Forecasts of Market Shares From VAR and BVAR Models a Comparison of Their Accurac

8/22/2019 Forecasts of Market Shares From VAR and BVAR Models a Comparison of Their Accurac

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International Journal of Forecasting 19 (2003) 95110

www.elsevier.com/locate/ijforecast

Forecasts of market shares from VAR and BVAR models: a

comparison of their accuracy

Francisco Fernando Ribeiro Ramos

Faculty of Economics, University of Porto, 4200Porto, Portugal

Abstract

This paper develops a Bayesian vector autoregressive model (BVAR) for the leader of the Portuguese car market to forecast the market

share. The model includes five marketing decision variables. The Bayesian prior is selected on the basis of the accuracy of the out-of-sample

forecasts. We find that BVAR models generally produce more accurate forecasts. The out-of-sample accuracy of the BVAR forecasts is also

compared with that of forecasts from an unrestricted VAR model and of benchmark forecasts produced from three univariate models.

Additionally, competitive dynamics are revealed through variance decompositions and impulse response analyses.

2002 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved.

Keywords: Automobile market; BVAR models; Forecast accuracy; Impulse response analysis; Marketing decision variables; Variance

decomposition; VAR models

1. Introduction related microeconomic variables, such as industry

and firm sales forecasting.

Multiple time series models have been proposed, The use of VAR models for economic forecasting

for some time, as alternatives to structural econo- was proposed by Sims (1980), motivated partly by

metric models in economic forecasting applications. questions related to the validity of the way in which

One such class of multiple time series models, which economic theory is used to provide a priori justifica-

has received much attention recently, is the class of tion for the inclusion of a restricted subset of

Vector Autoregressive (VAR) models. VAR models variables in the structural specification of each1

constitute a special case of the more general class of dependent variable. Sims (1980) questions the use

Vector Autoregressive Moving Average (VARMA) of the so-called exclusionary and identificationmodels. Although VAR models have been used restrictions. Such time series models have the

primarily for macroeconomic models, they offer an appealing property that, in order to forecast the

interesting alternative to either structural economet- endogenous variables in the system, the modeller is

ric, univariate (e.g., BoxJenkins/ARIMA or ex- not required to provide forecasts of exogenous

ponential smoothing) models, or multivariate (e.g.,1In the Marketing field these arguments are mutatis mutandisVARMA) models for problems in which simulta-

also valid. The lack of a generally accepted theory about aggregateneous forecasts are required for a collection of

market response and marketing mix competition means that there

is little a priori reason to support or reject any of a number of

E-mail address: [email protected] (F.F. Ribeiro Ramos). plausible model specifications.

0169-2070/02/$ see front matter 2002 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved.

P I I : S0169-2070(01)00125-X


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96 F.F. Ribeiro Ramos / International Journal of Forecasting 19 (2003) 95110

3explanatory variables; the explanatory variables in an simony may present serious problems when the

econometric model are usually no less difficult to model is used in a forecasting application. Thus, the

forecast than the dependent variables. In addition, the use of VAR models often involves the choice of

time series models are less costly to construct and to some method for imposing restrictions on the model

estimate. This does not imply, however, that VAR parameters: the restrictions help to reduce the num-models necessarily offer a parsimonious representa- ber of parameters and (or) improve their estimation.

tion for a multivariate process. While it is true that One such method, proposed by Litterman (1980),

any stationary and invertible VARMA process has an utilises the imposition of stochastic constraints,

equivalent representation as a VAR process of an representing prior information, on the coefficients of

eventual infinite order (see, for example, Fuller, the vector autoregression. The resulting models are

1976), generally the VAR representation will not be known as Bayesian Vector Autoregressive (BVAR)

as parsimonious as the corresponding VARMA repre- models.

sentation, which includes lags on the error terms as In the Marketing literature, applications of multi-

well as on the variables themselves. Despite this lack ple time series models (Transfer Functions, Interven-

of parsimony, and the additional uncertainty imposed tion and VARMA models) include Aaker, Carman

by the use of a finite-order VAR model as an and Jacobson (1982), Adams and Moriarty (1981),

approximation to the infinite-order VAR representa- Ashley, Granger and Schmalensee (1980), Bass and

tion, VAR models are of interest for practical fore- Pilon (1980), Bhattacharyya (1982), Dekimpe and

casting applications because of the relative simplicity Hanssens (1995), Franses (1991), Geurts and Whit-

of their model identification and parameter estima- lark (1992), Grubb (1992), Hanssens (1980a,b),

tion procedures, and superior performance, compared Heyse and Wei (1985), Heuts and Bronckers (1988),

with those associated with structural and VARMA Jacobson and Nicosia (1981), Krishnamurthi,2

models. Brodie and De Kluyver (1987) have re- Narayan and Raj (1986), Kumar, Leone, and Gaskinsported empirical results in which simple nave (1995), Leone (1983, 1987), Lui (1987), Moriarty

market share models (linear extrapolations of past and Salamon (1980), Moriarty (1985), Sturgess and

market share values) have produced forecasts as Wheale (1985), Takada and Bass (1988), Umashan-

accurate as those derived from structural econometric kar and Ledolter (1983).

market share models. Furthermore, the same paper In this paper, we develop a BVAR for the leader ofshows that using lagged market share often gives the Portuguese car market for the period 1988:1

better results than an econometric model, which throughout 1993:6 using monthly data. The rationale

incorporates marketing mix variables. Indeed, for the choice of a multiple time series technique is

Danaher and Brodie (1992) provided a criterion two-fold. Structural models of market share, as

which determines whether it is advantageous to use surveyed in Cooper and Nakanishi (1988), tend to be

marketing mix information for forecasting market based on a number of generalisations about the

shares. effectiveness and relative importance of advertising,

The number of parameters to be estimated may be price, and other elements of the mix, with little

very large in VAR models, particularly in relation to emphasis being placed on the correct determination

the amount of data that is typically available for of exogenous assumptions and on the appropriate

business forecasting applications. This lack of par- dynamic model specification. Secondly, and pre-

2Very few VARMA analyses of higher-dimensional time series

3(e.g., models with more than four series) are reported in the Apart from the multicollinearity between the different lagged

literature. The wider class of vector ARMA models were not variables leading to imprecise coefficient estimates, the large

considered because there was little evidence of moving average number of parameters leads to a good within-sample fit but poor

components and because both the identification and estimation of forecasting accuracy because, according to Litterman (1986a, p.

such models are relatively complicated. For a recent summary of 2), parameters fit not only the systematic relationships . . . but

the specification of VARMA models, see Tiao and Tsay (1989). also the random variation.


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F.F. Ribeiro Ramos / International Journal of Forecasting 19 (2003) 95110 97

sumably in part because of this, the forecasting processes with positive definite contemporaneous

performance of such models has been poor compared covariance matrix and zero covariance matrices at all

to that of time series models: for such evidence, see other lags, and the B s are (n 3 n) coefficientkBrodie and De Kluyver (1987), Danaher and Brodie matrices with elements b . This approximationij k

(1992), and Brodie and Bonfrer (1994). Out-of- assumption holds, in fact, if Y is a covariance-tsample one-through 12-months-ahead forecasts are stationary linearly regular process. Eq. (1) can be

computed for the leaders market share and their used to generate the forecast f at time t of Y ,t,h t1haccuracy is evaluated relative to that of forecasts with subsequent forecast error e 5 Y 2f andt,h t1h t,h

9from an unrestricted VAR model and from three error variancecovariance matrix V 5E(e ? e ).h t,h t,h4

best-fitting univariate models. Granger and Newbold (1986, chapter 7) show that

The paper is organised as follows. The first section the optimal (in terms of minimising the quadratic

briefly describes the VAR and the BVAR modelling form associated with V ) h period ahead forecast fh t,hmethodologies. The second section describes the ofY made at time t ist1hdatabase used and the rationale behind the choice of

pthe variables. The third section presents the selectedf 5

OB f (2)models, the main empirical results, and illustrates the t,h k t,h2k

k51use of impulse response analysis and variance de-

composition as a marketing tool in providing in-where f 5 Y for k5 h,h 1 1, . . . ,p, andt,h2k t2(k2h)

formation about the competitive dynamics of theB s are the coefficient matrices in Eq. (1).k

market. We conclude with a section on the limita-

tions of our research and possible extensions.2.1. The unrestricted VAR

In a VAR with n variables there is an individual2. VAR and BVAR modelling

equation for each variable. For the unrestricted case

there are p lags for each variable in each equation.The theory underlying VAR models has its founda-For example, the equation for the ith variable istion in the analysis of the covariance stationary

linearly regular stochastic time series. We assume p p9here that Y is (n 3 1) in dimension, i.e. Y 5t t Y 5Ob Y 1 ? ? ? 1Ob Y 1 e . (3)it i 1k 1, t2k ink n,t2k it

k51 k51(Y , . . . ,Y ). By Wolds decomposition theorem, Y1t nt t possesses a unique one-sided vector moving-average

As in the problem of seemingly unrelated regres-representation which, assuming invertibility, givessions, when the right-hand-side variables are therise to an infinite-ordered VAR. In empirical work itsame in all equations, the applications of the OLSis assumed that Y can be approximated arbitrarilytequation by equation is justified. The coefficientwell by the finite pth-ordered VAR:estimates are maximum likelihood estimates (MLE)

p only if the es are normally distributed, otherwiseY 5OB Y 1 e (1)t k t2k t quasi-MLE. The unrestricted VAR has been used

k51

extensively by Sims (1980), and in the initial stagesof model building by Caines, Keng and Sethi (1981),where e is a zero-mean vector of white noisetTiao and Box (1981), and Tiao and Tsay (1983).

The main problem with the unrestricted VAR is

the large number of free parameters that must be

estimated. Since the number of parameters increases4The basic metric of forecasting comparisons is the calculation of

quadratically with the number of variables, evenMAPE, RMSE and Theils U statistics across the range of

moderately sized systems can become highly over-forecasting horizons (out-of-sample forecasts) for the severalmodels. parameterised relative to the number of data points.


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5This over-parameterisation results in multicol- similar in many respects to the ridge and Stein

linearity and loss of degrees of freedom that can lead estimators. Since there is a ridge regression analogy

to inefficient estimates and large out-of-sample fore- to the BVAR, it is not surprising that BVAR solved

casting errors. While estimation of such a highly the multicollinearity problem. As is well known,

parameterised system will provide a high degree of from a Bayesian standpoint, shrinkage estimators candata fitting, the out-of-sample forecasts can be very be generated as the posterior means associated with

poor in terms of mean square error. Because of these certain prior distributions. While Littermans es-

problems, researchers have suggested imposing vari- timator can be justified as a posterior mean, the

ous types of parameter restrictions on VAR models. economic content of the prior information is not

Several types of these restrictions are described in strong.

the literature. One solution is to exclude insignificant To demonstrate Littermans procedure, consider

variables and (or) lags based on statistical tests. An the ith equation of the VAR model (3):

alternative approach to overcome over-parameterisa-p ption is to use a BVAR model as described in

Y 5 d 1Ob Y 1 ? ? ? 1Ob Y 1 eLitterman (1980), Doan, Litterman and Sims (1984), it it i 1k 1, t2k ink n,t2k itk51 k51

Todd (1984), Litterman (1986b), and Spencer(4)(1993).

where d is the deterministic component of Y and2.2. The Bayesian VAR it it can include the constant, trend, and dummies. Litter-

mans prior is based on the belief that a reasonableThe Bayesian approach starts with the assumptionapproximation of the behaviour of an economicthat the given data set does not contain informationvariable is a random walk around an unknown,in every dimension. This means that by fitting andeterministic component. For the ith equation theover-parameterised system some coefficients turn outdistribution is centred on the specificationto be non-zero by pure chance. Since the influence of

the corresponding variables is just accidental andY 5 d 1 Y 1 e . (5)does not correspond to a stable relationship inherent it it i,t21 it

to the data, the out-of-sample forecasting perform-ance of such models deteriorates quickly. The role of The parameters are all assumed to have means ofthe Bayesian prior can therefore be described as zero except for the coefficient on the first lag of theprohibiting coefficients to be non-zero too easily. dependent variable, which has a prior mean of one.Only if the data really provides information, will the All equations in the VAR system are given the samebarrier raised by the prior be broken. form of prior distribution.

In an attempt to reduce the dimensionality of In addition to the priors on the means, theVARs, Litterman (1980) applied Bayesian techniques parameters are assumed to be uncorrelated with eachdirectly to the estimation of the VAR coefficients. His other (the covariances are set equal to zero) and toprocedure generates a shrinkage type of estimator have standard deviations which decrease the further

back they are in the lag distributions. The standard

deviations of the prior distribution on the lag co-5An alternative to a VAR is a simultaneous equations structural efficients of the dependent variable are allowed to bemodel. However, there are limitations to using structural models

larger than for the lag coefficients of the otherfor forecasting since projected values of the exogenous variablesvariables in the system. Also, since little is knownare needed for this purpose. Further, Zellner (1979), and Zellner

and Palm (1974) show that any linear structural model can be about the distribution of the deterministic compo-expressed as a VARMA model, the coefficients of the VARMA nents, a flat or uninformative prior giving equalmodel being combinations of the structural coefficients. Under weight to all possible parameter values is used. Incertain conditions, a VARMA model can be expressed as a VAR

equation form the standard deviation of the priormodel and a VMA model. A VAR model can therefore bedistribution for the coefficient on lag k of variable jinterpreted as an approximation to the reduced form of a structural

model. in equation i is


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suggest minimising the log determinant of the sam-g] ifi 5jd ple covariance matrix of the one-step-ahead forecastk

s 5 (6) errors for all the equations of the BVAR.ij k g ? w ?si]]]

5ifij.d

k ?sj

3. DataIn Eq. (6), s is the estimated standard error ofjresiduals from an unrestricted univariate autoregres-

The database used for this study is a monthly timesion on variables j. Since the standard deviations of

series sample of market shares, and marketing mixlag coefficients on variables other than the dependent

variables, for the period 1988:1 to 1994:6 in the carvariables are not scale invariant, the scaling factormarket in Portugal. The marketing mix variables s /s is used. This ratio scales the variables toi jincluded retail prices, advertising expenditures by

account for differences in units of measurement andmedia (TV, radio, and newspapers), and an age

thus enables specification of the prior without con-variable for the brand leading the car market. The

sideration of the magnitudes of the variables. ThePortuguese car market consists of 25 imported car

term g the overall tightness of the prior is the priorbrands, but the top seven account, on average, for

distribution standard deviation of the first lag of the 82.3% of the total market, with a standard deviationdependent variable. A tighter prior can be produced

of 4.75%. The leader is a general brand, present inby decreasing the value of g. The term d the decay

all segments of the market and represents, on aver-parameter is a coefficient that causes the prior

age, 16.8% of the total market, with a standardstandard deviations to decline in a harmonic manner.

deviation of 3.56%.The prior can be tightened on increasing lags by

The time series variables used in this study areusing a larger value for d. The parameter w the

defined as follows: MS1 is the market share of thetrelative tightness is a tightness coefficient for vari-leader brand; A1 is the relative age of the leadertables other than the dependent variables. Reducingbrand; P1 is the relative price of the leader brand;tits value, i.e. decreasing the interaction among theTVS1 is the TV advertising expenditures in shares oftdifferent variables, tightens the prior. Note that thethe leader brand; RS1 is the Radio advertisingt

prior distribution is symmetric. The same prior expenditures in shares of the leader brand; and PS1tmeans and standard deviations are used for eachis Press (newspapers and magazines) advertising

independent variable in each equation and acrossexpenditures in shares of the leader brand.

equations, and the same priors are used for eachThe data on MS1 is calculated from the monthlytdependent variable across equations. Doan et al.

new automobile registrations. This data is published(1984) have also considered another type of prior,

by the Portuguese General Directorate of Transports.known as general. In a general prior the interaction

As can be seen, Fig. 1 shows that our brand is notamong the variables leads to the specification for the

successful in increasing its market share. This seriesweighing matrix, f(i,j), given byseems to be stationary and does not present seasonal

fluctuations.1 ifi 5jf(i,j) 5 .Hf ifij (0 ,f , 1).ij ij

The BVAR model is estimated using Theils

(1971) mixed-estimation technique which involves

supplementing data with prior information on the

distribution of the coefficients. To apply Littermans

procedure one must search over the parameters g, d,

and w until some predetermined objective function is

optimised. The objective function can be the out-of-

sample mean-squared forecast error, or some otherFig. 1. Market share of the Portuguese market car leader.measure of forecast accuracy. Doan et al. (1984)


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The marketing decision variable A1 measures thetage (in months) of the different brand car models

after their introduction in the market. This variable

represents the models life cycle of the brand, and can

be seen as the product decision variable. It wasobtained as follows:

for the leader, we measure the age (in months)

after the launch of the most representative model

of each market segment, i.e. the model with the Fig. 3. Advertising expenditures in share by media.highest segment share. We apply a pseudo-

segmentation method based on the horsepower,

and followed by the Portuguese Trade Au- plots the relative price and the relative age of our

tomobile Association. This segmentation creates leading brand. The overall trend of A1 is upward,tfour segments: S1 (lower), S2 (lower-middle), S3 and seems negatively related to the market share

(upper-middle), and S4 (upper); series. The price variable is stationary around the

for the competing brands, we calculate the simple value of 1, indicating that our brand price is approxi-

average age of the most representative model of mately equal to the competitors brand prices.

each segment; The data on TVS1 , RS1 , and PS1 are expressedt t t to obtain the brand average age of its competitors, as shares of total advertising expenditures by media

we calculate the weighted average age for the and are obtained from Sabatina. This Portuguese

models chosen on each segment. The weights are firm records on a monthly basis the advertising

given by the relative importance of each segment expenditures by media and brand. These advertising

on total demand (S11S21S31S4); expenditures represent only official or contractual

the relative age, called A1, is then calculated as prices, and we know in the industry that prices are

the ratio between the weighted average age of the frequently lower. As indicated in Fig. 3, the shares of

brand and the weighted average age of its com- the major media advertising expenditures of our

petitors. brand fluctuated widely over the observed timeperiod.

The variable P1 is obtained following the steps All variables are measured in logs to help reducetjust described for A1 . The weights are the same, and the problem of heteroscedasticity. The data set istthe price for each model is the consumer price (all included in Appendix A.

taxes included) of the most representative model of

each segment. The price data are published on a

monthly basis and are recorded in the Guia do 4. Empirical applicationAutomovel (The Portuguese Car Magazine). Fig. 2

4.1. Models selected

Three classes of models are included in ourempirical comparisons, each class being represented

by one or more specific models. The classes are

univariate, unrestricted VAR, and BVAR. The usual

criteria, for example stationary, autocorrelation, and

partial autocorrelation functions, significance of co-

efficients, and the Akaike Information Criterion, are

used to select the best models. In all computations

we have used the RATS program (RATS 386,Fig. 2. The brands age and price evolution. version 4.02).


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In the class of univariate models we have consid- limitations. However, these can be obtained from the

ered four models (an ARIMA, a NAIVE, a RE- author upon request.

GRESSION and an exponential smoothing, specifi- To determine the optimal lag length of the unre-

cally a HoltWinters, model). stricted VAR in levels [VAR(U)] we have employed

The best-fitting ARIMA model for MS1 is as the likelihood ratio test statistic (LR), suggested by6follows: Sims (1980). Given the number of observations, we

have considered a maximum lag of nine and then(1 2 0.916B)MS1 5 21.898 1(1 1 0.814B)et ta tested downwards. The LR test supports the choice(0.06)a a( 0.03 ) ( 0.07 )

of six lags. The model is then estimated in levels (so

that it is comparable to the BVAR model) with 37Standard errors are in parentheses and superscript aparameters (including the constant) in each equation.indicates significance at the 0.01 level. As Montgom-

In the class of BVAR models, the variables areery and Weatherby (1980, p. 306) note: The Boxspecified in levels because, as pointed out by Sims,Jenkins approach uses inefficient estimates of im-Stock and Watson (1990, p. 360), . . . the Bayesianpulse response weights which are matched against a

set of anticipated patterns, implying certain choices

of parameters . . . the analyst skills and experience 6If AR(m) is the unrestricted VAR and AR(l) the restricted VAR,often play a major role in the success of the modelwhere m and l are the respective lags, then the LR statistic forbuilding effort. For other criticisms of ARIMA, seetesting AR(l) against AR(m) is given by

Chatfield and Prothero (1973), Hillmer and Tiao

(1982), and Prothero and Wallis (1976). LR5(T2c)(lnuVu2lnuV u)l mThe NAIVE (MS 5 a1bMS ), the REGRES-t t21

where T is the number of observations, c is the correction factorSION (MS 5a1 bMS 1 marketing mixt t21which is equal to the number of regressors in each equation in ARcovariates) and the HoltWinters models are com-(m), and V is the covariance matrix of residuals of AR(l) and

pared in terms of forecasting accuracy with the other AR(m), respectively. The statistic LR is asymptotically distributed2four models in Table 1. The final specification of as chi-squared with k (m 2 l) degrees of freedom, where k is the

number of regressions.these models is not presented here due to space

Table 1

Accuracy of out-of-sample forecasts (1993:11994:6)

Month Accuracy N NAIVE Holt REG. ARIMA VAR(U) BVAR(S) BVAR(G)

ahead statistic Winters

1 U 18 0.965 0.901 0.904 0.979 0.999 0.813 0.831

MAPE 0.025 0.021 0.021 0.029 0.031 0.108 0.019

RMSE 0.184 0.170 0.172 0.187 0.191 0.155 0.159

3 U 16 1.102 0.841 1.013 0.846 0.894 0.761 0.747

MAPE 0.039 0.025 0.038 0.025 0.029 0.017 0.016

RMSE 0.228 0.173 0.210 0.175 0.185 0.158 0.149

6 U 13 0.985 0.685 0.919 0.665 0.596 0.65 0.633

MAPE 0.035 0.020 0.033 0.017 0.014 0.016 0.015RMSE 0.257 0.179 0.240 0.174 0.156 0.169 0.162

12 U 7 1.39 1.15 1.30 1.13 1.06 1.09 0.04

MAPE 0.049 0.040 0.045 0.039 0.032 0.034 0.031

RMSE 0.256 0.216 0.239 0.207 0.195 0.20 0.191

Average U 1.110 0.895 1.034 0.905 0.887 0.828 0.812

MAPE 0.037 0.027 0.034 0.028 0.027 0.021 0.020

RMSE 0.231 0.185 0.215 0.186 0.182 0.17 0.165

Note: N is the number of observations. The RMSEs, the MPEs and the U statistics are reported for log MSI. Average is the average of

the 1-, 3-, 6- and 12-months-ahead RMSEs and the U statistics.


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MS1 A1 P1 TVS1 RS1 PS1approach is entirely based on the likelihood function,MS1 1 0.75 0.75 0.5 0.5 0.5

which has the same Gaussian shape regardless of the A1 0.5 1 0.75 0.75 0.75 0.75presence of nonstationarity, [hence] Bayesian infer- f(i,j)5 P1 0.5 0.75 1 0.5 0.5 0.5 .

TVS1 0.5 0.5 0.5 1 0.75 0.75ence needs to take no special account of nonstation-

3 4RS1 0.5 0.5 0.5 0.75 1 0.75

arity (see also Sims, 1988, for a discussion on PS1 0.5 0.5 0.5 0.75 0.75 17Bayesian scepticism on unit root econometrics). The

4.2. Evaluation of accuracymodels are estimated with six lags of each variable.

Longer lags (up to nine) were also tried, but theThe accuracy of the forecasts for 1993:1 to 1994:6substantial results remained unchanged.

is measured by the MAPE, the RMSE and the TheilThe optimal Bayesian prior is selected by examin-U statistics for 1- to 12-months-ahead forecasts. IfAing the Theil U and the RMSEs values for the tdenotes the actual value of a variable, and F theout-of-sample forecasts. In a first step we assume a tforecast made in period t, then the MAPE, the RMSEsymmetric prior, i.e. f(i,j) 5 w, ij [BVAR(S)],and the Theil statistic are defined as follows:then we relax this assumption to take into account a

more general interaction between the variables K uA 2 F u1 t1j1k t1j1k[BVAR(G)]. ] ]]]]]

MAPE5

ON A t1j1kj51In our search for the symmetric prior we have

considered three values for w: 0.25, 0.5, 0.75. For k 0. 52the parameter g we have assumed a relatively loose RMSE5 O(A 2 F ) /NH Jt1j1k t1j1k

j51value of 0.3 and a tight value of 0.1. We set the

harmonic lag decay, d, to 1 as recommended byU5RMSE(model)/RMSE(random walk)

Doan et al. (1984). This has given us six alternative

specifications. The best values according to our where k5 1,2, . . . ,12 denotes the forecast step and N8

criterion function were obtained for g 50.3, w 50.5 is the total number of forecasts in the prediction

and d51. To specify the general prior we must period.

define g, d, and the interseries tightness parameters, The U statistic is the ratio of the RMSE for the

f(i,j). Using the information provided by impulse estimated model to the RMSE of the simple random

response analysis and variance decompositions, and walk model which predicts that the forecast simplyafter some initial search, the best values were equals the most recent information. Hence, ifU , 1,

obtained for d51, g 50.15 and the model performs better than the random walk

model without drift; if U . 1, the random walk

outperforms the model. The U statistic is therefore a7This kind of discussion about classical vs. Bayesian analysis of relative measure of accuracy and is unit-free. The

time series was the subject of a special issue of the Journal of forecasted value used in the computation of theApplied Econometrics (OctoberDecember 1991). One of the MAPE, the RMSE and the U statistics is the level (inmost interesting contributions seems to be the article of Phillips

logarithms) of the market share, so these statisticswhich is an answer to Sims (1988). The criticism of Phillips

can be compared across the different models.agrees with the view that the tool used by Sims to criticise thelegitimacy of unit roots tests is based on the mechanical use of flat The accuracy measures are generated using the

priors in a Bayesian analysis of time series models. Phillips Kalman filter algorithm in RATS. The models aredemonstrates that flat priors are not uninformative but unwittingly estimated for the initial period 1988:1 to 1992:12.introduce a tendency towards stationary models.8 Forecasts for up to 12 months ahead are computed.Instead of preselecting some values we could select the Bayesian

One more observation is added to the sample andhyperparameters by minimising the following function:n H forecasts up to 12 months ahead are again generated,

Min U( g,w,d)5OOuih and so on. Based on the out-of-sample forecasts,

i51h51

MAPEs, RMSEs and the Theil U statistics arewhere u is the Theil U for time-series i h-forecast steps ahead.

ih computed for 1- to 12-months-ahead forecasts.Because the functional relationship is highly nonlinear, numerical

The three accuracy measures for MS1 for sevenmethods must be used to minimise U. In particular, we could use agrid search over the arguments hg,w,dj. models are reported in Table 1. The table also


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reports the average of these statistics for the 1-, 3-, 6- is not well specified, an alternate model such as an

and 12-month-ahead forecasts. The conclusions from unrestricted VAR or an ARIMA model may have a

Table 1 are as follows: better performance.

1. MAPEs versus Theil U statistics: as both MAPE 4.3. Performance of alternative modelsand Theils U indicate, the forecasting perform-

ance of all models deteriorates with larger fore- While the BVAR models, in general, produce the

casting horizons (6 and 12 months). The RMSE most accurate forecasts, another way to evaluate the

do not follow a consistent pattern with an increase performance of alternative models is to examine their9

in the forecast horizon. ability in predicting turning points. We focus on the

2. BVAR versus univariate models: BVAR models performance of the BVAR models relative to that of

produce more accurate forecasts (for all horizons) the unrestricted VAR and the univariate ARIMA

than the corresponding univariate models models. As can be seen in Fig. 4, it seems that the(NAIVE, REG., HoltWinters, and ARIMA). ARIMA model is predicting a near immediate return

3. The HoltWinters model, on average, performed to the mean, the BVAR models are predicting a time

the same as the ARIMA, but is clearly superior to trend (differencing and a constant) and the unre-the NAIVE and REG. models. We did not confirm stricted VAR gives the best visual forecasts. The

in this study one of the main conclusions of the unrestricted VAR is better at picking turning points

Big Mac paper (Makridakis et al., 1982) that than is the BVAR model. In fact, the BVAR model

. . . exponential smoothing was more accurate appears to do a very poor job at forecasting turning

than ARIMA models on average . . . . points in the market share data.

4. BVAR versus the unrestricted VAR models: in all Besides generating excellent baseline forecasts,

cases, except for forecast horizon 6, BVAR(G) BVAR models can also be used to study the effects

outperforms the unrestricted VAR model. The of movements in one variable on movements in

comparison between BVAR(S) and VAR(U), others using impulse response analysis and variance

however, yields mixed results. VAR(U) seems to decompositions.

be better for longer horizons (6 and 12).

5. BVAR(G) versus BVAR(S): BVAR(G) always 4.4. Impulse response analysisprovides the most accurate forecasts. This is not

surprising since the prior for the model was Impulse responses are the time paths of one or

selected on the basis of minimisation of the more variables as a function of a one-time shock to a

average of 1- to 12-month-ahead RMSEs and the given variable or set of variables. Impulse responses

U statistics. are the dynamic equivalent of elasticities. For exam-

6. The Theils U for 12-month-ahead forecasts are ple, in a static multiplicative interaction model of theb

larger than one for all the models, which indicates form ms 5 a? p , price elasticity [5 (dms/dp)(p/

that the random walk model is an improvement ms) 5 b] is constant. However, in a dynamic system,

on all the time series models. changes in market share in 1 month are a function of

changes in price over several months. The net effect

The results, in general, show that there are gains must be represented as a convolution sum. Thefrom using a BVAR approach to forecasting. On graphic representation of this sum provides a com-

average, the BVAR models produce more accurate

forecasts than the alternative forecasts. Finally, in the9class of BVAR models, BVAR(G) always produces This topic is of vital importance, as statistical methods canperform extremely well in terms of forecasting overall levels, andthe most accurate forecasts except for 1-month-aheadyet still perform poorly in the prediction of turning points.forecasts.Unfortunately, the question of turning points does not easily lend

Finally, like other authors (Hafer and Sheehan,itself to quantitative analysis due to difficulties in, firstly, defining

1989) we found that the accuracy of the forecasts is turning points, and, secondly, knowing when a given method hassensitive to the specification of the priors. If the prior adequately predicted a turn.


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Fig. 4. Market share forecasts for 1993:71994:6 (made in 1993:6).

plete description of the dynamic structure of the RS1 , PS1 and partly on the timing of the availabili-t tmodel (it is incorrect to directly interpret estimated ty of data. For instance, information on prices and

coefficients for the variables of a BVAR model). market shares is released prior to the advertising

The innovations in the BVAR models have been expenditures.

orthogonalised via a Choleski decomposition, and To illustrate, Fig. 5 shows the impulse responses

unless the error terms of each equation are contem- of MS1 to a shock for each one of the six variables

poraneously uncorrelated it may not make sense to of the system and Fig. 6 shows the cumulative

assume that a shock occurs only in one variable at a responses over 12 months. Both figures illustrate10

time. The variables are ordered in the following that:

sequence:

1. The response of MS1 to a positive shock in MS1MS1 , A1 , P1 , TVS1 , RS1 , PS1 . decreases rapidly and is cancelled after 4 months.t t t t t t

The cumulative effect after 12 months is signifi-

This ordering is based partly on our prior belief that cant.

changes in MS1 precede those in A1 , P1 , TVS1 , 2. Over the sample period an unexpected increase int t t t A1 produces a longer decrease (over 6 months) of

MS1, which is not totally recuperated after 12

months, i.e. A1 has a negative permanent impact10

The correlation matrix of the residuals shows that the correlation on MS1.among the off-diagonal elements seems to be small, thus 3. Surprisingly, the response of the MS1 to a pricechanges in the order of the variables are likely to have minor

shock does not appear to be significant. A plaus-effects on the impulse response results:

ible explanation for this behaviour is that our

brand does not compete on a price basis and evenMS1 A1 P1 TVS1 RS1 PS1MS1 1 20.21 20.23 0 .07 0.30 20.23 if it changes its prices, these increases will beA1 1 20.19 20.22 20.18 0.08 associated with the launch of new models (ver-P1 1 90.24 20.25 0.15 .

sions).TVS1 1 0.17 0.261 2 4. The MS1 responses to shocks on advertisingRS1 1 0.04PS1 1 expenditures (TV, Radio, and Press) are positive,


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Fig. 5. Responses of MS1 to shocks on all variables.

Fig. 6. Cumulative responses of MS1 to shocks on all variables.

lagged and varying in magnitude. TV advertising the h-periods-ahead forecast error variance of a

effects begin after 3 months, but seem to rest for a variable that can be attributed to another variable.

period longer than that of Radio and Press. Our The pattern of the variance decomposition alsofindings are consistent with the theory of the indicates the nature of Granger causality among the

cumulative advertising effects on sales (e.g., variables in the system, and, as such, can be very

Palda, 1964; Clarke, 1976), and even the mea- valuable in making at least a limited transition from

surement and duration of these effects are easy to forecasting to understanding. If innovations in A1tcalculate. result in unexpected fluctuations in MS1 , thent

information on A1 would be useful in predictingt4.5. Variance decomposition MS 1 . In interpreting these variance decompositions,t

one should bear in mind Runkles (1987) criticism

Variance decompositions give the proportion of that the implicit confidence intervals attached to both


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variance decompositions and impulse response func- sense, i.e. if other variables in the model are not

tions are often so large as to render precise infer- useful in predicting it, a large proportion of that

ences impossible. variables error variance should be explained by its

The Choleski decomposition is used for the own innovations. How large is large? According to

BVAR(G) model. The variables are ordered in the Doan (1992), in a six-variable model such as ours,following sequence MS1 , A1 , P1 , TVS1 , RS1 , 50% is quite high. If another variable is useful int t t t t PS1 . This ordering is based partly on our prior explaining a left-column variable, that useful vari-tbelief that changes in MS1 precede those in A1 , able will explain a positive percentage of the predic-t tP1 , TVS1 , RS1 , PS1 and partly on the timing of tion error variance. In practice, it is difficult tot t t t the availability of data. For instance, information on distinguish between a variable that has no predictive

prices and market shares is released before that on value and one that has little predictive value. Some

advertising expenditures. conclusions, however, can be derived by comparing

The variance decompositions for our six-variable the magnitudes.

model for the period 1988:1 to 1993:6 are reported in Table 2 shows that, at a forecast horizon of 12

Table 2. For each variable in the left-hand column, months, only 35.6% of the forecast error variance in

the percentage of the forecast error variance for 1, 6 the MS1 is explained by its own innovations, sup-

and 12 months ahead that can be attributed to shocks porting the assumption that MS1 is not exogenous,

in each of the variables in the remaining columns is and that other variables such as A1, TVS1, and PS1

reported. Each row sums to 100% (ignoring rounding can be equally useful in forecasting MS1. The

errors) since all the forecast error variance in a market share is extremely stable from 1 month to the

variable must be explained by the variables in the next none of the other variables figure at all in its

model. If a variable is exogenous in the Granger 1-month-ahead forecast. Moreover, longer-term fore-

Table 2

Variance decompositions

Variable Step MS1 A1 P1 TVS1 RS1 PS1MS1 1 100 0 0 0 0 0

6 51.01 29.03 3.87 3.61 6.61 6.43

12 35.6 31.9 3.91 15.8 4.97 7.8

A1 1 5.87 0 0 0 0 0

6 3.95 8.09 8.09 3.74 2.45 2.94

12 4.08 19.85 19.85 6.06 1.34 3.45

P1 1 23.86 65.76 65.76 0 0 0

6 16.25 52.67 52.67 9.55 8.96 4.96

12 12.69 40.44 40.44 15.06 13.88 5.21

TVS1 1 0.54 5.94 5.94 83.88 0 0

6 9.34 4.56 4.56 56.87 3.9 3.68

12 9.7 7.75 7.75 50.79 4.79 5.9

RS1 1 27.64 0.32 0.32 1.22 68.3 0

6 21.79 1.18 1.18 14.52 36.05 8.99

12 18.15 9.49 9.49 18.23 29.01 9.19

PS1 1 10.47 0 0 12.16 4.4 72.96

6 10.3 5.42 5.42 19.01 10.14 39.75

12 13.98 8.56 8.56 17.17 8.91 34

Note: entries in each row are the percentages of the variance of the forecast error for each variable indicated in the rows that can be

attributed to each of the variables indicated in the column headings. Decompositions are reported for 1-, 6- and 12-month horizons.


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casts of market shares are heavily influenced by model such as an unrestricted VAR or an ARIMA

age, TV advertising and not much at all by price. model may perform better. Third, the prior that is

The exogenous behaviour of A1 seems to be re- selected on the basis of some objective function (e.g.

flected in the 65.2% error variance explained by its the Theils U) for the out-of-sample forecasts may

own innovations. The results for the price variable not be optimal for beyond the period for which itvary within the forecasting horizon. For instance, the was selected.

market share variable seems more important at This model, like all time series models, is best

shorter horizons (1 to 6 months), while the age and suited for stable environments (e.g. wide-sense

the advertising (TV and Radio) variables become the stationary processes) where sufficient numbers of

largest contributors for longer horizons (12 months). observations are available. Thus, BVAR is not a new

There are some interesting media differences in the product model and its forecasts may be unreliable in

advertising variables. TV advertising seems much markets characterised by frequent new entries or

more exogenous than the other media advertising dropouts.

(50.79% vs. 29.01% and 34%). Long-term forecasts We propose several extensions of this initial

of TV advertising are more explained by innovations application of BVAR to forecast brand market shares.

in A1 than in MS1. However, for RS1 and PS1 the First, the Bayesian approach can be improved by

innovations in MS1, A1, and TVS1 help explain putting more structure based on marketing theory

most of the forecast error variance attributable to into the prior, thereby abandoning the symmetric

other innovations. treatment of all variables. This would make the

approach more Bayesian in spirit, since the prior can

now reflect better the a priori beliefs of the inves-

tigator. On the other hand, the greater flexibility5. Limitations and extensions makes it more difficult to find the optimal forecasting

model. Second, the inclusion of the contemporaneousIn this paper, we demonstrate the utility of VAR values of some variables in some equations (using,

and BVAR methodologies as a marketing tool that for example, a Wold causal ordering) may result infulfills two requirements: it forecasts market shares, improved forecasting accuracy due to a simpler

and it provides insights about the competitive dy- model specification. Third, we used one specificationnamics of the marketplace. We compared the fore- of each model to forecast over the entire testingcasting accuracy of BVAR with several traditional period. More frequent specifications (e.g., a time-approaches. Using data, we establish that BVAR is a varying parameter BVAR model) would undoubtedlysuperior forecasting tool compared to univariate improve accuracy. An important problem deals withARIMA and VAR models. Because BVAR uses few how often a model should be re-specified. Finally,degrees of freedom and is easy to identify, it satisfies we used single equation procedures to estimate allthe practical requirements as a marketing forecasting models. Forecasting accuracy may improve by es-tool. Finally, using impulse response functions and timating all equations in each model simultaneouslyvariance decompositions, we illustrate that BVAR and exploiting the information in the cross-equationprovides important insights for marketing managers. residual covariance matrix.

Although BVAR is a promising and reliableforecasting tool, certain limitations should be pointed

out. First, BVAR models are highly reduced forms.

Structural interpretations based on the signs and Acknowledgements

magnitudes of estimated parameters should be avoid-ed. Hypotheses about effects should be tested using The author is grateful to Jose C. Ribeiro, the

impulse response analysis. Second, the accuracy of associated editor and two anonymous referees for

the forecasts is sensitive to the specification of the helpful and constructive comments. Responsibility

prior. If the prior is not well specified, an alternate for any error is solely mine.


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06-92 0.183 1.23 0.91 0.209 0.072 0.171Appendix A07-92 0.151 1.57 1.05 0.241 0.114 0.161

08-92 0.143 1.60 1.05 0.443 0.071 0.134

09-92 0.143 1.84 1.05 0.211 0.197 0.196Period MS1 A1 P1 TVS1 RS1 PS1

10-92 0.160 2.28 0.95 0.135 0.130 0.16401-88 0.273 0.97 0.81 0.325 0.176 0.385

11-92 0.164 2.18 1.04 0.166 0.143 0.16202-88 0.229 1.06 0.90 0.372 0.223 0.33312-92 0.161 2.25 0.95 0.145 0.111 0.171

03-88 0.217 1.14 0.90 0.246 0.132 0.25801-93 0.175 2.37 0.94 0.149 0.123 0.212

04-88 0.234 1.39 0.83 0.189 0.122 0.21402-93 0.147 2.19 1.02 0.159 0.061 0.180

05-88 0.234 1.35 0.86 0.257 0.001 0.16903-93 0.164 2.24 0.95 0.091 0.061 0.173

06-88 0.245 0.64 1.09 0.338 0.117 0.19304-93 0.159 2.27 0.91 0.170 0.089 0.166

07-88 0.169 0.86 0.98 0.389 0.068 0.31505-93 0.134 2.54 0.97 0.097 0.021 0.079

08-88 0.178 0.40 0.85 0.375 0.176 0.18906-93 0.121 2.57 0.98 0.069 0.025 0.089

09-88 0.126 0.77 0.96 0.265 0.214 0.19007-93 0.147

10-88 0.192 0.97 0.99 0.326 0.168 0.12808-93 0.125

11-88 0.220 0.73 0.99 0.285 0.208 0.20009-93 0.128

12-88 0.180 0.62 1.05 0.233 0.157 0.32010-93 0.121

01-89 0.215 0.84 1.00 0.193 0.253 0.37611-93 0.145

02-89 0.223 0.74 1.05 0.120 0.196 0.25612-93 0.184

03-89 0.181 0.56 1.02 0.083 0.143 0.185 01-94 0.12604-89 0.149 1.00 0.94 0.147 0.145 0.169

02-94 0.17505-89 0.203 0.62 1.11 0.170 0.064 0.124

03-94 0.13206-89 0.223 0.52 1.04 0.205 0.224 0.075

04-94 0.14707-89 0.181 0.74 1.03 0.260 0.154 0.100

05-94 0.11308-89 0.149 0.88 1.08 0.244 0.133 0.124

06-94 0.11909-89 0.142 0.54 1.17 0.198 0.148 0.201

10-89 0.202 1.05 1.15 0.280 0.196 0.143

11-89 0.209 0.91 1.13 0.216 0.184 0.097

12-89 0.120 1.44 1.20 0.185 0.164 0.151

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Forecasts of Market Shares From VAR and BVAR Models a Comparison of Their Accurac

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