computing level-impulse responses of log-specified var systems

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International Journal of Foreca

Computing level-impulse responses of log-specified VAR systems

Jaap E. Wieringaa,*, Csilla Horvathb

aFaculty of Economics, University of Groningen, Groningen, The NetherlandsbEconometric Institute, Erasmus University of Rotterdam, Rotterdam, The Netherlands

Abstract

Impulse response functions (IRFs) are often used to analyze the dynamic behavior of a vector autoregressive (VAR) system.

In many applications of VAR modelling, the variables are log-transformed before the model is estimated. If this is the case, the

results of the IRFs do not have a direct interpretation, since they are also log-transformed. In this paper, we present explicit

expressions for computing impulse response functions that are expressed in the levels of the variables, given a log–log

transformed model. We illustrate the methodology by an application in marketing.

D 2004 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Keywords: VAR models; Log-transformation; Impulse response functions

1. Introduction

Vector autoregressive (VAR) models are often used

to model the simultaneous dynamic behavior of

multiple economic variables. There are several appli-

cations of VAR models in marketing (see for example

Dekimpe, Hanssens, & Silva-Risso, 1999; Nijs,

Dekimpe, Steenkamp, & Hanssens, 2001), in finance

(e.g. Joseph, 2001; Park & Shenoy, 2002), and since

the seminal work of Sims (1980), there have been

numerous applications in economics (for recent

0169-2070/$ - see front matter D 2004 International Institute of Forecaste

doi:10.1016/j.ijforecast.2004.09.007

* Corresponding author. Tel.: +31 50 363 7093; fax: +31 50

363 7337.

E-mail addresses: [email protected] (J.E. Wieringa)8

[email protected] (C. Horvath).

URL: http://www.eco.rug.nl/wieringa.

publications, see e.g. Jung & Seldon, 1995; Ballab-

riga, Sebastian, & Valles, 1999).

In many applications, the VARmodel is specified in

terms of the natural logarithm of the variables of

interest (for recent examples in marketing, see Srini-

vasan, Leszcyzyc, & Bass, 2000; Nijs et al., 2001;

Horvath, Leeflang, Wieringa, &Wittink, 2003). This is

particularly useful in situations where the relationships

between the variables are assumed to be of a multi-

plicative nature, which is a common assumption in

marketing (see, e.g. Leeflang,Wittink, Wedel, &Naert,

2000, p. 74; Hanssens, Parsons, & Schultz, 2001, p.

102). The log-transformation linearizes amultiplicative

model, so that estimation is greatly simplified. Other

reasons for applying the log-transformation include

reducing the impact of outliers and reducing the

increasing variance of trending time series.

sting 21 (2005) 279–289

rs. Published by Elsevier B.V. All rights reserved.

J.E. Wieringa, C. Horvath / International Journal of Forecasting 21 (2005) 279–289280

An estimated VAR model can be used to compute

out-of-sample forecasts of the vector of log-trans-

formed variables. Arino and Franses (2000) argue that

if it is desired to obtain forecasts of the levels of the

time series, one cannot simply take exponentials of the

forecasts for logged data, since this procedure yields

substantially biased forecasts. They present explicit

expressions for computing unbiased forecasts of the

levels of the variables in the model. They also indicate

that a possible extension of their work is to employ it

for the computation of impulse response functions

(IRFs). In this paper, we provide this extension.

IRFs are commonly used to analyze the dynamic

behavior of a vector autoregressive system. They track

the changes in the system of equations that are induced

by a shock to one of the variables in the system. IRFs

can be constructed by forecasting the values of all

variables in the system at the time of the shock and in

the periods thereafter, and subtract the corresponding

forecasted values if the system is not shocked. Hence,

if the variables in the system are log-transformed, the

IRFs contain effects that are also log-transformed. In

many cases, this complicates the interpretation of the

effects. As an illustration of this point and as a prelude

to the application later in this paper, we refer to a recent

marketing publication in Section 2 that points out

three well-published misunderstandings in marketing

literature concerning the effects of sales promotions.

These misunderstandings could have been avoided if

the sales effects were (also) computed in levels.

The interpretation of IRF results of a log-

specified VAR model becomes even more difficult

when one is interested in the cumulative effect (i.e.

the net effect) of a shock over time. For example, in

Section 4 we discuss a marketing application, where a

log-transformed VAR model is used to capture the

dynamics in sales responses and price reactions of

multiple brands to price promotions. We employ IRFs

to track the effects of dprice shocksT over time. In

order to evaluate the profitability of a price promotion,

a marketing manager will be interested in a forecast of

the net revenues of the discount. Cumulating over

time the IRF that tracks the effects of a price-shock in

a log–log specified VAR model does not provide her

with this information, since it represents the net effect

of a price discount on the logarithm of the sales.

Instead, she would be interested in a forecast of the

cumulative additional unit sales, after accommodating

for important dynamic effects, such as competitive

reactions and (cross-)feedback effects.

Nijs et al. (2001) justify the use of such cumulated

IRFs by interpreting them as long-term elasticities. We

do not agree with this interpretation. Consider the IRF

that captures the dynamic response of own sales (St)

to a shock in the price variable (Pt). It can be shown

that the elements of the IRF of a log–log specified

model are elasticities. Hence, the cumulation of the

IRF over K periods equals

XKk¼1

BStþk

BPtþ1

Ptþ1

Stþk

;

where the price shock occurs at time t+1. In general,

this is not equal to the elasticity of the cumulative

sales with respect to the price shock, which equals

BPK

k¼1 Stþk

� �BPtþ1

Ptþ1PKk¼1 Stþk

:

A side benefit of computing level-IRFs is that these

can be used to compute proper long-term elasticities.

The key contributions of this paper are the

following. First, it draws attention to the usefulness

of computing level-IRFs for VAR models of log-

transformed variables. Second, it presents a procedure

to calculate level-IRFs for log-specified VAR models

that allows researchers to properly evaluate the

magnitude of dynamic effects of an exogenous shock

to the system. Finally, it illustrates the usefulness of

the proposed procedure by an empirical marketing

application. The computations in this paper build on

the work of Arino and Franses (2000).

This paper is organized as follows. In Section 2, we

discuss why it is important to compute effects on

level. In Section 3, we present explicit expressions for

the IRFs that are expressed in the levels of the

untransformed variables. In Section 4, we illustrate the

methodology by a marketing application. We con-

clude the paper in Section 5 with some remarks.

2. Motivation: the importance of unit sales effects

in marketing

In the marketing literature, it is common practise to

employ multiplicative models to capture the effects of

1 First, we assume that all elements of Yt are log-transformed

variables. At the end of this section, we consider the case when

some of its elements are not log-transformed.2 R is usually assumed to be diagonal.3 In the following calculations, we assume that the VAR model

is properly estimated and the system is identified.

J.E. Wieringa, C. Horvath / International Journal of Forecasting 21 (2005) 279–289 281

marketing instruments on sales. As stated in the

previous section, a multiplicative model can be

linearized by applying the log transformation. One

of the reasons why marketing researchers prefer

multiplicative models is that their parameters can be

interpreted as (constant) elasticities (Leeflang et al.,

2000, p. 74; Hanssens et al., 2001, p. 102). Elasticities

play an important role in marketing theory as they

have some advantages: e.g. elasticities are unit-less

and allow for comparison of promotion effects

between different types of products. However, in

recent research there is strong support for evaluating

the effects of sales promotions (also) in terms of unit

sales. Van Heerde (in press) identifies three well-

published misunderstandings in marketing that are

due to the misinterpretation of elasticities. He argues

that the effects of sales promotions can only be fully

understood if sales elasticities are accompanied by

unit sales effects. The three misunderstandings are:

! asymmetries in cross-price market share effects

that were found by Blattberg and Wisniewski

(1989) do not exist when evaluated in absolute

cross market share effects. That is, the finding that

higher-quality brands attract more switchers if

these are promoted than low-quality brands,

vanishes or is sometimes even reversed when

evaluated in terms of unit sales effects instead of

elasticities;

! the brand switching effects of a sales promotion

are much smaller than those reported by some

often-cited publications, based on improper inter-

pretation of Gupta’s (1988) seminal work on

decomposition of sales elasticities. The misinter-

pretation is due to the fact that the promoted brand

usually benefits more from category expansion

effects than the non-promoted competing brands.

The resulting shifts in market shares are often

incorrectly interpreted as brand switching effects

(for references, see Van Heerde, Gupta, and

Wittink, 2003);

! many marketing studies at the aggregate data level

(e.g. store- or chain-level) use market share models

without category sales models to investigate the

effects of sales promotions (for references, see

Cooper & Nakanishi, 1988). However, as category

expansion effects are much more important when

evaluated in terms of unit sales effects, a market

share model should be accompanied by a category

sales model (Van Heerde, in press).

Van Heerde (in press) stresses the importance of unit

sales effects, but only considers contemporaneous

effects of promotions. He refers to Pauwels, Hanssens,

and Siddarth (2002) for the assessment of long-term

effects of price promotions and notes that their assess-

ment is based on elasticities. The approach presented in

the present paper can be used to assess the long-term

unit sales effects of log-specified VAR models.

3. Computing the levels of the IRFs

Let Xt be an m-dimensional vector time series:

XtV=(X1,t, . . ., Xm,t), such that Yt with YtV=(Y1,t, . . .,Ym,t) with Yi,t=ln Xi,t, follows a structural VAR( p)

model:1

A0Yt ¼ a þ A1Yt�1 þ A2Yt�2 þ . . .ApYt�p þ et; ð1Þ

where aV=(a1, . . ., am) is an m-dimensional vector of

intercepts, Ar (r=0, . . .,p) are (m�m)-matrices con-

taining the immediate reaction coefficients if r=0, and

the lagged reaction coefficients if rz1. Furthermore,

EtV=(E1,t , . . ., Em,t) is an m-dimensional vector of

disturbances, with Et fi:i:d: N (0, R)2. Assuming that

A0 is nonsingular (this is certainly the case if the

immediate effects are identified), the reduced form of

Eq. (1) is3

Yt ¼ bþ B1Yt�1 þ B2Yt�2 þ . . . þ BpYt�p þ ut; ð2Þ

where b=A0�1a, Br=A0

�1Ar (r=1, . . ., p), and ut=A0�1Et,

so that ut fi:i:d: N(0, X), where X=A0�1RA0

�1V.

Given the linear additive structure of a VAR model,

the IRFs in terms of {Yt} can be computed independ-

ently of the values of the elements of Yt and of

preceding values when the shock is applied to the

system at period t+1 (see e.g. Lutkepohl, 1993); the

effects of a shock on the variables over time are additive

to the corresponding non-shocked values. However, if

ˆ


it is desired to compute the response to an exogenous

shock in one of the variables in terms of {Xt}, the

multiplicative interactions make the IRFs dependent on

the values of the elements in Xt, Xt�1, . . .. For thisreason, we propose to compute each level-IRF as the

difference of level-forecasts of the shocked system and

the level-forecast of a non-shocked system.

Define Xi,t+k |Xtas the forecast of the i-th element

of X at period t+k, conditional on the information

that is known at time t (i.e. conditional on the

vectors Xt, Xt�1, . . .), assuming none of the variables

is shocked. Similarly, X si;tþkjXt

denotes the condi-

tional forecast of Xi,t+k, assuming that the VAR-

system is shocked at time t+1.

Using this notation, we compute the level-IRFs of

the i-th variable as follows:4

(1) Compute X i;tþ1jXt; X i;tþ2jXt

; . . . ;(2) Compute X s

i;tþ1jXt; X s

i;tþ2jXt; . . . ;

(3) Compute the values of the IRF for variable i at

lags 0, 1, . . . as5

IRFi k � 1jXtð Þ ¼ X si;tþkjXt

� X i;tþkjXt

for k=1,2,. . . .

We now discuss how to compute X i;tþkjXtand

X si;tþkjXt

: Arino and Franses (2000) show that

X i;tþkjXt¼ exp

ei kð Þ2

þ c0;i kð Þ� �

jp

r¼1jm

j¼1X

ci;j;r kð Þj;t�rþ1

4 In cases where it is undesirable to have the IRF depending on a

particular set of realizations of Xt,Xt�1, . . ., one can compute the

elements of the IRF as IRFi(k�1|X) =X si;tþkjX � X i;tþkjX; where

all the computations assume that Xt,Xt�1, . . . are all equal to their

mean. Another option is to replace these values by an estimate of the

expected value of Xt, given the estimated VAR system. For this, the

expected value of Yt needs to be estimated first, which

can be done in the following way (utilizing Eq. (2)):

E Ytð Þ ¼ I � B1 � B2 � . . . � Bp

� ��1b. Subsequently, the expect-

ed value of Xt is estimated as: EðXi;tÞ ¼ expð E Yi;t� �

Þexpri;1þ...þri;k

2

� �; where r i,j is the element in the i-th row and in the

j-th column of X. This estimated expected value can be used to

compute the constant value of the naive forecast (i.e. the forecast of

Xi,t when we do not induce any shock) and can also be used for the

initial values for forecasting the system when shocks are induced.

We apply the latter procedure in Section 4. The IRFs that result

from employing either of the two approaches can be interpreted as

average impulse responses.5 In these expressions, we assume that the shock takes place in

period t+1. In line with standard notation, we express the immediate

responses as IRFi(0|Xt).

is an unbiased conditional forecast of the i-th

element of Xt+k, where

ei kð Þ ¼ ei k � 1ð Þ þ

di;1 kð Þ; . . . ; di;m kð Þ� �

Xðdi;1 kð Þ; . . . ; di;m kð ÞÞV:

In this expression, ei(0)=0 and di,j(k) is the element

in the i-th row and the j-th column of a matrix D(k)

that satisfies the recursion

D kð Þ ¼Xpr¼1

BrD k � rð Þ;

with initial conditions D(1)=Im and D( j)=0 for jV0,and where X is an estimate of the variance–covariance

matrix of ut. Furthermore, c0,i(k) is the i-th element of

a column-vector C0(k) that satisfies the recursion

C0 kð Þ ¼ bþXpr¼1

BrC0 k � rð Þ;

with initial conditions C0( j)=0 for jV0. Finally,

ci,j,h(k) is the element in the i-th row and the j-th

column of a matrix Ch(k) (h=1, . . ., p) that satisfies

Ch kð Þ ¼Xpr¼1

BrCh k � rð Þ;

with initial conditions for jV0

Ch jð Þ ¼ Im if j ¼ 1� h

0 otherwise:

For the computation of X si;tþkjXt

, we note that

shocking the l-th element of the X-vector with a

portion of d at time t+1 is equivalent to adding gl to

et+1, the disturbance term in the structural VAR model

of Eq. (1), where gl is an m-dimensional vector whose

elements are all equal to zero, except for the l-th

element, which equals ln(1+d). This shock in the

structural VAR model is equivalent to adding A0�1gl to

ut+1, the disturbance term of the reduced VAR model

of Eq. (2), so that

X si;tþkjXt

¼ exp

�ei kð Þ2

þ di;1 kð Þ; . . . ; di;m kð Þ� �

� A�10 gl

þc0;i kð Þ�

jp

r¼1jm

j¼1X

ci;j;r kð Þj;t�rþ1

is an unbiased conditional forecast of the i-th element

of the X-vector at period t+k when its l-th element is

shocked at period t+1.


In applications where a researcher desires to

include both log-transformed and non-log transformed

variables endogenously in the model (e.g. when some

of the variables are continuous variables that are zero

sometimes), the formulae above are also useful.6 For

the log-transformed variables, the levels of the IRFs

are computed exactly as discussed above. The

equation of a non-transformed variable (say, the j-th

element in Xt) is already specified on level so that for

such a variable the level-IRF can be written as

IRFj ðk � 1jXtÞ ¼ dj;1 kð Þ; . . . ; dj;m kð Þ� �

A�10 gl

for k = 1,2,. . ..

4. An empirical marketing application

In this section, we apply the methodology of the

previous section to a marketing problem. We study the

Chicago market of the three largest national brands in

the US in the 6.5-oz tuna fish product category. We

have 104 weeks of observations for each of 28 stores

of one supermarket chain in the metropolitan area,

covering unit sales, actual and regular prices, features,

and displays. This is the same data set that was used in

the study of Horvath et al. (2003). We use the data

from 26 stores.7 We first specify the model. Sub-

sequently, we present some estimation results, and

show the impulse responses in own unit sales to a

20% price discount of the three brands. Finally, we

compare the unit sales impulse responses obtained

with our approach to those of a naive approach.

4.1. Model specification and estimation

The variables of interest are the logarithms of unit

sales and logarithms of price indices (ratio of actual to

regular price) of three brands at the store level. We

define two types of price promotion variables: (1)

own- and other-brand temporary discounts without

support and (2) own- and other-brand temporary

6 We thank one of the anonymous reviewers for drawing

attention to this issue.7 In two stores, the brands are not promoted. We exclude these

stores from the analysis.

discounts with feature only, display only, or feature

and display support. Van Heerde, Leeflang, and

Wittink (2000, 2001) use this approach to allow for

interaction effects between discounts and support. In

addition, the promotion variables are minimally

correlated by definition.8

We treat the sales variables and the price variables

of the three brands endogenously in the system of

equations. We consider the non-price instruments to

be exogenous to the model (hence, in fact, we build a

VARX model9). Furthermore, we do not include

lagged non-price instruments but we allow for

dynamic effects indirectly through the inclusion of

lagged endogenous variables. For each brand, we

include a sales response function, and two price

reaction functions (one for supported price and one for

non-supported price).

This results in a nine-dimensional VARX model

with three sales response functions and six price

reaction functions for each of the 26 stores. In

Horvath and Wieringa (2003), we show that a Fixed

Effects Model is appropriate for the data, where all

parameters are equal for all stores, except for the

intercepts: each store has a specific intercept for each

equation. The sales response function for product i in

store q is specified as (Horvath et al., 2003):

ln Sqi;t ¼ aqi þXnj¼1

X2k¼1

Xpt4¼0

bPIijk;t4 ln PIqjk;t�t4

þXnj¼1

Xpt4¼1

uij;t4 ln Sqj;t�t4 þXnj¼1

bFijFqj;t

þXnj¼1

bDijDqj;t þXnj¼1

bFDijFDqj;t þ eqi;t;

q ¼ 1; . . . ;Q; i ¼ 1; . . . ; n; and t ¼ 1; . . . ; Tð Þ ð3Þ

where: ln Sqi,t is the natural logarithm of sales of

brand i in store q in week t; ln PIqik,t is the log price

index (actual to regular price) of brand i in store q in

week t (k=1 denotes prices that are supported and k=2

8 Van Heerde et al. (2000, 2001) use four different price

promotion variables. We use only two variables to reduce degrees-

of-freedom problems.9 The exogenous variables do not play a role in the computation

of the level-IRFs. They are all indicator variables and we evaluate

the IRFs assuming that all exogenous variables are zero.

Table 1

Immediate price elasticities

Price elasticities of

Brand 1 Brand 2 Brand 3

Supported price of brand 1 �6.10* 0.49* 1.04*

Non-supported price of brand 1 �3.90* �0.03 0.94*

Supported price of brand 2 0.46* �3.27* 0.78*

Non-supported of price brand 2 0.03 �1.97* 0.69*


denotes prices that are not supported); Fqj,t is the

feature-only dummy variable for non-price promo-

tions of brand j in store q at time t; Dqj,t is the display-

only dummy variable for non-price promotions of

brand j in store q at time t; FDqj,t represents the

combined use of feature and display for non-price

promotions of brand j in store q at time t;10 aqi is astore-specific intercept for the equation corresponding

to brand i and store q; bPIijk,t* is the (pooled across

stores) elasticity of brand i’s sales with respect to

brand j’s price index in week t�t*; uij,t* is the

(pooled) elasticity of brand i’s sales with respect to its

own sales in week t�t*; bFij, bDij, and bFDij are the

current-week effects on brand i’s log sales resulting

from brand j’s use of feature-only (F), display-only

(D), and feature and display (FD); n is the number of

brands in the product category; Q is the number of

stores; Eqi,t are the disturbances.

We specify the price reaction function for price

variable l of product i in store q as follows:

ln PIqil;t ¼ dqil þXnj¼1

X2k¼1

Xpt4¼1

ciljk;t4 ln PIqjk;t�t4

þXnj¼1

Xpt4¼1

gij;t4 ln Sqj;t�t4 þ vqil;t

ðq¼ 1; . . .;Q; i ¼ 1; . . . ; n l¼ 1;2 and t ¼ 1; . . . ; TÞ:ð4Þ

where the variables are defined as in Eq. (3).

Unit root tests proved that all the variables are

stable, so that the effect of a shock (e.g. a price

discount) is only temporary; the system will gradually

revert to its mean values. Using several information

criteria, we decided to estimate a VARX model of

order 2. We employ Generalized Least Squares to

estimate the model.

For the identification of the immediate effects, we

apply the SVAR approach, which is capable of

supplementing sample-based information with mana-

gerial judgement and/or marketing theory (Dekimpe

& Hanssens, 2000). We use the following assump-

tions for the identification. We allow the price

variables to have immediate effects on the sales of

10 The variables F, D and FD only deviate from zero when

there is feature and/or display activity and no price discount.

the brands but do not allow for immediate effects from

sales on the prices because feedback requires time. As

price reactions of competitive brands also take place

with a lag, we do not allow prices to be affected

immediately by other brand’s prices. We estimate the

immediate price elasticities with Full Information

Maximum Likelihood. In Table 1, we present a

selection of the elements A0, viz. the own- and

cross-price elasticities.

These results provide high face-validity of the

model since:

(1) All own-brand price elasticities have the right

expected sign and are significant.

(2) All but one of the cross-brand elasticities have

the expected sign and most of them are

significant. The single cross-brand elasticity

with a negative sign is close to zero and is not

significant at the 5% significance level.

(3) All but one of the supported own- and cross-brand

elasticities are larger (in absolute value) than the

corresponding non-supported elasticities.

(4) Brands with higher own-brand effects usually

have higher cross-brand elasticities. Brands that

have high own-brand elasticities are expected to

have a stronger impact on other brands as well

(Blattberg & Wisniewski, 1989). All brands

react strongest to a price promotion of brand 3,

whereas the effect of a price promotion of brand

2 is the smallest.

The elements of A1 and A2 are much harder to

interpret. We trace out the effect of a shock in a price

variable on all the endogenous variables in the system

over time by computing the level-IRFs for sales using

the procedure outlined above.

Supported price of brand 3 0.85* 0.33* �6.89*

Non-supported price of brand 3 0.49* 0.35 �4.39*

* Indicates a significant parameter estimate (a=0.05).


4.2. Unit sales impulse responses to price discounts

In marketing applications of VAR modelling, IRFs

are most commonly used to track the effects of a price

discount on own-brand performance while allowing

for feedback, purchase reinforcement, competitive

reactions, internal decisions, and dynamic effects

(see Horvath et al., 2003). Therefore, we present only

the IRFs that represent the effects of a price discount

on own sales. In order to conserve space, we only

present immediate effects (the IRF-values at the time

of the shock) and the net effects (the values of the

IRFs summed over time). The depth of the price

discount that we apply is 20%, i.e., each shock

consists of lowering one of the prices by 20%. This

percentage is close to the average depth of the price

promotions in the data set.

In Section 3, we argue that the unit sales IRFs

depend on the last p realizations of the X-vector prior

to the price-shock. However, as promotion campaigns

need to be planned in advance, these values are

usually not known at the time a manager needs a

efficacy forecast when she is planning promotional

activities. For that reason, we estimate the expected

levels of the variables, and use those as dinitial valuesT(see note 4). The resulting impulse responses can be

interpreted as average responses where the averaging

is done with respect to the effect of the initial values.

The model that we specify in Section 4.1 assumes

common response parameters across stores. However,

we allow for effects that are proportional to store-

specific intercepts. Hence, we have 26 store-specific

IRF results. In Table 2, we present the average results

of these IRFs.

The numbers in Table 2 can be interpreted as

follows. The entries in the row that is labeled

bImmediate effectsQ are the number of additional

units that are sold in an average store in the period

where the price variable is reduced with 20%. The

numbers in the row labeled bNet effectsQ are the

cumulative additional sales that result from the price

discount, after the system has reverted back to its

mean value (i.e. after the dust-settling period11). For

11 The net effects are determined by summing up the sales

effects over a horizon of 40 weeks after the price discount. As Fig. 1

shows, this is sufficient to account for all possible dynamic effects

of the discount: in all cases the effects die out within 15 weeks.

each of the point estimates in the table, a 95%

confidence interval is presented below. The intervals

are obtained by bootstrapping; we re-sampled the

residuals and re-estimated the immediate and net

effects a thousand times. All immediate effects and all

net effects appear to be different from zero at a

significance level of 5%.

Two interesting observations can be made from the

results in Table 2. Firstly, as expected, the effect of

communicating the price discount by a feature or a

display always leads to larger sales effect, both for the

immediate effects and for the net sales effects.

Secondly, we see that the point estimates of all the

net effects are larger than the corresponding immedi-

ate effects. This indicates that the dynamic relation-

ships are an important part of the model.12

Apparently, the effect of a price discount is not

restricted to the period of the price discount. However,

the fact that we find that the net effects are larger than

the immediate effects may seem somewhat counter-

intuitive. One might expect that negative sales effects

such as a post-promotion dip or competitive reactions

at least partly offset the initial positive effect on sales.

To investigate this result further, we depict the unit

sales responses to discounts of 20% in own supported

price and own non-supported price against time in

Fig. 1.

The IRFs show the characteristic sales response to

a price promotion over time: the sales decline rapidly

after a large initial peak, sometimes even followed by

a so-called dpost promotion dipT. Some smaller

additional effects (possibly due to competitive reac-

tions) occur before the effect of the price promotion

dies out (in all cases within 15 weeks). Interestingly,

the sales peak appears last longer than 1 week: in the

second week after the price shock, the sales effect is

also positive. It appears from further analyses that this

is caused by the fact that most price promotions in our

data last longer than one week (for details, see

Horvath et al., 2003). Negative sales effects (such as

post-promotion dips or competitive reactions) appear

to be much smaller in absolute sense so that these do

not offset the positive effect of the prolonged price

12 We note that the individual differences are not significantly

different from zero, but that they all have the same sign. Hence

using a sign test we conclude that the net effects are significantly

higher than the immediate effects.

,

Table 2

IRF results: average own-brand effects of 20% price discounts

Shocked variable

PIws,1a PIns,1 PIws,2 PIns,2 PIws,3 PIns,3

Immediate effects 821 [489,1234]b 393 [212,604] 273 [162,467] 140 [85,229] 651 [387,1014] 297 [177,471]

Net effects 951 [549,1446] 432 [180,761] 309 [153,563] 209 [56,422] 803 [462,1280] 405 [210,661]

aPIws,i: supported price index of brand i; PIns,i: non-supported price index of brand i. bThe numbers between square brackets indicate the 95%

confidence intervals.


promotion. As a result, we observe that the net effects

are higher than the corresponding immediate effects.

However, this phenomenon may be a-typical in the

sense that we may not observe this for promotions that

last for 1 week only.

To judge the usefulness of the computations, we

compare our results with a naive estimate of the level of

the immediate and the net effects. The naive forecast is

computed by taking the exponential of the forecasted

Fig. 1. Average level-IRFs (solid line) and 95% confidence intervals

value of the log-specified model. Arino and Franses

(2000) have shown that this naive forecast is biased and

that their bias-correction approach provides better

predictive validity. We investigate the size of the

differences between the naive approach and our

approach.

We compute the sales effects (both immediate

and net effects) of a 10%, 15%, and 20% price

discount using both approaches. The results can be

(dashed lines) of the own-brand effects of 20% price discounts.

Table 4

A comparison of long-term elasticities to other measures

Shocked

variable

Sum of IRFs of

transformed model

Long-term

elasticity

Net effects/mean sales

Relative price change

PIws,1a �7.01 �0.18 �7.33

PIns,1 �4.35 �0.11 �4.46

PIws,2 �3.48 �0.09 �3.57

PIns,2 �3.06 �0.08 �3.10

PIws,3 �8.80 �0.23 �9.16

PIns,3 �6.54 �0.17 �6.69

aPIws,i: supported price index of brand i; PIns,i: non-supported price

index of brand i.


found in Table 3. The results labeled dnaiveapproachT represent the estimates of the naive

computations; the results labeled dour approachTare determined using the approach of the present

paper.

Table 3 indicates that the differences between the

naive approach and our approach are certainly not

negligible. It appears that the naive approach

underestimates the sales effects for the present

application, and that size of the bias is positively

correlated with the size of the effect. The negative

bias can be explained by noting that the naive

approach does not take the error term into account.

The positive sign of the bias correction term can be

inferred from the expressions for ei(k) in Section 3.

Table 3

The effect of the bias correction on the IRF results

Shocked variable

PIws,1a PIns,1 PIws,2 PIns,2 PIws,3 PIns,3

10% price discount

Immediate effects—

our approach

255 144 104 59 190 105


naive approach

166 94 89 50 122 67

Net effects—our

approach

298 160 115 89 241 150

Net effects—naive

approach

192 102 95 72 153 92

15% price discount


our approach

480 251 178 96 368 186


naive approach

312 163 153 82 236 119

Net effects—our

approach

559 277 200 145 461 259

Net effects—naive

approach

360 178 168 117 292 160

20% price discount


our approach

821 393 273 140 651 297


naive approach

534 256 234 120 418 191

Net effects—our

approach

951 432 309 209 803 405

Net effects—naive

approach

610 277 259 169 508 250

aPIws,i: supported price index of brand i; PIns,i: non-supported price

index of brand i.

In Section 1, we noticed that a side benefit of

computing level-IRFs is that these can be used to

compute proper long-term elasticities. In Table 4, we

compare such elasticities to an alternative approach,

where the sum of the elements of the IRFs of the log-

transformed model are considered (see e.g. Nijs et al.,

2001).

From Table 4, we observe that the long-term

elasticities (third column) are all considerably smaller

than the sum of the IRFs of the log-transformed

model (second column). This is due to the fact that

for computing the long-term elasticity one is

interested in the percentage change inPK

k¼1 Stþk as

a result of a given percentage change in price. For

stationary VAR models, the effect of a temporary

shock to the system is bounded, so that the absolute

value of a long-term elasticity decreases to zero with

K. This property does not hold for the sum of the

IRFs of the log-transformed model.

In order to compare the sum of the IRFs of the log-

transformed model to our approach, we express the

net effects as a percentage of the mean sales level

during the dust-settling period, and divide this by the

percentage change in price. The fourth column of

Table 4 shows that these ratios are similar to the sum

of the IRFs of the transformed model (second

column). Interestingly, the absolute values of the

entries in the second column are always smaller than

those of the fourth column, consistent with the earlier

noted positive sign of the bias correction term. From

this table, it appears that the effect of the bias

correction term is small. However, Table 3 unambig-

uously shows that the bias is substantial when

evaluated in unit sales.


5. Concluding remarks

In many applications of VAR modelling, the system

is specified in terms of log-transformed variables. One

of the advantages of such a specification is that it

accommodates multiplicative interaction between the

variables of interest. A common tool to analyze the

dynamics in a VAR system is the impulse response

function (IRF). In this paper, we argue that the

responses induced by a shock to one of the variables

are much easier to interpret if the effects are expressed

in the same units as the variables. We present explicit

expressions for computing the levels—IRFs for a log-

transformed VAR system, and illustrate the proposed

computations by an application in marketing.

The IRF results can be used to answer questions

that relate to the absolute value of the effects; an

example is judging the profitability of a price

discount. Using a forecast of the number of additional

units sold, a marketing manager can compute the

(additional) revenue, and subtract the costs of the

promotional activities. Such an analysis is not

possible with the standard IRFs of log-transformed

VAR systems. We therefore think that our approach

facilitates the use of IRFs in practice.

We note that the assumption that the error term

follows a (multivariate) normal distribution is critical

for the expressions that we present in this paper.

The expressions that are presented in the text of this

paper can also be used to compute IRFs for nonsta-

tionary and cointegrated VAR models. However, using

themeanor theestimatedexpectedvalueas initial values

for the computation of daverageT IRFs, as described in

note 4 and applied in Section 4, is not appropriate in

such situations. Rather, one should then compute IRFs

conditional on specific prior realizations.

Acknowledgements

We thank the editor and two anonymous referees

for their helpful comments.

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