Random Level-Shift Time Series Models_ ARIMA Approximations_ and by dfgh4bnmu


									Random Level-Shift Time Series Models, ARIMA Approximations, and Level-Shift

         Chung Chen; George C. Tiao

         Journal of Business & Economic Statistics, Vol. 8, No. 1. (Jan., 1990), pp. 83-97.

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                                                                                                     Thu Jan 24 07:21:20 2008
O 1990 American Statistical Association                               Journal of Business & Economic Statistics, January 1990, Vol. 8, No. 1

Random Level-Shift Time Series Models,
ARIMA Approximations, and
Level-Shift Detection
Chung Chen
  Department of Quantitative Methods, Syracuse University, Syracuse, NY                               13244-2130

George C. Tiao
  Graduate School of Business, University of Chicago, Chicago, IL 60637

                      The main purpose of this article is to assess the performance of autoregressive integrated
                      moving average (ARIMA) models when occasional level shifts occur in the time series under
                      study. A random level-shift time series model that allows the level of the process to change
                      occasionally is introduced. Between two consecutive changes, the process behaves like the
                      usual autoregressive moving average (ARMA) process. In practice, a series generated from a
                      random level-shift ARMA (RLARMA) model may be misspecified as an ARlMA process. The
                      efficiency of this ARlMA approximation with respect to estimation of current level and forecasting
                      is investigated. The results of examining a special case of an RLARMA model indicate that the
                      ARlMA approximations are inadequate for estimating the current level, but they are robust for
                      forecasting future observations except when there is a very low frequency of level shifts or when
                      the series are highly negatively correlated. A level-shift detection procedure is presented to
                      handle the low-frequency level-shift phenomena, and its usefulness in building models for fore-
                      casting is demonstrated.

                      KEY WORDS: ARMA models; Estimation of current level; Forecasting; RLARMA models.

                   1. INTRODUCTION                                        be written as
  Consider the general Gaussian (autoregressive mov-                                 (1   +   .40B     +   .27B2)(1 - B)z, = a,,      (1.2)
ing average) ARMA(p, q ) model
                                                                                          Z, =   (1   -    .81B1')-'(1   -   B12)y
                                                                                                                                 17   (1.3)
where {z,) is the observable time series, q(B) = (1 -
q l B - ... - qpBP), 0(B) = (1 - BIB - ... - OqBq),                       {y,) is the observable logarithmic series, and {a,) is sup-
B is the backshift operator, all the zeros of q(B) and                    posed to be a series of homogeneous white noise. For
B(B) are on or outside the unit circle, and {a,) is a series              simplicity, we shall work with the seasonally filtered
of iid N(O, 02)variables. During the past 15 years, this                  series {z,) to motivate the problems considered in this
class of models has been widely used to represent real                    article.
time series in many areas, especially for forecasting in                     From (1.2), z, follows an AR(3) model in which the
business and economic applications. Model (1.1) as-                       autoregressive polynomial contains a differencing factor
sumes that observations are homogeneously generated                       (1 - B). On the other band, Figure 1 shows that a
from the same probabilistic structure. In practice, how-                  linear trend appears to exist in the data with an obvious
ever, time series data often contain structural changes                   level drop occurring in 1976. If this level drop is ignored,
or discrepant observations. It is critical to investigate                 one may entertain a model containing a linear-trend
to what extent such phenomena can make standard sta-                      term and an AR(3) noise term of the form
tistical methods less efficient or even invalid.
                                                                                                  Z, =     c   + mot + N,,            (1.4)
1.1 An Example
                                                                          where (1 - q l B - q2B2- q3B3)N,= a,. The estimation
  Figure 1 shows the seasonally filtered series of log-                   results of this model are given in Table 1. Standard
arithmic monthly variety-store sales from January 1968                    model checking based on residuals autocorrelations
to September 1979. The original logarithmic variety-                      does not indicate evidence of model inadequacy. The
store series was analyzed by Chang and Tiao (1982);                       estimated trend coefficient k0 = .24 x lo-' with a t
see also Bell, Hillmer, and Tiao (1983). In their anal-                   value of .02 is exceedingly small, however, which seems
ysis, the fitted model before adjustment for outliers can                 incongruent with the behavior of the series.
84              Journal of Business & Economic Statistics, January 1990

                                                                                               with the behavior of the series exhibited in Figure 1.
 am                                                                                            Now, examining the zeros of the fitted autoregressive
                                                                                               polynomial (1 - q , B - q2B2 - q,B3), we consider an
                                                                                               alternative parameterization ( 1 - v ; B ) ( 1 - @ B -
 00                                                                                            @ B 2 )for the noise term N , , where 9; is the zero of B - '
a05-                                                                                           closest to the unit circle. The factorization of fitted

 413            N-J      I           I         I            I         I          I       I'    results gives for (1.4), (1 - .97B)(1 + .35B + .24B2);
                                                                                               for (1.5), ( 1 - .93B)(1 + .48B + .31B2);and for (1.6),
       6   7        8    8          n         n                 n              m        81
                                                                                               ( 1 - .89 B)(1 + .51 B + .34B2).In particular, the es-
  Figure 1. Seasonally Filtered Logarithmic Monthly Variety-Store                              timate of v ; decreases from 4; = ,97 for (1.4) to 4;
Sales.                                                                                          = .93 for (1.5) and then to 4; = .89 for (1.6), and we
  This calculation shows the need for careful exami-                                           see that prior to level-shift adjustments the fitted model
nation of the homogeneity assumption underlying                                                (1.4) is in reasonably close agreement with the factor
Model (1.4). Indeed, applying the level-shift detection                                        (1 + .40B + .27B2)(1 - B ) in (1.2). Although one
                                                                                               cannot reject the hypothesis that there is a unit root in
procedure considered by Chen (1984), which is sum-
                                                                                               the A R polynomials for (1.4), (1.5), and (1.6) (Dickey
marized later in (4.17) and (4.18), to the residuals of
                                                                                               and Fuller 1979; Said and Dickey 1985), it is clear that
(1.4),we find strong evidence of a level drop occurring
                                                                                               the fitted A R part of Model (1.6) shows much less
at the time point t(l) = April 1976. To adjust for the
effect of this unusual event, we may entertain the fol-                                        evidence of nonstationarity compared with that of (1.4).
lowing modified model:                                                                         This is implying that the differencing operator ( 1 - B )
                                                                                               simply attempts to model the occasional level changes
                    z,       = c   + w,)t + w , L S , ( t ( ' ) )+ N,,                (1.5)    in this series. It should be pointed out that, as reported
where                                                                                          by Chang and Tiao (1982), the level shift at t'l) = April
                                                                                               1976 was associated with the fact that a major chain
                                                                                               store went out of business at that time, but the reasons
                                                                                               for the other two indicated shifts are unknown.
                                                                                                  The key point of the preceding example is that oc-
   The estimates of Model (1.5) are also reported in                                           casional level shifts in real time series can lead to models
Table 1. It shows that the adjustment of a single level                                        of a very different nature, depending on whether these
shift leads to appreciable changes in the estimates of                                         changes are appropriately taken into account. It is also
the parameters ( p i , y z , q 3 ,C , w,,)and a large reduction                                of interest to examine the potential influence of ad-
in the estimated variance a?.Repeating the level-shift                                         justing occasional level shifts on the forecasting per-
detection procedure based on the residuals of Model                                            formance. For illustration, let t, = September 1978 be
(1.5),two additional possible level shifts have been in-                                       the forecast origin and observations from October 1978
dicated at t(?) = September 1970 and t") = January                                             to September 1979 be reserved for calculating forecast
1972. If all three level shifts are taken into account, the                                    errors. A model of the form (1.2), which contains an
estimation results of Model (1.6) are again in Table 1:                                        exact differencing factor ( 1 - B ) , and Models (1.4) and
                                                   3                                           (1.6) are estimated, respectively, based on observations
               z,       =    c   + w,,t +              ~ , L s , ( t ( ~+ )N,.
                                                                         )            (1.6)    up to to. Table 2 lists the 12-period forecast errors. As
                                              k= 1                                             expected, the forecast errors of (1.2) and (1.4) are in
   We see that the estimated linear-trend coefficient oj,,                                     close agreement and are much worse than those of (1.6).
changes markedly from .24 x             for (1.4) to .17 X                                     For this particular forecast period, we find that adjust-
l o - ? for (1.6) after the level adjustments. The latter                                      ing occasional level shifts helps improve forecast ac-
estimate, with a t value of 3.84, is seen to be compatible                                     curacy.

                                                                Table 1. Time Series Modeling of Variety-Store Series

                                 Model                 C                  Wo           PI     P2      p3      103 x     c2   uc

                                 NOTE: Esttmated standard errors are In parentheses
                                                      Chen and Tiao: Random Level-Shift Models and a Detection Procedure        85

                              Table 2. Forecast Errors for Variety-Store Series (lead 1 = October 1978)

                                                              Number of lead
                 Model    1        2      3       4      5       6      7       8      9      10      11   12

                  (1.2) - ,023 - ,006 ,002 - ,052 - ,009 - ,031 - ,061 - ,046 - ,041 - ,063 - ,077 - ,086
                  (1.4) -.024 -.007 .004 -.053 -.011 -.033 -.064 -.050 -.045 -.068 -.083 -.092 

                  (1.6) -.015 ,001 -.008 -.041 ,003 -.Ole -.047 -.030 -.024 -.044 -.057 -.064 

1.2 The Random Level-Shift Problem                                   (1984). The random-effect formulation is intuitively
                                                                     appealing because economic and environmental time
   The preceding example motivates the study of the                  series are affected by many unusual events, the oc-
following problems: (a) What is an appropriate model                 currence and the impact of which may be described
to describe the phenomena of occasional level shifts in              by probability laws. Furthermore, it provides a con-
time series? (b) What would be the potential loss of                 venient framework to assess the performance of stan-
forecasting efficiency by ignoring the information of                dard time series method on series with level shifts by
such level shifts? (c) Is there an efficient way to detect           evaluating the contributions of the level shifts to fore-
such occasional level shifts? This article attempts to               cast (or level estimation) mean squared error (MSE).
address the issues in (a) and (b). A procedure for (c)               In Section 2, a random level-shift ARMA model is
will be given with illustrations of its usefulness in model          introduced. The connection between this model and
building, but its detailed properties will be discussed              standard autoregressive integrated moving average
elsewhere.                                                           (ARIMA) model-building procedures will be discussed
    General statistical approaches to problems of occa-              in Section 3. In Section 4, the efficiency of an ARIMA
sional stuctural changes so far consist of (a) developing            approximation to the random level-shift ARMA model
methods to detect changes or outliers at known or un-                is investigated. Section 5 outlines some possible gen-
known time point (e.g., Box and Tiao 1975; Chang and                 eralizations.
Tiao 1982; Hinkley 1970, 1972; Page 1955, 1957), (b)
devising robust prdcedures for data-analysis in the pos-              2. THE RANDOM LEVEL-SHIFT ARMA MODEL
sible presence of structure changes or outliers (Martin
1980), and (c) modeling the nature of the changing                     A time series { y , } is said to follow a random level-
behavior or outlying observations and deriving methods               shift ARMA (RLARMA) model with order ( p , q), if
according to the proposed models (Box and Tiao 1968;                 the stochastic structure of { y J is defined as
Chernoff and Zacks 1964; Kander and Zacks 1966; Tay-
lor 1980; Yao 1984). In particular, the models consid-
ered by Chernoff and Zacks, Taylor, and Yao share the                where p, = j,q, + pI- ( t = 2, 3, . . . , p,) is an unknown
characteristic that the structure of the series may change           initial level of the process; { x J follows a stationary
occasionally and the time periods of structural changes              Gaussian ARMA ( p , q) model such that 4(B)x, =
are determined by a stochastic process. The models                   O(B)a,, 4(B) = (1 - 4 , B - ... - 4pBp), and O(B) =
considered by Chernoff and Zacks and Yao, however,                   (1 - OIB - ... - 0,Bq); all the zeros of 4(B) and O(B)
are perhaps too simple to describe usual time series                 are outside the unit circle; and {a,} are iid N(0, a ? ) . It
data, and Taylor's model is so flexible that it needs an             is also assumed that {q,} are iid N(0, ~ 0 ~{ j , } ; iid
                                                                                                                         ) are
identification procedure to specify a suitable form of                                              .
                                                                     such that Pr( j, = 1) = 1 and Pr( j, = 0) = 1 - j; and .
the model through the data to gain any insight from the              {a,}, {q,}, and { j , } are mutually independent.
 model.                                                                 A random level-shift ARMA process { y , } can be re-
    In this article, we focus on the phenomena of occa-              garded as the sum of a level process b,} a stationary
sional level shifts in time series. It is known that sta-            ARMA process {x,}. In the level process bl,},the ran-
 tionary ARMA models can represent series that                       dom variable j, is an indicator variable for the occur-
 homogeneously fluctuate around a constant level and                 rence or nonoccurrence of level shift, the frequency of
 nonstationary ARMA models can describe series in                    the level shift is determined by I., and the magnitude
 which the level changes from one time period to an-                 of the shifts are given by the random variables qr If 1      .
 other. When the level of the series changes only oc-                 = 0, the level of { y , ) stays unchanged and the structure
 casionally, as illustrated by the preceding example, the            of { y , } reduces to that of a stationary A R M A ( p , q)
 usual ARMA model may not be appropriate. Now oc-                    process. If 3. = 1, the level changes from period to
 casional level shifts may be regarded as interventions              period. In this case, { y , } is the sum of a Gaussian ran-
 with fixed effects at unknown time points; this approach            dom-walk process and the stationary process {x,} and
 was taken by Hinkley (1970, 1972). An alternative way               hence follows a nonstationary A R I M A ( p , 1, q * )
 is to assume that the effect and the time point of the              model, where q* = max(p, q + 1). When 0 < 3. < 1,
 level shift are both subject to some probability rules,             the degree of nonstationarity of the { y , } process is be-
 as considered by Chernoff and Zacks (1964) and Yao                  tween that of a stationary A R M A ( p , q) process and a
 86          Journal of Business & Economic Statistics, January 1990

 nonstationary ARIMA(p, 1, q*) process. In this sense,                 the series is often signalled by the slowly decaying be-
 the RLARMA process can be used to represent series                    havior of the sample autocorrelation function (SACF).
 that exhibit nonstationarity between zero differencing                The following theorem gives the asymptotic properties
 and first-order differencing.                                         of the SACF for series generated from an RLARMA
             MODEL-BUILDING PROCEDURE                                     Theorem I . Let { y,) ( t = 1, . . . , T ) be observations
                                                                       of an RLARMA(p, q) process, as defined in (2.1). If
    In this article, we shall not be concerned with direct                                              .
                                                                       the probability of level shift i is strictly positive, then
 estimation of the parameters of the RLARMA model                      for any positive integer 1
 (2.1). Our main focus will be on assessing the perform-
 ance of an ARIMA approximation to the RLARMA
 model for forecasting and estimation of current level.
 For the model in (1.1), widely used specification pro-
 cedure proposed by Box and Jenkins (1976) is (a) to
 difference y, as many times as is needed to produce                   The theorem follows from the results of lemma 2.6 and
 stationarity and then (b) to identify the resulting sta-              corollary 2.6 of Tiao and Tsay (1983).
 tionary ARMA process. The necessity of differencing                     To illustrate the pattern of SACF for finite samples,

                    Table 3. Mean and Standard Deviation (SD) of SACF of an RLARMA (0, 0) Process: 500 iterations

i    = .05
i    = .02
i.   = .01

E.   = .05
i.   = .02
i    = .01

.    = .05
i    = .02
i.   = .01

i    = ,05
i   .02
i = .01
                                                       Chen and Tiao: Random Level-Shifl Models and a Detection Procedure    87

Table 3 gives the means and standard deviations of the
SACF from a simulation experiment based on an
RLARMA(0, 0) process with selected values of i., K,
and T. The averaged SACF behaves as one would ex-
pect from Theorem 1. The convergence rate of the sam-
ple autocorrelations to 1 depends on the magnitude of
E.K. When i . is small, the sample autocorrelations may
not be near 1, but the slowly decreasing pattern always
exists. This pattern of SACF usually leads to differ-
encing the series. Applying V = (1 - B) to both sides
of Equation (2.1) yields                                                          Figure 2. Time Series Plot of Series A.

                     WI =   jrqr + VX,,                  (3.2)       It could be interpreted that the ARMA model is able
where w, = Vy,. The differenced series {w,) is the sum               to explain the behavior of the first two moments of a
of a stationary ARMA process {Vx,) and a stationary                  suitably differenced RLARMA process. It is then nat-
process {j,q,): hence it is stationary but non-Gaussian.             ural to ask (a) whether the RLARMA formulation pro-
Let p,,(l) be the autocorrelation function (ACF) of se-              vides any advantage over the ARIMA model for the
ries {w,). It is easy to see that                                    purpose of data analysis and (b) whether, if one starts
                                                                     with the ARMA modeling, it is possible to discover the
                                                                     random level-shift ARMA structure.
where c = 	 + i ~ a ~ / v a r ( V x , ) ) -This shows that the
            (1                             '.                                     4.  PERFORMANCE OF THE
ACF of a differenced RLARMA series, {w,), will have                                  ARIMA APPROXIMATION
the same pattern as that of a stationary ARMA process,
{Vx,), except for the initial dropoff from p(0) to p(1).                Statistical models are built to explain phenomena that
                                                                     involve uncertainty. The choice of suitable models usu-
Hence, following the Box-Jenkins iterative-modeling
                                                                     ally depends on the purpose of the practitioners. In
procedure, an RLARMA process may easily be mis-
                                                                     Section 3, it was shown that a random level-shift ARMA
specified as an ARIMA process.
                                                                     model may often be misspecified as an ARIMA model.
  As an illustration, series A with 150 observations is
                                                                     Inferential procedures-for instance, forecasting-
generated by the model
                                                                     based on this misspecified ARIMA model may be re-
                                                                     garded as an approximation to those corresponding to
                                                                     the RLARMA process. In this section the efficiency of
                                                                     this approximation is investigated by comparing the
a,, j,, and q, are as defined in (2.1); E. = .02; K = 9; a 2         MSE's of forecasts and those of estimating the current
= 1; and the initial level p, is set to be p1 = 3. 'Figure           level of the process. We assume first that all of the
2 shows the plot of the series, exhibiting two level shifts          parameters in (2.1) are known, but this assumption will
at t, = 48 and t2 = 113. The SACF of the original and                be relaxed later.
the differenced series are given in Table 4. The SACF
                                                                     4.1 	 Minimum Mean Squared Error Estimates of
of the original series dies out slowly, and that of the
differenced series gives insignificant values after lag 2.                  YT+I and PT
With the preceding patterns of SACF's, one would ten-                   Let yT = ( y , , . . . , yT)I be the T x 1 vector of
tatively suggest an ARIMA(0, 1, 2) model for the                     observations of an RLARMA model (2.1) and J = ( j, ,
original series. The estimated model for series A is                 . . . ,jT)' be the T x 1 vector of indicator variables for
                                                                     level shifts. In practice, J is usually unknown. For es-
                                                                     timating (forecasting) the future observations yT+/ =(1
                                                                     1, 2, . . .) and the current level p T based on yT, the
                                                                     minimum MSE (MMSE) estimates are the conditional
                                                                     expectations E ( y T + // yT) and E(pT I yT). We shall de-
where the values in parentheses are the associated stan-             note the errors as
dard errors.
  Portmanteau tests on the SACF of the residuals does
not show evidence of inadequacy of ARIMA(0, 1, 2).

                                            Table 4.   Sample ACF' of Series A and {Cy,}
88        Journal of Business & Economic Statistics, January 1990

and the corresponding MSE of e(1) as                                where {q:) are iid with zero means, common variance
                                                                    Kl.o2,and Gaussian, then y, will follow a standard Gauss-
                                                                    ian ARIMA model. The covariance structure of the
where the expectation is taken over the joint distribu-             vector series { y,, p,) corresponding to the model in (2.1)
tion of (p,, y,,,, yT). When 1 > 0, e(1) is the lead-1              and that corresponding to the replacement (4.5) are
forecast error and e(0) is the error of estimating the              exactly the same, however. Thus the misspecification
current level. Now it will be useful to decompose e(1)              of an RLARMA model by an ARIMA model concerns
as the sum                                                          only the distributional differences between {q:) in (4.5)
                                                                    and {j,q,) in (2.1). Now with the replacement (4.5),
                   e(1)   =   el(/) + en(/),              (4.3)     minimum MSE estimates of y,+, and p, based on yT
where                                                               will be linear in y,. We can alternatively regard our
                                                                    comparison of the efficiency of forecasting y,,, and
                                                                    estimating p T between the RLARMA model and the
             -                                                      ARIMA approximation as a study of the loss in the
             - Yr+/ - E ( ~ T + yr, J ) ,        1 > 0,
                                                                    efficiency of linear estimation.
                                                                       Specifically, for estimating (forecasting) y,+, based
                                                                    on y,, note first that from (2.1) and (4.5), the overall
                                                                    ARIMA model for y, takes the form
and E(. / y,, J ) are expectations conditional on (y,, J).
Using the fact that
                                                                    where a(B) = (1 - a l B - ... - a,*B47, q* = max(p,
                                                                    q + I), {c,) are iid N(0, o f ) , and o f and a(B) can be
                                                                    obtained from the relation
we can then write MSE(1) as                                           cy(B)a(F)af                                   @(B)@(F)
                                                                                  , = Kl.   + (1 - B)(l    -   F)
                                                                      4(B)4(F)a-                                    4(B)4(F)'
where V1(l) = C, P(J) ~ [ e : ( l )I J ] and V"(1) = E[ei(l)].
Note that in (4.4) the expectation E[e:(l) J ] is taken             where F = B-' (e.g., see Cleveland and Tiao 1976;
over the joint distribution of (p,, y7+,, yT), conditional          Hillmer and Tiao 1982). In particular, for 1. > 0, a(B)
on J , and p(J) is the distribution of J . On the other             # 0 for 1 BI = 1. For large T, the results of Box and
hand, the expectation E[ei(l)] is taken with respect to             Jenkins (1976) can be used to obtain the optimal fore-
the joint distribution of (y,, J ) . As will be shown in            cast for y,+,, );,(/), whose MSE is
Appendix A, where explicit expressions of V,(l) and
                                                                       M S E A R M A = ) ( Y T +-~
V,,(l) are derived, the main reason for the decomposi-
tion in (4.3) and (4.4) is that E[e:(l) I J] can be readily                                 1- 1

calculated, but determination of E[e:(l)] is considera-                              = of   C v/5,
                                                                                                        v/,, = 1,      1 > 0,
bly more burdensome. Note that, since V,,(l) 2 0, we
can regard V1(l) as a lower bound for MSE(1). There-                                                                      (4.8)
fore, in our comparison of MSE(1) with the MSE from                 where the W,'s satisfy the relation 4(B)(l - B)(1 +
ARIMA approximations to be presented later in Sec-                  tylB + F 2 B 2 + ...) = ff(B).
tion 4.3, we shall mainly use simulation estimates of the              Next, for estimating the current level p, from (2.1)
lower bound V,(l). In a few cases, simulation estimates             and ( 4 . 9 , we may employ the signal-extraction theory
of V,(l) will also be given to illustrate the magnitude             for a nonstationary ARIMA model given, for example,
of V"(1).                                                           by Box, Hillmer, and Tiao (1976). Substituting the fu-
                                                                    ture y's with the forecast values based on (4.6) and { y , ,
4.2   The MSE of the ARIMA Approximation                            . . . , yT), it can be shown that the best linear estimate
  From the discussion in Section 3, we see that the                 of current level is
RLARMA model (2.1) may be misspecified as a Gauss-
ian ARIMA model following standard model-building
procedures. The reason is that these procedures assume
normality of the observations and consider mainly the               As shown by Beveridge and Nelson (1981), the right
covariance structure of {y,) and that of the differenced            side is, in fact, the limit lim,,, jT(1), and they have
series {w,). o see this misspecification clearly, note that
           T                                                        regarded this as an estimate of the permanent level of
in (2.1) the level component p, is such that for t 2 2,             y7-. It is shown in Appendix B that the MSE of fi,,
p, = p,-, + j,q,, where the j,q,'s are independent and              denoted as MSEARMA(0),     is
have zero means and common variance k-l.02. If we                   MSEARMA(O) E(PT - lll^~)?
assume instead that
                                                     Chen and Tiao: Random Level-Shift Models and a Detection Procedure          89

where 4'(1) and a'(1) are, respectively, the first deriv-             The MSE of the ARIMA approximation MSEARMA(l)
atives of 4(B) and a(B) with respect to B evaluated at             can be easily obtained from (4.8) and (4.10). From (4.4)
B = 1. The results in (4.8) and (4.10) can now be used             and applying the weak law of large numbers, a con-
to compare with the MSE(1) in (4.4).                               sistent estimate of V1(l) can be obtained by calculating
   Note that in the Gaussian ARIMA framework, de-                  2 E[ei(l) I J]/N,, where the summation is taken over
composition of an overall model for y, into a nonsta-              a random sample of J'S from the distribution ( A . l ) and
tionary component model for p, and a stationary                    E[e:(l) / J] is obtained for each J from (A.8). The length
component model for x, is, in general, not unique (e.g.,           of the observed series is set equal to 100; that is, T =
see Tiao and Hillmer 1978); also the estimation of p,              100. After some preliminary experimentation over var-
will depend on the component model assumed. For the                ious values of i.'s, we choose the number of samples N,
present problem, we are investigating the efficiency of            to be 1,000. An upper bound for the loss of efficiency
the ARIMA approximation to the RLARMA model                        by using the ARIMA forecasts is obtained by calculat-
defined in (2.1); here the random-walk formulation                 ing the ratio
(4.5) for p, seems to be the natural one to use, since,
as mentioned earlier, it leads to an identical covariance
structure of (y,, p,) for both the RLARMA and the
ARIMA models. One may raise the broader issue of
the uniqueness of the RLARMA component structure,                     For forecasting comparisons, the R(1)'s have been
but this topic is beyond the scope of this article. The            calculated for E. = (.01, .02, .05, .08, . l o , .15), 4 =
uniqueness issue discussed here only affects estimation            (-.9, -.5, .O, .5, .9), K = (9.0,25.0),and1 = 1 , . . . ,
of current level and has no impact on forecasting.                 10. Only partial results are reported here to show the
                                                                   pattern of relative forecasting performances. Specifi-
4.3 	 Efficiency of the ARIMA Approximation to                     cally, Table 5 gives R(1)'s when 4 = .O and K = 25.0
      the RLARMA Model                                             for various values of i. and 1, and Figure 3 shows plots
                                                                   of R(1) (1 = 1, . . . , 10) for i. = (.01, .05, .15), =
  To illustrate the loss in efficiency of the ARIMA                ( - .9, - .5, .O, .5, .9), and K = 25. The loss in MSE's
approximation to the RLARMA model, we consider                     by using the ARIMA approximation varies with the
the following simple model:                                        frequency of level shift I., the autoregressive coefficient
                                                                   4 of the noise term, and the lead time I. In general, the
                                                                   loss in efficiency decreases as E. is increased, and it
                                                                   becomes very small for large 1. For short-term forecasts
                                                                   (low values of I), the potential maximum loss by using
where {a,), {j,), and {q,) are defined in (2.1). Chernoff          the ARIMA approximation is about 25% when 4 2 0.
and Zacks (1964) considered methods of estimating the              Although this loss can be substantially higher for 1 =
current level for a special case of this model ( 4 = 0).           1 when 4 < 0, it declines rather precipitously for 1 2
The purpose of our study is to investigate the perform-            1. We remark that for K = 9, the same pattern of R(1)
ance of the ARIMA approximation to this RLARMA                     is found and, as would be expected, the loss is uniformly
model in terms of both forecasting and current-level               smaller than the cases reported here. For instance, for
estimation. Let w, = Vy,; it is straightforward to see             ( 4 = .O, I. = .01), R(1) is 1.173 for K = 9 versus 1.267
that                                                               for K = 25; for ( 4 = - .5, E. = .01), it is 1.341 for h-
                                                                    = 9 versus 1.487 for K = 25; for ( 4 = .5, 1. = .01), it
                                                                   is 1.087 for K = 9 versus 1.105 for K = 25. In sum, our
                                                                   results suggest that ARIMA approximation performs
                                                                   quite well for forecasting except when the frequency of
                                                                   shift 1. is very small andlor the autoregressive coefficient
                                                                   4 has a large negative value.
and p,,(l) = $p,(l - 1) for 1 2 2. From the ACF of                    Accuracy of the Upper Bound R (I) and Current Level
w,, the ARIMA approximation to {y,) would be the                    Estimation Loss. From (4.4), the exact loss of effi-
model                                                               ciency is

where a is a constant parameter and {c,) would be re-
garded as iid N(0, o f ) .The quantities a and afcan be
determined in terms of the parameters in (4.11) by
                                                                    To assess the sharpness of the upper bound R(1) of the
                                                                    exact loss, evaluation of V,(l) is required. From (A.13),
                                                      (4.14)        VO(l)may be estimated by the average of eZ(1) over,
where rn   =   ((1   +   4 2 ) i + 2)1(1
                                 ~         + $1.~)and 0 < a         say, N1 random drawings of ( J , yT) from the joint dis-
5 1.                                                                tribution P(J)P(yTI J ) given in ( A . l ) and ( A . l l ) . Now,
                                                                Chen and Tiao: Random Level-Shift Models and a Detection Procedure      91

                                                  Table 6. V,(I)I(V,(I)   + V,(I)):   =   25, T   =   100

 = 25 and selective values of ( 4 , I., 1). It shows that for                  given in (4.5) will lead to very inefficient results if in
the comparisons of forecasting performances the upper                          fact p, only changes occasionally.
bound R(1) (1 2 1) will be close to the exact ratio Eff(1)                        Now nonstationary models of nearly the forms (1 -
for measuring the loss by using the ARIMA approxi-                             B)dp, = yl: for d = 1, 2 or (1 - BS)p, = yl: for s = 4,
mation. The entries for 1 = 0 in Table 6 imply that the                        12 have been employed to represent the underlying
upper bound R(0) will substantially overestimate the                           stochastic structures for the trend and seasonal com-
loss in efficiency for estimating the current level p. It                      ponents of commonly used seasonal-adjustment pro-
is for this reason that the R(0)'s have not been included                      cedures such as the Census X-11 method (e.g., see
in Table 5. The estimated loss of efficiency Eff(0) is                         Burridge and Wallis 1984; Cleveland and Tiao 1976),
given in Table 7 for K = 25 and selected values of ( 4 ,                       model-based approach (Hillmer and Tiao 1982), and
i.). Although because of computational complexity we                           Kalman-filtering method (Engle 1976). Since the main
have only estimated Eff(0) for a much smaller number                           interest in these adjustment procedures is in estimating
of combinations of values of ( 4 , j) it is clear from
                                        .,                                     the components rather than forecasting, our findings
Table 7 that, in contrast to the forecasting comparison                        suggest that the procedures could be very sensitive to
shown in Table 5 and Figure 3, a very substantial gain                         occasional level shifts in the series, and appropriate
in efficiency will be obtained by knowing the true model                       statistical analysis should be performed on the data to
when estimating the current level. Moreover, this gain                         ascertain if such shifts indeed occur.
is higher when the noise term is negatively correlated
and there is a general tendency for Eff(0) to decrease
as E. is increased for fixed 4 . Notice that the results are                    4.4   A Level-Shift Detection Procedure
obtained from approximation and the accuracy is not
high enough to establish the property that Eff(0) is                              A full analysis of the RLARMA process (2.1) would
uniformly decreasing in E. for fixed 4 . The overall mag-                      involve specification of the model for x, and joint es-
nitude of Eff(0) is our major concern, however.                                timation of the parameters (I., K) for the level process
                                                                               p,, as well as those for the x, process. For series exhib-
   Discussion. When 3. = 1.0, the model in (4.11) be-                          iting occasional level shifts, accurate estimates of (I., K)
comes exactly a nonstationary ARIMA(1, 1, 1) model.                            would require a very large sample. The simulation re-
The findings reported in Table 3 and Figure 3 show the                         sults in Section 4.3 suggest that, at least for forecasting
remarkable degree of robustness of the ARIMA(1, 1,                             purposes, one needs only be concerned with the situ-
1) model to departure from i. = 1.0 for forecasting                            ation in which i. is very small or x, is highly negatively
future observations. The ARIMA approximation, how-                             autocorrelated, but in these cases the sample would
ever, becomes inadequate when the probability . of   ;                         contain much information concerning the time points
level change is very small or the series are negatively                        and the magnitudes of possible level changes. The rea-
correlated.                                                                    sons are: (a) when i. is very small, the changes are
   On the other hand, Table 7 shows that the ARIMA                             infrequent so that there will tend to be a large number
approximation is clearly inadequate for estimating the                         of observations available between changes, and (b) it
current level of an RLARMA model. This finding sug-                            is well known that negative autocorrelations increase
gests that for a component model of the form y, = p,                           the efficiency in estimating the mean of observations.
 + x,, where x, follows a stationary stochastic model,                         Now for series of moderate length with very infrequent
estimation of p, depends rather critically on the true                         level shifts of unknown magnitude at unknown time
distributional structure of p,. In particular, a Gaussian                      points, there is little to choose between a random-effect
random-walk type of approximation for p, such as that                          formulation such as that given by the RLARMA model
                                                                               in (2.1) and a fixed-effect formulation for practical pur-
            Table 7. Eff(0) for   ti   =   25 and T   =   100                  poses. A potentially useful modeling procedure to im-
                                                                               prove forecast performance might consist of (a) con-
                                                                               structing an ARIMA model for the observed series {y,},
                                                                               (b) detecting the possible level shifts (i.e., estimat-
                                                                               ing J by appropriate analysis of the residuals, and (c) re-
                                                                               modeling the data and making forecasts based on the es-
                                                                               timated information of J.
92        Journal of Business & Economic Statistics, January 1990

   Specifically, let y, = ( y , , . . . , y,)' be a set of ob-      Examining the estimated ARMA coefficients, there is
servations and this series be empirically described by a            a near cancellation between the MA and the AR parts.
general ARMA model q(B)y, = O(B)a,. Let a, = (dl,                   This leads us to consider a more parsimonious model,
. . . , 6,)' be the estimated residuals of the ARMA
fitting to y,. Based on the fitted model and the resid-
uals, we can define the following statistics:
           f,   =~;IIL(~)/~~~(IIL(~))'IIL(~), (4.17)
where     is the residual variance; L(i) = (LS,(i), . . . ,
LST(i))' is a T x 1 vector to indicate the occurrence
of a level shift at t = i; and II is a T x T lower triangular
matrix with elements rjj     such that z,, = 0 if j > i, r,, =      Note that the fitted noise structure (1 + .49B)a, is close
1 if j = i, and otherwise r,, = - 7r,-,, where (1 - 7r,B            to the structure of x, in (3.4). This example shows that
 - 7r2B2- ...) = q(B)O(B)-I. Notice that 2, is the like-
                                                                    the procedure not only detects the level shifts but also
lihood ratio statistic for testing the hypothesis of no             helps identify partially the true underlying structure of
level shift versus that of a fixed level shift at t = i. Chen       the process.
(1984) studied the properties and the performance of a
level-shift detection procedure based on                            4.5 	 A Study of ARMA Approximation to
                                                                          RLARMA When the Parameters
                                                                          Are Estimated
where rm,, = maxi t,l and C is a positive constant. Typ-               The results of Section 4.3 are obtained assuming that
ically, we choose C = 2.8 for high sensitivity level-shift          the parameters of the RLARMA model (4.11) and
detection, C = 3.0 for moderate sensitivity, and C =                those of its ARIMA approximation (4.13) are known.
3.3 for low sensitivity. It can be shown that the statistic         In reality, the model and the parameters are determined
?, is measuring difference of levels before and after time          from the nature of the data. The following Monte Carlo
period i. Consequently, the power of this statistic will            experiment is designed to investigate, in finite-sample
be less when i is near to either the beginning or the               situations, the relative performance of ARIMA model
ending of the sample. Sampling properties of this sta-              building with and without incorporating the level-shift
tistic and its relationship with the RLARMA model will              detection procedure (4.17) and (4.18) when the data
be discussed in another article. In practice, a more ef-            are generated from RLARMA models.
ficient approach is to apply the method in an iterative
fashion. Once a level shift is identified, the residuals               Step 1. For given values of E., K , and 4 , 200 obser-
are adjusted by removing the effect of this level shift.            vations of 2, are generated from an RLARMA (1, 0)
Then a new set of r, statistics as defined in (4.17) may            model in (4.11).
be computed based on the adjusted residuals. The it-                   Step 2. Standard Box-Jenkins techniques are ap-
eration stops when r,,, is smaller than or equal to C.              plied to the first 190 observations of the simulated series
    In Section 1.1, the statistic (4.18) was applied to the         to build an adequate ARIMA (or ARMA) model. To
filtered variety-store sales data. To further illustrate the        reduce the judgmental effect in the process of model
potential usefulness of the method, we apply the iter-              building, however, three models are sequentially con-
ative procedure as described with C = 2.8 to series A               sidered to fit the first 190 observations of the simu-
generated by the RLARMA model in (3.4). The pro-                    lated series. They are AR(1), ARIMA(1, 1, O), and
cedure successfully identifies two level shifts at t, = 48          ARIMA(1, 1, 1). We take the position that a simpler
and      = 113. To incorporate this information in the
                                                                    model is maintained unless there is strong evidence
model-building process, we consider the fixed-effect                against it. Model adequacy is tested by the Box-Ljung
model with level shifts at known time points t, = 48                Q statistics with 12 lags, Q12,at the 1% level of signif-
and t2 = 113 in (4.19) as an alternative to Model (3.4)             icance. If both AR(1) and ARIMA(1, 1,O) models are
and obtain the following estimates (numbers in paren-               rejected, then the ARIMA(1, 1, 1) model will be fitted
theses are standard errors of the estimates):                       to the data.
                                                                       Step 3. Lead-1 to lead-10 out of sample forecasts and
                                                                    the corresponding forecast errors are calculated at or-
                                                                    igin 190 based on the model specified in Step 2.
                                                                       Step 4. The level-shift detection procedure (4.18) is
                                                                    applied in an iterative fashion to the residuals of the
                                                                    fitted results in Step 2. In this study, the critical value
                                                                    C is chosen as 2.8. When level shifts are identified, an
                                                                    intervention model incorporating level shifts with
                                                                    AR(1) noise will be employed to estimate the magni-
                                                                    tudes of the shifts as well as the time series parameter
                                                                                 Chen and Tiao: Random Level-Shift Models and a Detection Procedure       93

in a manner similar to (4.20). Lead-1 to lead-10 out-                                               adjusting for the occasional level shifts, the estimated
sample forecasts are computed based on this modified                                                n weights are very close to the true structure of the
model, assuming that the effect of level shift will persist                                         underlying process. Figure 4 presents the ratios of mean
to the future.                                                                                      squared forecast errors from ARIMA (or ARMA)
  Step 5. Repeat Steps 1-4 500 times for each set of                                                models over those from models incorporating level
i K , and 4 . Mean squared forecast errors based on
  . ,                                                                                               shifts. The overall patterns are largely consistent with
ARIMA (or ARMA) models and AR(1) models in-                                                         those obtained in Figure 3, corresponding to the cases
corporated with the level-shift information in Step 4 are                                           of known parameters. In most cases, the magnitudes of
computed, respectively.                                                                             R(1)'s are about the same except for the case of (1. =
                                                                                                    .01, 4 = .5), in which the improvement of forecast
   Results. The averages of the Q12'sof the residuals-                                              accuracy is even higher when the parameters are esti-
sample ACF over the 500 iterations for various com-                                                 mated. With critical value set to C = 2.8, about 75.4%
binations of /. and 4 range from 7.52 to 13.83 for the                                              of level shifts are correctly identified, and the average
final model in Step 2 and from 9.06 to 12.75 for model                                              number of false level shifts found is 267 for each com-
fittings in Step 4. On the average, the results do not                                              bination of i. and 4 over the 500 iterations. To further
indicate any serious inadequacy in model specification.                                             illustrate the comparison, we compute the ratios of
Table 8 reports the average n weights of the fitted                                                 mean squared forecast errors when parameters are es-
ARIMA (or ARMA) models for Step 2, as well as those                                                 timated to that when parameters are known. The results
from the intervention models for Step 4. The average                                                of ARIMA approximation and intervention model
n weights without adjusting the level shifts match fairly                                           (RLARMA consideration) are reported in Figures 5
well with the theoretical ones calculated from (4.13)                                               and 6, respectively. For the same 6 , the magnitudes of
and (4.14), which also indicates that model specification                                           MSE ratios in both approaches are about the same; this
has been conducted properly. We also found that, after                                              suggests that the loss of accuracy when the parameters
                                                   Table 8.     Theoretical and Estimated n Weights:              k- =   25,500 Iterations

                                                                                      ARIMA models

                                                                                   Intervention models
                                                                            .5               ,042            .O
                                                                            .5             - ,012            .o
                                                                            .5             - ,036            .O

NOTE   n, IS the theoretical value and R,   IS   the mean estlmate of 11.Numbers In parentheses are the standard devlat~ons the estimates
94          Journal of Business & Economic Statistics, January 1990


                    0 0

                          2 0
                                . ..
                                   4 0
                                       +   ..     +

                                                6 0
                                                        ~~   . . * . ~ . . . ~ ~ + ~-
                                                               8 0          loo
                                                                                        - -     ...................... .................. . .
                                                                                              O D     2 0     4 0
                                                                                                                      6 0     (10    LOO
                                                                                                                                                D O
                                                                                                                                                         . - -- .
                                                                                                                                                        2 0   6 0     6 0
                                                                                                                                                                          - ~ ~ ~ * . ~ . .~. 

                                                                                                                                                                                8 0
                                                                                                                                                                                           l o o

                    Figure 4. Plots of R(1) When Parameters Are Estimated:                                                  K =     25; A: i = .01; B: i= .02; C: i
                                                                                                                                            .                                         =    .05.

are estimated is about the same in both approaches.                                                                     follows an RLARMA process. The first generalization
There is no obvious pattern in Figures 5 and 6 with                                                                     can be defined explicitly as
respect to A. In both cases there is a slight increasing
                                                                                                                                           yr=pr+N,,                      t = 1 , . . . , T,                        (5.1)
trend of MSE's with respect to the number of forecast
leads, which seems to indicate that the short-run fore-                                                                                 )
                                                                                                                        where U d ( B N , = x,, {p,)and {x,)are as defined in (2.I ) ,
casts suffer less due to the uncertainty of estimated                                                                   and U d ( B )is a polynomial of degree d, having all of its
parameters. Further studies are called for to understand                                                                zeros lying on the unit circle. In particular, if {N,) is a
the performance of the proposed detection procedure                                                                     random-walk process, the model becomes a discretized
and the sampling properties of the estimates in the in-                                                                 version of the compound events models considered by
tervention model.                                                                                                       Press (1967) and Clark (1973).That type of model has
                                                                                                                        been a candidate for describing the behavior of security
                  5. GENERALIZATIONS                                                                                    prices. The second possible generalization may be for-
   The idea of occasionally changing level in stationary                                                                mulated as Ud(B)y, = P, + & ( t = 1, . . . , T ) or
process can be extended to the situations that the ho-
mogeneous component {x,} in (2.1) is nonstationary.                                                                                                            -r
                                                                                                                                                      y , = - +P                      Xr
Two potentially useful models are (a) the observed se-                                                                                                        Ud(B)            I/d(B)'
ries is the sum of a level process {p,)and a nonstationary                                                              where {x,)and {p,)are as defined in (2.1) and U d ( B )is
ARMA process N, and (b) a suitably differenced series                                                                   that given in (5.1). This class of model may be used to

    Figure 5. Ratios of MSE's: Estimated Parameters Versus Known Parameters, ARlMA Approximation:                                                                   K =     25; A: A        =      .01; B: A   =   .02; C:
i   = .05.
                                                       Chen and Tiao: Random Level-Shift Models and a Detection Procedure                        95

    Figure 6. Ratios of MSE's: Estimated Parameters Versus Known Parameters, RLARMA Consideration: h-     =   25; A: i   =   .01; B: i   =   .02; C:
i   = .05.

represent series with various types of abrupt structural              suppose that, for t     =   1, 2 , . . . .
changes. For instance, if U d ( B ) = ( 1 - B ) in (5.2),
then { y,) may reveal occasional changes in trend. Notice
that ( 5 . 1 ) and ( 5 . 2 ) are very different in terms of the
effect of the changing p,. For example, if U d ( B ) = 1
 - B , then (5.1) implies y, has random level shifts and
                                                                      where the y,'s can be obtained from the relation
nonstationary error, whereas (5.2) implies that y, has                $ ( B ) y ( B ) = O(B) with y ( B ) = ( 1 + y I B + y z B 2 +
random linear-trend shifts and nonstationary error.                   ...). From (2.1) and ( A . 2 ) , we can write
Moreover, ( 1 - B ) y , has random point outliers and
stationary error under (5.1) and random level shifts and
stationary error under (5.2). Identifying these two
models requires careful analysis of both the original
series and suitably differenced data. In general, the                 where 0 is a T x 1 null vector, T and U are T x T
preceding models can be employed to represent series                  lower triangular matrices such that
with behavior of nonstationarity between d differencing
and ( d + 1 ) differencing. It would be of interest to
investigate the efficiency of A R I M A approximations to
such models and for low frequency shift in p, (small
values of /.) and to extend the level-shift detection pro-
cedure in (4.17) and (4.18) to cover these cases.
                                                                      D is a T x T diagonal matrix with diagonal elements
                                                                      ( j l r . . . . jr), +(I' = (yT+1-1, . . . . V I ) ' , rl = C5=l
                                                                      ~ T + ~ Y I T and? rz = 2:: W 5 a ~ + ~ - s
                                                                                    + ~                      Since { j,), { ~ r ) , and {a,)
  We thank the referees and an associate editor for                   are independent, it follows that, conditional on J , ( Y ~ ,
their helpful comments.                                               pr, y T + / )are jointly normal and their covariance matrix
                 V,(I) AND V,(I)
   We here give explicit expressions for V , ( l ) and V,,(l)
in (4.4). From (2.1), the distribution p ( J ) is
                                                                      A = K ~ "J"t" ' '
                                                                                  U         "
                                                                                            m         m

For notational convenience, we now assume that the
processes {p,) and {x,) start at t = 1. Specifically, we
96          Journal of Business & Economic Statistics, January 1990

m    =    J ' J is the number of level shifts, and q =                where p ( J ( yT)is the conditional distribution of J given
CJ=:/-l    tyf. It follows that, for the Vl(l) component in           yT. Moreover, by Bayes's theorem, we find that
                                                                      P(J 	I YT)= P ( J ) P ( Y ~J)/C                           I
                                                                                                                  P ( J ) P ( Y ~J).   (~.ll)

                                                                      where p(yT 1 J ) = 	(2n)-T'21MJ11'2a-Texp{-
                                                                      - r'wJ$m)}.
                                                                        It follows that
where R = (KUDU' + q Y 1 ) - ' .
   For actual computation of ( A S ) , the main burden
                                                                       0	 1 = 1 - J L          EJ            1= 0

lies in evaluating the T x T matrix R. Note that the                         =   (I,,    -   W;+(l))'$,,, 

rank of UDU' is m, the number of nonzero values in
J or the number of level shifts that occurred in the                             -      EJJ 1      T          -   )                    1 > 0,
observational period. For small m , computation of R
can be drastically simplified by employing the identity
(I, + PQ)-I = I, - P(I,, + QP)-lQ, where P is an                      and hence
n x n' and Q an n' x n matrix. Specifically, letting H
be the T x T lower triangular matrix

                                                                      From (4.4), (A. I), (A.8), (A.12), and (A. 13), it is clear
                                                                      that exact evaluation of either V1(l) or V,,(l) will be
                                                                      extremely burdensome for large T, because there are
                                                                      2Telements in the sample space of J . Since E[e:(l) I J ]
                                                                      is available from (A.8), however, estimates of VI(l)can
where the elements h,'s can be obtained from the re-                  be readily obtained by performing simulation experi-
lation B(B)(l - B)H(B) = 4(B) with H(B) = (1 +                        ments with respect to J only. Moreover, we see from
hlB + h2B2+          If t,, . . . , t,,,denote the positions

                                                                      (A.12) and (A. 13) that considerably more complex sim-
where a level shift occurs, let WJ = (h,,, . . . , h,) be             ulation experiments will be required to estimate the
a T x m matrix such that j,, + ... + jlm = m. An                      component V,,(l).
alternative expression for R is                                                   APPENDIX B: PROOF OF THE
                                                                                      MSEARMA(0) (4.10)
                                                                        From (2.1) and (4.9), we have that
Then Equation (AS) may be simplified as

                                                                      where w = a ( l ) / 4 ( l ) . Since {p,}and {a,} are indepen-
                                                                      dent, the covariance generating function of (pT - ,iT)
                                                                      G(B, F)     =     Kia2(l - B)-'(l - F ) - '

where 1 is an m x 1 vector of ones, Mj = (I +
KW;WJ)-I is an m x m matrix, and KM,W;WJ = I -
Mj. In (A.8), t ,, . . . , t, are.the positions at which a
level shift occurs. Expressions ( A . l ) and (A.8) can now           From (4.7), by setting B = F = 1, we obtain xE.a' =
be employed to evaluate the component V,(l) in (4.4).                 azw2, and hence in (B.2)
  Turning now to the component V,,(l)in (4.4), we first
consider the terms of e,,(l) in (4.3). From (A.3) and
(A.4), it is readily shown that E(pT1 y,, J ) = lk4, and

where $ = h-MJW;r, r = IIy,, and II              =   Y-l is a T
X T lower triangular matrix. Note that                                It follows that (B.2) can be written as

    E ( Y T +I IYT)=      2 p(J I YT)E(.YT+/ J),
                                          / YT,          (A.10)
                                                            Chen and Tiao: Random Level-Shift Models and a Detection Procedure                      97

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       Random Level-Shift Time Series Models, ARIMA Approximations, and Level-Shift
       Chung Chen; George C. Tiao
       Journal of Business & Economic Statistics, Vol. 8, No. 1. (Jan., 1990), pp. 83-97.
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    A Bayesian Approach to Some Outlier Problems
    G. E. P. Box; G. C. Tiao
    Biometrika, Vol. 55, No. 1. (Mar., 1968), pp. 119-129.
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    Unobserved-Components Models for Seasonal Adjustment Filters
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    Journal of Business & Economic Statistics, Vol. 2, No. 4. (Oct., 1984), pp. 350-359.
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    Distribution of the Estimators for Autoregressive Time Series With a Unit Root
    David A. Dickey; Wayne A. Fuller
    Journal of the American Statistical Association, Vol. 74, No. 366. (Jun., 1979), pp. 427-431.
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    An ARIMA-Model-Based Approach to Seasonal Adjustment
    S. C. Hillmer; G. C. Tiao
    Journal of the American Statistical Association, Vol. 77, No. 377. (Mar., 1982), pp. 63-70.
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    Inference About the Change-Point in a Sequence of Random Variables
    David V. Hinkley
    Biometrika, Vol. 57, No. 1. (Apr., 1970), pp. 1-17.
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    Time-Ordered Classification
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    Test Procedures for Possible Changes in Parameters of Statistical Distributions Occurring at
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    On Problems in which a Change in a Parameter Occurs at an Unknown Point
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    A Compound Events Model for Security Prices
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    Some Consideration of Decomposition of a Time Series
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    Consistency Properties of Least Squares Estimates of Autoregressive Parameters in ARMA
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    Estimation of a Noisy Discrete-Time Step Function: Bayes and Empirical Bayes Approaches
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    Stable URL:

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