BLS WORKING PAPERS
U.S. DEPARTMENT OF LABOR
Bureau of Labor Statistics
OFFICE OF PRICES AND LIVING
CONDITIONS
A State Space Model-Based Approach to Intervention
Analysis in the Seasonal Adjustment of BLS Series
Some Empirical Results
Raj K. Jain
Working Paper 228
June 1992
The views expressed in this paper are those of the author and do not necessarily represent an official position of the
Bureau of Labor Statistics or the views of other BLS staff members. Please do not quote without author’s permission.
A STATE
SPACE MODEL-BASED
ADJUSTMENT
APPROACH
TO INTERVENTION
OF BLS SERIES:
SOME EMPIRlCAL
ANALYSIS
IN nIE
SEASONAL
RESULTS
Raj K. lain
U.S. Bureau of Labor Statistics, 600 E Str~t, N.W., Room 4013, Washington, D.C. 20212
Keywords: Explanatory variables, Kabnan Filter, EM Algorithm
The Intervention and Explanatory variables are incorporated in the statistical models of seasonal
EM
Two
~
I. rnTRODUCIlON
In the lasl decade. two eventS OCCIDTed
which significantly affected the prices of aude and rermed
~D"Oleum products. In January 1981,lhe U.S. Price Allocation and EntitJement regulation were lifted which
effectiveJy decontrolled the oil prices. ImmediateJy,lhe price indexes or such products registtzed silDificanBt
upward jumps. Again, in January 1986, Saudi Arabia made I poli1icaJ d«:ision 10 aImosr double the ~on
of crude oil and the prices of crude oil fell sharpJy. ConsequantJy.
~liStered
I very sharp one month d~line.
Such events taIled
the BLS pice
~terveotiODS.
indexes of these pod~
in IWisticaJ
IenninolO8Y.
Ire
cxttmaJ to the market forces or demandand suppJywhich detcmine Ihe ~ces or Ihesepooduc1S.Now the
quality or se.a.sonaJ
adjustmentor any serieswhich is arr~ted by such interventions wi1Jbe adverseJyIfr~
if
Ihe eff~t of interventions is not accounted for by the method used to seasonally adjust. that se.ries. The X-II
AJUt..1A which is the method currently being used by BLS to ~na1]y
adjust 11]price index series. d~
not
have any me.chanism in it 10 eliminate the effects of interventions; hence their erre.ct is rlrst eliminated from a
seriesby ad.hoc me.lhodsand the scies is then seasonallyadjustedby the X.J J ARIMA method.
The state spacestructura] mode;1-based
me;thodof seasonaJadjustment. on the other band. can
simultaneously adjust for the interventions as well as for the seasonaJeffectS. This is done by inUoducing in the
SU1lCtura]models, in~rvention
and b"end components
variables (to be defined later) as explanatory variables in addition to the seasonal
and other explanatory
variables.
In section 2, the structuraI model for se.asonaladjustment with intervention and other explanatory
variables is presented. The state space form or Ihis model and its estimauon is also discussed. In section 3.
empirical
~uon
resultS from the estimation
or this model using three different
&lSOline series are discussed.
Fmal
includes the swnmary and conclusions.
-
2. STRUCI1JRAL
MODEL
OF SEASONAL
ADJUSTMENT
"
,~~,};
,;;,,-
unobserved components, intervenuon variables and other explanatory variables, a mode1 for each of iu
unobserved components, definitions or intervention and explanatory variables and the dynamic specification or
2
~
Ihe parameters
or the intervention
and other explanaU)ry
variab]es.
as follows:
The generaI fonn of Ibis SD'UcturaJ mode] is
~
kl
k2
Y1- I ~u Xu + !0 + 8tj Jtj + ~ + 'Y1+ Et
i=l
j=l
i 81.
Stj=
(I)
i = 1. 2
St-l.j+4>tj
JJ.L = 2 ~L-l
2
-JJ.L-2
kl
(2)
k2
(3)
(4)
+ T1L
5-1
'Y1 c
.!.
'Y1-j
+
(5)
(!)11
j=1
t = 1. 2
n
where yt is the observedseries.xti and Itj are explanatoryand intervention variablesresp«tive]y. which are
different for eachseriesfitted to the model; Pti and ~tj and the comsponding parametC'Swhich are StOChastic
varlables following a random walk without drift in equauons(2) and (3). Someother models of unobserved
c:ompon~nts~ and 'YLis equabons(4) and (5) w~ Iried but the mod~l aOOveprovided the best fit and forecasting
perfonnance for all series. The tJTOr5or all Ihe five equationsarecassumed10be mutualJyand serialJy
uncorrelate.d random variables having zero mean and conStant variance. This StructuraJ model or seasonal
adjusunentis cast into a statespaceform and Ka1manr1lte.ringand smoothingtt.chniqueis usedtOestimatean
d1eunobservedcompon~ntSincluding ~. '11and param~tD:SPu and &u. 1b~ Statespaceform or this model is:
)'1 = ~ a1 + Et
(6)
3
~
Ct = T Q1-1+ R;1
(7)
where
Jkl + k2 is the kl + k2 identify mattix.
R = zero (kl+ k2 + I + 1, kl + k2 + 3} except
R.(i,Jl = 1, i-l,2,
-.tl
+t2+
1,tl + t2+ 3;
.
f.t = [ell
e&k1. 4It1.-.
tlk2.1}l'
0. fOtl]
:
4
~
Equations (6) and (7) are respectiveJy the measurement and state transition equations. The variance 0'£2 and the
diagonal covariancematt"ixQ whose diagona1elementsare relative variances(relative to O't2)(a1socal1~ hyperparameters} of the random errors of the model componentS and expJanatmy and intervention variable equations
are unknown parame~
Donna1Jy distirbuted
10 be estima~.
random variable
To do this, it is assumed that ~ and f.t I.
and vector respectively.
Zt. R and T are liv~
I. 2.
are DncaJda~
matrices which ~
obtained
from the SD'Uctureof the model.
The estimation or the se.asona]adjusunent model in the state space fonn (6} and (7) is done in tWo pans:
(i) estimation or the State vector at and it's covariance mauix Pl' given the initiaJ vaJue or the state
vectOr {ao). and il'S covariance matrix {PO) and initiaJ vaJues or the matrix or hyper-pararnelers {QO>.by Kalman
fiJte.ring and smoothing. This technique is discussed in the literature:. See. for ex.ample. Kalman (1960).
Anderson
and Moore (1979), laswinski
(ii)
discussion
function
estimation
(1970), and Harvey
of Q and a£2 by EM algorithm
of EM algorithm,
see Dempster
(1981) and (1991).
and BFGS numerical
el 81. (1977) and Shumway
optimization
~hniqes.
and SlOffer (1982).
The likelihood
which is optimised
to estimate Q is obtained via prediction
The Kalman
is initia1iz~.d with ao and Po = k.J whert ,k is chosen to be a large but fmite number
and I is the identity
Filter
matrix.
To start the filter,
en'Or decomposition
For a
QO is set to equal the identity
using Kalman
Filter.
matrix.
3. EMPIRJCALRESULTS
The seasonaJadjustmentmodel presentedaboveis estimatedfrom Ihe following three price index series
0) a>I of gasoline. (ii) PPI or gasoline.and (w') PPI of domesticcrude petroleum. The Dmple period is from
January 1979 to December 1986 for each series. These data series we.re chosen because intervenuons of January
1981 and January 1986 had significantly affecled these series. Severa1 Sb'UCturaImodels with different
specificauons for unobserved component models for trend and seasonal were fitted to these series but the model
in section (2) gave the best fit and best forecasting performance.
s
~CES
And~son. B.D.O., and Moore. J.B. (1979), ODtimal Filt~rin&. Engtewood Cliffs. NJ.: Prentice Hall.
Box, G.E..P., and Tiao G.C. {197S), -Intervention AnaJysis with Applicabons to «;ooomic and Environmental
Problems,. Jouma] or the American Statistical Association, 70, 70-79.
Dempsttz , A.P ..NoM. Laird and D.B. Rubin (1977), wMaximum Likelihood from IncompIele Data Via Ihe EM
AJgoril.hm.-JoumaJofRoyaI Statistical Society. SerB.. 39.1-38.
Fletcher,R.. (1980), Practica1 Methods of Opumization, Vot I, New York, John Wiley & Sons.
Harvey, A.C. (1981), Times Series Models, Oxford, U.K. Philip A11anPublishers Limited.
(1984 ), ~A Unified View of Statistical For~g
Procedures,~ Journal of Forecasting, 3.245-
275.
(1990), FoT~astin~. Strnrtura1Time Serie~Mod~l~ and me J(8JmanFilter, Cambridge,
Cambridge University Press.
Harvey, A.C., and Todd, P.HJ. (1983), "Fo~ling
Economic Time SeriesWith SU'UctUraJ
and Box.Jenkins
Models: A CaseStudy." JoumaJor Business& Economic Statistics. 1.299.306.
Jazwinski, A.H. (1970), WStochasticProc~ss~s and Filt~rint! Th~nD:", New York., Academic Press.
Kalman, RoE. (1960), MA New Approach to Linear Filtering and Prediction ProbJems,MTrans. ASME, Journal of
Basjc Engineering, Ser. D, 80, 35-45.
Kjtagawa. G., and w. Gerscb (1984), ~A Smoothness Priors-State Sp~e Mode11ing or Time Series with Trend
and Se.a.sonality",
Journal or American StatiSticalAssociation,79, 378-389.
Shumway,R.H. and D.5. Storrer (1982), .An Approach IDTime S=ies Smoothingand FOr'eC;asting
Using sheEM
Algorithm.. JoumaJof Time SeriesAnalysis. 3. 253-264.
Shumway.R.H. (1988), .Ap;DliedStBtisticalTime Serie~Anal~i~.. Englewood Cliffs, NJ.: PrenuceHaIl.
Schweppe,F. (1985}, ~vaJuation or Likelihood Functions for GaussianSigna1s-.IEEETransactionson
1nf0000tion Theory. 11.61-70.
9
~
Table I. Estimatesor Hyper-parametersor the SeasonaJAdjusunent Model
Series
Model EJTor
Trend
Seasonal
I,
0.0 }00
1.2 466
PPI.Crude
Petroleum
)2
2.2464
XI
0.0003
I 7.3914
0.0006
X2
-
I
!
0.8457
0.0355
O.O(KX)
0.0 001
9.1639 \
0.0002
10.0000
)
,
10
I
I
=
.~
"C
<
~
"0
~
£
~
e
.-u
E
..~
~
i
&E~
:~
~r1:
..i?'5
(00"\
C)
~
c.§
iCI)
j~
~
.=
.~
l~
E~
!i."""
Q.~
3.
~~.
-~
9
~
~
l
u
=
~
~
~~
t-
0
&>
-
~
~
~
~
DO
C
..=
I
~
o
"C
C
~
~
-
J
i
B'
~
N
.2
i!
u
<
~g
[0
'.J
u
.c
1>
~
~
N
-
~
~
~
~
!J
-<
~
II
!]
=<
~
c.'=
~5
~~
f
B
~
.
t::
c
ii,
~
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t
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I
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-,
,
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I
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.,
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.-"
"
~
"
.to
E-
., ~
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!(I;;,
.i
.~--
Figure 2. The Producer Price Index of Gasoline
Figure 3. The Producer Price Index of Crude PetroJeum