An Empirical Investigation of the Spillover Effects of Advertising and Sales Promotions in Umbrella Branding Author(s): Tülin Erdem and Baohong Sun Source: Journal of Marketing Research, Vol. 39, No. 4 (Nov., 2002), pp. 408-420 Published by: American Marketing Association Stable URL: http://www.jstor.org/stable/1558554 Accessed: 28/01/2010 14:58 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ama. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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TULINERDEMand BAOHONGSUN* The authorsinvestigateand findevidencefor advertisingand sales
promotion spillover effects for umbrella brands in frequently purchased packaged product categories. The authors also capture the impact of advertising (as well as use experience) on both utilitymean and variance across two categories. They show that variance of the random component of utility declines over time on the basis of advertising (and use experience) in either category. This constitutes the first empirical evidence for the uncertainty-reducing role of advertising across categories for umbrella brands.
An
of the Spillover Empirical Investigation Sales of Advertising and Effects Promotions in Umbrella Branding may be insufficientfor umbrellabrandingto generate savings in marketingcosts and enhance marketingproductivity over time. To increasesavings in marketingcosts over time, the marketingmix, such as advertising and sales promotions, must be more effective for umbrellabrandsthan for other brands. For example, Tauber (1981) suggests that umbrellabrandingcan create advertisingefficiencies. Leigh (1984) finds some experimentalevidence for better recall and recognition performance for umbrella branding than other branding strategies. Yet there is surprisingly little empirical work on advertisingand/or any other marketingmix synergies for umbrellabrands. The objectives of this study are twofold. First, we test for marketing-mixstrategyspillover effects of umbrellabrands in frequently purchased packaged product categories. We are especially interestedin testing for own- and cross-effects of advertising,that is, whether advertising in one product category affects sales in another product category for umbrella brands. However, we investigate the crosscategory effects of all marketing-mix effects, including price, coupon availability, display, and feature. Another relatedobjective is to shed some light on the magnitudeof any effects that may exist. We seek to answer the following questions: (1) What is the impact of the cross-effects on sales? (2) How large are the cross-effects of the marketing mix elements in relationto the own-effects? and (3) Which marketing-mix elements seem to reveal higher crosseffects? Second, we explore the dynamics behind the advertising spillover effects. Similar to use experience, advertising spillovereffects occur for variousreasons,but they may also transpirebecause of the uncertaintyreduction.More specifically, we examine whetherthere is a reductionin consumer uncertainty in one category due to advertising of the
Many companies widely practice umbrella branding. Also, a fair amount of managerial research argues that umbrellabrandinggeneratessavings in branddevelopment and marketing costs over time (e.g., Lane and Jacobson 1995; Tauber 1981, 1988) and enhances marketingproductivity (e.g., Rangaswamy,Burke, and Oliver 1993). Wernerfelt(1988) has shown analytically that a multiproductfirm can use its brandname as a bond for quality when it introduces a new-experience product. Umbrella brandingis posited to both increaseexpected quality(Wernerfelt 1988) and reduce consumer risk (Montgomery and Wernerfelt1992). Consumers'use experience in one product category needs to affect their perceptionsof quality in anotherfor umbrellabrandingto serve as a credible signal of a new-experienceproduct'squality,becausea false signal would be costly if the quality of the extension turnedout to be poor. Experimentalresearch has shown some evidence that the parent brand'sperceived quality affects the extension evaluations (Aaker and Keller 1990) and vice versa, which indicatesthatbrandequity dilutionmay occur (Loken and RoedderJohn 1993). Although the cross-categorylearningeffects for umbrella brandsare a necessary condition for umbrellabrandingto functionas a signal (Erdem 1998), the learningeffects alone *TulinErdem is an associate professor,Universityof California,Berkeley (e-mail:
[email protected]).Baohong Sun is an assistant professor, Kenan-Flagler Business School, University of North Carolina, Chapel Hill (e-mail:
[email protected]).The authorsthankthe three anonymous JMR reviewers for their constructive feedback. The authors also thank Andrew Ainslie and Michael Keane for their suggestions in regardto the updating mechanism of the stochastic component of utility employed in this article. This researchwas supportedby NationalScience FoundationgrantSBR-9812()67 grantedto Tiilin Erdelm.
Joi)rltl o1!f Marketing Research Vol. XXXIX (Novcelbhcr 200(2), 4()8-420
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Advertising and Sales Promotions in Umbrella Branding umbrellabrandin anothercategoryover time. If uncertainty exists and advertisingreduces consumer uncertaintyabout products,advertisingis expected to shift both utility mean and variance.To model and test for both utility mean- and utility variance-shiftingcross-effects of use experience and advertising,we develop a novel and parsimoniousapproach. Our approachenables us to test whether there is any evidence for the uncertainty-reducing role of advertisingacross categories. We use ACNielsen scanner panel data on two categories-toothpaste and toothbrushes-to test for marketingmix synergies for umbrellabrands.For the first time in the literature,on the basis of our analysis of revealedpreference data, we find evidence for advertisingspillover effects for umbrellabrands.Furthermore,we find that advertising(as well as use experience) in one category serves as a mechanism of reducing uncertainty in the other category for umbrellabrands.In addition, our results indicate that there are spillover effects of price, coupon availability,and display. Feature cross-effects exist only in the toothbrush category. The rest of the article is organizedas follows: In the first section, we briefly discuss the relevantliterature.In the second section, we outline the proposedmodel and the estimation method. In the third section, we briefly describe the data. We then outline the results and conclude the article with implicationsand a discussion of furtherresearch. LITERATURE REVIEW Previous researchin marketinghas studied the effects of sales promotionsacross categories for complementaryand substitute products (e.g., Mulhern and Leone 1991) and multicategory choices (e.g., Bell and Lattin 1998; Manchanda,Ansari,and Gupta 1999). A separatestreamof literature focuses on cross-categorytraitsof consumerbehavior and finds that consumer price sensitivities (Ainslie and Rossi 1998; Erdem 1998) or sensitivities to state dependence (Erdem 1998; Seetharaman,Ainslie, and Chintagunta 1999) can be correlatedacross categories. However, there is surprisingly little published research with respect to cross-categoryeffects in choice for umbrella brands. Erdem and Winer (1999) have shown that brand preferencescan be correlatedacross categoriesfor umbrella brands.Erdem (1998) has studied how consumers'experience in one category may affect theirquality perceptionsin another category for umbrella brands. In a two-category context, Erdem has shown that consumer mean quality beliefs, as well as the varianceof these quality beliefs, may be updated through experience in either category for umbrella brands.Thus, Erdem studied only cross-category use experience effects. Specifically, Erdem(1998) does not study any promotion or advertising effects, neither own-category nor crosscategory, which is the topic of this article. Therefore,she does not study and answerany questionsaboutthe existence and magnitude of cross-advertisingand promotioneffects for umbrellabrands.Furthermore,her model does not incorporatepurchaseincidenceand coincidence in choice, thatis, the possibility that the stochastic component of the utility function could be correlatedacross categories. In addition, Erdem'sapproachonly allows for use experienceeffects due to learning, whereas the approach we take in this article
allows for both learning effects and other sources through which use experience may affect currentchoices. Furthermore, the approachErdemtakesto incorporatethe impactof use experienceon utility varianceis difficult to apply to settings in which cross-categoryvariancereductioneffects can be due to other variablesas well (such as advertising).The approachtaken in this articleprovidesa more parsimonious and novel way to incorporatesuch effects. Finally,we allow for more complex heterogeneity structuresin this article than Erdemdoes. Manchanda,Ansari,and Gupta(2000) also find some preliminaryevidence for spillover effects in store promotional activity for umbrella brands;however, they have not analyzed advertising and they have not allowed for choice dynamics. THEMODEL UtilitySpecification Considera set of consumersI = {ili = 1, 2, ..., I that,on each purchaseoccasion, makes purchasesfrom none, one, or both of two distinct productcategories m = 1, 2, where I is the numberof consumers.A nonemptysubset, but not necessarily all, of these consumersmakes purchasesfrom both of the categories. Suppose that for both productcategories, the purchasesof the consumersare observedover the period T = tit = 1, 2, ..., T}, whereT is the time span of the period and t denotes the week. Let J = (jlj = 1, 2, ... J} be the set of brand options available, where j = 1 ..., J brands are available in at least one of the two categories and J is the numberof brands.Note that we set j = 0 to indicatethe nopurchaseoption (choice). Let Knj,m = 1, 2, be an indicatorvariablesuch thatK,-j = I if brandj is available in category m, and K,n.= 0 otherwise. Also let Dmijt,m = 1, 2, be an indicatorvariablesuch that if consumeri purchasesbrandj at time t in categorym, D,iji = 1, and D,,jt = 0 otherwise.Finally,denote the utility consumeri derives from purchasingbrandj in categorym at time (week) t by Uijt, which is postulatedto be (1)
Umijt
= (tmij +
,miPRmiit
+ ymiADmiit + XmiDlSPmijt
+ TmiFEATmijt + OmiCAVmijt+ KmiUEmijt + K3-m,j(miPR3-
m,i,j,t + tmiAD3-m,ij,t
+ lmiDISP3 m,i,j,t + VmiFEAT3,- ,i,j,t + ,nmiCAV3 - m,i,j,t +(On, i UE3 m,i,j,t) + ?Emijt,
where PR,,ijtis the price paid for brandj in category m by consumer i at t, AD,n,jtis the advertisingstock variablefor consumer i with respect to brand j in category m at t, DISP,nijtis the display dummy, FEAT,nitis the in-store ad dummy, CAV,ijt is the coupon availability variable, and UE,nijtis the use experience of brandj in category m consumer i has at t. Equation 1 suggests that if brandj is available in both categories, the utility of purchasingbrandj in one of the categories will also depend on the price, coupon availability,advertising,display,feature,and use experience associated with the same brandname in the othercategory. INote that in Equation I, cross-effects are specified for umbrellabrands
only. In the empirical analysis, to ensure that the effects exist for only
umbrella,we also tested whethersuch effects exist for otherbrandsas well, and we did not find evidence for such effects.
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JOURNAL OF MARKETINGRESEARCH, NOVEMBER 2002
The parametersin Equation I are as follows: a,ij is the consumer- and brand-specificconstant (brand preference), and imi is the consumer-specific price coefficient. Moreover, Ymi,Xni, tmi, Omi, and Kmi are the consumer-specific
advertising, display, feature, coupon, and use experience coefficients in category m (we label these own-effects response coefficients), whereas, ini, tmi,
mi, Vmi, cmi, and
are the price, advertising,display,feature,coupon availability, and use experience coefficients of the price, advertising, display, feature, and use experience variablesin the other category, respectively (we label these cross-effects response coefficients). Similar to Erdemand Keane (1996), we specify i's utility from not purchasingany brandin category m at t as2 ()mi
(2)
Umi0t = (mo + TRENDmt + emot,
where TRENDMis the time trendcoefficient, and for identification purposeswe set (3)
(mO
= 0,
m = 1,2.
The use experience variable, UE,ijt, in Equation I is defined as the exponentiallysmoothed weighted averageof past purchases, as in Guadagni and Little's (1983) brand loyalty variable. Its decay parameteris denoted by DUm. Similarly,AD,nijtis defined as the exponentially smoothed weighted average of past aggregate gross rating points (GRPs).3 Its decay parameter is denoted by DAm. The AD,,,ijtis updatedon a weekly basis. Note thaty,i (andr1mi), the coefficient on the advertising stock variable, embeds both a consumer'stelevision commercialviewing habitsand responsivenessto the advertisements,conditionalon having seen them. Because advertisingeffectiveness in the context of television advertisementsdependson television and commercial viewing habitsof consumers,we can justify thaty,, (and tl,) representsadvertisingresponsivenessat the individual level.4 We define the DISP variablesuch that DISP,nij,= I if a display is availablefor brandj in categorym in the store that consumer i visits at t, whereas DISP,,ijt= 0 otherwise.Similarly,we define the FEATvariablesuch thatFEAT,ijt= 1 if a feature is available for brandj in category m in the store = 0 otherwise. that consumer i visits at t, whereasFEAT,nijt of is the value the store's and manaverage Finally, CAV,,ijt 2Similarto Erdem and Keane (1996), our purposeis to control for purchase incidence ratherthanexplicitly model purchaseincidence decisions. 3An alternativeto GRP data is television exposure data;however,television exposure data are not available because ACNielsen discontinuedthe collection of such data in the early 1990s. Televisionexposuredatahave the advantageof havingthe estimatedadvertisingcoefficient reflectadvertising responsivenessonly because the advertisingvariableitself will capturethe television/commercial viewing habits. However, the television exposure data are noisy as well, because we only knew that the television was tuned to that channel duringthe airingof a particularcommercial.One advantage of the GRP data, in contrast,is that GRP is underthe direct control of the firm whereas television exposure is not. Therefore, firms are more interested in models that use GRP data (Abe 1997). Finally, Tellis and Weiss (1995) find that advertising effectiveness results are not sensitive to the types of advertisingmeasures. 4Wealso have GRP data for different"dayparts"(e.g., primetime versus late morning).When we used these dataandestimatedEquationI with separate coefficients for each daypart,we obtained largeradvertisingcoefficients for certain daypartsthan others, as we would expect, but the main results are the salmewhetherthe daypall data are used or not.
ufacturer'scoupons availablefor brandj in categorym in the store thatconsumeri visits at t. It includes the zero value for nonavailable coupons (Keane 1997b). We introduce this variableto avoid the endogeneityproblemcaused by including coupons redeemedin the utility specification.Last, Emijt in Equation 1 denotes the time-varyingstochastic component of utility that is knownby the consumerbutnot observable by the analyst. We should note that the cross-effects specified in Equation 1 may exist for several reasons. First, price cuts, coupons, advertising,displays, and featuresin one category may remind the consumer about the umbrella brand and thereforeincreasethe likelihoodof theirchoosing thatbrand in anothercategory, conditional on productcategory decision. In the case of promotionaltools such as displays, the promotionof the umbrellabrandin one category may help promote the umbrella brand in consumers' consideration sets in the second category, in which the brandis not promoted, as well. Advertisingcan be expectedto increasegeneral brandawareness of individualproductsthat share the same brand name. Cross-category use experience effects may be due to habit formation. Second, the effects of use experience and advertisingon consumeruncertaintymay cause use experienceand advertising spillover effects. If use experience affects quality beliefs, both the mean and variance of these beliefs may change on the basis of use experience. Therefore, if use experienceaffects consumeruncertainty,we shouldexpect it to affect both the utility mean and variance in a reducedform model. The same rationale holds for advertising. Erdemand Keane (1996) have shown that in the context of one category, both use experience and advertising affect both mean and varianceof qualityperceptions.Therefore,if advertisingspillover effects are partly due to the effect of advertisingon consumer uncertainty,we should expect utility variance(that is, the varianceof the randomcomponent of utility) to decline over time on the basis o' advertising. We capture these utility variance effects by adopting a novel approach.Specifically,we allow for not only the "random" components of utility to be correlatedacross categories but also the variancesof the randomcomponentsfor umbrellabrandsto be updatedon the basis of past purchases and advertising associated with umbrella brands in either category.We cover this issue in more detail subsequently.It may suffice to state that we attemptto capturethe effect of use experience and advertisingas a shifter of both utility mean and variance; the latter effect should occur mainly because the consumer uncertaintydecreases over time on the basis of past purchasesand advertising. Heterogeneity Specification
It is well established that not accounting for unobserved heterogeneitycan bias parameterestimates(Heckman 1981; Rossi and Allenby 1993). In this article, we account for unobservedheterogeneityas well. To do so, we assume that
brand preferences for each category A,mi= (a,n,i, aii2, .. a1ij)T; own-effects response coefficients B,,i =(P:- i, ni, , lmi, K,lni)T;and cross-effects response coefficients mi' mti t li, Cmi = (i, lli, Vm,I, i O,,1i)T, m = i, 2, where the
superscriptT denotes the transpose,are normallydistributed with the following mean vectorand covariancematrix:
411
Advertising and Sales Promotions in Umbrella Branding -A7 ii
-Al-
Bli
B, Ci
C,i
(4)
A2i
to be more advertisingsensitive than the averageconsumer in the toothpaste category. The corresponding crosscategorycorrelationscan be calculatedfrom the covariance matrixn. We denotethese cross-categorycorrelationsby A.6
- Nc
A2
B2i
B2
-C2i
-C2.
Choice Probabilities Let us rewriteEquation1 for each categorym, wherem = 1, 2, as -'
-E;A
"A1B, YBB ]EBI
"AlC,
nBC B11C, ?C|
!AIB2 HAB IIAIC,
nAC2
BBIA2
!CiA2
nClB !BiB, nBC
-CC2
nAA2
nBB2 I"AB2
nciA2 ?A2
"CIB
A,B2
mmijt
Vmijt
+
?mijt'
A,C2
nBiA,
nA2C2
(5)
i"A2B2
EB2 A 2C2
nB,C2
nCC2 r[
A2C2
A2C2
ZC2
The vectors Al, B , Ci are the means in Category I and the vectors A2, B2, C2 are the means in Category 2. The block diagonal matrices EAI, E?B,ICI, IA2, ?B2, and ?C2 are the covariancematricesof A l, Bij, Cli, A2i,B2i,and C2i, respectively.The diagonal elements ofA AI, B 1,ECi, IA2, ZB2, and ZC2themselves containthe variancesof the corresponding heterogeneity distributions. Finally, F! are the appropriatecovariancematrices. Within-CategoryCorrelations Given the large numberof covariancesto be estimatedas shown in Equation4, we allow only a subsetof these covariances to be nonzero for parsimony.5We allow heterogeneity in all brand-specific constants and allow own and cross marketing-mix response and use experience coefficients; however,we estimate and reportin the "Results"section of the article only the following covariances (correlations) among the heterogeneity distributions: (I) correlations among the brand specific constants and price (as well as advertising)sensitivities, (2) pairwise correlationsbetween own price (and advertising) sensitivities and all other marketing-mixsensitivities (e.g., display), and (3) pairwise correlationsbetween own price (and advertising)sensitivities and use experience. Cross-CategoryCorrelations Ainslie and Rossi (1998) suggest that consumersensitivities to marketing variables (own-effects of the marketing mix in our model) may be correlated across categories. Therefore, a person who is more price sensitive than the average consumer in one category can be also more price sensitive than the average consumer in another category. Therefore,we also estimate nBI B2 (the covariancematrixof Bl and B2). This means that we permit consumer price, advertising, display, feature, coupon, and use experience sensitivities to covary across the two categories. For example, a consumer who is more advertisingsensitive than the averageconsumer in the toothbrushcategory may also tend sHowever, in the empirical analysis, we tested the sensitivity of our results to the assumptions made about covariance structureby allowing more covariances(correlations)to be nonzero. For example, we estimated the covariancesbetween brand-specificconstants,a few covariancesamong the cross-responsecoefficients, and so forth,but these did not improvethe model fit significantly and did not change the parameterestimiates.
whereVmijtis the "deterministic"partof the utility. Let 0 denote the vector of parameters(Al, A2, BI, B2, Cl, C2, d(ZAI), d(EA2), d(ZBI), d(IB2), d(ECI), d(IC2), w(n))T. The vectors d(EAI), d(AA2), d(ECI), and d(IC2) containthe diagonalelements of the correspondingmatrices (i.e., the variances of the heterogeneity distributions of brandpreferencesand cross-effects response coefficients). The vectors d(EBi) and d(EB2) contain the diagonal elements(variancesof own-effects responsecoefficients)of the correspondingmatrices, as well as the off-diagonal elements, or covariances, that we allowed to be nonzero. Finally,the vector w(n) contains the covariancesamong the brand-specificconstantsand advertisingand price sensitivities, as well as cross-categorycovariancesto be estimated. Note that though we used the covariancematricesin defining the parametervectorfor notationalconvenience,we estimate the correspondingstandarddeviationsand correlations introduced previously, instead of the variances and covariances. Recall thatall coefficients in EquationI are allowed to be heterogeneousacross consumers,and their distributionsare given in Equation4. Let (pidenote the multivariatestandard normalvectorthatgeneratesthese coefficients.To be able to write down the choice probabilities,we need to specify a distribution for Emijt We assume that ?,mijtj e J is given by
(6)
?miit
= Ymiit -Et[Ymiit],
where y,ijt is the generator of ?,ijt, whose prior distribution
is
(7)
Y,io
Y2ij0_
N([o]P p I1
,
Note that Et[.] indicates expectation conditional on the informationat time t. Equations 6 and 7 indicate that the means of the randomcomponentsare always zero, whereas the assumption is that the initial variancesare one and the initial covariancebetween the randomcomponentsis p. A parsimonious and new way of capturing the utility variance-shiftingrole of use experienceandadvertisingis to allow the initial utility variance,which we set to unity,to be updatedconditionalon past purchases.Neitheradvertising's 6Finally,note that we adopt the recent classical inferencetechniquesto model and estimate unobserved heterogeneity. An alternative approach would have been Bayesiantechniques(e.g., Rossi, McCulloch,andAllenby 1996). However, the main advantageof such techniques is the ability to providehousehold-levelparameterestimates, which is particularlyuseful if the research objective includes micromarketingimplications, which is beyond the scope of this article.
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JOURNAL OF MARKETINGRESEARCH, NOVEMBER 2002
impact on the consumer utility mean nor its impact on the consumer utility variance across categories for umbrella brandshas been empiricallyverified in previouswork. To capturethe effects of both use experience and advertising on the varianceof the randomcomponent,thatis, utility variance,let the informationobtainedby consumeri by purchasingbrandj before time t fromcategorym be denoted by xmijt.This informationcan be about attributes,including quality,or match informationbetween consumertastes and productofferings and the like, which are not observedby the analyst. Let x,ijt be given by (8)
Xmijt
=
Ymijt
+
6mijt'
where 6i.jt is an i.i.d. random error term that reflects the level of noise in the informationobtained.We suppose that (9)
-
,mi.t
N(O, o62).
Note that additionalinformationcomes from weekly advertisements. Denoting this informationwith z,ijt , we let (10)
mijt
=
Ymijt
+ 1
Given these, we then assume that the randomcomponent of utility is updated according to Bayesian updatingrules and invoke the Kalmanfilter for updatingthe distributionof Y,nijtand therefore of ?,j1.t There is no updatingfor j = 0, which correspondsto no purchase.To derivethe Kalmanfilter, we regress Ymiit- Et- I[Ymiit]
(14)
b2mijt
am,, m,i,j,t-
b3mijt 2 b 12 _ 4mijt
i,3-m
- Et( ,mi,2 = E{[Ymijt 12i.jt=
_
12,i,j,t-I
m = 1,2
t)]2}
E {[Yijt- Et (ylijt)][Y2ijt - Et(Y2ijt)]}
Note that the precedingequationssuggest that the higher the precisionof informationfromuse experience(l/(' n)and advertising (I/'2 n) are (or the lower the use experience variability, 'n,, and advertising variability,oG2, are), the more consumerswill update. Given Equations 12 to 14, the distributionof ?1ijt, m = I, 2 at time t is given by 2 0
-.
O _101 (T 12ijt ijt
j=
1,2,...,J.
11t2 02i 12ijt
We then can write the choice probabilitiesconditional on 0 and (pias (16)
Probjt(D!ijt
= 1;p)=
= 1,Dt
)
tjk(0,(Pi),
where q)tjk(-)is the appropriatecumulativedistributionfunction whose probabilitydistributionfunction is determined from Equation 15. Note that this is a 2J + 2 dimensional integral.Therefore,the probabilityof consumer(household) i makingthe sequenceof purchasesgiven by D,,,jt, m = 1, 2, j = 0, , ..., J, t = 1, ..., Ti, conditionalon 0 and (pi,is T
(17)
J
J
Prob (0, i) = n
DlijtD2Ijtt ( pi).
t=l j =O k=0
m = 1, 2 Xmijt- Et_ I[xmijt]and Zmijt- Et_ i[Zmijt],
by suppressing the intercept,and we obtain the following updatingformulasfor the expectations: = Et[Ymiit] Et _[Ymijt] + blmijt{xmijt Et,- [mijt]} +
- Et
+ b4mijt{z3 -m,i,j,t
[nmi,j,t ]) Et[Z3
m,i,j, t]'
where bk,,lit, k = 1, 2 and m = 1, 2 are solved from 2 2 , t- I + 6m
-
Dm,i,jt-j
D3- m,i,j,t - 1o12,i,j,t -I 2 DDm i jt - I(Y m,i,j,t - I I Dm,i,j,t 11(12,i,j,t
(13)
D - Im,,j,
-
CY- 11,i,,-- II.. IniJDi.,t- I 12I,i,j,t
2 ++ 03 G
i,,t
D3 1- m i.j.t - lI312,i,,jt
D3 -m.i.,l t -
I
