Modeling seasonal effects in the Bass Forecasting Diffusion Model

Technological Forecasting & Social Change 88 (2014) 251–264

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Technological Forecasting & Social Change

Modeling seasonal effects in the Bass Forecasting Diffusion Model J.J. Fernández-Durán ⁎ School of Business, Instituto Tecnológico Autónomo de México, Río Hondo No. 1, Col. Progreso Tizapán C.P. 01080, México D.F., Mexico

a r t i c l e

i n f o

Article history: Received 10 February 2014 Accepted 9 July 2014 Available online xxxx Keywords: Forecasting Seasonal effects Nonnegative trigonometric series

a b s t r a c t The Bass Forecasting Diffusion Model is one of the most used models to forecast the sales of a new product. It is based on the idea that the probability of an initial sale is a function of the number of previous buyers. Almost all products exhibit seasonality in their sales patterns and these seasonal effects can be influential in forecasting the weekly/monthly/quarterly sales of a new product, which can also be relevant to making different decisions concerning production and advertising. The objective of this paper is to estimate these seasonal effects using a new family of distributions for circular random variables based on nonnegative trigonometric sums and to use this family of circular distributions to define a seasonal Bass model. Additionally, comparisons in terms of one-step-ahead forecasts between the Bass model and the proposed seasonal Bass model for products such as iPods, DVD players, and Wii Play video game are included. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Almost all products exhibit seasonality in their sales patterns. For example, the sales of many products increase in December or the fourth quarter of the year due to the bonuses paid by employers to employees or due to religious traditions, such as Christmas. Other seasonal human activities, for example, the beginning of the school year, increase the sales of certain products related to these activities. Finally, weather is one of the main causes of observed seasonal effects on the sales of different products. The identification and estimation of seasonal effects on the sales of a product are very important for marketing decisions, such as determining the best time to introduce a new product, when to increase spending on advertising, the rotation of products in a store, and the amount to produce, among others. For a previous analysis of seasonal effects in marketing problems, the interested reader can consult Radas and Shugan (1998). The Bass Forecasting Diffusion Model (Bass, 1969) is one of the most frequently used models for forecasting the sales ⁎ Tel.: +52 55 56 28 40 84; fax: +52 55 56 28 40 86. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.techfore.2014.07.004 0040-1625/© 2014 Elsevier Inc. All rights reserved.

of a new product. The main idea behind the Bass model is that the sales of a product at a given point in time depend on the number of buyers since the introduction of the product. In this sense, the buyers of the new product are classified as innovators and imitators. Innovators are buyers who buy the product without considering the recommendation of previous buyers, only buying in response to external influences, such as advertising. The timing of an innovator's initial purchase is not influenced by the number of buyers who have already bought the product. Imitators are buyers who buy the product because they have been influenced by the number of previous buyers; i.e., they buy in response to internal influences, such as word of mouth. This model is determined by three parameters, m, p, and q where m is the total number of people who eventually buy the product, p is the coefficient of innovation, and q is the coefficient of imitation. We refer to the original Bass model as the classical Bass model. Later, Bass et al. (1994) extended the classical Bass model to include the effects of explanatory variables such as price and advertising, on the sales of a new product. For reviews of the classical Bass model, its modifications, and its applications, one can consult Rao (1985), Mahajan et al. (1990), Parker (1994), Meade and Islam (2006), Mahajan et al. (2000), Chandrasekaran and Tellis (2007) and Peres et al. (2010).

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The inclusion of seasonal effects in the Bass model is related to the problem of temporal aggregation, as discussed by Putsis (1996) (see also Non et al., 2003). Putsis criticizes the fact that the classical Bass model invariably used annual data, not taking into account the available quarterly or monthly data, with the exception of the paper by Tigert and Farivar (1981). Putsis recommends that studies should use higher-frequency data whenever possible because the models presented in the paper, adjusted with quarterly data, outperform those estimated with annual data. Unfortunately, Putsis (1996) analyzed monthly and quarterly data series that were seasonally adjusted using the census X-11 moving average method prior to estimating the parameters. In contrast to Putsis (1996), one of the main objectives of this paper is the estimation of seasonal effects. Rao (1985), when comparing the forecasting performance of different diffusion models, considers that an important goal for future research is a comparative study using more refined data sets with more closely spaced points because such reports would provide more opportunities to forecasts and more precise and useful conclusions about model performance in predicting the time to peak sales. For certain products, the modeling of seasonal effects in diffusion models is very important; this is true, for example, among pharmaceutical products designed to cure seasonal diseases e.g., the flu diseases and electronic products with higher sales during the fourth quarter. From a statistical point of view, working with aggregated annual data when quarterly or monthly data are available is equivalent to throwing away important information from the time series data. For example, when making a forecast with only 2 years of data, in monthly terms, 24 data points are available, but in aggregated annual terms, just two data points are available, which is not enough to estimate the three parameters of the classical Bass model. As indicated by Radas (2001, 2005), due to increasing global competition and the shortening of product life cycles, managers are not allowed to wait for several yearly data points; thus, models that account for seasonal effects are very relevant in practice. Additionally, many authors have reported that the parameter estimates of the classical Bass model are highly sensitive to the number of observations (see Chandrasekaran and Tellis, 2007). Parker (1994) provides a warning about the problem in which the classical Bass model will inadvertently interpret early variations in adoption, such as those due to seasonality in monthly or quarterly sales data, as inflection points or peaks that can produce unreasonable parameter estimates and forecasts. One of the main modifications of the classical Bass model has been the inclusion of marketing mix variables to allow the parameters of the Bass model to vary with time. The Generalized Bass Model (Bass et al., 1994) considers the use of price and advertising to modify the p and q parameters, while other authors have considered the modification of the market potential and/or the diffusion parameters through the use of variables such as price, advertising, detailing, distribution, and direct-to-consumer advertising, among others (see the review by Ruiz-Conde et al., 2006). In this sense, the proposed seasonal Bass model in this paper can be considered a modified Bass model in which the diffusion parameters p and q are constant but modified through the use of a positive function x(t) during a full cycle to model the seasonal effects. However, additional variables, such as price or advertising, are not included in this

model. Again, this is an advantage when dealing with short time series to make forecasts and variables such as price has not changed since the introduction of the new product. The classical Bass model is a particular case of the proposed seasonal Bass model. Putsis (1998) compared models with constant parameter values to those with parameters following a stationary or a nonstationary stochastic process. Recently, Peers et al. (2012) used seasonal dummies to modify a closed-form innovation model in discrete time, in particular the Bass model, by considering that the seasonal peaks are the effect of postponed and accelerated sales in previous and future months. By working in discrete time, this model becomes too complicated in the case of multiple seasonal peaks, it is not able to deal with data given by combining different time frequencies and, requires many data points to make the estimation. Also, it is difficult to use this model in analogy forecasting. Contrary to Peers et al. (2012), the proposed model in this paper is derived in continuous time by the use of circular distributions based on nonnegative trigonometric sums (see Fernández-Durán, 2009). By considering continuous circular distributions, many of the disadvantages of the Peers et al. model are avoided and the proposed methodology can be applied to sales data observed in any frequency or even in different frequencies. The estimated circular distribution can be used in analogy forecasting for similar products to be launched in the future. From meta-analysis studies of the classical Bass model (see the references in Chandrasekaran and Tellis, 2007), one finds that the mean value of the coefficient of innovation, p, lies between 0.0007 and 0.03, while the coefficient of imitation, q, lies between 0.38 and 0.53. Sultan et al. (1990) analyzed 213 applications of diffusion models and found that p is equal to an average of 0.03, and q is equal to an average of 0.38. To address seasonality in monthly and quarterly sales data, it is common practice to use seasonally adjusted time series. The seasonal adjustments are performed using statistical methods, such as the X11 (see Ladiray and Quenneville, 2001), X12-ARIMA (see Findley et al., 1998), or TRAMO-SEATS (TRAMO: Time series Regression with ARIMA noise and SEATS: Signal Extraction in ARIMA Time Series, see Gómez and Maravall, 1998) procedures. Once the sales series has been adjusted for seasonality, then the classical Bass model is applied to the seasonally adjusted sales series to estimate the parameters (p and q) of the model (see Putsis, 1996). These parameter estimates are then used to forecast future sales. In this paper, the proposed Seasonal Bass model based on NNTS circular distributions is compared with models using seasonally adjusted data obtained by the X12-ARIMA and TRAMO-SEATS procedures. Statistical procedures for seasonal adjustment consider that a time series, Yt for t = 1, …, T, can be decomposed into Y t ¼ T t þ St þ I t

ð1Þ

for an additive decomposition, or Y t ¼ T t St I t

ð2Þ

for a multiplicative decomposition, where Tt is the trendcycle, St is the seasonal component, and It is the irregular

J.J. Fernández-Durán / Technological Forecasting & Social Change 88 (2014) 251–264

component. The objective of a seasonal adjustment is to obtain the deseasonalized time series Y t −St ¼ T t þ It

ð3Þ

for an additive decomposition or Yt ¼ T t It St

ð4Þ

for a multiplicative decomposition. A multiplicative decomposition is analyzed by transforming the original time series into its logarithm and using an additive decomposition. Before applying X12-ARIMA or TRAMO-SEATS, one must test for stationarity of the time series. When stationarity is rejected, one must work with the differenced time series. By default, in X12-ARIMA and TRAMO-SEATS, the original series (additive model) is compared with its logarithm (multiplicative model). The first stage of the X12-ARIMA, RegARIMA, TRAMO-SEATS, TRAMO procedures consists of fitting a regression model for the original time series Yt, 0

Y t ¼ x t β þ Zt

ð5Þ

where the vector of explanatory variables x t includes variables related to the calendar effects and dummy variables for the 0 identification of outliers. The time series of errors, Z t ¼ Y t −x t β for t = 1, …, T, is assumed to follow a seasonal ARIMA process, SARIMA(p,d,q)(P,D,Q) s, defined as
s
s d D ϕðBÞΦ B ð1−BÞ ð1−BÞ Z t ¼ θðBÞΘ B ϵt

ð6Þ

where ϵt is a white noise process and (p, d, q) are the orders of the lag polynomials defining the ARIMA process: p is the order of the autoregressive polynomial ϕ(B) = 1 − ϕ1B − ϕ2B2 − … − ϕpBp; q is the order of the moving average polynomial θ(B) = 1 + θ1B + θ2B2 + … + θqBq; and d is the order of differentiation. Note that B is the backward operator defined as BkYt = Yt − k, where k is an integer. The values of (P, D, Q) represent the orders of the seasonal lag polynomials: p is the order of the seasonal autoregressive polynomial Φ(B) = 1 − ϕs,1Bs − ϕs,2B2s − … − ϕs,pBsp; q is the order of the seasonal moving average polynomial Θ(B) = 1 + θs,1Bs + θs,2B2s + … + θs,qBsq; and D is the order of seasonal differentiation, where s is the frequency of the series (s = 4 for quarterly data and s = 12 for monthly data). By default, the RegARIMA routine only considers SARIMA models among the set (p, d, q)(P, D, Q) = {(0, 1, 1)(0, 1, 1), (0, 1, 2)(0, 1, 1), (2, 1, 0)(0, 1, 1), (0, 2, 2)(0, 1, 1), (2, 1, 2)(0, 1, 1)}, and the best model is identified by the Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), or minimum quadratic prediction error criteria. The TRAMO routine, by default, only considers SARIMA models that satisfy d ≤ 2, D ≤ 1, p ≤ 3, q ≤ 3, P ≤ 2, and Q ≤ 2 and selects the best model according to the BIC criterion. Once the outliers and calendar effects are removed, the second subroutine of X12-ARIMA applies nonparametric linear filters (weighted averages of the time series over moving windows) that are based on the X11 algorithm to identify and estimate the seasonal component, St, from ^ ¼ T t þ St þ I t , where Z^ t is the time series of Z^ t ¼ Y t −x0 t β

253

residuals reported by RegARIMA (X12-ARIMA) or TRAMO (TRAMO-SEATS). The second subroutine of TRAMO-SEATS, called SEATS, considers parametric ARIMA models for each component Tt and St in Z^ t . Finally, the seasonally adjusted time series, SAt, is obtained as SAt ¼ Z^ t −^St ¼ T^ t þ It . The routine X11 contains 11 different goodness of fit test statistics M1, …, M11 that check for the identification and stability of the seasonal effects. The M statistics, which have values between 0 and 3, with a value above 1 indicating poor fit, are combined to obtain a general statistic for goodness of fit. Additionally, there are statistics to test for the presence of residual seasonality in the seasonally adjusted time series. The diagnostics of SEATS are mainly based on the graphical comparison of the empirical autocorrelation and partial autocorrelation functions and spectrums with those of the fitted ARIMA models. Once one obtains the seasonally adjusted time series using X12-ARIMA or TRAMO-SEATS, the classical Bass model can be fitted to obtain deseasonalized sales forecasts. We call this type of approach a two-step forecasting algorithm because the time series is first seasonally adjusted, and then the parameters of the classical Bass model are estimated. In the proposed methodology of this paper, the seasonal effects are modeled using circular distributions based on nonnegative trigonometric sums (NNTS distributions, from now on) developed by Fernández-Durán (2004). We call this model the seasonal Bass model (see Fernández-Durán, 2009). 2. The Bass Forecasting Diffusion Model Let T be a random time when a consumer purchases the product of interest and fT(t) and FT(t) be the density and cumulative distribution functions of T, respectively. The classical Bass model is defined through the hazard rate function of T, hT(t), as follows:

hT ðt Þdt ¼ P ðtbTbt þ dtjT ≥t Þ ¼

f T ðt Þdt 1−F T ðt Þ

ð7Þ

where dt is an infinitesimal time period. Then, hT(t)dt is the probability that a customer will buy the product at a time between t and t + dt, given that he/she has not bought the product before. The original Bass model was specified in such a way that

hT ðt Þ ¼ p þ q F T ðt Þ ¼ p þ

q Y ðt Þ m

ð8Þ

where m is the number of eventual adopters, p is the coefficient of innovation, and q is the coefficient of imitation. Note that Y(t) = mFT(t) is the total number of buyers who bought the product in the time interval from 0 to t. The quantities m, p, and q are the parameters of the model. The distribution function of T is equal to

F T ðt Þ ¼

1−e−ðpþqÞt 1 þ qp e−ðpþqÞt

ð9Þ

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model in terms of an NNTS density function of a circular random variable, Θ:

and the density function of T is equal to 

 ðpþqÞ e−ðpþqÞt p f T ðt Þ ¼  2 q −ðpþqÞt 1þ e : 2

ð10Þ

f Θ ðθÞ ¼

p

Later, Bass et al. (1994) modified the original model by redefining the hazard rate function as follows: hT ðt Þ ¼ ðp þ q F T ðt ÞÞxðt Þ

ð11Þ

where x(t) is a positive function that can be expressed as a function of explanatory variables. This model was known as the Generalized Bass Model. In this model, the distribution function of T is defined as follows: 1−e−ðpþqÞðX ðtÞ−X ð0ÞÞ F T ðt Þ ¼ 1 þ qp e−ðpþqÞðX ðtÞ−X ð0ÞÞ

ð12Þ

and the corresponding density function is the following:   ð Þð ð Þ−X ð0ÞÞ Þ xðt Þ ðpþq e− pþq X t p f T ðt Þ ¼  2 q −ðpþqÞðX ðt Þ−X ð0ÞÞ 1þ e

ð13Þ

p

where X(t) is the accumulated value of x(t) from 0 to t: Z

t 0

xðsÞds:

ð14Þ

In particular, Bass et al. (1994) considered the effects of price and advertising spending on sales: xðt Þ ¼ α

ΔP t Δ Advt þβ Pt Advt

ð15Þ

where Pt is the price of the product at time t, Advt is the spending on advertising at time t, and ΔPt and ΔAdvt are the incremental changes in price and advertising. Note that any other specification of x(t) could also be used. Other applications and extensions of the Bass model are included in Mahajan et al. (2000). Alternatively, instead of specifying the original and generalized Bass models in terms of the number of eventual adopters, m, one can work with the number of potential adopters, denoted by M, specifying the model in terms of the probability of eventually adopting, r, defined as follows: r¼

m : M

ð18Þ

This family of density functions is very flexible and allows the modeling of situations in which the population distribution presents asymmetry, many modes, etc. Because the support of a circular random variable is the interval (0, 2π], the function x(t) is defined as follows: xðt Þ ¼ 1 þ

n
X

   ak cosð2πkt Þ þ bk sinð2πkt Þ :

ð19Þ

k¼1

which corresponds to the density function of the random Θ with the support interval (0, 1]. variable T ¼ 2π Note that by defining the generalized Bass model in this way, the new coefficients of innovation and imitation can be expressed as functions of time as follows: pðt Þ ¼ pxðt Þ and qðt Þ ¼ qxðt Þ:

2

X ðt Þ ¼

n 1 1X ða cosðkθÞ þ bk sinðkθÞÞ: þ 2π π k¼1 k

ð20Þ

These new coefficients take into account seasonal effects. We refer to this model as the seasonal NNTS Bass model. In the next section, the foundations of the NNTS circular distributions are presented. 3. Nonnegative trigonometric sums and circular distributions for modeling seasonal patterns The support set of a circular random variable is the circumference of a circle. In this sense, the density probability function, f ðθ; c Þ , of a circular random variable Θ must be nonnegative, must integrate to one on the interval (0, 2π] and must be periodic with the period 2π, i.e., f ðθ þ 2kπ; c Þ ¼ f ðθ; c Þ for any integer k, where c is the vector of the parameters. Fejér (1915) expressed a univariate nonnegative trigonometric sum for a circular variable θ as the squared norm of a sum of complex numbers, 2   X n  ikθ   ck e   

ð21Þ

k¼0

pffiffiffiffiffiffiffiffi where i ¼ −1. From this result, the parameters (ak, bk) for k = 1, …, n of the trigonometric sum of order n, T(θ), T ðθÞ ¼ a0 þ

ð16Þ

n X ðak cosðkθÞ þ bk sinðkθÞÞ

ð22Þ

k¼1

are expressed in terms of the complex parameters in Eq. (21), The distribution function for the time of adoption of a randomly selected member of the entire population, GT(t), is defined as follows: GT ðt Þ ¼ r F T ðt Þ:

ð17Þ

The main objective of this paper is to study seasonal sales effects by specifying the function x(t) of the generalized Bass

n−k

ck, for k = 0, …, n, as ak −ibk ¼ 2∑ν¼0 cνþk cν . The additional constraint,

n 2 ∑k¼0 jck j

¼

1 2π

¼ a0 is imposed to make the

trigonometric sum integrate to 1. By using the previous result, the probability density function for a circular random variable is defined as (see Fernández-Durán, 2004) f ðθ; c; nÞ ¼

n 1 1X ða cosðkθÞ þ bk sinðkθÞÞ: þ 2π π k¼1 k

ð23Þ

J.J. Fernández-Durán / Technological Forecasting & Social Change 88 (2014) 251–264

In this case, the vector of complex parameters, c , has a dimension of 2n + 2, with 2n free parameters. For parameter identifiability, c0 is considered a real number, i.e., cc0 = 0. Additionally, the order of the trigonometric sum, n, is considered an additional parameter. Note that the uniform circular distribution is a particular case in the family by considering ak = bk = 0 for k = 1, …, n or, equivalently, n = 0. For example, for n = 1, a1 ¼ 2πðcr1 cr0 þ cc1 cc0 Þ b1 ¼ 2πðcr1 cc0 −cc1 cr0 Þ 2

2

2

2

with the constraint cr0 þ cc0 þ cr1 þ cc1 ¼ 2π1 . For n = 2, we have four free c parameters, and a1 ¼ 2πðcr1 cr0 þ cc1 cc0 þ cr2 cr1 þ cc2 cc1 Þ b1 ¼ 2πðcr1 cc0 þ cr2 cc1 −cc1 cr0 −cc2 cr1 Þ and

4. Estimation of the seasonal NNTS bass model In practice, the estimation of the parameters of the (Generalized) Bass model can be performed using the maximum likelihood method, by regressing sales on cumulative sales and cumulative sales squared (originally employed by Bass (1969), non-linear least squares (NLS) Srinivasan and Mason (1986), or analogy, which consists of fixing parameter values to the values of a previous similar product. Generally, the observed sales data consist of accumulated sales during regular time intervals (weeks, months, quarters, or years). It is important to note that the seasonal NNTS Bass model can be fitted to the accumulated sales observed during irregular time intervals or intervals that are not of the same length, contrary to applying seasonal adjustment routines, such as X12-ARIMA and TRAMO-SEATS. 4.1. Maximum likelihood estimation

a2 ¼ 2πðcr2 cr0 þ cc2 cc0 Þ b2 ¼ 2πðcr2 cc0 −cc2 cr0 Þ with the constraint c2r0 þ c2c0 þ c2r1 þ c2c1 þ c2r2 þ c2c2 ¼ 2π1 . The extension for trigonometric sums of higher orders is direct. The accumulated distribution function of an NNTS density is easily calculated as: Z F ðθ; c; nÞ ¼

255

 n  X θ ak b þ f ðs; c; nÞds ¼ sinkθ þ k ð1−cosðkθÞÞ : 2π k¼1 k k 0 θ

In the case of accumulated sales data y1, y2, …, yN, where yk is the total number of sales of the new product in the interval (tk − 1, tk], then the likelihood when working with the number of eventual adopters, m, is given by the following: y

N

Lðy1 ; y2 ; …; yN jc; n; m; p; qÞ ¼ ð1−F T ðt N ÞÞ ∏ ð F T ðt k Þ−F T ðt k−1 ÞÞ

ð27Þ

ð24Þ If one is interested in analyzing the yearly seasonal patterns in the sales of a product, one can fit an NNTS model to the time series of accumulated sales, y1, y2, …, yN, where yk is the observed accumulated sales during the interval (tk − 1, tk], where tk is measured in yearly terms. For example, if one observes the quarterly sales of a product 90 273 , t 2 ¼ 181 during a year, then t0 = 0, t 1 ¼ 365 365, t 3 ¼ 365, and t4 = 1. To work with these data, one has to transform the circular Θ , taking values in the interval (0, 1]. variable to Θ ¼ 2π The likelihood function for grouped (accumulated) data y1, y2, …, yN observed on N time intervals is given by N

yk

Lðy1 ; y2 ; …; yr jc; nÞ ¼ ∏ ð F Θ ðt k Þ−F Θ ðt k−1 ÞÞ

ð25Þ

k¼1

where  n  X
 
ak b
   F Θ θ ; c; n ¼ 1 þ sin2πkθ þ k 1−cos 2πkθ : k k k¼1 ð26Þ In this case, the estimation of the c parameters can be accomplished by the maximum likelihood method. In particular, Fernández-Durán and Gregorio-Domínguez (2010) present a Newton-like algorithm to implement the maximum likelihood method for grouped and ungrouped data by assuming nonnegative trigonometric sum models. The package CircNNTSR (Fernández-Durán and Gregorio-Domínguez, 2012) of the freely available statistical software R (R Development Core Team, 2012) includes computational routines to implement the maximum likelihood method in nonnegative trigonometric sum models.

yk

k¼1

where y∗ = m − ∑ N k = 1yk are the adopters that will buy the product after time tN, and t0 corresponds to the date at which the new product was launched. Now, when specifying the total number of potential adopters, M, the likelihood is given by the following: y



N

y

Lðy1 ; y2 ; …; yN jc; n; p; q; r Þ ¼ ð1−GT ðt N ÞÞ ∏ ðGT ðt k Þ−GT ðt k−1 ÞÞ k ; k¼1

ð28Þ where y∗ = M − ∑ N k = 1 yk. The properties of the maximum likelihood estimators were studied by Schmittlein and Mahajan (1982). 4.2. Nonlinear least squares (NLS) estimation The main idea behind applying NLS is to express the accumulated sales in the interval (tk − 1, tk], yk, as yk ¼ mð F T ðt k Þ− F T ðt k−1 ÞÞ þ uk for k ¼ 1; …; N;

ð29Þ

where uk is an additive error term with variance σ2, in the case of working with the number of eventual adopters, and yk ¼ MðGT ðt k Þ−GT ðt k−1 ÞÞ þ uk for k ¼ 1; …; N;

ð30Þ

in the case of working with the number of potential adopters. NLS estimators are obtained by minimizing the sum of squared errors, SS, with respect to the parameters SSðc; n; m; p; qÞ ¼

N X k¼1

2

ðyk −mð F T ðt k Þ−F T ðt k−1 ÞÞÞ

ð31Þ

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J.J. Fernández-Durán / Technological Forecasting & Social Change 88 (2014) 251–264

and N X

SSðc; n; p; q; r Þ ¼

likelihood function of Eq. (27). If using the eventual number of adopters, this term is equal to 2

ðyk −MðGT ðt k Þ−GT ðt k−1 ÞÞÞ :

ð32Þ

k¼1

!2 N X m− yk −mð1−F T ðt N Þ

ð37Þ

k¼1

The properties of NLS estimators were studied by Srinivasan and Mason (1986). Boswijk and Franses (2002) included heteroskedastic additive error terms. For a comparison and references for the different estimation methods that have been implemented for the Bass model, the interested reader can consult Venkatesan et al. (2004). 4.3. Considerations in the estimation of the parameters of the seasonal Bass model Many authors, among them Meade and Islam (2006) and Van den Bulte and Lilien (1995, 1997) have reported some biases in the estimation of the parameters of the classical Bass model: m, p and q. In particular, when using NLS, the approximate value of m is underestimated. This bias is reduced when using the maximum likelihood estimation method, but a tendency is observed for the estimate to approximate the cumulative number of adopters observed in the last observed period, MFT(tN). This bias is explained by considering the derivative of the loglikelihood function (the log of the function in Eq. (27)) for the classical Bass model with respect to the parameter r ¼ Mm lðy1 ; …; yN jp; q; r Þ ¼ lnðLðy1 ; …; yN jp; q; r ÞÞ ¼

N X yk lnðr F T ðt k Þ−r F T ðt k−1 ÞÞ k¼1

þ

! N X M− yk lnð1−r F T ðt N ÞÞ:

ð33Þ

k¼1

Then, the derivative of l with respect to r, ∂r∂l , is given by ∂l ¼ ∂r

XN

y k¼1 k r

−

  XN M− k¼1 yk F T ðt N Þ 1−r F T ðt N Þ

:

ð34Þ

Obtaining the score equation, ∂r∂l ¼ 0, one finds XN ^F ðt Þ ¼ T N

y k¼1 k m

ð35Þ

or, equivalently, y k¼1 k

M

!2 N X M− yk −Mð1−GT ðt N Þ :

ð38Þ

k¼1

In this sense, Van den Bulte and Lilien (1995) consider that because the Bass model is a two-state hazard model with one absorbing state (adoption) and no population heterogeneity, then m = M, that is, the entire population (potential market) ultimately adopts the product, and then r = 1. In practice, it is easier to have a good idea of the value of the size of the potential market, M, than of the number of eventual adopters, m. In our experience, in terms of estimation and forecasting, the best results are obtained when considering r = 1 for a suitable population size, M. When using the Bass model to make forecasts with no observations (prior to launch) or with few observations, one of the most commonly used methods is to consider the parameter estimates of a previous similar product. This estimation and forecasting by analogy can also be applied to the seasonal Bass model, particularly in the seasonal pattern defined by the NNTS circular distribution. Then, using the pattern of sales of a similar previous product, an estimate of the parameters of the NNTS model can be obtained and plugged-into the likelihood or NLS objective function of the new product. Note that the sales of the previous product can be observed on a different frequency than the new product. For example, the quarterly sales of a previous product can be used to estimate the parameters of the NNTS model to forecast the monthly sales of a new product. When applying seasonal analogy forecasting, it is important to distinguish between the product time and the calendar time. The Bass models are expressed in terms of product time, i.e., the time that has elapsed since the introduction of the product. Contrary to this, seasonal effects are commonly defined in terms of calendar time to identify different seasons, holidays, etc. In this case, Eq. (11) has to be modified to hT ðt Þ ¼ ðp þ q F T ðt ÞÞxðt 0 þ t Þ

ð39Þ

where t0 is the time of the introduction of the new product and Eq. (12) has to be modified to

XN ^ ðt Þ ¼ G T N

and, when considering the size of the potential market,

ð36Þ

^ and ^r satisfy ^, q implying that the maximum likelihood estimates p that the supposition estimate of the distribution function in the last observed time point is equal to the percentage of the (potential) market that has already adopted the product. When seasonal effects are present, this restriction can produce biased estimates of the classical Bass model parameters. This fact is a consequence of the presence of right censoring in the observation of the diffusion process. In cases using NLS estimation, an additional term can be introduced in Eq. (31) to take into account the right-censoring process analogous to the last term in the

F T ðt Þ ¼

1−e−ðpþqÞðX ðt 0 þtÞ−X ðt 0 ÞÞ 1 þ qp e−ðpþqÞðX ðt 0 þt Þ−X ðt 0 ÞÞ t þt

ð40Þ

where X ðt 0 þ t Þ−X ðt 0 Þ ¼ ∫t 0 xðsÞds with x(s) representing 0 the NNTS circular density in calendar time. Taking into consideration the calendar time and the periodic nature of NNTS circular distributions, the seasonal Bass model is an accelerated time model in which the product time in the classical Bass model, t, is transformed into a new time, t ∗, defined by t ∗ = X(t0 + t) − X(t0). Fig. 1 presents the plot of t vs. t ∗ for the estimated seasonal pattern of the monthly sales of

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0.6 0.4

t*

2.0

Fitted NNTS Density

2.5

0.8

3.0

3.5

1.0

of Figs. 2 and 3) is shown with large increases in the periods with large sales. In this section, we compare the one-step-ahead forecasts of the seasonal Bass model against the forecasts obtained first by using the X12-ARIMA and TRAMO-SEATS algorithms and then by fitting the classical Bass model to the deseasonalized sales time series. We make this comparison in the dataset of the quarterly sales of iPod devices. In the second example, the classical and seasonal Bass models are compared in a high-frequency dataset that consists of the weekly sales of the Wii Play video game. To compare the different forecasting models, the following one-step-ahead forecasting errors will be considered in the examples (see Hyndman and Koehler (2006)). The forecast error is defined as et = Yt − Ft where Y1, Y2, …, is the observed time series and Ft is the forecasting value given by the model. The percentage forecasting error is defined as pt ¼ 100Ye %, and the relative forecasting error is defined as r t ¼ ee , where e∗t is the forecasting error of a benchmark

0.2

1.5 1.0

257

t

0.0

0.5

t

0.0

0.2

0.4

0.6

Calendar Time

0.8

1.0

t  t

0.0

0.2

0.4

0.6

0.8

1.0

t

Fig. 1. Wii console monthly sales data. The first plot shows the fitted NNTS seasonal pattern for a whole calendar year. The vertical dashed line is the time of the introduction of the product. The second plot shows the accelerated time of the seasonal Bass model, t∗ (solid line), versus the time of the classical Bass model, t. The dashed line is the identity line.

the Wii console (see also the first row of Fig. 3), clearly showing the effect of the sales increase on the last quarter of the year. When considering a classical and a seasonal Bass model with the same parameters r, p, and q both models give the same forecasts at integer values of the product time, t = 1, 2, 3, …. If the year is taken as the unit of time, then both models provide the same forecasts at the end of each product year. Of course, when considering real data, the problem is that the parameter estimates of r, p, and q will be different for the classical and the seasonal Bass models; this difference could be large in the presence of seasonal effects. 5. Examples Figs. 2 and 3 present plots for the sales, the accumulated sales, and the histogram of the sales during a calendar year with a superimposed NNTS density function estimated from the data for different products. Fig. 2 includes the plots for the quarterly sales of iPod devices and the monthly sales of DVD devices. Fig. 3 includes plots for the monthly sales of Wii consoles and the weekly sales of the Wii Play video game. In the case of iPod devices, an NNTS model with n = 1 was fitted. For DVD devices, an NNTS model with n = 5 terms was fitted and included in the third column of Fig. 2. In the case of Wii consoles, an NNTS density with n = 5 was fitted, and for the Wii Play video game, an NNTS model with n = 10 was fitted. The fitted NNTS densities are included in the third column of Fig. 3. In each of these four time series, a peak in sales was observed at the end of the year due to the special November sales and the Christmas season. The NNTS circular densities were fitted to the sales in whole years to capture the seasonality in calendar time. The effect of this observed seasonality in the plots of accumulated sales (second column

model, which in this paper, is the classical Bass model. The following measures of forecasting accuracy based on the forecasting errors, et, will be considered: the Root Mean qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 Square Error (RMSE), defined as RMSE ¼ mean e2t , and the Mean Absolute Error (MAE), defined as MAE = mean(|et|). In terms of the percentage forecasting errors, pt, the Root qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 Mean Square Percentage Error (RMSPE), RMSPE ¼ mean p2t , and the Mean Absolute Percentage Error (MAPE), MAPE = mean(|pt|), were considered. Finally, the Mean Relative Absolute Error, which is a measure based on the relative errors, rt, with respect to the classical Bass model, MRAE = mean(|rt|), was used. Note that MRAE is not robust to outliers, then MdRAE = median(|rt|) was also used. In the examples that follow, the estimation of the parameters of Bass models was achieved by NLS (nonlinear least squares) using the function lsqnonlin in Matlab. The terms in Eqs. (37) and (38) were added to the objective function. When estimating the parameters of an NNTS model, the maximum likelihood method explained in Fernández-Durán and Gregorio-Domínguez (2010) and included in the library CircNNTSR (Fernández-Durán and Gregorio-Domínguez, 2012) of the free available statistical software R (R Development Core Team, 2012) was applied. 5.1. Application: forecasting accumulated quarterly sales of iPod electronic devices The first iPod electronic device was launched by Apple Inc. on October 20, 2001. The first row of Fig. 2 presents the quarterly sales in the U.S. of iPod devices from their introduction until the third quarter of 2010 with a total of 36 observations. Clearly, the quarterly sales reflect strong seasonal effects in the final quarter of the year that are likely the result of the Christmas season and the special November sales. In the second column of the first row of Fig. 2, the accumulated quarterly sales are plotted. A potential market size that is equal in size to 261,713,220 was calculated by applying the least-squares estimation originally developed by Bass (1969). Note that this size of a potential market is similar to the total U.S. population. The probability of adoption was considered equal to one,

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1.0

Density

0.5 4Q04

1Q

4Q07

2Q

3Q

4Q

2000

2003

2006

0.5 0.0

0.0e+00

0e+00 1997

1.0

Density

8.0e+07

Accumulated Sales

4.0e+07

2e+06 1e+06

1.5

1.2e+08

2.0

DVD Monthly Sales

3e+06

4e+06

0.0 4Q01

2001 2004 2007 2010

Sales

1.5

2.0e+08

Accumulated Sales

1.0e+08 0.0e+00

1.0e+07 0.0e+00

Sales

2.0e+07

2.0

IPOD Quarterly Sales

D97

D00

D03

D06

J

M

M

J

S

N

Fig. 2. Quarterly sales of iPod devices (first row) and monthly sales of DVD devices (second row) in the U.S. The plots in the first column are the bar plots of the sales of the products. The second column contains the plots of the accumulated sales. The third column includes the histograms of the frequencies of sales in each period of the year and a fitted NNTS density with n = 1 for the iPod sales and n = 5 for the DVD sales.

implying that the potential and eventual markets sizes are equal (see Van den Bulte and Lilien, 1995). Table 1 includes the original data from the fourth quarter of 2001 to the fourth quarter of 2007. Table 1 includes the one-step-ahead forecasting results for five different models: 1. The classical Bass model. 2. The X12-ARIMA model: The original data are deseasonalized using the X12-ARIMA algorithm; then, the classical model is fitted to the deseasonalized time series. The time series is reseasonalized using the seasonal factors given by the X12-ARIMA algorithm, including the times to be forecasted. 3. The TRAMO-SEATS model: The original data are deseasonalized using the TRAMO-SEATS algorithm; then, the classical model is fitted to the deseasonalized time series. The time series is reseasonalized using the seasonal factors given by the TRAMO-SEATS algorithm including the times to be forecasted. The X12-ARIMA and the TRAMO-SEATS procedures were implemented in E-Views,

and the goodness of fit statistics, M statistics for the X12-ARIMA, and autocorrelation functions for the TRAMOSEATS were acceptable. 4. The NNTS for the iPod pattern: A circular NNTS distribution is fitted to the data in complete calendar years, i.e., from 2002 to 2007 (see the first row of Fig. 2). The estimated parameters of the NNTS distribution are included as fixed parameters in the seasonal Bass model; then, the parameters p and q of the Bass model are estimated. 5. The NNTS DVD pattern model: An NNTS circular distribution is fitted to the quarterly sales of DVD devices in the complete years from 1998 to 2006 (see the second row of Fig. 2). Again, the estimated parameters of this NNTS circular distribution are included as known parameters in a seasonal Bass model to estimate the parameters p and q. In Table 1, the different models are compared in terms of their one-step-ahead forecast errors, percentage, and relative errors. The methodology based on X12 requires at least 16 observations to produce a forecast while the methodology

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259

2006

2008

3.0 2.5 2.0

Density

0.5

1.0

1.5

2.0e+07

2010

0.0

0e+00

0.0e+00

1.0e+07

Accumulated Sales

2e+06 1e+06

Sales

3e+06

3.0e+07

3.5

4e+06

Wii Console Monthly Sales

D06 D07 D08 D09 D10

J

M

M

J

S

N

2007

2008

2009

0

0e+00

0e+00

1

2

Density

3

8e+06 4e+06

Accumulated Sales

2e+05 1e+05

Sales

3e+05

4

4e+05

Wii Play Weekly Sales

07−08

08−09

w1

w18

w36

w53

Fig. 3. Monthly sales of Wii consoles (first row) and monthly sales of Wii Play video games (second row) in the U.S. The plots in the first column are the bar plots of the sales of the products. The second column contains the plots of the accumulated sales. The third column includes the histograms of the frequencies of sales in each period of the year and a fitted NNTS density with n = 5 for the Wii console sales and n = 10 for the Wii Play video game sales.

based on TRAMO-SEATS requires at least 17 observations. For the three NNTS models, at least three observations are required. Given these restrictions, we compared the NNTS models against the classical Bass model for the period from 2002Q3 to 2007Q4. The previous models were also compared with the X12-ARIMA model for the period from 2005Q3 to 2007Q4. Finally, all five of the models were compared for the period from 2005Q4 to 2007Q4. Based on the forecast accuracy measures in Table 1, all of the seasonal models presented, in general, smaller accuracy measures than the classical Bass model. This fact reflects the importance of including seasonal patterns in the forecasting models. When comparing the NNTS models with the X12-ARIMA and TRAMO-SEATS models, it is clear that the NNTS models have smaller values in the accuracy measures, in general. In a real application of these models to forecast sales before launching the product, only the NNTS models based on the seasonal patterns of DVD sales could be implemented for forecasting purposes. It is important to note that the model based on the DVD pattern delivers satisfactory results that are better than the X12-ARIMA and TRAMO-SEATS

methodologies in RMSE, MAE, RMSPE, and MAPE and similar in MRAE and MdRAE. Further, this model clearly outperforms the classical Bass model. 5.2. Application: forecasting accumulated weekly sales of the Wii Play video game Data on the weekly sales in the U.S. of the Wii Play video game were obtained from the VGCchartz website (www. vgchartz.com). This dataset covers the period from the release of the video game, February 12, 2007, to the week ending on September 5, 2009. There were a total of 134 weekly observations. The seasonal pattern obtained from fitting an NNTS model with n = 5 to the series of complete calendar years of monthly sales of the Wii console in the U.S. from November 2006 to May 2011 (see the first row of Fig. 3) was used to fit the seasonal Bass model to the Wii Play video game data. Table 2 presents different forecasting accuracy measures that compare the classical Bass model and the seasonal Bass model in terms of their one-step-ahead forecasts. The size of

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Table 1 Quarterly sales of iPod devices. One-step-ahead forecasting measures of accuracy considering five models: the classical Bass, the X12-ARIMA, the TRAMO-SEATS and two NNTS seasonal Bass models. The first NNTS seasonal Bass model uses an NNTS seasonal pattern estimated from the complete calendar years of the iPod sales (NNTS iPod Pattern). The second uses the estimated NNTS seasonal pattern of quarterly DVD sales (NNTS DVD Pattern). Quarter

Sales (000)

Cumulative sales (000)

2001Q4 2002Q1 2002Q2 2002Q3 2002Q4 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4 2005Q1 2005Q2 2005Q3 2005Q4 2006Q1 2006Q2 2006Q3 2006Q4 2007Q1 2007Q2 2007Q3 2007Q4

125 57 54 140 219 78 304 336 733 807 860 2016 4580 5311 6155 6451 14,043 8526 8111 8729 21,066 10,549 9815 10,600 22,121

125 182 236 376 595 673 977 1313 2046 2853 3713 5729 10,309 15,620 21,775 28,226 42,269 50,795 58,906 67,635 88,701 99,250 109,065 119,665 141,786

From

To

2002Q3

2007Q4

2005Q3

2007Q4

2005Q4

2007Q4

Classical bass pt (%)

X12-ARIMA pt (%)

TRAMO-SEATS pt (%)

NNTS iPod pattern pt (%)

NNTS DVD pattern pt (%)

16.92 −2.24 −8.90 19.32 4.40 −5.52 9.13 −0.35 14.74 −13.76 9.02 −2.11 −9.61 −11.85 −3.64 −4.72 −4.54 −1.13 −2.47 −2.63 −2.19 −0.70

6.02 4.71 − 0.60 19.14 −5.99 −0.14 15.50 −1.07 7.91 −3.47 17.60 −8.67 −19.93 −5.42 3.26 −5.88 −9.11 0.46 1.76 −2.87 −4.60 −0.19

15.81 18.21 −30.73 16.03 1.02 15.02 −5.83 −8.77 12.90 18.46 −13.91 −11.60 −10.77 10.45 −14.94 −10.19 −6.38 9.58 −6.23 −5.49 −3.60 5.58

−29.11 −8.38 −7.89 −3.34 −6.74 1.62 −7.18 −1.01 −0.82 0.61

−18.32 −15.12 −4.62 −8.18 1.11 −1.68 8.54 −2.06 −2.34

Accuracy measure

Classical Bass

X12-ARIMA

TRAMO-SEATS

NNTS iPod pattern

NNTS DVD pattern

RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE

4177.82 3042.64 13.06 11.43 1 1 6069.66 5825.27 8.93 8.32 1 1 6317.18 6134.65 8.70 8.05 1 1

5421.84 4603.03 9.06 6.89 0.83 0.57

1920.96 1375.27 8.73 6.81 0.77 0.54 2759.22 2535.45 5.55 4.35 0.52 0.47 2764.36 2515.77 4.90 3.76 0.48 0.46

2563.78 1659.27 8.92 6.56 0.91 0.52 3645.73 3025.10 7.66 5.35 0.68 0.52 3354.34 2736.14 4.60 3.73 0.54 0.52

the potential market was equal to 114,139,898, similar to the total number of households in the U.S. in 2007–2010, according to the Census Bureau (114,761,359). This value for the size of the potential market was obtained using the least squares approximation of Bass (1969) on the Wii console monthly sales. By using r = 1, as shown in Table 2, for the years of 2007, 2008 and 2009, and for the whole period from 2007 to 2009, the seasonal Bass model presents better RMSE, RMSPE, and MAPE accuracy measures for all the considered cases. For the case of the MAE measure, only during the one-step-ahead forecasts in the weeks of 2007 did the seasonal Bass model has a slightly larger value than the classical Bass model. The results including the MRAE are affected by outliers. By considering the median of the absolute relative errors instead

4201.86 3381.00 10.45 6.67 0.73 0.43 3480.95 2843.77 5.22 4.18 0.51 0.33

of the mean, one finds an MdRAE of 0.71 for the complete data set. These results confirm the utility of including seasonal effects in the Bass model when working with high-frequency data, such as weekly sales data. Furthermore, in Table 2, the forecasting accuracy measures are included to compare the classical and seasonal Bass models when r is estimated from the data with the restriction 0.05 ≤ r ≤ 0.2. In practice, the restriction 0.05 ≤ r ≤ 0.2 could arise when there are bounds on the number of eventual adopters as a percentage of the size of the whole population. For example, in the case of the Wii Play video game, one must consider that the size of the potential market is restricted by the number of Wii consoles that has already been sold. In terms of the RMSE, MAE, and RMSPE, the seasonal Bass model presents lower values than the classical Bass model in all of

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261

Table 2 Weekly sales of the Wii Play video game. Accuracy measures to compare the one-step-ahead forecasts of the classical and seasonal models when the probability of adoption, r, is fixed at r = 1 and when it is estimated, but under the restriction of 0.05 ≤ r ≤ 0.20. To

Accuracy measure

2007w9

2007w52

2008w1

2008w52

2009w1

2009w36

2007w9

2009w36

RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE RMSE MAE RMSPE (%) MAPE (%) MRAE MdRAE

r estimated with 0.05 ≤ r ≤ 0.20

r=1 Classical Bass

Seasonal Bass

Classical Bass

Seasonal Bass

54,486.65 27,840.52 3.32 2.04 1 1 81,672.47 51,980.89 1.24 0.89 1 1 69,330.22 67,752.65 0.69 0.67 1 1 70,307.89 48,391.18 2.09 1.21 1 1

46,895.71 27,914.71 2.77 1.69 6.84 1.03 54,277.50 44,132.06 0.93 0.78 3.04 1.13 35,219.47 28,254.09 0.36 0.28 0.56 0.33 47,245.92 34,445.40 1.70 0.95 3.60 0.71

55,548.13 27,579.68 3.12 1.89 5.27 0.92 84,080.42 48,949.78 1.23 0.82 1.46 0.89 50,411.83 48,233.93 0.50 0.48 0.72 0.70 62,711.87 41,738.44 1.97 1.08 2.51 0.78

42,497.40 20,628.42 2.48 1.35 11.38 0.77 56,745.02 47,022.05 0.98 0.84 3.41 1.21 32,643.76 24,595.83 0.34 0.25 0.49 0.22 46,516.95 32,195.56 1.56 0.84 5.22 0.69

0.5

1.0

Parameter q estimates

0.04 0.03

Mar−07 May−07 Jul−07 Sep−07 Nov−07 Jan−08 Mar−08 May−08 Jul−08 Sep−08 Nov−08 Jan−09 Mar−09 May−09 Jul−09

Mar−07 May−07 Jul−07 Sep−07 Nov−07 Jan−08 Mar−08 May−08 Jul−08 Sep−08 Nov−08 Jan−09 Mar−09 May−09 Jul−09

0.0

0.01

0.02

Parameter p estimates

1.5

0.05

2.0

0.06

From

Fig. 4. Monthly sales of Wii Play video games: One-step-ahead estimates of the p (left plot) and q (right plot) parameters when r = 1 for the seasonal Bass model (solid line) and the classical Bass model (dashed line).

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period than the classical Bass model estimates; these stabilize at a value near 0.2. Contrary to this pattern, the classical Bass model q parameter estimates are more variable and present a seasonal pattern that does not stabilize around a fixed value. Fig. 5 includes the r, p, and q parameter estimates when r is estimated but with the constraint 0.05 ≤ r ≤ 0.2. Again, the classical Bass model parameter estimates (dashed line) present seasonal patterns and do not stabilize around a fixed value. In contrast to this, the seasonal Bass r parameter estimates stabilize at the maximum allowed value of 0.2; the p parameter estimates, at approximately 0.14; and the q parameter estimates, beginning at zero and maintaining this value for a longer period than the classical Bass estimate before stabilizing at approximately 0.60. In the case of estimating the r parameter with the constraint 0.05 ≤ r ≤ 0.2, the p and q parameter estimates of the seasonal Bass model stabilize around values that are larger than the case for r = 1. The weeks in which q = 0 imply that the classical Bass model corresponds to an exponential model. The initial zero values for the q parameter

1.0

Parameter q estimates

0.8 0.6

Parameter p estimates

Mar−07 May−07 Jul−07 Sep−07 Nov−07 Jan−08 Mar−08 May−08 Jul−08 Sep−08 Nov−08 Jan−09 Mar−09 May−09 Jul−09

0.0 Mar−07 May−07 Jul−07 Sep−07 Nov−07 Jan−08 Mar−08 May−08 Jul−08 Sep−08 Nov−08 Jan−09 Mar−09 May−09 Jul−09

Mar−07 May−07 Jul−07 Sep−07 Nov−07 Jan−08 Mar−08 May−08 Jul−08 Sep−08 Nov−08 Jan−09 Mar−09 May−09 Jul−09

0.0

0.00

0.2

0.5

0.05

0.4

0.10

Parameter r estimates

1.5

0.15

1.0

1.2

2.0

0.20

1.4

the considered years and over the whole period. In terms of the MdRAE and MAPE, only in the data from 2008 does the classical Bass model present lower values than the seasonal Bass model. Over the whole period, 2007w9 to 2009w36, the seasonal Bass model presents lower values in all of the considered forecasting accuracy measures. Fig. 4 presents the p and q parameter estimates when considering r = 1. The solid line corresponds to the seasonal Bass model parameter estimates; the dashed line, to the classical Bass parameter estimates. Note how the classical Bass parameter estimates are more variable than the seasonal Bass model parameter estimates. The classical Bass model parameter estimates present a seasonal pattern that reflects the original seasonal pattern observed in the data. For the p parameter estimates of the seasonal Bass model, the estimates stabilize around a value near 0.035, which is similar to the mean value of the analysis reported by Sultan et al. (1990). For the case of the q parameter estimates, in both models, the estimates are initially equal to zero. However, note that the seasonal Bass model q parameter estimates are equal to zero for a longer

Fig. 5. Monthly sales of Wii Play video games. One-step-ahead estimates of the p (first plot), q (second plot), and r (third plot) parameters when r is estimated with the restriction 0.05 ≤ r ≤ 0.2. The seasonal Bass model parameter estimates are represented by a solid line, and the classical Bass model parameter estimates, by a dashed line.

1e+07

5.0e+06

1.0e+07

Accumulated Sales

1.5e+07

8e+06 6e+06 4e+06

Fig. 6. Wii Play video game weekly sales. Analogy forecasting using the classical Bass model (dashed line) and the seasonal Bass model (solid line). The points represent the observed accumulated sales.

estimates imply exponential diffusion that can be related to heterogeneity rather than lack of contagion (see Bemmaor and Lee, 2002). In practice, the Bass model is commonly used to forecast sales of a new product when no or few observed sales periods exist. The seasonal Bass model can be applied in this situation by considering similar past products or expert opinions that specify the percentage of sales expected in each period (e.g., a quarter), indicating in this way the parameter values of the circular density, x(t), included in the seasonal Bass model. The same (analogous) approach can be followed to specify the parameters of the potential-buyers seasonal Bass model M, r, p, and q. In this case of forecasting before launch, Fig. 6 presents the forecasts of the weekly sales of the Wii Play video game over the whole period and uses a seasonal Bass model considering M = 114, 761, 359 (the approximate total number of households in the U.S.), r = 1, p = 0.035, q = 0.2, and the fixed seasonal pattern derived from the monthly sales of the Wii console. For the purposes of comparison, the forecasts of a classical Bass model with M = 114, 761, 359, p = 0.035, and q = 0.2 are also included. Because the classical and seasonal Bass models have the same values of M, r, p, and q, their forecasts are equal at the end of each calendar year. Contrary to the classical Bass model, the seasonal Bass model is able to reproduce the seasonal patterns. To compare the classical and seasonal Bass models in multiple period forecasts, Fig. 7 includes the forecasts of the classical and seasonal Bass models when the parameters p and q are fitted in the period from the introduction of the product until the last week of 2007. The parameters M and r are fixed at the values of 114,761,359 and 1, respectively. The NNTS Wii console seasonal pattern with n = 5 is used (see first row of Fig. 3). The ^ ¼ 0.013734 and classical Bass model has parameter estimates p ^ ¼ 1.996476. The seasonal Bass model has parameter estimates q ^ ¼ 0. Note in Fig. 7 how the classical Bass ^ ¼ 0.035856 and q p

Jul−09

May−09

Jan−09

Mar−09

Nov−08

Jul−08

Sep−08

May−08

Jan−08

Mar−08

Nov−07

Jul−07

Sep−07

Mar−07

May−07

Jul−09

May−09

Jan−09

Mar−09

Nov−08

Jul−08

Sep−08

May−08

Jan−08

Mar−08

Nov−07

Jul−07

Sep−07

Mar−07

May−07

0e+00

0.0e+00

2e+06

Accumulated Sales

263

2.0e+07

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Fig. 7. Wii Play video game weekly sales. Multiple period forecasts. The classical and seasonal Bass models were fitted to the sales data from the introduction of the product to the last week of 2007 (dashed vertical line). The forecasts of the classical Bass model are represented by a dashed line, and those of the seasonal Bass model were represented by a solid line. The points represent the observed accumulated sales.

model forecasts are unreasonable, given the strong seasonal effect at the end of the year; this is reflected in the parameter estimates (see Eqs. (35) and (36) and Parker, 1994). The seasonal Bass forecasts are adequate for approximately 12 weeks into the future. All of the future forecasts of the classical Bass model are much larger than the observed sales. For the purposes of comparison, the Gamma/Shifted Gompertz (G/SG) model (see Bemmaor and Lee, 2002) given by 1−e−ðpþqÞt α F T ðt Þ ¼  q −ðpþqÞt 1þ e

ð41Þ

p

is a modification of the classical Bass model that includes an additional parameter α, which was also fitted. When α = 1, the G/SG reduces to the classical Bass model; when α = 0, it is equivalent to the exponential distribution. The G/SG model is able to capture more skew in the data than the classical Bass model. This model can also be modified to include seasonal effects by considering ð

Þð ð

Þ−X ð ÞÞ

t0 1−e− pþq X t 0 þt α F T ðt Þ ¼  q −ðpþqÞðX ðt0 þt Þ−X ðt 0 ÞÞ 1þ e

ð42Þ

p

where X(t) is the distribution function of an NNTS model and t0 is the time at which the product was launched. The G/SG was fitted to the weekly sales of Wii Play video games, including and not including seasonal effects. In both cases, the one-stepahead forecasting errors presented much higher values than the corresponding classical and seasonal Bass model forecasting errors.

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6. Conclusions The majority of time series for the sales of products present seasonality. When using the classical Bass model without considering seasonal effects, it is possible to obtain biased forecasts that can affect company decisions concerning the launch of a new product. However, seasonal effects can be included by considering the use of flexible circular distributions, such as the NNTS family of circular distributions. The introduction of seasonal effects into the Bass model is equivalent to working with accelerated time models. The proposed seasonal Bass model was fitted to the quarterly sales of iPod devices and the weekly sales of Wii Play video games and demonstrated a large improvement in terms of forecasting accuracy when compared with the classical Bass model, highlighting the importance of taking these effects into account. The proposed methodology can be applied to the forecasting of product sales prior to their launch using the seasonal pattern of a previous analogous product and allowing for the inclusion of seasonal effects into this situation. Contrary to the application of seasonal adjustment techniques, such as X12-ARIMA and TRAMO-SEATS, the NNTS seasonal Bass model can work with data observed during irregular periods of time or at intervals of different lengths. Acknowledgments The author wishes to thank the Asociación Mexicana de Cultura, A.C. for its support. Also, the author thanks Prof. Victor Cook for all his support during the elaboration of this paper while the author was a student of the AACSB Postdoctoral Bridge to Business Program in the Freeman School of Business of Tulane University in 2008. References Bass, F.M., 1969. A new product growth for model consumer durables. Manag. Sci. 15–5, 215–227. Bass, F.M., Krishnan, V., Jain, D.C., 1994. Why the bass model fits without decision variables. Mark. Sci. 13–3, 203–223. Bemmaor, A.C.,Lee, J., 2002. The impact of heterogeneity and ill-conditioning on diffusion model parameter estimates. Mark. Sci. 21–2, 209–220. Boswijk, P.H.,Franses, P.H., 2002. The econometrics of the Bass diffusion model. ERIM Report Series Research in Management (ERS-2002-66-MKT). Chandrasekaran, D., Tellis, G.J., 2007. A critical review of marketing research on diffusion of new products. Rev. Mark. Res. 2, 39–80. Fejér, L., 1915. Über trigonometrische Polynome. J. Reine Angew. Math. 146, 53–82. Fernández-Durán, J.J., 2004. Circular distributions based on nonnegative trigonometric sums. Biometrics 60, 499–503. Fernández-Durán, J.J., 2009. Modeling seasonal effects in the Bass forecasting diffusion model. Working Paper DE-C09.2, Department of Statistics, ITAM. Fernández-Durán, J.J., Gregorio-Domínguez, M.M., 2010. Maximum likelihood estimation of nonnegative trigonometric sums models using a Newton-like algorithm on manifolds. Electron. J. Stat. 4, 1402–1410. http://dx.doi.org/10.1214/10-EJS587. Fernández-Durán, J.J., Gregorio-Domínguez, M.M., 2012. CircNNTSR: An R Package for the Statistical Analysis of Circular Data Using Nonnegative Trigonometric Sums (NNTS) Models. R Package Version 2.0. (http:// CRAN.R-project.org/package=CircNNTSR). Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C., Chen, B.-C., 1998. New capabilities and methods of the X-12-ARIMA seasonal-adjustment program. J. Bus. Econ. Stat. 16, 127–177. Gómez, V., Maravall, A., 1998. Guide for using the programs TRAMO and SEATS (beta version: December 1997). Banco de España Working Papers, 9805, Banco de España.

Hyndman, R.J., Koehler, A.B., 2006. Another look at measures of forecast accuracy. Int. J. Forecast. 22–4, 679–688. Ladiray, D., Quenneville, B., 2001. Seasonal Adjustment with the X-11 Method. Lecture Notes in Statistics 158. Springer Verlag. Mahajan, V., Muller, E., Bass, F.M., 1990. New product diffusion models in marketing: a review and directions for research. J. Mark. 54, 1–26. Mahajan, V., Muller, E., Wind, Y., 2000. New-Product Diffusion Models. Kluwer Academic Publishers, Norwell, Massachusetts. Meade, N., Islam, T., 2006. Modelling and forecasting the diffusion of innovation — a 25-year review. Int. J. Forecast. 22, 519–545. Non, M., Franses, P.H., Laheij, C., Rokers, T., 2003. Yet another look at temporal aggregation in diffusion models of first-time purchase. Technol. Forecast. Soc. Chang. 70, 467–471. Parker, P.M., 1994. Aggregate diffusion forecasting models in marketing: a critical review. Int. J. Forecast. 10, 353–380. Peers, Y., Fok, D., Franses, P.H., 2012. Modeling seasonality in new product diffusion. Mark. Sci. 31–2, 351–364. Peres, R., Muller, E., Mahajan, V., 2010. Innovation diffusion and new product growth models: a critical review and research directions. Int. J. Res. Mark. 27, 91–106. Putsis, W.P., 1996. Temporal aggregation in diffusion models of first-time purchase: does choice of frequency matter? Technol. Forecast. Soc. Chang. 51, 265–279. Putsis, W.P., 1998. Parameter variation and new product diffusion. J. Forecast. 17, 231–257. R Development Core Team, 2012. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria 3-900051-07-0 (URL http://www.R-project.org/). Radas, S., 2001. Forecasting the life cycle of a new seasonal durable. Ekon. Pregl. 52 (1–2), 206–218. Radas, S., 2005. Diffusion models in marketing: how to incorporate the effect of external influence? Privredna kretanja i ekonomska poliyika 105, 31–51. Radas, S., Shugan, S.M., 1998. Seasonal marketing and timing new product introductions. J. Mark. Res. XXXV, 296–315. Rao, S.K., 1985. An empirical comparison of sales forecasting models. J. Prod. Innov. Manag. 4, 232–242. Ruiz-Conde, E., Leeflang, P.S.H., Wieringa, J.E., 2006. Marketing variables in macro-level diffusion models. J. Betriebswirtsch. 56, 155–183. Schmittlein, D.C., Mahajan, V., 1982. Maximum likelihood estimation for an innovation diffusion model of new product acceptance. Mark. Sci. 1–1, 57–78. Srinivasan, V., Mason, C.H., 1986. Non-linear least squares estimation of new product diffusion models. Mark. Sci. 5, 169–178. Sultan, F., Farley, J.U., Lehmann, D.R., 1990. A meta-analysis of applications of diffusion models. J. Mark. Res. 27–1, 70–77. Tigert, D.,Farivar, B., 1981. The Bass new product growth model: a sensitivity analysis for a high technology product. J. Mark. 45, 81–90. Van den Bulte, C., Lilien, G., 1995. Why Bass model estimates may be biased (and what it means). Proceedings of the 24th EMAC Conference, Paris, pp. 2045–2051 (May). Van den Bulte, C., Lilien, G., 1997. Bias and systematic change in the parameter estimates of macro-level diffusion models. Mark. Sci. 16–4, 338–353. Venkatesan, R., Krishnan, T.V., Kuman, V., 2004. Evolutionary estimation of macro-level diffusion models using genetic algorithms: an alternative to nonlinear least squares. Mark. Sci. 23–3, 451–464. Juan José Fernández-Durán (b. 24th of June, 1969, in Mexico City) holds a BSc in Actuarial Science from the Instituto Tecnológico Autónomo de México (ITAM, 1989–1993), a PhD in Statistics and Operational Research from the University of Essex (1994–1998) and a Postdoctorate in Marketing from the Freeman School of Business of Tulane University (AACSB Postdoctoral Bridge to Business, 2009–2010). For 14 months, during 1993 and 1994, he worked as a statistical analyst in A.C. Nielsen, México. From August 1998 he joined the Department of Statistics of ITAM as a full-time professor being named Head of Department from August 2004 to August 2010. From August 2010 he joined the ITAM School of Business as Director of the Research Center. He is a member of the National System of Researchers (Sistema Nacional de Investigadores — a government agency founded to promote the scientific research in Mexico) as Researcher Level I. He is an Associate of the Society of Actuaries (ASA-2005), member of the Mexican Actuarial Board (Colegio Nacional de Actuarios) and member of the Mexican Statistical Association (Asociación Mexicana de Estadística). He has acted as a consultant in statistical projects for the United Nations (FAO) and the Mexican Government.