Forecasting long-term
sales based on
early demand
An Empirical Study in Collaboration
With Pepperminds Nederland
June 29, 2015
Philip van den Donk (10462775)
philipvdd@hotmail.com
Bachelor Thesis in Econometrics
Faculty of Economics and Business
Supervisor Maurice Bun
1 INTRODUCTION
Forecasting is widely used in the corporate sector, because it can greatly improve the quality of decisions that need to be made before the necessary information is available. Lapide (2002) describes three types of forecasting in companies: Firstly there is operational forecasting, in which forecasts are generated for a very short term (daily, weekly, etc.). This type of
forecasting is useful in areas such as customer service, inventory management, and transportation. Next there is tactical forecasting, in which medium-term forecasts are generated (fortnightly, monthly, quarterly, etc.). These forecasts support tactical planning such as sales, marketing and labour planning. And lastly there is strategic forecasting, in which long-term forecasts are generated (annually, etc.). These can be used in long-term investments and decision-making. This paper only makes use of tactical forecasting, as this best complements the dataset and provides the most useful information for the firm with which is collaborated.
A prime example of a model used to forecast time series is the Bass diffusion model (Bass, 1969). It is a model that predicts the development of sales for durable goods over time. It is frequently used for its relative simplicity and due to the fact that it can capture a wide variety of diffusion patterns observed in practice (Boswijk & Franses, 2005). The basic Bass diffusion model contains three parameters, namely: the internal effect (coefficient of
imitation), the external effect (coefficient of innovation), and the saturation level (coefficient of maturity). This model is particularly useful for out-of-sample forecasting.
The goal of this research is to forecast long-term demand of a single firm by
implementing its short-term demand information in the Bass diffusion model, in which the demand for a project is characterized by the number of sales. More precisely, the coefficient of innovation, the coefficient of imitation, and the potential market size, and the timing and magnitude of the sales peak are estimated. The Bass model is further explained in the
theoretical framework ahead. The corresponding central research question is given by: can the future sales of a project be accurately forecasted based on early sales using the Bass diffusion model?
To achieve this goal, data provided by Pepperminds Nederland B.V. are used. Pepperminds B.V. is one of the market leading field marketing bureaus of the Netherlands. The company raises 300.000 subscriptions, memberships, donors, leads, and sales each year. In addition, each year Pepperminds B.V. reaches ten million potential consumers through its
recruitment and its promotion projects1. This research uses sales data from January 2011 to May 2015 for four of the bestselling projects from Pepperminds B.V. within this period.
Note that consumer behaviour has become more volatile than ever in recent years (Fisher, Hammond, Obermeyer & Raman, 1994). Because of this Fisher et al. (1994)
acknowledge that demand forecasting is not always accurate in practice and there is a growing need to face this demand uncertainty. However, this does not mean that there is no use in forecasting demand. Demand-related forecasts in organizations mostly include a considerable amount of company data and are likely to be accurate, given the research is unbiased
(Armstrong & Green, 2011). Together this means the results of this research are theoretical and should not be accepted as the entire truth, although they still give a decent prediction of what will happen in practice.
The rest of this paper is structured as follows. In the following section background literature on forecasting using the Bass model is discussed in order to provide a solid theoretical framework. Section 3 gives a proposition for how the forecast is executed. This proposition also includes the regression model that is used. The fourth section contains the forecasting results and other empirical analysis. In section 5 the findings and the implications they have for tactical planning are briefly discussed and a suggestion for further research is given.
2 BACKGROUND ON THE BASS DIFFUSION MODEL
This section provides a theoretical framework for the Bass diffusion model, to better understand what it does, what the problems are, and why it is used in this paper.
The Bass diffusion model (Bass, 1969) is widely applied in econometric research (see Dodds (1973), Schmittlein & Mahajan (1982), Srinivasan & Mason (1986), Venkatesan (2002), Wenrong et al (2006), and Aytac & Wu (2011) for different applications). However, there is still a lack of theoretical background in the sector Pepperminds B.V. is in, promoting subscription services. Therefore most of the literature discussed in this section either focus on the development of mobile subscriptions or the introduction of some other tech product into the market. The research in these sectors provide the most parallels to this research.
Bass (1969) has developed a model of initial purchaser activity. It aims at modelling the growth pattern for a product. The common growth pattern is described by an exponential growth in sales at the start followed by a peak in sales and subsequently a decline in sales (see Figure 1). The Bass model works best on products with little repeat purchases, so that there is a fixed potential market size. Subscription services fall into this category. One of the advantages this gives the Bass model is that it permits a forecast of a turndown in sales during a period of rapid sales growth, whereas naïve models tend to project indefinite sales growth (Dodds, 1973).
2.1 The S-Curve
The Bass diffusion model is based on the basic S-curve (see Figure 2). This curve is well known in market research. It describes the cumulative initial purchaser activity (the adoption rate) over time, after a new product is introduced (Rogers, 1962). The adoption rate stands for the cumulative number of people that have adopted divided by the potential market size. The product goes through slow take-off at the beginning when only innovators and early adopters are in the market. A rapid growth follows in the middle of the cycle when the early majority and then the later majority enter the market. Finally it slows down when the market reaches a saturated state (Meade & Islam, 2006).
Sal e s Time T* S(T*)
Because the Bass diffusion model relies upon a product following the S-curve, it is mainly used to predict the product cycle of durable goods. Examples of such durables are tech products and subscription services. Wenrong, Xie & Tsui (2006) find significant results using the Bass model to predict the diffusion of innovations for mobile subscription services in five major Asian countries. Wenrong et al. (2006) compare the full dataset with the Bass model estimates that are found by extrapolating the first part of the dataset. They find that the estimates follow the actual data very closely, albeit with a large sum of squared residuals. However, this is a logical consequence of doing research on continental scale.
2.2 Formulating the Bass Model
So, how is the Bass diffusion model formulated? Rogers (1962) describes five classes of adopters: (1) innovators, (2) early adopters, (3) early majority, (4) late majority, and (5) laggards. Apart from innovators, adopters are influenced in the timing of adoption by the pressure of the social system (Bass, 1969). As the number of previous adopters increases in time, so does the pressure on later adopters. Bass (1969) therefore only distinguishes between (1) innovators and (2)-(5) imitators. To formulate the Bass model, some mathematical and behavioural assumptions need to be made first.
2.2.1 Assumptions of the Bass diffusion model
The Bass model assumes that every purchase made is an initial purchase. Over the period of interest there are m initial purchases and there are no repeat purchases.
A d o p tion r ate ( Yt /m ) Time (T) Early ad o p te rs Late Majo ri ty Lag gard s In n o vat o rs Early m aj o rity
Figure 2: The Basic S-curve: how the Bass model is defined. p
Consumers in the Bass model are divided into two categories: innovators, who make decisions independent of others and imitators, whose choices are influenced by the number of existing adopters. The forces of innovative and imitative behaviour are assumed to exert different effects on the rate of initial purchase. These behavioural forces are represented by parameters p and q respectively in the model. Innovators make decisions independent of others. The choices of imitators however are influenced by the number of existing adopters. The number of previous adopters before time T is represented by the Yt variable in the model.
2.2.2 Derivation of the regression model
Now the probability of a purchase at time T, given that the assumption of no repeat purchases is not violated, is hypothesised to be:
𝑃𝑡 = 𝑝 + 𝑞
𝑌𝑡−1
𝑚 (1)
Since Pt is a probability, and the adoption rate (
𝑌𝑡−1
𝑚 ) is a number between zero and one, p and
q are also numbers between zero and one. Furthermore it is evident that m should be a
positive integer, as it describes the number of people in the potential market size. Assuming that sales are entirely compromised of initial purchases, the initial sales at time T can be written as:
𝑆𝑡 = Δ𝑌𝑡 = 𝑃𝑡[𝑚 − 𝑌𝑡−1] (2)
In words this is the probability of a purchase multiplied by the number of potential adopters that have not yet adopted. Substituting (1), equation (2) can be written as:
𝑆𝑡= 𝑝𝑚 + (𝑞 − 𝑝)𝑌𝑡−1− 𝑞
𝑚𝑌𝑡−12 (3)
𝑃𝑡 Where:
= Probability of adoption at time T
𝑝 = Coefficient of innovation
𝑞 = Coefficient of imitation
𝑚 = Potential market size
𝑌𝑡 = Cumulative number of existing adopters at time T
The assumptions of the theory are formulated in terms of a continuous model and a
distribution function of time to initial purchase. This equation is found by using differential equations (see Appendix). The solution in which time (T) is the only variable is given by:
𝑌(𝑇) = 𝑚 ⋅ [ 1 − 𝑒−(𝑝+𝑞)𝑇
1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇] (4)
The density function is then found by deriving to T:
𝑆(𝑇) = [𝑚(𝑝 + 𝑞)
2
𝑝 ] ⋅
𝑒−(𝑝+𝑞)𝑇
[1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇]2 (5)
This equation gives a modified S-curve (see Figure 2) after summing over t from one to T. Adding control variables and an error term to equation (3), this results in the following regression equation:
𝑆𝑡= 𝛽0+ 𝛽1𝑌𝑡−1+ 𝛽2𝑌𝑡−12 + 𝛽3′𝐷𝑉𝑡+ 𝜀𝑡 (6)
Where 𝑡 = 1,2, … and 𝐷𝑉 stands for a vector of control variables and εt an error term.
Equation (6) is an extended version of the original Bass model, which does not include a vector of decision variables. This regression gives OLS estimates 𝛽0, 𝛽1, and 𝛽2, which is used to derive the coefficient of innovation (p), the coefficient of imitation (q), and the potential market size (m) as:
𝑝 =𝛽0
𝑚 𝑞 = −𝑚𝛽2 𝑚 =
−𝛽1− √β12− 4𝛽 0𝛽2
2𝛽2 (7)
Since 0 < 𝑝, 𝑞 < 1 and 𝑚 > 0, it follows that 𝛽0 and 𝛽1 should be positive, and 𝛽2 should be negative. Thus to predict sales, these coefficients are substituted into (4).
It can be derived algebraically (see the Appendix) that the peak value of sales (St) and the predicted time (T) of this peak are given by:
𝑆(𝑇∗) =𝑚(𝑝 + 𝑞)2
4𝑞 (8)
𝑇∗ =ln 𝑞 − ln 𝑝
2.3 Problems of the Bass Model
Although the theoretical framework on the Bass diffusion model is quite solid, its translation into practice is not straightforward. Boswijk & Franses (2005) describe two of the main deficiencies. First, the model is formulated in continuous time, whereas the observed empirical data are often discrete. Second, the model can be written in multiple ways and a decision on where to put the stochastic error term has to be made. Satoh (2001) brings up a way to overcome the first problem, by formulating the model in discrete time. He finds in his research that applying OLS to this discrete model gives more accurate parameter estimation than applying more difficult methods to the continuous model. The second problem is less difficult to overcome, as there are a limited number of ways to formulate the model and the decision can be made in retrospect.
In addition to these deficiencies, there are some problems with the methodology of the OLS approach. Srinivasan & Mason (1986) outline the main issues to consider. First, the OLS approach has the disadvantage of improper time aggregation of the continuous time model and the Bass parameters cannot be expressed as linear functions of the regression coefficients. Standard errors are easier to estimate and moreover more likely to be valid when using an NLS approach for example. Even then, the standard errors are based on asymptotic
approximations. For the small amount of data, one may wonder about the validity of these approximations. Heeler & Hustad (1980) researched the number of years of data required and recommended at least ten years of input data. This problem is overcome in this paper by using smaller time intervals.
Another possible problem is that the assumption of homoscedasticity is made, while this assumption may be violated. One method to overcome this problem is to use White standard errors to allow for heteroscedasticity. A more elaborate method is used by Boswijk & Franses (2005), who extend the original Bass model with an autoregressive variable and a heteroskedastic error term. Using simulations, they examined the consequences of excluding these additional terms. They found that the estimates obtained from the method of Bass (1969) and Srinivasan & Mason (1986) are far from reliable, in terms of their values and the distribution of their t-ratios. They do however find that the differences across the models are little for highly aggregated data. Again, smaller time intervals are chosen in this paper to emulate such data that are eligible for research.
Furthermore, the serial correlation between the residuals in successive time periods should be small. If the serial correlation is significant, econometric procedures for handling autocorrelation problems need to be applied (Heij et al. 2004, p. 354). Some of these problems are discussed in the next sections.
2.4 Estimating market potential
Because the coefficients of the Bass model are nonlinear functions of the OLS estimates, it is difficult to determine the standard errors for these coefficients. Heeler & Hustad (1980) report a great improvement in the quality of forecasting by constraining the value of the potential market size (m) to an intuitive estimate. The methodology they use is based on an iterative estimation procedure with a prespecified value of m. The Bass parameters are first estimated via the original method by Bass, then the value of m is determined iteratively to see which m provides the best fit. However, Heeler & Hustad (1980) do state that the model may lose value as a managerial forecasting tool because of this. Tigert & Farivar (1981) propose a more efficient model to turn the coefficients of innovation and imitation into linear functions of the OLS estimates, thus giving p and q the same standard error as 𝛽0 and 𝛽2 respectively. The
model they use is acquired by dividing both sides of equation (3) by m and then substituting 𝑆𝑡′=𝑆𝑡
𝑚 and 𝑌𝑡 ′= 𝑌𝑡
𝑚. The modified basic model is then obtained:
𝑆𝑡′= 𝑝 + (𝑞 − 𝑝)𝑌
𝑡−1′ − 𝑞𝑌𝑡−1′2 (10)
Using the standard least squares regression formula, values of 𝛽0, 𝛽1, and 𝛽2 can be derived from the discrete form of the model:
𝑆𝑡′= 𝛽0+ 𝛽1𝑌𝑡−1′ + 𝛽2𝑌𝑡−1′2 (11)
Now the coefficients of innovation and imitation are indeed linear functions of the OLS estimates and it is trivial to determine the standard errors of p and q.
2.5 The NLS approach
A more intuitive approach to the Bass diffusion model is to determine the parameters directly by estimating non-linear least squares coefficients. The foundation of this method lies with the research of Srinivasan & Mason (1986). In their paper they wrongly credit Schmittlein & Mahajan for equation (4), which was already derived by Bass (1969) in the original work. The NLS regression equation is given by:
𝑆𝑡 = 𝑌𝑡− 𝑌𝑡−1+ 𝑢𝑡 = 𝑚 ⋅ [ 1 − 𝑒−(𝑝+𝑞)𝑡 1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑡− 1 − 𝑒−(𝑝+𝑞)(𝑡−1) 1 +𝑞𝑝 𝑒−(𝑝+𝑞)(𝑡−1)] + 𝑢𝑡 (12)
Where 𝑡 = 1,2, … and ut is an additive2 error term with variance 𝜎2.
The Bass parameters follow directly from this equation, as do their standard errors. The non-linear regression model does not feature a vector of control variables. With the NLS method, the expectation is that control variables would badly influence the fit of the model. The next paragraph discusses the influence of control variables in general.
2.6 Including decision variables
Questions could be asked whether other decision variables need to be included in the model, such as promotion activity. However Bass, Krishnan & Jain (1994) show that this is not necessary, by formulating a generalized Bass model including decision variables and
comparing it with the normal Bass model. Bass et al. (1994) find that if the period-to-period percentage changes of decision variables are approximately constant, the generalised Bass model and the normal Bass model provide approximately the same fit. Also they find that decision variables can shift the Bass curve in time, but the shape of the curve is always similar. In the empirical analysis of this paper however, some decision variables are included as it is not evident these are constant over time.
3 RESEARCH DESIGN
In this paper a forecast of long-term sales is made with the Bass diffusion model using early sales data. This section discusses the methods used in this research.
The data are provided by Pepperminds B.V., one of the market leading field marketing bureaus in the Netherlands. The results of this research can be used by Pepperminds B.V. for their tactical planning, such as sales, marketing and labour planning.
2
Srinivasan & Mason (1986) also estimated a formulation with a multiplicative error term, but the additive error term structure provides better results in terms of predictive validity. The additive error term does however have the theoretical problem of leading to negative sales under the assumption of normality of u. Though they claim that the problem of negative sales is only a minor issue in practice.
Pepperminds B.V. divides their workers into three categories: talents, promotors, and captains. The talents are new to the company, whereas the captains are often experienced and have more responsibilities. The type of employment and other worker traits (age, tenure, gender, education, etc.) are included in the dataset. These decision variables may be useful to take into account, as they may explain some of the different variations in the sales model.
Pepperminds B.V. recruits subscriptions for several companies, such as charities, newspapers, and multiple lotteries. As subscription services are durable goods, it should be possible to make long-term projections using the Bass model. The data that are provided for this paper also cover the sales data, for all running projects within Pepperminds B.V., from January 2011 to May 2015. Yet, only a selection of four of the best-selling projects through this period are used, namely HelloFresh, KWF, NRC, and Oxxio3. This leads to four separate regression models, since the sales data are different for every project.
Also it should be noted that the starting date of all these projects are different. For example, HelloFresh started selling mid-2013, whereas KWF and NRC have been selling since before the start of the dataset. This deficiency in the data could lead to less accurate results. Normally a project shows exponential growth in sales at the start, then reaches a sales peak, and then declines (see Figure 1). When the beginning of the dataset is missing, it is possible that the model cannot predict at which stage of selling the project is. Thus it may also not be able to predict the timing and magnitude of the sales peak well. The regression models of the projects are analysed empirically in the next section.
Sales data from the first eighteen to thirty-six months are used to forecast the rest of the available sales data accurately using the Bass model. Regressions are executed on daily, weekly, and fortnightly basis, then these results are analysed to determine which time frames give the best OLS estimations. The main regression model that is used has been discussed in the literature review above, and is given by:
𝑆𝑡= 𝛽0+ 𝛽1𝑌𝑡−1+ 𝛽2𝑌𝑡−12+ 𝛽3′𝐷𝑉
𝑡+ 𝜀𝑡 (5)
Where Yt-1 is the number of existing adopters before time T, equal to ∑𝑇−1𝑡=1𝑆𝑡. DV is a vector
of relevant decision variables. These variables include promotion activity (expressed in the number of shifts of a project in a certain time period) and various worker traits (age, gender,
3
HelloFresh is a subscription service that delivers a box of fresh ingredients and recipes to people’s homes every week. KWF is a charity dedicated to fighting cancer. NRC is a daily newspaper published in the Netherlands by NRC media. Oxxio is a power company that provides sustainable energy.
tenure, and type of employment). Furthermore εt is a stochastic error term, which is assumed to be homoscedastic unless stated otherwise. Regressions are also run without decision
variables for comparison, as proposed by Bass, Krishnan & Jain (1994). The OLS estimates of the model that provides the best fit are then used to derive the coefficient of innovation (p), the coefficient of imitation (q), and the potential market size (m) by equation (7). The alternative estimates using an intuitive value of m (see equation (11)) are also given. Subsequently these coefficients are implemented in equation (5), the density function for initial sales. In addition, White standard errors are used to allow for heteroscedasticity, and an NLS regression is executed based on equation (12). In this paper an attempt is also made to combine the methodology of Tigert & Farivar (1982) and Srinivasan & Mason (1986), and run an NLS regression using an intuitive value of m.
These estimates are then compared to the observed data to see if the forecast is
accurate. The properties that are compared include the timing and magnitude of the sale peak and the value of (adjusted) R-squared, as well as the p-values of the coefficients. The first two properties are estimated using equations (8) and (9). The value of (adjusted) R-squared is evaluated in conjunction with residual plots.
Schmittlein & Mahajan (1982) note that applying OLS to the Bass model with so little data points can result in multicollinearity between 𝑌𝑡−1 and 𝑌𝑡−12 . The variance inflation
factors (VIF), which indicate the extent to which multicollinearity is present in a regression analysis, are therefore taken into consideration. A VIF value greater than ten is reason to be concerned about multicollinearity. In this case one might obtain parameter estimates that are unstable or possess wrong signs, and then the results have no practical meaning. The NLS approach does not suffer from multicollinearity and thus provides an outcome in this situation. However, the NLS method may suffer more from the lack of data.
4 EMPIRICAL ANALYSIS
In this section, firstly the Pepperminds sales data that are used in this research are described. After that, the regression results and the matching forecasts are given, and the implications these results have for tactical planning are discussed.
0 50000 100000 150000 200000 250000 2011 2012 2013 2014 2015 2016 N u m b e r o f e xi sti n g ad o p te rs Year
KWF
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 2013 2014 2015 2016 N u m ve r o f e xi sti n g ad o p te rs YearHelloFresh
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 2011 2012 2013 2014 2015 2016 N u m b e r o f e xi sti n g ad o p te rs YearNRC
0 5000 10000 15000 20000 25000 30000 2011 2012 2013 2014 2015 2016 N m b e r o f e xi sti n g ad o p te rs YearOxxio
4.1 Data DescriptionThe sales data used for this research are provided by Pepperminds B.V. Sales data belonging to a selection of four of the best-selling projects within Pepperminds are analysed, to forecast long-term sales. These data are given numerically in table 1 and graphically in Figure 3.
Year (T) HelloFresh KWF NRC Oxxio
2011 - 40879 7861 7357
2012 - 92291 14155 18156
2013 2275 148926 26531 22885
2014 29369 196691 34422 24728
20154 41738 218822 38483 24771
Table 1: Number of existing adopters up to and including time T
4 Data available until May 2015.
0 5000 10000 15000 20000 25000 30000 35000 40000 2011 2012 2013 2014 2015 N u m b e r o f ad o p te rs at tim e T Year HelloFresh KWF NRC Oxxio
The first thing that should be noted is that HelloFresh is the only one of these projects to have started during the period in which data are available. It is not known when the take-off was for KWF, NRC, or Oxxio. This lack of data is likely to lead to less accurate estimates, for example because the model underestimates the market potential. When looking at the shape of the cumulative curve, a quite constant increase over time can be seen for KWF and NRC. This suggests that these projects have been running for an amount of years before 2011 and are not near a saturated state, where the entire potential market size has adopted. Also, KWF stopped selling mid-2013, which can be seen by a stagnation of the cumulative curve at that time. The sales data for Oxxio however do show a slowdown in sales, this suggests that this project is in a phase past the sales peak and is likely to reach a saturated state in the near future.
It is also possible to look at the discrete sales data at time T (see Figure 4), the
theoretical pattern for the discrete sales is given by Figure 1. Again it is difficult to determine a pattern within the data. The data do suggest a slight growth in sales for KWF before the interlude mid-2013, which might mean that the sales peak has yet to occur. However, the growth can also be the consequence of some external factor. The sales peak in mid-2012 for example is also noticeable in the data of NRC as well as Oxxio. The decline in sales in the second quarter of 2015 give a distorted view of reality, since these data only cover one month of sales instead of an entire quarter. By taking this into account the data for HelloFresh again seem to fit the theoretical model, with a steady, almost exponential increase. It appears HelloFresh provides the best fit for forecasting with the Bass diffusion model.
4.2 Data Analysis
The empirical results are given in table 2. In this table a distinction is made between regressions on daily, weekly, and fortnightly basis, as well as between excluding and including decision variables. Furthermore the estimates of the Bass model parameters (p, q, and m) are given, but the regression coefficients and its standard errors themselves are not. These are not included, because the parameters are nonlinear functions of the OLS estimates in this case, which makes the standard errors difficult to interpret.
Daily data Project↓\Var→ Excluding DV Including DV p q m T* S(T*) p q m T* S(T*) HelloFresh 1.98E-04 6.55E-03 61249 519 106 5.66E-04 1.62E-03 41007 482 30 KWF *1 * * * * 3.96E-04 9.60E-05 (403739)2 (2880) (254) NRC 1.81E-04 6.70E-04 102395 1542 28 * * * * * Oxxio 5.51E-04 3.64E-03 24818 451 30 * * * * * Weekly data Project↓\Var→ Excluding DV Including DV p q m T* S(T*) p q m T* S(T*) HelloFresh 1.34E-03 4.56E-02 61486 76 743 * * * * * KWF * * * * * 1.43E-03 5.65E-04 (527170) (465) (924) NRC 1.30E-03 4.95E-03 97808 214 193 * * * * * Oxxio 3.51E-03 2.65E-02 24809 68 211 * * * * * Fortnightly data Project↓\Var→ Excluding DV Including DV p q m T* S(T*) p q m T* S(T*) HelloFresh 2.514E-03 9.09E-02 61825 39 1483 * * * * * KWF * * * * * * * * * * NRC 2.75E-03 1.09E-02 90057 101 385 * * * * * Oxxio 6.30E-03 5.49E-02 24807 36 423 5.60E-03 2.89E-02 27435 48 282
Table 2: Variable estimates for multiple time periods
1
Regression returns wrong signs for Bass parameters.
From this table a few things can be derived. One is that the project KWF is not a good
candidate for forecasting with the Bass diffusion model. In all regressions that were executed the OLS estimates returned wrong signs, making it impossible to determine the value of the Bass parameters. This complication is most likely due to the starting date of this project. KWF has been part of Pepperminds’ portfolio for some time before 2011. Also because KWF is a charity, the assumption of no repeat purchases may be violated.
Secondly it shows that although including decision variables increases the percentage of explained variation (the R-squared excluding decision variables averages about 0.41, whereas the R-squared including decision variables averages 0.86), it makes the majority of the results unrealistic. Bass, Krishnan & Jain (1994) have shown however that this is not a problem, as the model without decision variables should provide a similar fit. Therefore after this, only the model excluding decision variables is regarded.
Lastly it is noticeable that the estimates of the potential market size (m) and the timing of the sales peak (T*) are quite similar between the time periods (see table 3). However, the coefficients of innovation (p) and imitation (q), and the magnitude of the sales peak are completely different. This shows that these coefficients are sensitive for new data.
Project↓\Property→
Estimates of m Estimates of T* (in days)
Mean Std. Deviation 95%-Confidence Interval Mean Std. Deviation 95%-Confidence Interval HelloFresh 61520 289.50 [60952.58,62087.42] 532.33 13.50 [505.87,558.80] NRC 96753.33 6236.25 [84530.28,108976.36] 1484.67 65.03 [1357.20,1612.13] Oxxio 24811.33 5.86 [24799.85,24822.82] 477 26.51 [425.03,528.97]
Table 3: Averages and Std. Deviations of m and T* estimates
Using these estimates to generate an intuitive value of the potential market size (m), it is possible to acquire standard errors for p and q from equation (11). Also regressions using only the first 24 months of the dataset and These regression results are shown in table 4.
Daily data β0 β1 β2 p 95%-CI p q 95%-CI q HelloFresh 2.02E-04 (4.82E-05)1 6.35E-03 (4.60E-04) -6.42E-03 (7.40E-04) 2.02E-04 (4.82E-05) [1.10E-04, 3.00E-04] 6.42E-03 (7.40E-04) [4.97E-03, 7.87E-03] HelloFresh(*)2 2.02E-04 (2.81E-05) 6.35E-03 (4.76E-04) -6.42E-03 (8.28E-04) 2.02E-04 (2.81E-05) [1.47E-04, 2.57E-04] 6.42E-03 (8.28E-04) [4.80E-03, 8.04E-03] NRC 1.91E-04 (3.00E-05) 4.90E-04 (3.70E-04) -6.33E-04 (9.05E-04) 1.91E-04 (3.00E-05) [1.30E-04, 2.50E-04] 6.33E-04 (9.05E-04) (0, 2.41E-03]
NRC (**)3 3.47E-04 (4.27E-05) -3.87E-03 (1.22E-03) 2.07E-02 (7.93E-03) *4 * * * Oxxio 5.51E-04 (7.45E-05) 3.09E-03 (3.40E-04) -3.64E-03 (3.00E-04) 5.51E-04 (7.45E-05) [4.10E-04, 7.00E-04] 3.64E-03 (3.00E-04) [3.05E-03, 4.23E-03] Oxxio (*) 5.51E-04 (6.65E-05) 3.09E-03 (4.04E-04) -3.64E-03 (3.79E-04) 5.51E-04 (6.65E-05) [4.20E-04, 6.80E-04] 3.64E-03 (3.79E-04) [2.90E-03, 4.38E-03] Oxxio(**) 3.31E-04 (1.30E-04) 5.01E-03 (8.60E-04) -6.15E-03 (1.09E-03) 3.31E-04 (1.30E-04) [7.60E-05, 5.90E-04] 6.15E-03 (1.09E-03) [4.01E-03, 8.28E-03] Oxxio(*+**) 3.31E-04 (1.21E-04) 5.01E-03 (1.03E-03) -6.15E-03 (1.25E-03) 3.31E-04 (1.21E-04) [9.40E-05, 5.68E-04] 6.15E-03 (1.25E-03) [3.70E-03, 8.60E-03] Weekly data5 β0 β1 β2 p 95%-CI p q 95%-CI q HelloFresh 1.34E-03 (4.02E-04) 4.43E-02 (3.88E-03) -4.57E-02 (6.43E-03) 1.34E-03 (4.02E-04) [5.50E-04, 2.13E-03] 4.57E-02 (6.43E-03) [3.31E-03, 5.83E-02] NRC (*) 2.42E-03 (3.79E-04) -2.67E-02 (1.70E-02) 1.42E-01 (6.90E-02) * * * * NRC 1.32E-03 (2.03E-04) 3.65E-03 (2.50E-03) -4.90E-03 (6.12E-03) 1.32E-03 (2.03E-04) [9.20E-04, 1.72E-03] 4.90E-03 (6.12E-03) (0, 1.69E-02] Oxxio (*) 1.91E-03 (1.90E-03) 3.66E-02 (1.25E-02) -4.40E-01 (1.57E-02) 1.91E-03 (1.90E-03) (0, 5.64E-03] 4.40E-02 (1.57E-02) [1.31E-02, 7.48E-02] Oxxio 3.51E-03 (1.04E-03) 2.30E-02 (4.68E-03) -2.65E-02 (4.16E-03) 3.51E-03 (1.04E-03) [1.47E-03, 5.55E-03] 2.65E-02 (4.16E-03) [1.83E-02, 3.46E-02]
Table 4: Regression results with predetermined m, as given in table 3. 1
Standard errors are between parentheses. 2
(*) means White standard errors are used, allowing for heteroscedasticity. 3
(**) means only the first 24 months of the dataset are used for parameter estimation. 4
Regression returns wrong signs for Bass parameters. 5
Fortnightly data are not taken into account because these have too little data points to give a good forecast based on part of the dataset.
The question remains whether there is serial correlation present in the model. To determine this, the Durbin-Watson statistic is taken into account. The DW significance tables show that the DW-statistic should be larger than 1.789 to reject the null hypothesis of no autocorrelation with 95% certainty. Another point of focus is the presence of multicollinearity between Yt-1 and Yt-12. To check this the VIF are taken into account. As a rule of thumb, the
VIF should not be larger than ten to be able to discard multicollinearity. An output for daily data of the main projects and their test statistics is given in table 5 below. These results suggest that the dataset is too small and there is still multicollinearity present. This means that even though the p-values for all the projects were significant, the OLS estimates are not necessarily accurate. The DW-statistic furthermore shows significant serial correlation for HelloFresh and Oxxio. This means the reported standard errors are likely smaller than the true standard errors.
Project↓\Test statistic→ VIF (>10) DW-statistic (<1.789)
HelloFresh 13.27629 1.197112
NRC 19.52588 1.823187
Oxxio 24.96543 0.722438
Oxxio (*)1 20.86696 0.657785
Table 5: Test statistics for serial correlation and multicollinearity. 1
(*) means that only the first 24 months of the dataset are used for parameter estimation.
From the results in table 4 it appears that the sales of NRC cannot be forecasted based on part of the dataset, because the OLS coefficients possess wrong signs, due to
multicollinearity. Instead the full dataset is needed to show a sensible result. As Heeler & Hustad (1980) mention however: it is easy to provide post hoc estimates of plausible
parameters, therefore such estimates have no added value on its own. These estimates can be considered for similar projects, but only in conjunction with a good intuition, for example to estimate the potential market size.
For Oxxio however, it is possible to make a forecast based on data from January 2011 to January 2013. A confidence interval of the timing of the sales peak is given by 𝑇∗ ∈
[299,970], which means the sales peak is estimated to be between the fourth quarter of 2011 and the third quarter of 2013 with 95% certainty. As expected, the sales peak for Oxxio has likely already occurred. This still seems like a wide interval, but this can be explained by the lack of data. When comparing the theoretical forecasts to the observed data, the theoretical cumulative curve (see Figure 6) appears to follow the actual data quite decently. The
theoretical point of saturation matches the observed point and the observed data stays within the theoretical bounds. The confidence bounds are stated with the 95% confidence level using White standard errors.
Figure 5: OLS Theoretical discrete sales vs. Observed discrete sales for Oxxio
0 50 100 150 200 250 2011 2012 2013 2014 2015 2016 Nu m b er of ad op te rs at t im e T Year Theoretical UB LB Observed Theoretical
In regards to HelloFresh, there are too little data to make a forecast without using the full dataset, as there would be multicollinearity between Yt-1 and Yt-12. Therefore a forecast is only made by using the full dataset, as shown in Figure 7 and Figure 8. The sales peak is
estimated to occur between the first quarter of 2012 and the first quarter of 2013, thus giving a narrower estimate than with Oxxio. Since HelloFresh is still relatively new and there is not a substantial amount of data, the forecast is likely to be an underestimation of reality (Heeler & Hustad, 1980).
Figure 6: OLS Theoretical cumulative sales vs. Observed cumulative sales for Oxxio
Figure 7: OLS Theoretical discrete sales vs. Observed discrete sales for HelloFresh
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2011 2012 2013 2014 2015 N u m b e r o f e xi sti n g ad o p te rs Year Theoretical UB LB Observed Theoretical Observed 0 50 100 150 200 250 300 350 400 2013 2014 2015 2016 2017 2018 N u m b e r o f ad o p te rs at tim e T Year Theoretical Observed UB LB Theoretical
An estimation was also made using the NLS approach. This approach overcomes problems such as multicollinearity and serial correlation. It is even possible to combine the methods of Tigert & Farivar (1982) and Srinivasan & Mason (1986), and run an NLS
regression with an intuitive value of m. The regression results, using equation (12) and White standard errors, are given in table 6 below.
#Years of data used p q m T* S(T*) HelloFresh 21 2.99 E-07 (1.54E-07) 6.93E-03 (4.66E-04) 60548 (3.84E03) 1450 105 2 2.81 E-07 (9.28E-08) 6.99E-03 (2.58E-04) 615202 - 1448 43 KWF 2 3.08 E-04 (5.59E-04) 6.40E-04 (1.12E-03) 378106 (7.09E05) 771 132 2 3.03 E-04 (1.91E-05) 6.28E-04 (1.67E-04) 3850002 - 783 133 5 2.87 E-04 (3.98E-05) 8.78E-04 (2.62E-04) 378106 (6.76E05) 960 146 5 2.83 E-04 (1.49E-05) 8.58E-04 (8.68E-05) 3850002 - 972 146 NRC 2 ** 4 - ** - ** - ** - ** - 2 * 3 - * - 967532 - * - * - 5 ** - ** - ** - ** - ** - 5 1.90 E-04 (1.63E-05) 7.03E-04 (9.99E-05) 967532 - 1465 27 Oxxio 2 5.14 E-04 (5.50E-05) 4.11E-03 (5.05E-04) 248112 - 450 32 2 4.02 E-04 (6.41E-05) 5.55E-03 (6.54E-04) 21249 (8.50E02) 441 34 5 5.47 E-04 (5.40E-05) 3.82E-03 (4.03E-04) 248112 - 445 31 5 5.34E-04 3.97E-03 24182 445 31
Figure 8: OLS Theoretical cumulative sales vs. Observed cumulative sales for HelloFresh
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2013 2014 2015 2016 2017 2018 2019 N u m b e r o f e xi sti n g ad o p te rs Year Theoretical UB LB Observed Theoretical
(5.24E-05) (3.64E-04) (7.59E02)
Table 6: NLS regression results using White standard errors to allow for heteroscedasticity. 1
There are only two years of data available for HelloFresh. 2
Regressions run with predetermined m. 3
Regression returns negative value for (one of) the Bass parameters. 4
The estimation of the market potential does not converge for NRC using the NLS method.
The NLS approach gives mixed results. For HelloFresh, the model does not seem to be able to determine the coefficient of innovation, which should not lie so close to zero. With regards to Oxxio however, the NLS approach provides a visual improvement to the fit compared to the OLS method, as shown in Figure 9. For KWF it is now possible to get a realistic result with this method, albeit with large standard errors. On the other hand the model does not react well to the data of NRC. A possible explanation for this difference is that the inflection point does not lie in sample (Mahajan, Müller & Bass, 1990). In this case the timing of the sales peak, lies within the dataset for Oxxio and KWF, whereas it lies out of sample for HelloFresh and NRC. This lack of information makes it harder for the model to fit to the observed data.
Overall the Bass diffusion model visually does a decent job of forecasting long-term sales, but it is difficult to validate the results when one takes all the deficiencies of the model into account. Moreover, using the model correctly requires the knowledge of a few key turning points, namely the take-off and the slowdown of the sales for the OLS method, and the inflection point for the NLS approach. When at least one of these is known, a sensible forecast can be made, but the restriction of needing this information makes the analysis with the Bass model somewhat retrospective rather than predictive (Wenrong et al., 2006).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2011 2012 2013 2014 2015 N u m b e r o f e xi sti n g ad o p te rs Year Theoretical UB LB Observed Theoretical Observed
5 CONCLUSIONS: IMPLICATIONS FOR TACTICAL PLANNING
In this paper the central question was: can the future sales of a project be accurately
forecasted based on early sales using the Bass diffusion model? After conducting empirical research, it is still difficult to answer this question. To be able to make an accurate forecast, it is necessary to have data from the beginning years of market diffusion. Without these data, it is impossible to determine the potential market size. Without knowing the potential market size, it becomes difficult to determine other values of interest. It is therefore not possible to exclude the Bass diffusion model as a viable forecasting tool, but its utility was not evident in this research.
From a planning viewpoint, the central interest in long-term forecasting lies in the timing and the magnitude of the sales peak. While the Bass diffusion model gives a decent prediction for the timing of the peak, more data is needed to get a good forecast for the magnitude thereof.
No forecasting model should be a replacement for other elements in the tactical planning process. The same goes for the Bass diffusion model. With the right intuition, the Bass forecasts could be a great managerial tool. By rightly estimating market potential, the nonlinearity of the Bass parameters in the OLS estimates is eliminated and the standard errors of the coefficients of innovation and imitation can be determined. With these, a forecast can be made within a certain confidence interval. The NLS method eliminate this problem by estimating the Bass parameters directly, however this approach has its own shortcomings in the fact that one needs a dataset that contains much information to get accurate forecast, while this information should be what you want to know in the first place.
Altogether the Bass model has its limitations. The estimation from the Bass model could be implausible if the data is limited and the parameter estimates of the Bass model are sensitive to new data. Most importantly, a good estimate of the Bass model requires knowing at least one of a few key turning points, namely the take-off (as with HelloFresh) and
slowdown (as with Oxxio) for OLS, and the inflection point for NLS, which makes the analysis retrospective rather than predictive to a certain extend. Some of the problems have been resolved by extensions of the original Bass diffusion model, whereas others remain. Further studies may focus on addressing these challenges.
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APPENDIX
Derivation of equation (3)
Starting with equation (2) and substituting (1), we get: 𝑆𝑡 = Δ𝑌𝑡 = (𝑝 + 𝑞 𝑚𝑌𝑡−1) [𝑚 − 𝑌𝑡−1] = 𝑝𝑚 + 𝑞𝑌𝑡−1− 𝑝𝑌𝑡−1− 𝑞 𝑚𝑌𝑡−12 = 𝑝𝑚 + (𝑞 − 𝑝)𝑌𝑡−1− 𝑞 𝑚𝑌𝑡−12 Derivation of equation (4)
Bass (1969) uses an alternative notation for the adoption rate: 𝐹(𝑇) = 𝑌𝑡
𝑚 and its derivative
𝑓(𝑇) =𝑑[𝐹(𝑇)]𝑑𝑇 =𝑆𝑡
𝑚 . Equation (3) can then be rewritten as:
𝑓(𝑇) =𝑑[𝐹(𝑇)]
𝑑𝑇 = 𝑝 + (𝑞 − 𝑝)𝐹(𝑇) − 𝑞𝐹(𝑇)2 This gives the following differential equation to be solved:
𝑑𝑇 = 𝑑𝐹(𝑇)
𝑝 + (𝑞 − 𝑝)𝐹(𝑇) − 𝑞𝐹(𝑇)2
Integrating both sides over T and rewriting gives:
⇒ 𝐹(𝑇) = 𝑞 − 𝑝𝑒−(𝑝+𝑞)(𝑇+𝐶)
𝑞(1 + 𝑒(−𝑝+𝑞)(𝑇+𝐶)) (*)
The additional constraint that the adoption rate should be zero for T=0 allows the integration constant C to be evaluated (analogous to derivation of equation (8) below):
−𝐶 =ln(𝑞) − ln (𝑝) 𝑝 + 𝑞
Substituting this expression for C into (*) and using 𝑌𝑡 = 𝑚𝐹(𝑇) results in equation (4).
Derivation of equation (7)
Starting with equation (3) and (6) the derivation of p and q are trivial and m follows from: 𝑞 − 𝑝 = 𝛽1 = −𝛽2𝑚 −
𝛽0 𝑚
Multiplying both sides by m and bringing all variables to the left side, we can write: 𝛽2𝑚2+ 𝛽
1𝑚 + 𝛽0 = 0
Derivation of equation (8) and (9)
Starting with equation (5) and deriving to T, we get: 𝑑 𝑑𝑇[ 𝑚(𝑝 + 𝑞)2 𝑝 ] ⋅ 𝑒−(𝑝+𝑞)𝑇 (1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇)2 = 0 ⇒ [𝑚(𝑝 + 𝑞)2 𝑝 ] ⋅ [ −(𝑝 + 𝑞)𝑒−(𝑝+𝑞)𝑇 (1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇)2 + 2(𝑝 + 𝑞)𝑞𝑒−(𝑝+𝑞)𝑇 (𝑝 (1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇)) 3 ] = 0 ⇒ 2(𝑝 + 𝑞)𝑞𝑒−(𝑝+𝑞)𝑇 (𝑝 (1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇)) 3 = (𝑝 + 𝑞)𝑒−(𝑝+𝑞)𝑇 (1 +𝑞𝑝 𝑒−(𝑝+𝑞)𝑇)2 ⇒ 𝑒−(𝑝+𝑞)𝑇 = 𝑝 2𝑞(1 + 𝑞 𝑝𝑒−(𝑝+𝑞)𝑇) ⇒ 𝑒−(𝑝+𝑞)𝑇 = 𝑝 2𝑞+ 1 2𝑒−(𝑝+𝑞)𝑇 ⇒𝑞 𝑝= 𝑒−(𝑝+𝑞)𝑇 ⇒ 𝑇∗ = ln(𝑞) − ln (𝑝) 𝑝 + 𝑞 Substituting 𝑇∗ into equation (5) then yields:
[𝑚(𝑝 + 𝑞)2 𝑝 ] ⋅ 𝑒ln(𝑝)−ln(𝑞) (1 +𝑞𝑝 𝑒ln(𝑝)−ln(𝑞))2 = [ 𝑚(𝑝 + 𝑞)2 𝑝 ] ⋅ 𝑝 𝑞 (1 +𝑞𝑝𝑝𝑞)2 = 𝑚(𝑝 + 𝑞)2 4𝑞
