Series Simulated with Temporary Structural
Change (S.C) at the Start, Middle, and End of the Series.
(1  0.95B)(Yt  10)  (1  0.95)a t
Method
mean
(s.e.)
% diff
phi1
(s.e.)
%
diff
theta1
(s.e.)
% diff
MAPE1
MAPE2
31.11
16.51
91.22
2.11
36.69
99.71
6.34
30.93
96.43
(a) 1 year temporary structural change
CLS
BS1
BS2
11.23
(0.617)
13.40
(1.572)
10.40
(1.002)
12.33
34.05
3.98
0.94
(0.009)
0.95
(0.002)
0.95
(0.004)
1.16
0.25
0.27
-0.65
(0.019)
-0.93
(0.013)
-0.89
(0.042)
32
BS3
10.51
(1.37)
5.10
0.94
(0.009)
0.56
-0.84
(0.072)
11.45
25.72
97.03
-0.61
(0.02)
-0.88
(0.02)
-0.88
(0.043)
-0.89
(0.059)
35.78
7.13
27.58
7.59
8.26
42.55
7.45
8.66
42.66
6.17
8.89
43.54
-0.66
(0.019)
-0.86
(0.019)
-0.89
(0.036)
-0.92
(0.023)
30.30
7.34
54.39
9.37
8.13
58.09
5.99
8.48
55.78
2.68
8.86
50.68
(b) 5 years temporary structural change
CLS
BS1
BS2
BS3
12.24
(0.678)
13.56
(1.844)
11.06
(0.957)
10.73
(1.132)
22.43
35.56
10.58
7.31
0.95
(0.008)
0.95
(0.002)
0.95
(0.005)
0.94
(0.013)
0.38
0.11
0.13
1.14
(c) 10 years temporary structural change
CLS
BS1
BS2
BS3
10.55
(0.595)
8.93
(0.6)
9.09
(1.079)
9.89
(1.558)
5.54
10.69
9.11
1.10
0.94
(0.009)
0.94
(0.006)
0.91
(0.022)
0.86
(0.038)
1.54
1.54
3.83
8.95
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 year
BS2 – overlapping segments moving a length of 5 years, BS3 – non-overlapping 10-year block
Table 8
Parameter Estimates for the Short ARMA(1,1) Series Simulated
with Temporary Structural Change (S.C) at the Start, Middle, and End of the Series.
(1  0.4 B)(Yt  10)  (1  0.5)a t
Method
̂
(s.e.)
CLS
BS1
BS2
10.70
(0.47)
9.88
(0.255)
11.03
(0.452)
% diff
ˆ
% diff
(s.e.)
6.98
1.20
10.26
0.55
(0.166)
0.29
(0.03)
0.73
(0.148)
37.27
27.44
81.55
ˆ
(s.e.)
-0.45
(0.155)
-0.32
(0.031)
-0.13
(0.317)
% diff
MAPE
MSE
9.08
8.55
1.30
35.99
9.03
1.57
74.18
8.68
1.38
33
(1  0.95B)(Yt  10)  (1  0.95)a t
Method
̂
(s.e.)
CLS
BS1
BS2
6.55
(1.681)
10.78
(4.311)
5.32
(1.374)
% diff
ˆ
% diff
(s.e.)
34.47
7.83
46.78
0.74
(0.118)
0.43
(0.04)
0.11
(0.245)
ˆ
% diff
MAPE
MSE
83.54
94.05
9.83
46.72
190.85
15.72
43.37
145.00
13.00
(s.e.)
22.55
55.24
88.29
-0.16
(0.165)
-0.51
(0.046)
-0.54
(0.192)
Note: CLS – Conditional Least Squares, BS1 – overlapping segments moving a length of 1 month, BS2 – nonoverlapping 1-year block
34
Table 9
Forward Search Estimates of the Series From (1  0.5 B)(Yt  10)  a t
with 5 Years Contamination at the Start, Middle, and End of the Series
Time
Points
̂ FS
Distance
from CI
ˆFS
Distance
from CI
1-120
121-240
241-360
361-480
481-600
601-720
721-840
841-960
961-1080
1081-1200
1201-1320
1321-1440
1441-1560
11.63
10.09
9.93
9.72
10.37
10.17
9.58
9.83
10.04
10.24
10.16
9.96
11.01
1.12
0.81
0.50
0.35
0.61
0.41
0.38
0.96
0.50
0.48
0.50
0.40
0.56
0.93
0.13
-0.03
-0.24
-0.38
-0.13
0.51
-0.11
-0.05
-0.08
0.28
-0.06
0.25
Table 10
Bootstrap Confidence Intervals for the Parameters of the Series Generated from
(1  0.5 B)(Yt  10)  a t with 10-year S.C. at the Start, Middle, and End of the Series.
Method
BS1
BS2
BS3
Method
BS1
BS2
BS3
95% BCI for
10.3170
10.1927
10.0259
10.7174
11.1016
11.4186
99% BCI for
10.2615
10.1208
9.9272


10.8047
11.3090
11.7415
95% BCI for
0.5026
0.4507
0.4464
0.5727
0.5891
0.6176
99% BCI for
0.4926
0.4287
0.4282


0.5786
0.6139
0.6355
Proportion of segments w/
estimate(s) outside the BCI
120/121
25/25
10/13
(99.2%)
(100.0%)
(76.9%)
Proportion of segments w/
estimate(s) outside the BCI
120/121
25/25
8/13
(99.2%)
(100.0%)
(61.5%)
35
Table 11
Bootstrap Confidence Intervals for the Parameters of the Series Generated from
(Yt  10)  (1  0.5B)a t with 10-year S.C. at the Start, Middle, and End of the Series.
Method
BS1
BS2
BS3
Method
BS1
BS2
BS3
95% BCI for
10.3318
10.2193
10.0126
10.6532
11.0058
11.4009
99% BCI for
10.3066
10.0979
9.9606

95% BCI for
-0.6103
-0.6367
-0.6554

10.6963
11.1565
11.6474
Proportion of segments w/
estimate(s) outside the
BCI
-0.5643
-0.5427
-0.5117
99% BCI for
-0.6134
-0.6490
-0.6757


120/121
24/25
9/13
(99.2%)
(96.0%)
(69.2%)
Proportion of segments w/
estimate(s) outside the
BCI
-0.5592
-0.5318
-0.4980
118/121
22/25
9/13
(97.5%)
(88.0%)
(69.2%)
Table 12
Bootstrap Confidence Intervals for the Parameters of the Series Generated from
(1  0.4 B)(Yt  10)  (1  0.5 B)at with 10-year S.C. at the Start, Middle, and End of the Series.
Method
BS1
BS2
BS3
Method
BS1
BS2
BS3
95% BCI for
10.3364
10.2274
9.9758
10.8172
11.7590
12.9525
99% BCI for
9.9178
10.0733
9.9758


13.5397
11.8561
12.9525
95% BCI for
0.4128
0.3552
0.3544
0.4891
0.5457
0.6057
99% BCI for
0.3241
0.3163
0.3544


0.6392
0.5871
0.6057
95% BCI for
-0.5450
-0.6075
-0.6507
-0.4992
-0.4969
-0.4817
99% BCI for
-0.6889
-0.6258
-0.6507


-0.4434
-0.4876
-0.4817
Proportion of segments w/
estimate(s) outside the BCI
121/121
25/25
12/13
(100.0%)
(100.0%)
(92.3%)
Proportion of segments w/
estimate(s) outside the BCI
121/121
24/25
12/13
(100.0%)
(96.0%)
(92.3%)
36