Forecasting model with asymmetric market response and its application to pricing of consumer package goods

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRYAppl. Stochastic Models Bus. Ind., 2005; 21:541–560Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.605

Forecasting model with asymmetric market response and itsapplication to pricing of consumer package goods

Nobuhiko Terui*,y and Yuuki Imano

Graduate School of Economics and Management, Tohoku University, Sendai 980-8576, Japan

SUMMARY

This paper presents a dynamic forecasting model that accommodates asymmetric market responses tomarketing mix variable}price promotion}by threshold models. As a threshold variable to generate amechanism for different market responses, we use the counterpart to the concept of a price thresholdapplied to a representative consumer in a store. A Bayesian approach is taken for statistical modellingbecause of advantages that it offers over estimation and forecasting. The proposed model incorporates thelagged effects of a price variable. Thereby, myriad pricing strategies can be implemented in the timehorizon. Their effectiveness can be evaluated using the predictive density. We intend to improve theforecasting performance over conventional linear time series models. Furthermore, we discuss efficientdynamic pricing in a store using strategic simulations under some scenarios suggested by an estimatedstructure of the models. Empirical studies illustrate the superior forecasting performance of our modelagainst conventional linear models in terms of the root mean square error of the forecasts. Usefulinformation for dynamic pricing is derived from its structural parameter estimates. This paper develops adynamic forecasting model that accommodates asymmetric market responses to marketing mixvariable}price promotion}by the threshold models. Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: asymmetric market response; Markov chain Monte Carlo; sales forecasting; store data;threshold models

1. INTRODUCTION

This paper proposes a dynamic model to accommodate asymmetric market responses to salesforecasting of a brand in a store. We first introduce an established class of non-linear time seriesmodels}threshold autoregressive (TAR) models}in dynamic market response models andmodify the models in conformity with sales forecasting at the brand level. Afterwards, weexamine dynamic pricing in terms of Bayesian forecasting through Markov chain Monte Carlo(MCMC) integrations.

For modelling, we use the counterparts of three concepts}reference price, price threshold,and latitude of price acceptance}all of which have been investigated in studies of consumer

Received 13 August 2004Revised 25 April 2005

Copyright # 2005 John Wiley & Sons, Ltd. Accepted 21 June 2005

*Correspondence to: Nobuhiko Terui, Graduate School of Economics and Management, Tohoku University,Kawauchi, Aoba-ku, Sendai 980-8576, Japan.yE-mail: [email protected]

Contract/grant sponsor: Japanese Ministry of Education; contract/grant numbers: (C)(2)12630024, (C)(2)15530137

behaviour. In fact, many related studies have been conducted using mainly scanner panel, i.e.disaggregated, data to elucidate these concepts within the analysis of consumer’s choicebehaviour to profile a segment of consumers. For example, some recent studies, References[1–8], have used these concepts to investigate the asymmetric market response of consumersfrom rather different perspectives. In relation to our present approach, Han et al. [9] recentlybuilt a threshold into a logit choice model under homogeneity of consumers. Subsequently,Terui and Dahana [10] extended it to accommodate consumer heterogeneity.

In this study, using scanner data aggregated at the store level, we apply these concepts to arepresentative consumer in a store. Although the different names of these concepts}quasi-reference price, quasi-price threshold and quasi-latitude of price acceptance}should be given,we use equivalent terminology as a convention. We ultimately derive some implications for pricemanagement for the store manager. By testing the linearity of time series data in a store, weobserve non-linearity in both sales and price series, suggesting a class of non-linear TAR modelsusing the price threshold. Some empirical studies, including one by Man and Tiao [11], showthat the price threshold disappears after aggregating over consumers, but our study shows thatnon-linearity in the sense of time series remains in data that are aggregated at the store level.Empirical studies show better forecasting performance of our model against conventional linearmodels in terms of root mean squares error of forecasts. We observe that useful information forthe dynamic pricing is derived from its structural parameter estimates.

More specifically, in Section 2, we first survey the recent development of non-linear time seriesmodels, particularly focusing on TAR models. Subsequently, we develop a dynamic forecastingmodel as an extended TARmodel. The proposed model is motivated by empirical findings of non-linearity of sales and prices time series in a store. Section 3 describes the structure of the proposedmodel and its dynamic characterization such as long run and delayed responses to current andpast pricing. Subsequently, we discuss the associated inferential procedures: model specification,forecasting, and strategic simulations. Section 4, reports empirical findings applied to actual dailyscanner data. Finally, Section 5 concludes the paper and provides discussion for future research.

2. MODEL FOR MEASURING ASYMMETRIC DYNAMIC MARKET RESPONSE

Non-linear time series models have been developed over the last two decades and have beensubsequently presented in the statistical literature. They include the bilinear models by Grangerand Andersen [12], the threshold autoregressive models by Tong [13,14] and Tong and Lim [15],the exponential autoregressive model by Haggan and Ozaki [16], the state-dependent model byPriestley [17], and the Markov switching model by Hamilton [18], among others.

Denote by Yt ¼ f ðYt�1;Yt�2; . . . ; et�1; et�2; . . .Þ þ et � f ðFt�1Þ þ et a general non-linear timeseries model for univariate stationary time series fYtg: Then, as a class of non-linear time seriesmodels, the threshold autoregressive (TAR) model is defined as

Yt ¼bð1ÞYt�1 þ eð1Þt if Yt�d4r

bð2ÞYt�1 þ eð2Þt if Yt�d > r

8<: ð1Þ

We define the key parameters d and r as delay and threshold parameters, respectively. Themodel above is known as self-excited TAR (SETAR) because the non-linearity is induced by itsown past value: Yt�d :

Copyright # 2005 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind., 2005; 21:541–560

N. TERUI AND Y. IMANO542

As shown in model (1), TAR models use piecewise linear models to obtain a approximation ofthe non-linear conditional mean equation, mt ¼ EðYt j Ft�1Þ: The basic concept is the localapproximation over the state by introducing regimes, in terms of which a complex system isdecomposed into simpler linear system. They employ piecewise linear models in which the linearrelationship changes with the values of the process. For that reason, they are particularly usefulwhen the process exhibits different dynamic characteristics in some regimes. This model wasmotivated by several non-linear characteristics that are commonly observed in practice, such asasymmetry in rising and declining patterns of the series, and limit cycle phenomena for processesencountered in the natural sciences. In contrast to the traditional switching model, which allowsfor model change to occur in time space, the TAR model uses ‘threshold space’ to improvelinear approximation. Tong and Lim [15] developed the TAR models for univariate series.Subsequently, Tong [14] extended it to multivariate series by incorporating explanatory series.

In this paper, as a class of TAR model for a multivariate time series, we employ the open loopthreshold autoregressive models introduced as ‘OLTAR’ models by Tong [14]. We modify it toconform to market response models in the marketing literature. The term ‘open loop’ used inengineering means ‘one way causal’ in econometrics and marketing. OLTAR models mean oneway causal multivariate TAR models. For that reason, we call it CMTAR for short hereafter.The models use some controllable threshold series fDt�dg corresponding to the target seriesfYtg of sales quantity of some specific brand in a store.

First, designate fðXt;YtÞ; t ¼ 1; . . . ;Tg as the price and the sales time series after sometransformations for stationarity. Then the CMTAR model for a bivariate stationary time seriesis defined as a non-linear time series model which switches from one linear model to anothercorresponding to the level of threshold series Dt�dð04d4UÞ (U indicates a upper bound ofpossible delay which is specified prior to data analysis). Dt�d should be a function of Xt: Themodels assume that the direction of causality is one-way from X to Y and that no feedbackrelation exists between them. In the marketplace, it corresponds to the observation that theconsumer looks at the price first and then makes a purchase decision. Few empiricalapplications of these models exist particularly in business, but some applied works on economictime series have been done in general. For example, GNP series by Geweke and Terui [19] andTiao and Tsay [20], unemployment series by Montgomery et al. [21], stock price series by Caoand Tsay [22], and Granger [23] discusses its use in macroeconomic models.

2.1. Model

Next, we formally define the model for our analysis. By introducing some threshold series fDtg;the model is described as

Yt ¼bð1Þ0 þ

Pky1i¼1 að1Þyi Yt�i þ

Pks1i¼0 bð1Þxi Xt�i þ eð1Þt if Dt�d4r

bð2Þ0 þPky2

i¼1 að2Þyi Yt�i þPks2

i¼0 bð2Þxi Xt�i þ eð2Þt if Dt�d > r

8<: ð2Þ

where we assume that the innovations for each regime feðjÞt g; j ¼ 1; 2; follow an independentlynormal distribution with zero mean and variance s2j : Key parameters are denoted as f ¼ ðd; rÞhereafter. In our analysis, it might be reasonable to set d ¼ 0 a priori because it does not makesense to assume that a consumer responds to the past price without considering the currentprice. However, we discuss the model in a general way so that non-zero d is included in themodel.


FORECASTING MODEL WITH ASYMMETRIC MARKET RESPONSE 543

By using a lag operator, LdYt ¼ Yt�d and lag polynomial representations AjðLÞ ¼1�

Pkyji¼1 aðjÞyi L

i and BjðLÞ ¼ bðjÞ0 þPkxj

i¼1 bðjÞxi Li; the CMTAR model (2) can be written as

A1ðLÞYt ¼ B1ðLÞXt þ eð1Þt if Dt�d4r

A2ðLÞYt ¼ B2ðLÞXt þ eð2Þt if Dt�d > rð3Þ

Respectively, inverting A1ðLÞ and A2ðLÞ generates the final form

Yt ¼M1ðLÞXt þ A1ðLÞ

�1eð1Þt if Dt�d4r

M2ðLÞXt þ A2ðLÞ�1eð2Þt if Dt�d > r

8<: ð4Þ

where MiðLÞ ¼ AiðLÞ�1BiðLÞ; i ¼ 1; 2: The sequence fMðjÞ

0 ;MðjÞ1 ;M

ðjÞ2 ; . . .g of M

ðjÞj ðLÞ ¼M

ðjÞ0 þ

MðjÞ1 LþM

ðjÞ2 L2 þ � � � is termed as the set of dynamic multipliers for regime j; and MðjÞ

u means theuth lagged effect from Xt�u to Yt at that regime. The sum of dynamic multipliers M

ðjÞ1 ¼P1

k¼0 MðjÞk is called the total multiplier. It indicates the final effects from X to Y in the long-run

at regime j: Bronnenberg et al. [24] and Terui [25] discuss these concepts for dynamic response inmarketing. We have a recursive formula to get these multipliers from the structural parametersin the next section.

On the other hand, using this model, we combine the concept of the latitude of priceacceptance with threshold parameter r below. The latitude of price acceptance is recognized as ‘aregion of price insensitivity around a reference price’ in marketing. Numerous studies havesought a region for efficient pricing. For example, the price discount promotion incurs a loss if amanager sets the discounting level as falling inside this region because consumers do not actuallyfeel as though the product has been discounted. For a price hike, the opposite is true. See, forexample, Kalyanaram and Little [6], Han et al. [9] and their references. Under the assumption ofrepresentative consumers in a store, we define threshold series fDtg as a consumer’s perceivedprice gain or loss at time t; which is formulated as the percentage change of the retail price ðPtÞfrom the reference price ðRPtÞ; Dt ¼ logðPtÞ � logðRPtÞ: Then, corresponding to the sign ofestimated r; latitude A of price acceptance as a region of price insensitivity is represented in twoways:

A ¼A1 : ½0; r� if r50

A2 : ½r; 0� if r50

(ð5Þ

A1 represents the upper reference range: the consumer will recognize the prices inside that rangeas lower. In the case of A2; the lower reference range, a consumer perceives the price as highereven if the shelf price is actually decreasing. In the literature of consumer behaviour studies, theexistence of two thresholds is assumed for the latitude of price acceptance, as shown in Figure 1.The two-regime model (2) specifically focuses on some region restricted to the neighbourhood ofa price threshold. In empirical analysis of the two-regime model, we have negative estimates ofthresholds; the model represents the case of having a lower reference range in (5). The lowerregime signifies the regime of perceived gain. The upper regime contains the latitude of priceacceptance and the region of perceived loss. Model (2) has a meaning of local approximation toglobal modelling of a (representative) consumer’s behaviour, which formally corresponds to athree-regime model in the literature. For three regimes, the latitude of price acceptance isA ¼ ½r1; r2�:



Several types of conceptualization with respect to reference price RPt; particularly focusingon memory-based definitions. We employ the following six types of formulation and choose oneduring modelling. RPt and Pt denote the reference price and the retail price at the time t;respectively. Then the reference prices proposed so far can be grouped as follows: the price atthe consumer’s last purchase RP1 : RPt ¼ Pt�1 [4,26,27]; the minimum or maximum prices ofthe consumer’s preceding purchases RP2 : RPt ¼ minðP1; . . . ;Pt�1Þ or RP3 : RPt ¼ maxðP1;. . . ;Pt�1Þ [28]; the average of the prices with and without several competing brands in theconsumer’s past consideration sets RP4 : RPt ¼ ð1=t� 1Þ

Pt�1j¼1 Pj and RP5 : RPt ¼

ð1=nðt� 1ÞÞPn

i¼1

Pt�1j¼1 Pij ; where n is the number of brands [29]; and finally, the adaptive

expectation model RP6 : RPt ¼ lRPt�1 þ ð1� lÞPt�1 [2,6].From these model settings, we can formally examine the following two hypotheses. P1: There

exists a reference price appropriate to the corresponding market, and P2: There exists a pricethreshold r=0: P1 can be tested by comparing our CMTAR model with linear models. Theposterior distribution of r is used to test P2: We describe those procedures in the subsequentsections.

3. INFERENCE ON THE MODEL AND FORECASTING

3.1. Model estimation

In general, the proposed model engenders the difficulty of calibration because the estimation ofkey parameter f ¼ ðd; rÞ is not well defined. First, the range of d is limited to a finite number ofintegers, and the profile likelihood function of r is a step function with steps occurring at thesample values of Dt�d : Conventional maximum likelihood estimation can be implemented, but

Figure 1. Price thresholds and market response.



the classical asymptotic distribution theory is not operative because the likelihood function isnot differentiable in f and the range of r further depends on the observation Dt�d ; which breaksthe regularity condition of maximum likelihood estimation of r: Then the inference of otherparameters proceeds essentially as that in the Gaussian linear models if the parameters f andthe orders of lag for each regime are known. Therefore, one approach is the conditional leastsquares method proposed in Tong and Lim, 1980 [15], where conditioned on some specific valueof f; the model specification as well as the estimation of f; is conducted by the extensive use ofAIC. However, this approach does not allow statistical inference methods such as testinghypotheses and confidence intervals for f:

Another approach is Bayesian inference proposed by Geweke and Terui [19,30]. They findthat the unconventional likelihood function mentioned above proves to be no barrier torecovery of the posterior density and that statistical inference on f can be conducted using theposterior density. Denote by Y the data information. Then under non-informative diffuse prior,they derived the posterior density pðf jYÞ of f as

pðf jYÞ ¼ pðfÞY2j¼1

2�ðnj=2þ1ÞðpÞ�ðnj=2ÞGnj2

� � njs2j2

!�ðnj=2ÞjZ0jZj j�1=2 ð6Þ

where nj ¼ Tj � kyj � kxj � 1; s2j ¼ ðZj �Wj#CjÞ0ðZj �Wj

#CjÞ=nj ; #Cj ¼ ðW 0jWjÞ

�1W 0j Zj ; and j is

the index of each regime, Tj is the number of Yt in the regime j; Zj means Tj dimensional vectorcomposing from dependent variable Yt; and Wj means Tj � ðkyj þ kxj þ 1Þ matrix forexplanatory variables ðYt�1; . . . ;Yt�kyj ;Xt; . . . ;Xt�kxj Þ: We assume a prior pðfÞ to be specified,and conditional on f; we take independent priors on Cj and sj ; of standard Jeffreys’ conjugate,pðCj ; sj ; j ¼ 1; 2;fÞ ¼ p1ðCj ;sj ; j ¼ 1; 2 j fÞp2ðfÞ; where p1ðCj ;sj ; j ¼ 1; 2 jfÞ ¼ 1=ðs1s2Þ andp2ðfÞ could be essentially any form as long as

PU�1d¼0

R1�1 pðfÞ ¼ 1: A non-informative prior

density for p2ðfÞ is employed here. In the above, we used the notation Cj ¼ ðaðjÞ1 ; . . . ; a

ðjÞkyj; bðjÞ0 ;

. . . ;bðjÞkxj Þ of Tj ¼ kyj þ kxj þ 1 dimensional vector of structural parameters.Apart from the inference problem on f; we employ a Bayesian inference procedure for TAR

modelling because of the use of full predictive density, which can be generated throughapplication of MCMC methods. Conditional on some controlled price orbits, we constitute amulti-step-ahead sales forecast and conduct strategic simulations for dynamic pricing.

3.2. Order determination of the model

The TAR modelling from Bayesian approach was investigated under the assumption that theorder of the model is known, e.g. References [19,30,31]. We employ the information criterionstrategy to determine the orders of a linear time series model of each regime. The informationcriteria in time series models have been recognized as an appropriate strategy for modelselection, particularly with respect to forecasting performance. Noting that the BIC value has ameaning of approximation to the Bayes factor, we take an information theory approach by BIC.Modelling is constructed from the following steps.

We select a candidate reference price from RP1 through RP6 described in previous sectionsand construct a threshold series fDtg: Denote a set of integers representing orders of the linearARX model of each regime by fk0sg � ðky1; kx1; ky2; kx2Þ; then for given fk0sg;

1. Fit the separate ARX models to the appropriate data subset.2. Evaluate the posterior density pðf j fk0sg;YÞ in terms of (6) and find the modal value *f:



3. Let BIC1 and BIC2 denote the BIC criteria for the individual regime models which areevaluated at the maximum likelihood estimates for model parameters ð #C1; #C2; #s1; #s2Þ:Then define BIC ¼ BIC1 þ BIC2:

4. Choose the model with the orders fk0sg that have the minimum BIC value.

These steps are applied to other candidates of reference price to find the best model.

3.3. Structural parameter estimates, multipliers and predictive density

After the model specification is finished, the Bayesian posterior distribution of pðCj j *f; fk0sg;YÞis evaluated by Markov chain Monte Carlo method to get the Bayesian estimatesEðCj j *f; fk0sg;YÞ and Eðsj j *f; fk0sg;YÞ: Furthermore, the dynamic multipliers for regime iare defined from its coefficient lag polynomial MðLÞðiÞ ¼ AðLÞðiÞ�1BðLÞðiÞ; and they arerecursively determined from the structural parameters by

MðjÞq ¼ bðjÞq þ *aðjÞq M

ðjÞ0 þ *aðjÞq�1M

ðjÞ1 þ � � � þ *aðjÞ1 M

ðjÞq�1 ð7Þ

where

*aðjÞq ¼aðjÞq ðp5qÞ

0 ðp5qÞ

(

On the other hand, the predictive density of Bayesian h step ahead predictor, conditional onthe future orbit of the price fXn

tþh;Xntþ2; . . . ;X

ntþhg; is represented by

pðYtþ1;Ytþ2; . . . ;Ytþh jXn

tþ1;Xn

tþ2; . . . ;Xn

tþh;YÞ

¼Z� � �Z Z

pðYtþ1;Ytþ2; . . . ;Ytþh;G;S;f jXn

tþ1;Xn

tþ2; . . . ;Xn

tþh;YÞ dG dS df ð8Þ

where G ¼ ðC1;C2Þ;S ¼ ðs1;s2Þ; f ¼ ðd; rÞ; and the integral is taken over the region�15G51; 05S51; �15r51 and d is summed over integers 0 up to a bound U: Theconcept of the predictive distribution has been well recognized and investigated from Bayesianapproach. With respect to the evaluation of (8), the multiple integration does not have ananalytical expression in a closed form. Monte Carlo integration by synthetic random numberscan be applied to generate a synthetic random sample from this predictive density. Based on thissynthetic sample, expected values of functions of interest of the future values can beapproximated.

Define #Yt ¼ Yt; t ¼ 1; . . . ;T ; and then for h ¼ 1; 2; . . . ; generate recursively

#Ytþh ¼bð1Þ0 þ

Pky1i¼1 að1Þyi Ytþh�i þ

Pkx1i¼0 bð1Þxi Xtþh�i if Dn

tþh�d4r

bð2Þ0 þPky2

i¼1 að2Þyi Ytþh�i þPkx2

i¼0 bð2Þxi Xtþh�i if Dntþh�d5r

8<: ð9Þ

For the formula of recursions (7) and (9), see for example, Lutkepohl, 1991 [32, Chapter 10].The posterior distribution of dynamic multipliers and predictive density are produced in theprocess of MCMC iterations by exact sampler. We leave this algorithm in Appendix A.



4. MODEL APPLICATION

4.1. Description of data and preliminary examinations

This study applies the model to daily store-level aggregated sales and price data series fornational brand milk in a store for an analysis extending from 28 February 1994 to 22 April 1995(total 400 daily samples for estimation). We take the holdout sample for the evaluation offorecasting performance. Video Research Inc. in Japan supplied these data. Qt and Pt denotethe sales quantity and the price at day t; respectively. We transformed each series to adaily percentage change, Xt ¼ log Pt � log Pt�1 and Yt ¼ logQt � logQt�1; to achieve theirstationarity. Seasonal adjustment was conducted using six binary seasonal dummy variables inthe regression model to remove weekly seasonal effects. These transformed series are plotted inFigure 2. As a threshold variable, we take the perceived price as Dt ¼ log Pt � log RPt: Forthe delay parameter, we formally set d ¼ 0; 1; 2; 3 in our model, although it might not bereasonable to consider that the consumer purchases the goods by merely considering the pastprice for the positive d: We set the range for the orders as 14kxi; kyi47 ði ¼ 1; 2Þ:

4.1.1. Time series non-linearity. The proposed model is based on non-linearity of the stationarytime series. We first test the linearity of our data in advance of modelling. We apply four well-known linearity tests to our data set: (i) the Ori-F test by Tsay [33], (ii) the Aug-F test byLuukkonenn et al. [34], (iii) the TAR-F test by Tsay [35], and (iv) the New-F test by Tsay [36].All tests use a linear model as a null hypothesis and set some specific non-linear model as thealternative. For a general autoregressive non-linear time series model with an i.i.d. process fetg;Yt ¼ hðYt�1;Yt�2; . . . ;Yt�pÞ þ et; the Ori-F and Aug-F tests specify a function hð�Þ as thesecond- and third-order polynomials, respectively; the TAR-F and New-F tests assume

Figure 2. Price and sales time series in a store: change(log-difference).



threshold type non-linear alternatives. In particular, the New-F test covers the most extensiveset of alternatives of non-linearity, including the ExpAR model. Detailed procedures anddistributional properties regarding these tests are described in Granger and Terasvirta [37].

The test statistics and their p-values for linearity tests are given in Table I; the null hypo-thesis of linearity for each series of Xt and Yt is rejected. The linearity test means a Gaussianitytest if we assume innovation et as Gaussian. We apply Jarque–Bera’s Gaussianity test as areference. This test is a combined test of skewness and kurtosis compared with those ofGaussian distribution. Results of these tests are provided in the lower panel of the table. Theunivariate analysis tests the Gaussianity of residual after fitting linear AR models for eachseries. The bivariate analysis tests the Gaussianity of residuals after fitting linear ARXmodel. The null hypotheses of Gaussianity for each series (univariate) as well as both series(bivariate) are rejected. These preliminary observations provide a background with ourframework.

4.2. Empirical results

4.2.1. Model estimation. Corresponding to each candidate reference price RP1–RP6, weestimate the six types of CMTAR models. Table II shows their results. Each column shows keyresults for respective reference prices. The first four rows show the model orders of each regimechosen by BIC. The next two rows below represent the number of samples for each regimeallocated at modal value of f ¼ ðd; rÞ: We note that the posterior distribution of d has a masswith probability of almost one at d ¼ 0 for all specifications, as expected. The values of BIC andR2 are reported in the next two rows. For parameter l appeared in the model with RP6, we takea grid search approach and estimate it to minimize the BIC value.

Table I. Linearity and Gaussianity tests for store data.

Linearity test

Ori-F Aug-F TAR-F New-F

Univariate Y 2.077 1.956 3.292 1.998(0.001) (0.001) (0.001) (0.004)

X 2.973 2.901 3.051 2.263(0.000) (0.000) (0.004) (0.001)

Gaussianity test

Skewness Kurtsis J–B test

Univariate Y 2.077 1.956 3.292(0.000) (0.000) (0.000)

X 2.973 2.901 3.051(0.000) (0.000) (0.000)

Bivariate ðY ;XÞ 2.077 1.956 3.292(0.000) (0.000) (0.000)

ðX ;YÞ 2.973 2.901 3.051(0.000) (0.000) (0.000)

The number inside parenthesis means the p value.



From Table II, every model clearly captures the non-linear property of the series. In terms ofthe minimum BIC criterion, we finally choose the model with RP4. The result for the linearARX model is given in the last column for comparison. The estimated CMTAR models havemuch smaller BIC values than the linear model. Following our experience with applied works ofTAR models using actual economic data, for example in References [19,38], we sometimesencounter a situation in which one or both regimes fail to contain a sufficient number of samplesto allow an inference. That is, the number of samples allocated in either regime is the minimumnecessary to estimate because the posterior distribution of r is concentrated on the edge of thepossible domain. This fact implies the degenerate special case of the linear model. However, thatdoes not occur here. We consider it as further evidence for favouring the threshold model overthe linear model for those specifications. The posterior distribution of r is relativelyconcentrated for every model. In particular, in the case of model 4, which is ultimately chosen,the posterior density of r concentrates almost on some region.

The estimated structure of model 4 is

Yt ¼

0:0458� 0:3837Yt�1 � 5:5162Xt if Dt4� 0:0412

ð0:0887Þ ð0:1132Þ ð0:6548Þ ð0:0001Þ

�0:0176� 0:8913Yt�1 � 0:6970Yt�2 � 0:4890Yt�3

ð0:0815Þ ð0:2190Þ ð0:2010Þ ð0:1720Þ

�0:2965Yt�4 � 0:1907Yt�5 � 0:1037Yt�6 � 0:0403Yt�7 if Dt > �0:0412

ð0:1815Þ ð0:0469Þ ð0:0354Þ ð0:0325Þ ð0:0001Þ

�3:8894Xt � 2:6808Xt�1 � 1:6954Xt�2 � 1:0178Xt�3

ð0:2866Þ ð0:2191Þ ð0:1958Þ ð0:1538Þ

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:T1 ¼ 35; T2 ¼ 351; P=RP ¼ 0:9596; BIC ¼ �24:18; R2 ¼ 0:75

Next, implications derived from the estimated model are summarized below.

Table II. CMTAR model estimation: the model value.

Model 1 2 3 4 5 6 Linear ARX

k1y 1 1 1 1 1 1 5k2y 7 7 7 7 7 7 }k1s 0 0 0 0 0 0 3k2s 3 3 3 3 3 3 }d 0 0 0 0 0 0 }l } } } } } 0.70 }r �0.1014 0.0521 �0.1718 �0.0412 0.0668 �0.0864 }s.d.(r) 0.0029 0.0006 0.0020 0.0001 0.0001 0.0005 }P/RP 0.9036 1.0535 0.8421 0.9596 1.0691 0.9172 }s1 1.428 1.352 1.281 1.269 1.274 1.298 1.005s2 0.989 0.968 0.849 0.840 0.843 0.921 }BIC 14.25 �0.82 �17.56 �24.18 �18.87 �5.08 66.84R2 0.6573 0.7491 0.7502 0.7501 0.7489 0.6966 0.7550



1. Threshold parameter r is estimated as �0:0412; which is the posterior mean. We note thatthe posterior density of r is the step function over the region ðD½s�1�t ;D½s�t Þ; the probabilityjumps at the order statistic D

½s�t : Then the posterior probability is concentrated on that

region at the point D½s�t ¼ �0:0412: The Bayesian odds ratio test is inappropriate to test the

hypothesis r ¼ 0 because of the diffuse prior employed in this paper to make a statisticalinference on r: Instead, we use highest probability density (HPD) region R1�a with the levelð1� aÞ: R1�a is defined as a region satisfying the next two conditions:

ðaÞ for positive a less than 1, Prfr 2 R1�ag41� a;ðbÞ for r1 2 R1�a and r2 =2 R1�a; pðr1 jYÞ5pðr2 jYÞ:

The approximate 95% HPD region does not include r ¼ 0: Consequently, we reject the nullhypothesis of non-existence of a latitude of price acceptance. Estimate of r; aftertransformation to real money term, turns out to be Pt=RPt ¼ 0:9596: Each regime isseparated at the retail price discounted by 4% from the reference price.

2. Regime 1 represents the dynamic response corresponding to a retail price with less than(4% + the average of past prices). Regime 2 indicates a response with a higher retail pricemore than this. Regime 2 includes a zero in the threshold space and the latitude of priceacceptance is measured as A2 ¼ ½�0:0412; 0�: Regime 1 can be interpreted as the price gainregime. On the other hand, regime 2 contains not only the price loss regime, but also thelatitude of price acceptance as shown in Figure 1.

3. The asymmetric response across regimes are evident from the estimates (posterior mean) ofmultipliers, M

ðiÞ0 ;M

ðiÞ1 ; . . . ;M

ðiÞ6 as well as M

ðiÞ1 in Table III. The current and the long-run

effects of regime 1,Mð1Þ0 ;M

ð1Þ1 ; are larger than those of regime 2. Delayed responses with lags

of 1 and 3 are estimated as positive for both regimes; we interpret them as the rebounds ofthe current price promotion. Posterior distributions of the current and long-run multipliersare drawn in Figure 3, which indicate the frequency distributions of MCMC draws of M

ðjÞk

described in Appendix B. Pricing below the threshold encourages consumers to be moresensitive. In turn, the consumer becomes insensitive to pricing unless the pricing getsover the threshold. We note also that the pricing of two, four and five days previousengender different effects to today’s sales on each regime because of the opposite sign ofthe estimates, ðMð1Þ

2 ¼ �0:8030;Mð2Þ2 ¼ 0:3176Þ; ðMð1Þ

4 ¼ �0:1199;Mð2Þ4 ¼ 0:4421Þ and ðMð1Þ

5

¼ 0:0467;Mð2Þ5 ¼ �0:2541Þ:

4.2.2. Forecasting. Then we examine the out of sample performance of multi-step forecasts. Oneadvantage of a Bayesian approach is forecasting because it offers a complete operational

Table III. Dynamic multipliers and their s.d.’s.

k 0 1 2 3 4 5 6 1

Mð1Þk �5.5163 2.0980 �0.8030 0.3093 �0.1199 0.0467 �0.0183 �3.9983

s.d. 0.0554 0.1596 0.1244 0.0722 0.0374 0.0183 0.0087 0.1127

Mð2Þk �3.8856 0.7697 0.3176 0.1992 0.4421 �0.2541 �0.0864 �2.5024

s.d. 0.0341 0.0962 0.0993 0.0981 0.1158 0.1032 0.0395 0.0262



concept of predictive distribution. Furthermore, it does not distinguish estimation of parametersand forecasting. In fact, Terui [25,39] suggests using the Bayesian predictive density for linearmarket-share time series models. The optimal predictor with the minimum mean squared erroris known as the conditional expectation *Ytþh ¼ EfYtþh jYt�1;Yt�2; . . .g: This predictor isgenerally difficult to evaluate for non-linear time series models. However, Bayesian inferenceproposed by Geweke and Terui [19,30] allows constitution of a predictive distribution ofCMTAR models. We explain the details of Bayesian predictive density for the CMTAR modelin terms of MCMC method in Appendix A. As a measure of forecasting performance, we usethe root mean squared error (RMSE) of the step ahead prediction, which is defined as

RMSEðhÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

m

Xm

j¼1ð #Ytþj j tþj�h � YtþjÞ

2

rð10Þ

Table IV shows the RMSE for the CMTAR in next one-week forecast. We set m ¼ 20 andh ¼ 1; . . . ; 7 in (10), and the optimal models were reselected whenever we roll over the samples.The number of pðr1Þ in the table means the percentage of which the predictor has been generatedfrom regime 1 for all predictions.

Figure 3. Posterior distribution of multiplier-store data.



4.2.3. Comparative study. We compare the forecasting performance of the CMTAR model withthree other kinds of models: the linear models, self-excited AR (SETAR) models, and anotherCMTAR model with a competitive brand variable. The RMSEs for these models are given inTable IV. In that table, ARXðp; qÞ indicates the model with the order of AR part of order p; andan exogenous part with the order of q: The linear ARX models with five least BIC values areexamined for comparison. For other CMTAR models, TB2 designates a CMTAR model thatuses other brands’ prices as the threshold variable and SB2 is the CMTAR model with theadditional explanatory variable of other brands’ prices for respective regimes.

Results show better performance of CMTAR than other models for seven steps ahead. Inparticular, the CMTAR model is superior to linear models as expected from the observation oflarge difference between their BIC values in Table II. However, the SB2 model shows betterperformance only at the seven days ahead than the CMTAR, but we expect that the SB2 modelgenerates a better predictor overall than that of the CMTAR because it contains additionalinformation in the structure. It turns out that the forecasting gain was not obtained more thanthe penalty induced from introducing more complicated structure.

4.2.4. Strategic simulations. Next, we conduct strategic simulations under several scenarios.Based on the knowledge of estimated model structure, the scenarios are constructed in twoways.

Table IV. The RMSE for multi-step forecasting.

Group Step 1 2 3 4 5 6 7

Main CMTAR 0.2591 0.2796 0.2765 0.2681 0.2820 0.3700 0.5129s.d. 0.049 0.061 0.131 0.046 0.094 0.108 0.143pðr1Þ (0.60) (0.60) (0.60) (0.60) (0.60) (0.53) (0.53)

Linear ARX(5,4) 0.3065 0.3263 0.3274 0.3214 0.3546 0.4056 0.6047s.d. 0.085 0.095 0.095 0.092 0.109 0.137 0.231

ARX(4,4) 0.3050 0.3273 0.3277 0.3251 0.3540 0.4067 0.6090s.d. 0.084 0.095 0.095 0.094 0.109 0.138 0.233

ARX(6,4) 0.3092 0.3311 0.3290 0.3228 0.3522 0.4066 0.6103s.d. 0.086 0.097 0.096 0.093 0.108 0.138 0.233

ARX(5,3) 0.2834 0.2956 0.3011 0.2938 0.3124 0.3763 0.5729s.d. 0.073 0.079 0.082 0.078 0.088 0.121 0.220

ARX(4,3) 0.2810 0.2965 0.3014 0.2973 0.3120 0.3775 0.5772s.d. 0.072 0.080 0.082 0.080 0.087 0.122 0.222

Original TAR SETAR 0.2892 0.3108 0.3182 0.3111 0.3422 0.3830 0.5796s.d. 0.075 0.084 0.081 0.118 0.101 0.187 0.210pðr1Þ (0.07) (0.07) (0.07) (0.13) (0.13) (0.13) (0.13)

Other brand TB2 0.3044 0.3189 0.3241 0.3117 0.3263 0.3982 0.5087s.d. 0.086 0.085 0.128 0.091 0.123 0.120 0.233pðr1Þ (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)SB2 0.2961 0.2976 0.3403 0.2866 0.3318 0.3761 0.6076s.d. 0.070 0.086 0.176 0.082 0.100 0.128 0.167pðr1Þ (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)



1. Knowledge of estimated threshold:

ðaÞ During the next seven days, we keep the same price (PTþ1 ¼ PTþ2 ¼� � � ¼ PTþ7 ¼ #r� a% discount); then it stays in the threshold space of regime 1.Respective values of a are chosen for 0.5, 1, 2, 3, and 4 under scenarios S1, S2, S3, S4,and S5.

ðbÞ During the subsequent seven days, we maintain a price (#rþ a% discount) that stays inthe threshold space of regime 2. Values of a are chosen for 0.5, 1, 2, 3, and 4 underscenarios S10; S20; S30; S40; and S50; respectively.

2. Knowledge of estimated dynamic multipliers:

ðaÞ Focusing on pricing PTþ1 of the next day, we set the price at each succeeding step,which switches the regime to that with a larger dynamic multiplier M

ðiÞk ; i ¼ 1; 2 when

both are negative. Therefore, we choose a regime with larger proper market response toprice promotion. In cases for which both are positive, we switch to a regime with asmaller rebound. For example, we set PTþ1 ¼ #r� a; which belongs to regime 1 becauseits current market response M

ð0Þ1 is larger than M

ð0Þ2 of regime 2. Next, on the 2nd day

of the forecast, we set a price so that the regime switches to regime 2, where the positivedelayed response M

ð1Þ2 representing the rebound of PTþ1 is smaller than positiveM

ð1Þ1 at

regime 1. According to the magnitude of dynamic multipliers, this operation isundertaken throughout the period. The pricing variation is fixed as (0.5, 1, 2, 3, 4%).Their corresponding strategies are denoted by S6–S10.

ðbÞ For reference, we set the price independently at every prediction step and conduct all-round competitions with respect to the pattern of regime switching over the next sevendays. That is, as two possibilities of regime switching on each day exist, we consider allcombinations of regime switching}27 ¼ 128 patterns in all}and generate predictorsfor every pattern. Then we search the model with the largest accumulated sales number(amount). We fix the price variation as constant a (0.5, 1, 2, 3, 4%) and thereby createscenarios S11–S15.

We denote by S0 the default scenario in which we maintain the current price throughout thesubsequent seven days. Table V shows the predicted sales number at each prediction step forscenarios S0–S15. The accumulated sales amount (number � price) and profit (amount �25%(assumption)) for the next seven days is also given in the table. Firstly, we have a thresholdwith a negative sign, meaning that we gave the lower reference range A2 ¼ ½�4:12%; 0� in (5).Thus, we expect that the price promotion is ineffective for pricing inside A2: Consequently, thepricing has to surmount this threshold. Results show that the difference between predicted salesof S1(R1: regime 1, outside A2) and S10(R2: regime 2, inside A2) is quite large, but the differencebetween their prices is only 1%ð¼ 0:5%þ 0:5%Þ: This does not change when the price variationis large: S2(1%)–S5(4%) and S20ð1%Þ–S5ð4%Þ0: Secondly, results of S6–S10 show that theinformation of dynamic multipliers could help increase sales. In comparison with scenarios ofr� a% (S1–S5), we have a 2.5–6% increase of accumulated forecasted sales. Finally, the bottomof the table shows forecasts of independent pricing. The selected pattern of regime change withthe largest sales was the same as the pricing using multipliers knowledge except at seven daysahead for every discount level. That is, the predicted sales of each step are the same as the



pricing by dynamic multipliers until six days ahead. The effect of the regime chosen at sevendays ahead brought a great difference of predicted sales. We note that the difference betweenMð1Þ6 and M

ð2Þ6 was small.

These results from our limited experiments cannot be generalized to dynamic pricing.However, we suppose that pricing based on dynamic multipliers might provide some usefulinsights. We leave this important problem for future research.

Figure 4 shows the predictive density for some scenarios: the default S0, S5, S10 and S15. Wecan see how pricing under the scenarios controls future sales. In particular, price changes willnot always bring a reasonable response of sales because of the unexpected direction of dynamicresponse for positively estimated multipliers. This inference is consistent with the predictivedensity, by which a discount (hike) produces an increase (decrease) in sales on the same day, butbrings a decrease (increase) on the next day, three days after, and so on. We note that thepredictive density derived from a linear model always gets more diffuse as the prediction stepproceeds. However, it does not happen for the CMTAR model because the predictive density isconstructed at each step from different regimes with different precisions of estimation. In ourcase, the predictive density generated from regime 1 has relatively larger variance than that ofregime 2; it is observed that the confidence bounds for the predictor do not become widermonotonically over time.

Table V. Forecasted sales under scenarios.

Prediction step Sales Sales Profit

Pricing Scenario 1 2 3 4 5 6 7 (number) (amount) (0.25)

Default(R2) S0 56 57 52 56 56 55 54 387 88206 30752

r� a% S1ð0:5%) 144 87 112 107 114 117 122 803 156659 37353(R1) S2(1%) 148 89 114 109 117 119 124 820 159156 37341

S3(2%) 156 92 119 113 121 124 129 855 164274 37278S4(3%) 165 95 124 118 126 129 135 892 169564 37161S5(4%) 174 98 130 122 131 134 140 930 175032 36986

rþ a% S10ð0:5%Þ 98 89 78 84 78 78 78 584 115062 28307(R2) S20ð1%Þ 96 88 77 83 77 77 77 576 114010 28477

S30ð2%Þ 93 85 75 81 75 76 75 560 111937 28795S40ð3%Þ 89 83 73 79 74 74 73 544 109904 29084S50ð4%Þ 86 80 71 76 72 72 72 529 107912 29346

Mk S6(0.5%) 144 52 193 58 231 70 74 823 168846 46661(R1–R2) S7(1%) 148 52 200 58 241 70 75 844 172531 47122

S8(2%) 156 52 215 58 262 71 76 890 180256 48038S9(3%) 165 51 231 58 285 72 77 940 188479 48940S10(4%) 174 51 249 58 309 73 78 993 197235 49823

Independent S11(0.5%) 144 52 193 58 231 70 278 1026 206029 53658(R1–R2) S12(1%) 148 52 200 58 241 70 291 1061 211910 54414

S13(2%) 156 52 215 58 262 71 320 1134 224324 55929S14(3%) 165 51 231 58 285 72 351 1214 237666 57442S15(4%) 174 51 249 58 309 73 386 1300 252007 58945



5. CONCLUDING REMARKS

This study developed a sales forecasting model that incorporates asymmetric market responses byintroducing a popular class of non-linear time series model}a TAR model. We observe that theprice threshold can describe a set of store data exhibiting some non-linear and non-Gaussianproperties. We also introduce the CMTAR models and show how it could be modified for use ina sales forecasting model. Furthermore, our experiments show that two-regime CMTAR modelsproduce better forecasting performance than simple linear time series models and even betterthan self-excited AR (SETAR) models. Bayesian predictive density in terms of MCMC methodallows the implementation of miscellaneous strategic simulations of future price management; theinformation of dynamic multipliers provides a store manager with insight into price management.

The two-regime CMTAR models specifically focus on the neighbourhood of a threshold inthreshold space. It implies that our modelling conducts a local approximation to global

Figure 4. Predictive density for strategic similations.



modelling of (representative) consumer behaviour, which formally corresponds to the three-regime model in the literature. We developed a three-regime model to extend the analysis. Thesame procedures as those used for the two-regime model are followed for this model. Detailedresults of empirical applications are not given herein because of space limitations, but weprovide the following summary: (1) The reference price chosen is the average price. (2) We haveestimates of price thresholds, #r1 ¼ �4:95% and #r2 ¼ 14:33%: (3) The approximate 95% HPDregions do not contain zero for these parameters. Practical implications are clear from theseestimates. (4) Regarding forecasting performance, the two-regime model shows much betterperformance than the three-regime model does in terms of RMSEs of forecasts. Details of theseresults are available from the author upon request.

Furthermore, we conducted some diagnostic check tests to our data below.(a) Heteroscedasticity and CMTAR model: We note that time plots of Figure 2 show

some heteroscedasticity in sales and price series. However, after fitting CMTAR model,we confirmed the homoscedasticity in each regime from the results of heteroscedasticitytests (Lagrange multiplier (LM) test, White’s test, and RESET tests) using the standardeconometric package TSP. We observe that the CMTAR models explained the hetero-scedasticity in the data.

(b) Non-linearity and linear VARX model: For the purpose of comparative study with linearmodels including VAR models, we applied Gaussianity tests to residuals of estimated linearmodels whose orders were chosen by the minimum BIC criterion. The AIC suggested the samelinear models. The Gaussianity of single series does not imply joint Gaussianity. For thatreason, we applied the Bispectrum test by Wong [40] to verify the multivariate Gaussianity ofour store data. Gaussianity tests for the residual of fitted linear VARX models were conductedto show that some non-Gaussianity characteristics remain after applying a VARX model to thedata. This is partial evidence of the necessity of a non-linear modelling instead of simple vectorlinear modelling. These results are also available from the author upon request.

We note that our modelling is conditional on the selection orders of the model and Bayesestimate f ¼ ðr; dÞ:We understand that the conditional inference of f is not so limited in our dataset. In fact, we confirmed that full Bayesian analysis does not bring any difference to the resultsbecause the posterior density of f is quite concentrated on the modal values throughout the models.On the other hand, model orders might be more influential on the result and a model-averagingapproach would be the next step to be explored. However, a more elaborate set-up is necessary. Forthat reason, we will address these issues in future work. Finally, the pricing suggested from ourlimited experiments cannot be generalized to dynamic pricing and some theoretical investigations ofdynamic pricing by the CMTAR models are also left for future research.

APPENDIX A: EXACT MCMC SAMPLER}SYNTHETIC RANDOM NUMBERGENERATION

Following Geweke and Terui [19,30], we use Markov chain Monte Carlo method by syntheticrandom number generation, called exact MCMC sampler, to implement the CMTAR modelestimation and prediction. Conditional on f; we use the property that the joint posteriordistribution is decomposed into marginal posterior and conditional posterior distributionsas pðC1;C1;s1; s2 jf;YÞ ¼ pðC1;C2 j s1;s2;f;YÞ � pðs1; s2 jf;YÞ; and further the syntheticrandom numbers from each density of right-hand side can be generated.



Under the assumption of independent normal distribution ejt � Nð0; s2j Þ; j ¼ 1; 2 forinnovations of each regime, and the standard Jefferys’ non-informative diffuse priors on theparameters B and sj

pðC1;C1; s1; s2;fÞ ¼ pðC1;C1; s1; s2 j fÞpðfÞ ¼1

s1s2ðA1Þ

the posterior distribution of these parameters, conditional on the parameter f; can be given as

pðC1;C1; s1;s2 j f;YÞ

¼ pðC1;C2 j s1;s2;f;YÞ � pðs1;s2 j f;YÞ ðA2Þ

¼Y2j¼1

pðCj j sj ;f;YÞ � pðsj jf;YÞ ðA3Þ

where pðCj j sj ;f;YÞ takes a form of normal distribution Nð #Cj ;s2j ðX0jXjÞ

�1Þ and pðsj j f;YÞ is theinverted Gamma distribution IGðnjÞ:

APPENDIX B: ALGORITHM: BAYESIAN ESTIMATES OF STRUCTURALPARAMETERS, DYNAMIC MULTIPLIERS AND PREDICTIVE DENSITY

The following procedures are repeated M times. (We used M ¼ 5000 in empirical application,after discarding 1000 burn-in samples. We confirmed no changes after that number ofiterations.) For each regime j ¼ 1; 2; generate random numbers from conditional posteriordistributions as follows:

1. Generate sðmÞj jf; fk0sg;Y� IGðnjÞ: Inverted gamma distribution with the d.f.nj :

2. Generate CðmÞj j s

ðmÞj ;f; fk0sg;Y � Nð #Cj ; s

ðmÞ2

j ðX 0jXjÞ�1Þ: Multivariate normal distribution.

3. Generate the multiplier MðjÞk ½m� for regime j by using (7).

4. Generate the predictor Y½m�tþh by using (9).

Then, the empirical distributions of fCðmÞj ;sðmÞj ;m ¼ 1; . . . ;Mg approximate the posteriordensity pðCi j f;YÞ and pðsi jf;YÞ and we obtain the point estimate

*Cj ¼1

M

XMm¼1

CðmÞj ; *sj ¼

1

M

XMm¼1

sðmÞj ðB1Þ

Similarly, the posterior density and Bayes estimate of multipliers of regime j are constructedform the sequence fMðjÞ

ðmÞ

k ;m ¼ 1; . . . ;Mg; and the predictive density and point predictor arealso constituted form the sequence f #Y ðmÞtþsðs ¼ 1; . . . ; hÞ; i ¼ 1; . . . ;Mg:

The point estimates which is the optimal under quadratic loss functions, defined asconditional expectations, and the sample standard deviation gives a numerical approximate ofposterior standard deviation. They have a property, as M !1

1

M

XMi¼1

Y½i�tþh!

a:s:EðYtþh j f;YÞ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

M

XM

i¼1ðY½i�tþh � %YÞ2

r!a:s:

SDðYtþh j f;YÞ ðB2Þ

The same formula applies to other statistics.



The estimates of dynamic and long-run multipliers after taking expectation with respect tothese posterior distributions, EfðEðMk jf;YÞÞ ¼

RMkpðf jYÞ df and EfðEð #Ytþh jf;YÞÞ ¼R

#Ytþhpðf jYÞ df; leads to full Bayesian estimates. In empirical applications, we confirmed thatfull Bayesian analysis does not bring any difference to the results because the posterior densityof f is quite concentrated on the modal values throughout the models.

ACKNOWLEDGEMENTS

We sincerely acknowledge the comments from the editor-in-chief, associate editor, and two referees. Theircomments were extremely helpful in improving our paper. The first author acknowledges the financialsupport by the Japanese Ministry of Education [Scientific Research Grant No.(C)(2)12630024 and(C)(2)15530137].

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Forecasting model with asymmetric market response and its application to pricing of consumer package goods