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 Please do not quote without author’s permission.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 APPROACH TO INTERVENTION ANALYSIS IN nIE SEASONAL

ADJUSTMENT OF BLS SERIES: SOME EMPIRlCAL 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. the BLS pice indexes of these pod~

~liStered I very sharp one month d~line. Such events taIled ~terveotiODS. in IWisticaJ IenninolO8Y. Ire

cxttmaJ to the market forces or demand and suppJy which detcmine Ihe ~ces or Ihese pooduc1S. Now the

quality or se.a.sonaJdjustment or any series which is arr~ted by such interventions wi1J be adverseJy Ifr~ 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

series by ad.hoc me.lhods and the scies is then seasonally adjusted by the X.J J ARIMA method.

The state space structura] mode;1-based me;thod of seasonaJ adjustment. on the other band. can

simultaneously adjust for the interventions as well as for the seasonaJ effectS. This is done by inUoducing in the

SU1lCtura] models, in~rvention variables (to be defined later) as explanatory variables in addition to the seasonal

and b"end components and other explanatory variables.

~

In section 2, the structuraI model for se.asonal adjustment 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 resultS from the estimation or this model using three different &lSOline series are discussed. Fmal

~

~uon 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. The generaI fonn of Ibis SD'UcturaJ mode] is

as follows:

~

~

(I)

kl k2

Y1- I ~u Xu + !0 + 8tj Jtj + ~ + 'Y1 + Et

i=l j=l

i 81. 2 kl (2)

~

i = 1. 2 k2 Stj= St-l.j+4>tj (3)

(4) JJ.L = 2 ~L-l -JJ.L-2 + T1L

~

(5)

5-1

'Y1 c .!. 'Y1-j + (!)11

j=1

~

t = 1. 2 n

~

where yt is the observed series. xti and Itj are explanatory and intervention variables resp«tive]y. which are

different for each series fitted to the model; Pti and ~tj and the comsponding parametC'S which are StOChastic

varlables following a random walk without drift in equauons (2) and (3). Some other models of unobserved

c:ompon~nts ~ and 'YL is equabons (4) and (5) w~ Iried but the mod~l aOOve provided the best fit and forecasting

perfonnance for all series. The tJTOr5 or all Ihe five equations arec assumed 10 be mutualJy and serialJy

uncorrelate.d random variables having zero mean and conStant variance. This StructuraJ model or seasonal

adjusunent is cast into a state space form and Ka1man r1lte.ring and smoothing tt.chnique is used tO estimate an

d1e unobserved compon~ntS including ~. '11 and param~tD:S Pu and &u. 1b~ State space form 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 covariance matt"ix Q whose diagona1 elements are relative variances (relative to O't2)(a1so cal1~ hyperparameters} of the random errors of the model componentS and expJanatmy and intervention variable equations

are unknown parame~ 10 be estima~. To do this, it is assumed that ~ and f.t I. I. 2. are DncaJda~

Donna1Jy distirbuted random variable and vector respectively. Zt. R and T are liv~ matrices which ~ obtained

from the SD'Ucture of 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 (1970), and Harvey (1981) and (1991).

(ii) estimation of Q and a£2 by EM algorithm and BFGS numerical optimization ~hniqes. For a

discussion of EM algorithm, see Dempster el 81. (1977) and Shumway and SlOffer (1982). The likelihood

function which is optimised to estimate Q is obtained via prediction en'Or decomposition using Kalman Filter.

The Kalman Filter 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 matrix. To start the filter, QO is set to equal the identity matrix.

3. EMPIRJCALRESULTS

~

The seasonaJ adjustment model presented above is estimated from Ihe following three price index series

0) a>I of gasoline. (ii) PPI or gasoline. and (w') PPI of domestic crude 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'UCturaI models 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.- JoumaJ ofRoyaI 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 A11an Publishers Limited.

(1984 ), ~ A Unified View of Statistical For~g Procedures,~ Journal of Forecasting, 3.245-

275.

(1990), FoT~astin~. Strnrtura1 Time Serie~ Mod~l~ and me J(8Jman Filter, Cambridge,

Cambridge University Press.

Harvey, A.C., and Todd, P .HJ. (1983), "Fo~ling Economic Time Series With SU'UctUraJ and Box.Jenkins

Models: A Case Study ." JoumaJ or Business & Economic Statistics. 1.299.306.

Jazwinski, A.H. (1970), WStochastic Proc~ss~s and Filt~rint! Th~nD:", New York., Academic Press.

Kalman, RoE. (1960), M A New Approach to Linear Filtering and Prediction ProbJems,M Trans. 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 StatiStical Association, 79, 378-389.

Shumway,R.H. and D.5. Storrer (1982), .An Approach ID Time S=ies Smoothing and FOr'eC;asting Using she EM

Algorithm.. JoumaJ of Time Series Analysis. 3. 253-264.

Shumway. R.H. (1988), .Ap;Dlied StBtistical Time Serie~ Anal~i~.. Englewood Cliffs, NJ.: Prenuce HaIl.

Schweppe, F. (1985}, ~vaJuation or Likelihood Functions for Gaussian Signa1s-.IEEE Transactions on

1nf0000tion Theory. 11.61-70.

9
Table I. Estimates or Hyper-parameters or the SeasonaJ Adjusunent Model

I,

0.0

1.2

PPI.Crude

Petroleum 0.8457 0.0355 O.O(KX) 0.0

~

-

466

Series Model EJTor Trend Seasonal

)2 XI X2

}00 2.2464 0.0003

I 7.3914 0.0006

I

!

001 9.1639 \ 0.0002 10.0000

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=

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DO ~~-~C-~o

i!

.2N~

B' iJ

u

.c

1> ~

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[0

~g<u=-u~~

C)

t-

~~

~

~

.= (0-

.~ 0"\

Q.~

!i."""

l~

E~

~~. l

-~

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~

=<

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c

t:: ~Bf

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Figure 2. The Producer Price Index of Gasoline
Figure 3. The Producer Price Index of Crude PetroJeum