combining related and sparse data in linear regression...

"COMBINING RELATED AND SPARSE DATA INLINEAR REGRESSION MCIDELS"

by

Wilfried VANHONACKERDonald LEHMANN**

andFareena SULTAN***

N° 89 / 19 (Revised March 89)

Associate Professor of Marketing, INSEAD, Boulevard de Constance,77305 Fontainebleau, France

George E. Warren Professor of Business, Graduate School of Business,Columbia University, New York, NY10027

Assistant Professor, Graduate School of Business Administration,Harvard University, Cambridge, MA02163

Director of Publication:

Charles WYPLOSZ, Associate Deanfor Research and Development

Printed at INSEAD,Fontainebleau, France

COMBINING RELATED AND SPARSE DATA IN LINEARREGRESSION MODELS

Wilfried R. Vanhonacker*Donald R. Lehmann**

andFareena Sultan***

Revised March 1989

* Associate Professor of Marketing, INSEAD, Boulevard de Constance,77305 Fontainebleau, France

** George E. Warren Professor of Business, Graduate School of Business,Columbia University, New York, NY10027

*** Assistant Professor, Graduate School of Business Administration, HarvardUniversity, Cambridge, MA02163

We would like to thank Gerry Tellis who was kind enough to share hisextensive data base on price elasticity estimates.

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COMBINING RELATED AND SPARSE DATA IN LINEAR

REGRESSION MODELS

ABSTRACT

Meta-analysis has become a popular approach for studying

systematic variation in parameter estimates across studies. This

paper discusses the use of meta-analysis results as prior

information in a new study. While hierarchical prior distributions

in a traditional Bayesian framework are characterized by complete

exchangeability, meta-analysis priors explicitly incorporate

heterogeneity in prior vectors. This paper discusses the nature of

the meta-analysis priors, tFieir properties, and shows how they can

be integrated into a familiar recursive estimation framework to

enhance the efficiency of parameter estimates in linear regression

models. This approach has the added advantage that it can provide

such estimates when (a) the design or data matrix is not of full

rank, or (b) when observations are too few to allow independent

estimation. The methodology is illustrated using published and new

meta-analysis results in market response and diffusion of

innovation models.

Keywords: META-ANALYSIS; PRIOR INFORMATION; RECURSIVE ESTIMATION;

RANDOM COEFFICIENT REGRESSION; EXCHANGEABILITY.

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1. INTRODUCTION

Meta-analysis has been promoted in a variety of fields as a way to

integrate research findings across various studies (Glass, McGaw, and Smith

1981, Hunter, Schmidt, and Jackson 1982, Peterson, Albaum, and Beltramini 1985,

Farley and Lehmann 1986, Hite and Fraser 1988, Sultan, Farley, and Lehmann

1988). Within an analysis-of-variance framework, the various studies are

viewed as imperfect replications of one overall but unplanned experiment. This

replication framework generalizes and enhances the understanding of systematic

variations in research findings based on design characteristics. Some

applications have appeared in the literature in recent years and a number of

interesting generalizations have been established and discussed (see, e.g.,

Assmus, Farley, and Lehmann 1984, Farley, Lehmann, and Ryan 1981, 1982,

Houston, Peter, and Sawyer 1983, Reilly and Conover 1983).

Meta-analysis findings, however, have rarely been incorporated as a source

of prior knowledge or information in subsequent studies. Given the design

characteristics of a new study, it should be rather simple to derive some prior

parameter estimates using meta-analysis results. When prior knowledge is

updated with sample information on the new situation, resulting parameter

estimates tend to be more stable and efficient (see, e.g., Lilien, Rao, and

Kalish 1981); hence, an attempt should be made to include prior information in

the estimation procedure.

Gain in efficiency from the use of prior information has been the reason

for extensive use and development of Bayesian statistical methods. The work by

Lindley and Smith (1972) which advocates the use of hierarchical prior distri-

butions has become the foundation for numerous applications. Early research on

integrating independently generated research findings into priors adopted this

approach (DuMouchel and Harris 1983, Kuczera 1983). However, the strong prior

assumption of exchangeability underlying hierarchical prior distributions has

been criticized as an oversimplification (Kass 1983). The assumption, usually

captured in a normal prior (Leaner 1978, p.49), states that the research

findings are related and that the data carry information about that relation-

ship. Because of differences in the research studies, heterogeneity is almost

certain to enter that relationship. Since prior information on heterogeneity

is often sparse, neither grouping nor the use of conditional priors has been

practical, and the traditional approach with its strict assumption of equal

information in all the prior studies has survived valid criticisms.

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In attempting to uncover systematic variations, meta-analysis results

explicitly recognize parameter heterogeneity across independently executed

studies. Hence, using the results as prior information circumvents the strict

exchangeability assumption underlying hierarchical Bayes priors and addresses

the valid concern often raised in empirical research relying on such priors.

Using a known recursive estimator in a linear regression framework, this paper

discusses how these priors can be integrated with sample information. Besides

the established gain in parameter efficiency, the approach has the added

advantage that it can provide updated parameter estimates in instances where

sample information is missing or insufficient to allow for independent

estimation. The reported empirical illustrations are very encouraging in that

respect.

2. DERIVATION OF META-ANALYSIS PRIORS IN A MULTIPLE REGRESSION FRAMEWORK

Published meta-analysis focus on individual parameters analyzed

independently using ANOVA. As the parameters in each of these studies are not

estimated independently, this approach to recovering systematic variance in

them has ignored their covariance structures. Casting meta-analysis within a

multivariate regression framework allows the incorporation of those covariances

which, in a familiar Generalized Least Squares (GLS) framework, enhances the

efficiency of the results and sharpens the statistical power of the

significance tests. Although conceptually not different from an ANOVA

approach, the regression approach is adopted here as efficient priors are

naturally preferred over inefficient ones. Furthermore, this approach

simplifies comparisons to the traditional hierarchical Bayes method which it

improves upon by assuming partial exchangeability. To set the tone for

subsequent discussion and pinpoint inherent limitations of the suggested

approach, the meta-analysis priors are derived here within a multiple

regression framework.

Meta-analysis attempts to capture and explain systematic variations in

parameter estimates across various studies. Hence, a set of original studies

are available for which linear regression models can be specified as

y. = X i + u.

1 1 1for i 1, 2, ... m (1)

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where y i is a (n x 1) vector containing observations on the dependent variable

in study i, X i is a (n x p) matrix containing observations on p non-stochastic

predictors in study i, Si is a (p x 1) vector containing the parameters, u i is

a (n x 1) disturbance vector, and m denotes the number of previous studies.

With respect to the disturbance terms, the assumptions are made that the

expected error is zero and that errors are independent both within and across

studies, or

E(u i ) = 0 for all i

2and E(u. 3 1= a. I for i j (2)1

0 otherwise.

In the current development of the methodology described hereafter, the

assumption of linearity in (1) is fundamental and represents a limitation.1

To the extent that prior studies cannot be cast in models which are linear in

parameters, the approach cannot be adopted. However, the methodology developed

subsequently is not confined to cases where disturbances are characterized by

scalar variance-covariance matrices as defined in (2). We confined ourselves

to this case here for simplicity.

The fundamental assumption of meta-analysis is that systematic variation

in the parameter vectors Oi can be related to study characteristics. For

example, systematic differences might be attributable to differences in product

category, industry, sample size, estimation method, etc. These study

characteristics define a multidimensional classification within which similar

studies can be grouped together. Within each cell, 1, of the classification,

the random coefficients assumption

0. + e.1 3

for all j (3)

holds with first and second moments

E(e.) = 0

for all j E k1 , 1 = 1, 2, ... k,

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and E(e. = Ej 1

= 0

if i j and i, j"

otherwise.

In these expressions, k 1 denotes the set of studies in group 1 of k groups. In

hierarchical Bayes priors (see, e.g., Kuczera 1983), the parameter vectors Oi

for all i would be assumed to be drawn from a prior distribution (most likely

normal) with a mean vector and covariance matrix. Accordingly, the O i 's are

exchangeable and this prior distribution captures the "connectedness" between

all studies (DuMouchel and Harris 1983). In other words, each parameter vector

of a previous study contributes information about any single parameter in the

basic linear model. This strict assumption can be tested (see, e.g., Kuczera

1983), and lead to grouping of studies which could be analyzed separately.2

Capturing and describing the heterogeneity explicitly as achieved in meta-

analysis enables simultaneous analysis which enhances efficiency and

understanding.

The parameter vectors of the various studies can be combined as

0 = 0 + e

where 0' (13', R2 . ), P'' = ... ) and e' (el e' )

assuming studies similar in terms of design characteristics (i.e., partially

exchangeable) are grouped together. Note that vector a has m vector entries,

one for each original study. The vector entries corresponding to similar

studies are identical N vectors. In other words, if the first three studies

belong to group one, then /1 = /11 ...).

Of specific interest in meta-analysis are the variations in the mean

vectors al for 1 1, 2, Systematic variations arising from design

characteristics imply that a Zc and, hence,

S = Zc e (4)

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That is, the mean parameter vectors can be expressed as a linear combination,

c, of specific design vectors which constitute the columns of matrix Z. Within

the multidimensional classification scheme adopted above, the design vectors in

matrix Z describe or explain group membership of the m original studies in the

k groups. These vectors could contain observations on continuous, categorical,

or dummy variables. Note that expression (4) not only describes the heteroge-

neity among the previous studies but at the same time captures the systematic

link among them.

In practice, the parameter vectors are not observed. Traditional meta-

analysis relies on parameter estimates in attempting to estimate vector c.

Hence, the basic model postulated is

0 = Zc + E (5)

whereW=(010',...0')with.containing unbiased least squares estimatesS.

of the parameters in study i, and E = e + u with u = 0 - 0 or the difference

between the estimated parameters and their true values (i.e., u i = (XiXi)Vu.

given (2)).

Given the assumptions about the random components in (2), (4), and (5), it

is evident that

E(E) = 0

2 _1E l + a l (X1X,) 0

E (EE') = Q =

2

(6)

0 E + a (X'X )k

m

Invoking Generalized Least Squares (GLS) principles given the non-scalar

variance-covariance matrix in (6), BLUE estimates of vector c in (4) can be

obtained which equal 3

and

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_1 _1-c (Z'S? Z) Z'Q 0

(7)

Stated simply,

_1 _1 _1c DO with D (Z'Q Z) Z'Q (8)

In a new study, a regression model similar to the ones specified in (1)

can be postulated. Specifically, we have

yo = X 0 +u

oo o

2 4with E(u

o) = 0 and E(u u') = a

o I.

oGiven the meta-analysis results in (7),

a prior estimate of 00 can be obtained from

b = Z 00

where Zo defines the design of the current study in terms of the

characteristics incorporated in the design matrix Z of the meta-analysis.

As shown in Appendix 1, the prior estimate b can be expressed aso

b = Z DM(3, + Z Dd + ZoDy

o o o o (9)

where y = E + u with D as defined in (8), M being a matrix of m identity

matrices stacked on top of one another, and d being a vector containing the

difference between the mean parameter vector of the corresponding study and the

mean parameter vector for the studies belonging to the same set as the new

study. This is the meta-analysis prior which will be updated with sample

information in the new study using a familiar recursive estimator. The use of

this estimator fits nicely into the advocated regression approach and, as will

be shown shortly, makes the approach applicable when data in the new study are

either incomplete or insufficient for independent estimation.

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3. RECURSIVE ESTIMATION

Updating the prior on 00 in (9) with sample information contained in yo

and Xo can be accomplished using the augmented system

Yo

=

Xo

uo

(10)

b - Z oDd

0 ZDM Z

oDy

When the prior information is with respect to a subset of the parameters in 00

(e.g.,°o1)'

the augmented

Yo

system

Xol

equals

Xo2

uo

°01 + 13o2

bol

- Zol

D1d1

Zol

D1M 0 Z

olD

1y1

where bol denotes the meta-analysis prior on 001 , and all other elements are

similar to the corresponding ones in (10) with 13,) = [g31 0,k] and X0 ,4X01 Xo2](where X

ol contains the columns in X

o corresponding to the parameters in S

ol ).

The system could be expanded further if independent prior information (e.g.,

separate meta-analysis results) was available for different subsets of 00.

This approach is similar to the Goldberger-Theil approach (Theil 1971)

with the apparent difference that (a) the prior values for 00 , b o , are adjusted

by ZoDd and (b) the matrix associated with vector 00 in the prior equation

(i.e. ZoDM in (10)) is different from an identity matrix. It is worth noting

that

b - Z Dd = Zo jc - Dd]0 o

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and, hence,

_1

b - Z Dd = Zoc - Z

o(Z 1 4 Z) Z'9 d.

o o

Accordingly, Dd can be interpreted as a least squares estimate of the design

matrix Z on the differences in group means. If no groups were identified

(i.e., complete exchangeability), the prior would equal Z oc. Moreover, ZoDd is

subtracted from this prior to adjust for the partial exchangeability captured

in the profiles contained in design matrix Z. In essence, b o is adjusted in

the direction of the mean parameter vector of the group to which the new study

belongs.

Expressing (10) compactly, we have:

y = X00 + v (11)

where y' = [y' (b - Z Dd)']o o o

X' = [X'

and v' = [u' (2.0Dy)']•

With the variance-covariance matrix of v, g , as derived in Appendix 2 thev

recursive estimator of 0 equalso

1 _1 _1

0 = (X'Qv X) X'Q

v y

(12)

which has BLUE properties.

Some interesting observations can be made with respect to the estimator

shown in (12). The development thus far assumed that all information for the

new study was available. This is not always the case in practice. For

example, data on some predictors contained in X could be missing. Hence, the0

data matrix Xo would have columns containing zeros and a direct estimate of 0

o

could not be obtained.5 Note, however, that the augmented matrix X in (11) is

still of full rank; hence, a recursive estimate of all parameters in 0o is

obtainable despite the missing data. This is essentially an approach to data

transferability (Aigner and Learner 1984) which, using completely exchangeable

priors, has been studied in Vanhonacker and Price (1988).

Recursive estimates can also be obtained when the number of observations

in yo and X0 are smaller than the number of predictors specified in the basic

model (i.e., the number of columns in Xo). Again, direct least squares

estimates of 0o could not be obtained since X' X would not be of full rank.

o o

The augmented system in (10), however, does provide estimates for 0o in this

case with degrees of freedom equal to the number of observations actually

available on the dependent and independent variables.

4. SOME ESTIMATION ISSUES

2The above discussion assumed that E. (for i = 1, 2, ..., k), a. (for i =

0, 1, 2, ..., m), and d are known. In practice, these terms will have to be

2estimated. a

o can be estimated as the average of the variances for the

corresponding studies in the meta-analysis,

2 m, 2a = (l/m ) .E a

1 i= 1i

2wherea.=[(y. - X.0.)' (y. - X. .)( n - 1)) and 0i contains the parameter

estimates of the ith

study, and m1 denotes the number of previous studies

belonging to the first set (to which the new study belongs).

ThematricesE.for i 1, 2, ..., k can be estimated as the average of

the covariance matrices for the corresponding studies in the meta-analysis,

(13)

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m.

.E1 (0. - 6.)(0. - (14)

m.

whereil.,..e1 (13./m.)withm.denoting the number of studies contained in the

.th1-- set. Note that assuming normality and invoking the Lindley and Smith

(1972) results, the estimates in (13) and (14) are related to the empirical

Bayes estimates using non-informative priors (see Aigner and Learner 1984).

2Estimatesfora.(i = 1, 2, ...m) can be obtained from the estimation

results of the previous studies. Specifically, the variance-covariance

matrices of the parameter estimates from the previous studies are estimates for

a. (X!X.)-1

(i = 1, 2, ...m) as incorporated in matrix 4, (see Appendix 2). The

difference between the mean parameter vectors contained in d can be derived

straightforwardly from the averaged estimates -6 i = j E l (0i /m i ) for i=1,2,...k.

5. EMPIRICAL ILLUSTRATIONS

Two different empirical illustrations are discussed. The first example

relies on two widely cited meta-analyses in the market response literature.

Both focus on a single but different parameter and were executed independently.

The second example relies on the meta-analysis of both parameters of the

linear, discrete analog of a popular model in the diffusion literature.

Market Response Models

The market response literature contains two meta-analyses: one on

advertising elasticities (Assmus, Farley, and Lehmann 1984) and one on price

elasticities (Tellis 1988). Accurate estimation of these parameters is

mandatory for optimal (e.g., profit maximizing) advertising spending and

pricing decisions. Elasticities are most often estimated using multiplicative

model specifications. Given that these can be linearized in parameters, the

procedure developed above can be implemented to enhance parameter efficiency

and stability in case of limited samples.

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In this illustration, time-series observations for a major brand in a low-

priced consumer product category were used together with the published meta-

analyses. Bimonthly observations were available on market share (the response

variable), advertising share, and relative price. Relative price was defined

as the brands price over the mean price of the three major competitors in the

category. These three competitors together account for over two-thirds of the

volume sold in each time period.

A multiplicative response model was specified as

In MSta + 0 In AS

t + y In RP

t + S In MS

t-1 u

t-1 t

where MSt denotes market share at time t, AS

t denotes advertising share at time

t, and RPt denotes relative price at time t. The parameters 0 and y denote,

respectively, the short-term advertising and price elasticities. Using the

meta-analysis prior on 0 from Assmus, Farley and Lehmann (1984) and the meta-

analysis prior on y from Tellis (1988), these parameters were estimated

recursively using five, six, seven, etc. sample observations. Before reporting

and discussing the results, a number of practical observations on the

implementation of the recursive estimator are in order.

Empirical studies seldom report standard errors of estimated parameters

(or exact t-values which would enable derivation of standard errors). The

data-base underlying Assmus, Farley, and Lehmann (1984) does not contain any

standard errors for the reported advertising elasticities. Tellis (1988)'s

data base contains some t-values but not for all price elasticities. Given

that published results are generally statistically significant, the standard

errors were derived assuming the corresponding t-values equaled three.6 With

this assumption, the entries for the matrices Q in (6) and T in (2-1) could be

derived. This assumption affected the weighting of the prior studies

incorporated in the meta-analysis data bases as well as the efficiency of the

derived parameter estimates.

Although the original data were used (with the original coding schemes),

the designs of both meta-analyses were reduced. This was done mainly to ensure

that the new study had a design profile which was matched in the data by at

least one study. Note that if that study belonged to an empty set, d in (9)

cannot be computed and, hence, no adjustment can be carried out as shown in

(10). The design reduction was done based on statistical significance. The

Assmus, Farley, and Lehmann (1984) design matrix Z was reduced from 26

Number of

Observations Meta-Analysis Prior on Singled

Parameter

Parameter Estimates (Estimated Variance)b

Based on

Sample observationsc Meta-Analysis Prior on Both

Parameters

Table 1

aELASTICITY ESTIMATES FOR MARKET SHARE RESPONSE MODEL

Advertising Price Advertising Price Advertising Price

5 0.2310(0.00) 1.7210(7.48) 0.0727(0.01) -2.3355(0.00) 0.0707(0.01) -2.3356(0.01)6 0.1530(0.01) -3.4815(12.8) 0.0633(0.00) -2.3363(0.00) 0.0639(0.00) -2.3359(0.01)7 0.1521(0.01) -3.4374(5.39) 0.0677(0.00) -2.3365(0.00) 0.0660(0.00) -2.3362(0.00)8 0.1544(0.00) -3.4914(4.67) 0.0684(0.00) -2.3365(0.00) 0.0672(0.00) -2.3363(0.00)9 0.1549(0.00) -4.0184(4.70) 0.0689(0.00) -2.3367(0.00) 0.0662(0.00) -2.3365(0.00)10 0.1520(0.00) -4.1684(4.05) 0.0679(0.00) -2.3368(0.00) 0.0650(0.00) -2.3366(0.00)11 0.0950(0.01) -3.5544(6.43) 0.0531(0.00) -2.3366(0.00) 0.0511(0.00) -2.3364(0.00)12 0.0462(0.01) -1.6317(10.7) 0.0372(0.00) -2.3357(0.01) 0.0383(0.00) -2.3356(0.01)13 0.0457(0.00) -3.5097(11.0) 0.0370(0.00) -2.3367(0.01) 0.0351(0.00) -2.3367(0.01)14 0.0554(0.01) -3.5970(9.87) 0.0408(0.00) -2.3367(0.01) 0.0390(0.00) -2.3367(0.01)15 0.0635(0.01) -4.2443(7.81) 0.0439(0.00) -2.3373(0.01) 0,0422(0.00) -2.3373(0.01)16 0.0582(0.01) -1.3461(5.05) 0.0420(0.00) -2.3348(0.01) 0.0431(0.00) -2.3348(0.01)17 0.0645(0.01) -1.0044(4.21) 0.0445(0.00) -2.3342(0.01) 0.0475(0.00) -2.3341(0.01)18 0.0548(6.00) -1.0553(3.78) 0.0452(0.00) -2.3342(0.01) 0.0558(0.00) -2.3340(0.01)19 0.0566(0.00) -1.0573(3.53) 0.0475(0.00) -2.3342(0.01) 0.0592(0.00) -2.3339(0.01)20 0.0610(0.00) -1.2296(3.26) 0.0504)0.00) -2.3344(0.00) 0.0605(0.00) -2.3341)0.00)

30 0.0739(0.00) -0.9645(1.45) 0.0640(0.00) -2.3320(0.00) 0.0642(0.00) -2.3320(0.00)

40 0.0937(0.00) -1.0894(0.99) 0.0806(0.00) -2.3294)0.01) 0.0832(0.00) -2.3293)0.01)

53 0.0493(0.00) 0.2452(0.55) 0.0473(0.00) -2.3083(0.01) 0.0563(0.00) -2.3112(0.01)

aMultiplicative response model with predictors advertising share, relative price, and lagged market share

bMeta-analysis priors derived from Assmus, Farley, and Lehmann (1984) and/or Tellis (1988)

cOLS estimates

dThe advertising elasticities shown were estimated using a prior on only that elasticity;the price elasticity estimates were estimated using a prior on only that elasticity

- 15 -

variables to the 11 most significant variables (including the intercept). The

Tellis (1988) design matrix Z was reduced from 21 variables to 10 variables

(including the intercept). The derived [bo - Z

o Dd] estimates equaled 0.0325

for the advertising elasticity and -2.3360 for the price elasticity.

It is important to note that the design profile of the new study matched

two studies out of 128 in the advertising-elasticity data base but only one out

of 421 in the price-elasticity data base. This limited representation has a

dual effect: first, the information contained in that subgroup will have

limited weight in the meta-analysis; second, as the adjustment of b o towards

the subgroup estimate is entirely a function of its members, if these are not

representative an inadequate adjustment can arise. These issues will have to

be kept in mind when evaluating the results, particularly for the price

response parameter.

The results are summarised in Table 1. Adding a single observation at a

time, the table shows the sample OLS estimates (no prior information), the

estimates using prior information on one of the two parameters, and the

estimates incorporating prior information on both parameters simultaneously.

The advertising parameter is very significant in the model and its estimate

gets adjusted very well in the direction of the "long-run" estimate (i.e., the

highly significant estimate over the entire sample observation period). For

example, the sample estimate with only five observations equals 0.2310;

incorporating prior information the estimate becomes, respectively, 0.0727 and

0.0707 which is much closer to the "long-run" estimate of 0.0493 and the

apparent equilibrium level between 0.05 and 0.06. For price, the sample

estimate is insignificant. Incorporating the corrected prior value of -2.336

draws the recursive estimate close to that value and makes it insensitive to

sample information. As the prior is adjusted towards a single study with a

highly significant price elasticity (i.e., a reported t-value of -8.33), the

significance of the recursive estimate is intuitively unreasonable given

limited representation. This raises an issue worth further investigation given

the otherwise very encouraging results.

New Product Diffusion: The Bass Model

In the modeling literature on the diffusion of innovations over time, the

Bass model has received considerable attention (Bass 1969). The basic premise

of the model can be stated as

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P(t) = p + (q/m)Y(t) (15)

where P(t) denotes the probability of adoption at time t, Y(t) denotes the

cumulative number of adopters by time t, and p, q, and m are the three model

parameters.7 Accordingly, the probability of adoption at one point in time is

specified as a linear function in the number of adopters by that time. Hence,

the model describes the adoption process of an innovation as a contagion-type

process.

Following the algebra in the original paper by Bass (1969), it can be

shown that

2S(t) = a + bY(t) + c[Y(t)] (16)

where S(t) denotes the number of adopters at time t, and the parameters a, b,

and c are functions of the three original parameters as follows: a = pm,

b = q-p, and c = (-q/m). In discrete terms, model (16) is expressed as

St = a + bY

t-1 + cY

t-1 (17)

where t refers to a discrete time period (e.g., a month, quarter, year, etc.).

This quadratic equation was suggested by Bass (1969) as the basic model for

empirical analysis.

Since the original paper by Bass (1969), substantial research on the

specification and estimation of the model has been done. The specification has

been extended to incorporate explanatory variables such as price, advertising,

etc. (see, e.g., Mahajan and Peterson 1985). OLS estimation of the parameters

of model (17) has been shown to have a time aggregation bias (Srinivasan and

Mason 1986). Maximum likelihood estimation (Schmittlein and Mahajan 1982) as

well as Nonlinear Least Squares (NLS) of the continuous time Bass model

specification have been shown to be preferable over OLS with NLS being the

preferred estimator in terms of fit and predictive validity (Mahajan, Mason and

Srinivasan 1986, Srinivasan and Mason 1986). The linearity-in-parameters

constraint of the methodology suggested above confines us to the OLS estimation

of a, b, and c in the discrete analog model (17). Despite the shortcomings of

OLS, the estimates are easier to obtain and provide good starting values for

NLS. Within these constraints, a number of product/process innovations were

analyzed empirically.

- 17 -

A meta-analysis of Bass diffusion model estimates is contained in Sultan,

Farley, and Lehmann (1988). However, that study is very much in the tradition

of Assmus, Farley, and Lehmann (1984) and Tellis (1988) and is confined to an

independent analysis of individual parameters. In order to illustrate the full

power of the regression framework adopted above (which allows simultaneous

analysis of all parameters), a small meta-analysis was performed on the Bass

model results in Table 2.8

For simplicity of the illustration and in order to

preserve enough observations for the estimation of the variance-covariance

matrices in (14), only a single dummy predictor was specified in (5). In other

words, the various innovations listed in Table 2 were broken down into 2

groups. The variable was whether or not the innovation was a consumer durable

product innovation. A distinction was made between consumer and industrial

durables because clear differences exist in their diffusion process

characteristics. The absolute magnitudes of the potential adopter populations

are very different with consumer durables generally having much larger

population figures than non-consumer durables. This results in striking

differences in the values of, for example, the parameter "a" in model (17)

which captures the number of adopters in the initial period of market

introduction.

As indicated in the derivation of expression (7), the covariances of the

estimated parameters should be incorporated in order to secure efficient

estimation of the meta-analysis parameters (i.e., vector c in (5)). However,

these covariances are not reported in the studies whose results are shown in

Table 2. Only the variances or standard deviations of the parameter estimates

are given. Accordingly, vector c in expression (5) was estimated with

variance-covariance matrices in which all off-diagonal elements were zero.

The intercept estimates for parameters a, b, and c (representing the

coefficients for an industrial product) were equal to, respectively, 8.52905,

0.56810, and -0.0049512; the additive effect for a consumer durable innovation

equaled, respectively, 641,196.0, -0.25537, and 0.0049511. These additive

effects essentially indicate that (as discussed above) the potential adopter

populations for consumer durables are much larger than those for industrial

innovations, and consumer durable innovations diffuse slower on their own than

industrial innovations. The R equaled 0.36. Given cross-sectional observa-

tions and a single predictor, this fit is impressive.

The methodology incorporating the meta-analysis results discussed above

was applied to forecasting sales of color televisions. The sales data were

obtained from Wind and Mahajan (1985, p.215) and consist of annu 1 observations

Table 2

Bass Model Regression Results for Consumer/Industrial Innovation

2

PRODUCT/TECHNOLOGY Time 11 of ob-

Period servations

3 -7

Electric Refrigerators '20-'40 21 104.67(10 ) 0.21305 -0.053913(10 ) 0.903

3 -7Home Freezers '46-'61 16 308.12(10 ) 0.15298 -0.077868(10 ) 0.742

3 -7

Black and White TV's '46-'61 16 2,696.20(10 1 0.23317 -0.025957(10 1 0.576

3 -7

Room Airconditioners '46-'61 16 175.69(10 ) 0.40820 -0.247770(10 ) 0.911

3 -7

Clothers Dryers '48-'61 14 259.67(10 ) 0.33968 -0.236470(10 ) 0.896

3 -7Power Lawnmowers '48-'61 14 410.98(10 ) 0.32871 -0.075506(10 ) 0.932

3 -7

Electr. Bed Coverings '49-'61 13 450.04(10 ) 0.23800 -0.031842110 ) 0.976

3 -7

Automatic Coffee Makers '48-'61 14 1,008.20(10 ) 0.28435 -0.051242(10 1 0.883

Steam Irons '49-'60 12 1,594.70(103) 0.29928 -0.058875(1 -7) 0.828

3 -7

Record Players '52-'61 10 543.94(10 ) 0.62931 -0.298170(10 ) 0.899

Holiday Inn Motels '54-'68 15 9.27866 0.35521 -0.000258 0.918

Howard Johnson '56-'69 11 10.64810 0.23687 -0.000502 0.759

Motor Lodges

Ramada Inns '62-'69 8 14.29150 0.14904 0.004850 0.746

Howard Johnson, '56-'69 11 24.58870 0.32699 -0.000150 0.930

Holiday Inn,

Ramada Inns

McDonald's Restaurants '55-'65 11 15.00440 0.52016 -0.000646 0.964

Rapid Bleach Proc. '59-'68 10 1.86071 0.54948 -0.014050 0.919

Conversion to 70% HO2 2

'62-'68 7 5.26971 1.02542 -0.006090 0.974

Deliv. Syst.

Continuous Bleach '44-'51 8 2.63906 0.90471 -0.014600 0.827

Ranges

Hybrid Corn '13-'41 9 3.51145 0.96654 -0.011107 0.926

aPublished results from Bass (1969) end Nevers (1972)

- 19 -

from 1963 until 1970.Some observations are in order with respect to the product

and the data. The Bass model has been fitted quite successfully to the color

television data (Bass 1969, Nevers 1972). Predictions of peak sales (which

occur within the period of observation) and the magnitude of the peak have been

quite accurate. Comparison of the results derived here with the published

independent Bass model predictions will constitute a relatively strong test of

the power of the recursive estimator.

The recursive estimation results and their implied estimates for the

parameters of model (15) are shown in Table 3.9 Three sets of results are

shown: one set of prior estimates derived from the meta-analysis estimates, one

set derived from the priors and the first annual sales observations for color

televisions, and one set derived from the priors and the first two annual sales

observations. Estimation results with more data points converged to the least

squares estimates based exclusively on color television sales observations

published in Bass (1969) and Nevers (1972). This rapid convergence indicates

that the sample observations contain much more information than the prior

information. This is intuitively reasonable since color televisions, relative

to the innovations listed in Table 2, are more an improvement of an existing

product (i.e., black/white television) than a discontinous innovation.

Accordingly, one might expect the faster rate of diffusion observed in the

recursive estimates in Table 3 as sample observations are added.

The corresponding unit sales forecasts are shown in Table 4. The pattern

of rapid diffusion is evident when comparing actual sales to the predictions

based on the prior estimates. Adding the first year sales figure adjusts the

forecasts in the right direction but there is still a significant

underprediction of actual sales. When the second year is added, the pattern is

adjusted further. A reasonable forecast is obtained with a slight

overprediction in the post sample years. This early 1965 forecast predicts

sales to peak in 1967 below the 7 million unit point. This forecast is more

acurate and much more reasonable than the industry expectations at that time

10(and even later; see Bass 1969, p.224). With a better-fitting meta-

analysis model than the one relied on in this illustration, the early forecast

is likely to improve further.

- 20-

Table 3

Estimation Results for Color Televisions

Parameter

Recursive Estimation

Meta-Analysis

Prior

Number of Observations Incorporated

1 2

a(103 )

b

7c(10

m(106

)

P

q

T*a

S(T*)(106

) b

)

641.205

0.31273

-0.10810

30.852

0.021

0.334

7.834

2.903

747.000

0.41389

-0.13918

31.445

0.024

0.438

6.315

3.824

747.000

1.01309

-0.42612

24.489

0.031

1.044

3.289

6.579

a Time of peak sales, or T* . [1/(p+q)7[1n(q/p)] (Bass 1969).

bMagnitude of peak sales, or S(T*) = [m(p+q) 2

]/4q (Bass 1969).

- 21 -

Table 4

Forecasts for Color Televisions

Forecast based on

Recursive Estimation

Number of Observations

Meta-Analysis

Incorporated

Actual

Prior

1 2

Time 1968 1972 1970 1967

Peak Sales

Magnitude 5.982 2.903 3.824 6.579

(106

)

Sales (106

) 1963 0.747 0.641 0.747 0.747

1964 1.480 0.837 1.048 1.480

1965 2.646 1.077 1.445 2.792

1966 5.118 1.370 1.942 4.758

1967 5.777 1.702 2.518 6.579

1968 5.982 2.059 3.109 5.915

-22-

6. SUMMARY AND CONCLUSION

Meta-analysis has been postulated in a variety of fields as a way to

integrate research findings across studies. Within an analysis-of-variance

framework, the parameter estimates of the various studies are viewed as

imperfect replications of one overall but unplanned experiment. This

replication framework generalizes and enhances the understanding of systematic

variations in research findings.

Meta-analysis results have never been used, however, as a source of prior

information in subsequent studies. Within the design underlying the meta-

analysis, it is rather straightforward to obtain prior estimates for the

parameters of a new study. This paper described a methodology to update this

prior information with sample information for the new study. The methodology

relies on a random coefficients regression framework and accomplishes the

updating task using a recursive estimator. The procedure is versatile and

allows for efficient estimation even when independent estimation is infeasible

because the data are either incomplete or too sparse. Empirical examples in

the market response modeling literature and in the diffusion modeling

literature are encouraging.

-23 -

FOOTNOTES

1. Linearity is an implicit assumption in ANOVA analyses.

2. Prior grouping is, of course, also possible but general lack of priorinformation on group membership makes this impractical.

3. The result shown in (7) suggests that the most efficient estimator of c inmodel (5) is the GLS estimator shown in (7).

4. Whether or not the similarity in model specification is justified dependson the problem studied and is largely determined by theoretical issues.Note, however, that if there is little similarity or analogy betweenprevious studies (incorporated in the meta-analysis) and the new study,the meta-analysis prior will be a non-informative prior.

5. Replacing missing values with zeros is the traditional approach in datatransferability (see e.g., Aigner and Learner 1984). Other approaches havebeen suggested in the literature on missing data (see, e.g., Dunsmuir andRobinson 1981, Gourieroux and Monfort 1981, Steward and Sorenson 1981).These approaches rely on likelihood functions and are often impractical.Working with zeros enables recursive estimation as a simple, practical,and conceptually appealing method to the problem investigated.

6. The importance of incorporating the estimated standard errors isemphasized in Montgomery and Srinivisan (1989).

7. At the aggregate level, parameter p in (15) denotes that fraction of thepopulation who adopt at a particular time whose decision to adopt was notinfluenced by the social environment (hence, the labeling of p as thecoefficient of "innovation"). Parameter q in (15) is labeled thecoefficient of "imitation" as it captures the effect of units sold on thelikelihood of adoption. Parameter m in (15) denotes the size of thepopulation which will ultimately adopt the innovation.

8. Water softeners and boat trailers from, respectively, Bass (1969) andNevers (1972) were not incorporated because of unrealistic (perhapsmisprinted) values.

2 29. Since a. for i = 1,2,...,m were not available for previous studies, a

o as

2shown in (13) could not be derived. The value for a

o incorporated in the

variance-covariance matrix 4 was set equal to the unit sales figure inv

year one.

10. With more than two sample observations, the results (not shown) converge tothe accurate forecasts shown in Bass (1969) and Nevers (1972).

- 24 -

Appendix 1: Derivation of Prior Estimate

Given expression (8), the prior estimate equals

b = Z DOo o

and, since 0 = 0 + u with terms as specified above,

b Z DO + ZoDu.

o o

Depending on the design of the new study, it will belong to one of the k

groups specified in terms of design characteristics in the meta-analysis. If

we arbitrarily assume that the study belongs to the first group (k 1 ), we can

write the parameter vector as

i3o = 1

eo

.

Hence,

0. = 0 + e. for all j k, (1-2)

o io

wheree.=e.-eo. For any other study 1 not belonging to set k 1 , we have

o

- 130(6i - 61) elo (1-3)

with 0i denoting the mean parameter vector for the set of studies containing

study 1.

Combining expressions (1-2) and (1-3) for all studies, we can specify the

augmented model

0 = M00 + d +

where E = [e'[eo e'

oe' I, and M denotes a matrix of m identify matrices1 2 MO

stacked on top of one another. Vector d contains zeros in the entries

corresponding to the parameter vectors for all studies belonging to the same

set as the new study, and the difference in mean parameter vectors (i.e.,

1) in the entries corresponding to parameters vectors for all studies

61

which do not belong to the same set as the new study. Accordingly, the prior

estimate bo in (1-1) can be expressed as shown in (9).

- 25-

Appendix 2: Computation of Variance-Covariance Matrix Qv.

Given expression (11)

21

a 0

E(vv') = 4v=

0 ZoDE(yy')D'Z'

o

21

a 0

0 ZoDyD'Z'

o

where E(yy') = y. Accordingly, y = ER + u)(E. + u)']

Assuming that E and u are independent and knowing that the new study belongs to

the first set, it can be shown rather easily that

22E 1+ a,(X1X,)

El

E,

E1

2E,

E,

2+ a 2 (X'2 X 2 )

E1

E1

E,

E,

E,

(2-1)

2

(E i +Ek) + a (X'X )-1

M M M

or simply y + E6) E,] where E denotes an (m x m) matrix containing all ones

and C)denotes a Kronecker product. Note that from the expression of the first

two diagonal elements in Y, it is evident that the corresponding prior studies

belong to the set which also contains the new study.

El E,

- 26 -

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INSRAD WORK/NC PAPERS SERIES

"The R i D/Production interface".

"Subjective estimation in integrating

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• Sponsorship and the diffusion oforganisational innovation: a preliminary viev'.

"Confidence intervals: an empiricalinvestigation for the series in the N-

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• note on the reduction of the workweek",

July 1985.

"The real exchange rate and the fiscalaspects of • natural resource discovery",

Revised version: February 1986.

'Judgmental biases in sales forecasting',

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"Forecasting political risks forinternational operations", Second Draft:

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"Conceptualizing the strategic process indiversified firma: the role and nature of thecorporate Influence process • , February 1986.

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'Les prime■ des offres publiques, le noted'informalion et le marche des transferts deconical* des societies'.

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Revised Version: March 1987

1986

86/01

Arnoud DE MEYER

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07/06 Arun K. JAIN,Christian PINSON andNaresh K. MALMOTKA

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87/37 Landis GABEL



1987

87/40 Carmen NAMES andPierre REGIBEAU

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"The dark side of CEO succession", November

'Product compatibility and the scope of entry',November 1987

87/41 Cavriel HAVAVINI andClaude VIALLET

87/42 Damien NEVEN andJacques-P. THISSE

87/43 Jean GABSZEVICZ andJacques-F. THISSE

87/44 Jonathan HAMILTON,Jacques-F. THISSEand Anita VESKAMP

87/45 Karel COOL,David JEMISON andIngemar DIERICKY

87/46 Ingemar DIERICKAand Karel COOL

1988

88/01 Michael LAVRENCE andSpyros MAKRIDAKIS


88/03 James TEBOUL


88/05 Charles WYPLOSZ

88/06 Reinhard ANGELMAR

88/07 Ingemar DIERICKX

and Karel COOL


and Susan SCHNEIDER

88/09 Bernard SINCLAIR-DESCAGN4

88/10 Bernard SINCLAIR-

DESCAGN6

88/11 Bernard SINCLAIR-DESCACNi

"Seasonality, size premium and the relationshipbetween the risk and the return of Frenchcommon stocks", November 1987

"Combining horizontal and verticaldifferentiation: the principle of max-mindifferentiation", December 1987

'Location", December 1987

"Spatial discrimination: Bertrand vs. Cournotin a model of location choice", December 1987

"Business strategy, market structure and risk-return relationships, a causal interpretation",December 1987.

"Asset stock accumulation and sustainabilltyof competitive advantage", December 1987.

"Factors affecting judgemental forecasts andconfidence intervals', January 1988.

"Predicting recessions and other turningpoints", January 1988.

"De-industrialize service for quality', January1988.

'National vs. corporate culture: implicationsfor human resource management", January 1988.

'The slanging dollar: is Europe out of step?",January 1988.

"Les conflits dans les canaux de distribution',January 1988.

"Competitive advantage: a resource based

perspective", January 1988.

"Issues in the study of organizational

cognition", February 1988.

'Price formation and product design throughbidding', February 1988.

'The robustness of some standard auction game

forms", February 1988.

"When stationary strategies are equilibrium

bidding strategy: The single-crossingproperty', February 1988.



08/14 Alain NOEL

'Business firms and managers in the 21st

century', February 1988

"Alexithysie in organizational life: theorganization man revisited', February 1988.

me interpretation of strategies: a study ofthe impact of CEOs on the corporation',March 1988.

88/29 Naresh K. MALHOTRA,Christian PINSON andArun K. JAIN

88/31 Sumantra CMOS/1AL andChristopher BARTLETT

'Consumer cognitive complexity end thedimensionality of aultidisensional scalingconfigurations', May 1988.

'Creation, adoption, and diffusion ofinnovations by subsidiaries of multinationalcorporations', June 1988.

88/30 Catherine C. ECKEL "The financial fallout from Chernobyl: riskand Theo VERMAELEN perceptions and regulatory response', May 1988.

88/24 B. Espen DC1010 and

Hervig LANGOHR

'The production of and returns froe industrialinnovation: an econometric analysis for a

developing country', December 1987.

'Market efficiency and equity pricing:international evidence and laplications forglobal investing', March 1988.

"Monopolistic competition, costs of adjustmentand the behavior of European employment',

Septenber 1987.

'Reflections on *Veit Unemployment' inEurope", November 1987, revised February 1988.

'Individual bias in judgements of confidence',March 1988.

'Portfolio selection by mutual funds, anequilibrium model", March 1988.

'De-industrialize service for quality',March 1988 (88/03 Revised).

'Proper Quadratic Functions with an Applicationto MDT", May 1987 (Revised March 1988).

'Equilibres de Nash-Cournot dans le arch!

europien 4u gel: un cis 6 les solutions enboucle ouverte et en feedback coincident',

Mars 1988

'Information disclosure, means of payment, and

takeover premix. Public and Private tenderoffers In Prance', July 1985, Sixth revision,

April 1988.

88/15 Anil DEOLALIKAR endLars-Hendrik ROLLER

88/16 Gabriel HAVAVINI

88/17 Michael BURDA

88/18 Michael BURDA

88/19 M.J. LAVRENCE andSpyros MAKRIDAKIS

88/20 Jean °ERMINE,

Damien NEVEN andJ.F. TIIISSE

88/21 James TEBOUL

88/22 Lars-Hendrik ROLLER

88/23 Sjur Didrik FLAN

and Georges 2.ACCOUR

88/32 Kasra FERDOVS and

'International manufacturing: positioningDavid SACKRIDER

plants for success', June 1988.

88/37 Mihkel M. TOMBAK

'The importance of flexibility inmanufacturing', June 1988.


'Flexibility: en important dimension inmanufacturing', June 1988.


"A strategic analysis of investment in flexibleeanufacturing systems', July 1988.

88/36 Vlkas TIBREVALA and 'A Predictive Test of the N80 Model thatBruce BUCHANAN

Controls for Non-stationarlty", June 1908.

08/37 Murugappa KRISHNAN 'Regulating Price-Liability Competition ToLars-Hendrik ROLLER Improve Velfere', July 1988.

88/38 Manfred KETS DE VRIES 'The Motivating Role of Envy : A ForgottenPatter in Management, April 88.

88/39 Manfred PETS DE VRIES "the Leader as Mirror : Clinical Reflections",

July 1988.

88/40 Josef LAKONISBOK and 'Anomalous price behavior around repurchaseTheo VERMAELEN tender offers', August 1988.

88/41 Charles VIPLOSZ

• Assysetry in the ENS! intentional or

systesicI", August 1988.

88/42 Paul EVANS

'Organisational development in thetransnational enterprise', June 1988.

88/43 B. SINCLAIR-DESGAGNE 'Croup decision support systems implementRayesian rationality', September 1988.

88/25

Everette S. GARDNER

and Spyros MAKRIDAKIS

88/26 Sjur Didrik FLAP'

and Georges ZACCOUR

88/27 Murugappa KRISHNAN

Lars-Rendrik ROLLER

88/28 Sunantra GHOSRAL andC. A. BARTLETT

'The future of forecasting', April 1988.

'Semi-competitive Cournot equilibrium in

multistage oligopolies', April 1988.

'Entry tame with resalable capacity',

April 1988.

'The multinational corporation as a netvork:

perspectives from interorganizational theory'',

May 1988,

88/44 Essaa MAHMOUD andSpyros MAKRIDAKIS

88/45 Robert KORAJCZYKand Claude VIALLET

88/46 Yves DOZ andAmy MEN

'The state of the art and future directionsin combining forecasts', September 1988.

'An empirical investigation of internationalasset pricing', November 1986, revised August1988.

'From intent to outcome: a process frameworkfor partnerships', August 1988.

88/47 Alain INATEZ,Els CIJS8RECRTS,Philippe NAERT and/let VANDEN A_AEELE

88/48 Michael BURDA

88/49 Nathalie DIERKENS

88/50 Rob VEITZ andArnoud DC MEYER

88/51 Rob VEITZ

88/52 Susan SCHNEIDER endReinhard ANGELMAR

88/53 Manfred KITS DE VRIES

88/54 Lars-Hendrik RULERand Mlhkel M. TOMBAK

88/55 Peter BOSSAERTSand Pierre MILLION

88/56 Pierre MILLION

88/59 Martin KILDUFF

88/60 Michael BURDA

88/61 Lars-Rendrik ROLLER

88/62 Cynthia VAN RUE,Theo YERMAELEN andPaul DE vouttas

'Asywaetslo cannibalise between substitutekite, listed by retailers'. September 1988.

"Reflections on 'Veit unesploysent' inEurope, II', April 1988 revised September 1988.

"Information asymmetry and equity issues*,September 1988.

'Managing expert systess: from inceptionthrough updating", October 1987.

"Technology, work, and the organisation) thelopact of expert systems", July 1988.

'Cognition and orgenisationel mmalyaielminding the store', September 1988.

'Vhatever happened to the philosopher - kings theleader's addict ion to power, September 1988.

'Strategic choice of flexible productiontechnologies and velfare implications',October 1988

'Method of moments tests of contingent claimsasset pricing models', October 1988.

'Sire-sorted portfolios and the violation ofthe random valk hypothesise Additionalempirical evidence and Implication for testsof asset pricing models', June 1988.

'The interpersonal structure of decisionmaking: a social cosparison approach toorganisational choice', November 1988.

'Is mismatch really the problem/ Some estimatesof the Chelvood Cate II model vith US data',September 1988.

"Modelling Cost structures the sell Systemrevisited', November 1988.

'Regulation, taxes and the market for corporatecontrol in Belgium', September 1988.

88/63 Fernando NASCIMENTOand Wilfried R.VANHONACKER

88/64 Kasra FERDOWS

88/65 Arnoud DE MEYERand Kasra FERDOWS

88/66 Nathalie DIERKENS

88/67 Paul S. ADLER andKasra PERDOWS

1989

89/01 Joyce K. BYRER andTavfik JELASSI

89/02 Louis A. LE BLANCand Tavfik JELASSI

89/03 Beth H. JONES andTavfik JELASSI

89/04 Kasra PERDOVS andArnoud DE MEYER

89/05 Martin KILDUFF andReinhard ANCELMAR

89/06 Mihkel M. TOMBAK andB. SINCLAIR-DESGAGNE

89/07 Damien J. NEVEN

89/08 Arnoud DE MEYER andHellmut SCHUTTE

89/09 Damien NEVEN,Carmen MATUTES andMarcel CORSTJENS

89/10 Nathalie DIERKENS,Bruno GERARD andPierre BILLION

"Strategic pricing of differentiated consumerdurables in a dynamic duopoly: a numericalanalysis", October 1988.

"Charting strategic roles for internationalfactories", December 1988.

"Quality up, technology dovn", October 1988.

"A discussion of exact measures of informationassymetry: the example of Myers and Majlufmodel or the importance of the asset structureof the fir.", December 1988.

"The chief technology officer", December 1988.

"The impact of language theories on DSSdialog", January 1989.

"DSS softvare selection: a multiple criteriadecision methodology", January 1989.

"Negotiation support: the effects of computerintervention and conflict level on bargainingoutcome", January 1989."Lasting improvement in manufacturingperformance: In search of a nev theory",January 1989.

"Shared history or shared culture? The effectsof time, culture, and performance oninstitutionalization in simulatedorganizations", January 1989.

"Coordinating manufacturing and businessstrategies: I", February 1989.

"Structural adjustment in European retailbanking. Some viev from industrialorganisation", January 1989.

"Trends in the development of technology andtheir effects on the production structure inthe European Community", January 1989.

"Brand proliferation and entry deterrence",February 1989.

"A market based approach to the valuation ofthe assets in place and the grovthopportunities of the fir.", December 1988.

88/57 vilfried VANHONACKER 'Clete transferability: esti.ating the responseand Lydia PRICE effect of future events based on historical

analogy", October 1988.

88/58 B. SINCLAIR-OCSGAGNE 'Assessing econoalc inequality', November 1988.and Mlhkel M. TOMBAK

89/11 Manfred KETS DE VRIESand Alain NOEL

89/12 Wilfried VANHONACKER




89/16 Wilfried VANHONACKER,

Donald LEHMANN andFareena SULTAN

89/17 Gilles AMADO,Claude FAUCREUX andAndre LAURENT

89/18 Srinivasan BALAK-RISHNAN andMitchell KOZA

"Understanding the leader-strategy interface:application of the strategic relationshipinterview method", February 1989.

"Estimating dynamic response models when thedata are subject to different temporalaggregation", January 1989.

"The impostor syndrome: a disquietingphenomenon in organizational life", February1989.

"Product innovation: a tool for competitiveadvantage", March 1989.

"Evaluating a firm's product innovationperformance", March 1989.

"Combining related and sparse data in linearregression models", February 1989

"Changement organisationnel et realtiesculturelles: contrastes franco-americalns",March 1989

"Information asymmetry, market failure andjoint-ventures: theory and evidence",March 1989

combining related and sparse data in linear regression...

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