In principle, trends in an ARMA model imply possibly unbounded behavior of the time series. For many macroeconomic data this may be sensible, but for business economic data, we may sometimes expect that there is some upper bound to the time series. An example of such nonlinear trends is a growth curve, describing product sales or adoption over time. In principle, the ST and DT models allow the yt series to be unbounded. This may imply that these models cannot be useful for variables such as the unemployment rate, because this variable is bounded by the values 0 and 100. However, in small samples we may still find that ST and DT models do yield approximately adequate data descriptions, although we should then exercise care when forecasting many periods ahead.
For many marketing economic time series, it is conceivable that the time series converge to a certain maximum or minimum as time goes by. The market penetration of durable consumer goods such as personal computers or mobile phones may be close to 100 percent in the long-run. Sales may have been small initially, while they rise through the adoption of the new product by the majority, and ultimately sales may decline such that penetration converges to some saturation level. Hence, for time series variables that characterize some product or market life-cycle, we may wish to modify the above trend models to allow for a saturation level.
An overview of such so-called growth curves is given inMahajan et al.(1993), see alsoMeade and Islam(1995). Two frequently used growth curves in practice are the
0 20 40 60 80 100 25 50 75 100 125 150 Gompertz Logistic
Figure 4.5: Example of a Gompertz curve (4.28) and a logistic curve (4.29) withα = 100, β = 4 and γ = 0.05.
Gompertz growth curve, given by
yt = α exp[−β exp(−γ t)], (4.28)
and the logistic growth curve, given by
yt = α/[1 + β exp(−γ t)], (4.29)
whereα is the saturation level, and α, β and γ are all positive parameters. An example of a typical growth curve pattern is given in Figure4.5, showing two time series yt of length T = 150, where one series is generated using a Gompertz curve (4.28) with
α = 100, β = 40, and γ = 0.05, while the other is generated using a logistic curve
(4.29) with the same parameter values. Standard normal shocksεt ∼ N(0, 1) are added to both series.
The expressions in (4.28) and (4.29) show that as time proceeds (that is, when t increases), the yt value approaches α. The key difference between the two growth curves is the rate of increase towards this saturation level. This can be understood most easily by considering the point of inflection, say,τ, where growth is fastest. It is easily shown that for both the Gompertz curve and for the logistic curve it holds that
∂2y
t/∂2t = 0 at time τ = log β/γ . The level of ytat the inflexion point differs though. For the Gompertz curve, it holds that yτ = α/e, while for the logistic curve yτ = α/2. Hence, growth is slower (faster) before (after) the inflexion point for the Gompertz curve than for the logistic curve, see also Figure4.5.
A growth curve model that is frequently applied in marketing is the Bass model, seeBass(1969). The model assumes a population of m potential adopters of a new product. For each adopter, the time to adoption is a random variable with a distribution
F (τ) and density f (τ) , such that the hazard rate satisfies f (τ)
1− F(τ) = p + q F(τ). (4.30)
The parameters p and q are associated with innovation and imitation. The cumulative number of adopters at timeτ, denoted as N(τ), is a random variable with mean
N (τ) = E[N(τ)] = mF(τ). (4.31) The function N (τ) thus satisfies the differential equation
n(τ) = d N (τ)
dτ = p[m − N(τ)] + q
mN (τ)[m − N(τ)]. (4.32)
The solution of this differential equation for cumulative adoption is
N (τ) = mF(τ) = m 1− e−(p+q)τ 1+ qpe−(p+q)τ (4.33)
and for adoption itself it is
n(τ) = m f (τ) = m p( p+ q)2e−(p+q)τ ( p+ qe−(p+q)τ)2 . (4.34)
Like the logistic and Gompertz curve, N (τ) has a sigmoid pattern and n(τ) has a hump-shaped pattern.
In practice one has discretely observed data, like per year or per quarter. Denote
Xt as sales and denote Nt as cumulative sales, where t corresponds to the discretely observed data.Bass(1969) proposes the regression model
Xt = pm + (q − p)Nt−1− q mN 2 t−1+ εt = α1+ α2Nt−1+ α3Nt2−1+ εt, (4.35) whereεtis white noise with varianceσ2.Bass(1969) recommends use OLS to estimate the parameters in (4.35), where non-linear least squares [NLS] is needed to estimate the standard errors of ˆp, ˆq and ˆm. Of course, for forecasting only estimates for ˆα1, ˆα2
and ˆα3are required and OLS can be used.
Boswijk and Franses(2005) have proposed an alternative to the expression in (4.35), which is based on the notion that N (τ) resembles an equilibrium path around which the actual cumulative adoptions fluctuate. The stochastic features of the diffusion process originate from the tendency of the data to revert to that equilibrium path in
an error-correction-type of way. The model to be fitted to actual data would then become
Xt = α1+ α2Nt−1+ α3Nt2−1+ α4Xt−1+ εt (4.36) Boswijk and Franses(2005) also propose to make the error process heteroskedastic, so that uncertainty around the diffusion path is largest around the sales peak.
Another convenient empirical version of the Bass model is proposed inSrinivasan and Mason(1986). These authors propose to apply NLS to
Xt = m[Ft( p, q) − Ft−1( p, q)] + εt, (4.37) where Ft( p, q) = 1− e−(p+q)t 1+ qpe−(p+q)t . (4.38)
The Bass model is very often used to forecast future sales data. It is important to recognize Van den Bulte and Lilien (1997) that with data available only before the inflection point reliable estimates of p and q cannot be obtained. Practical sales forecasting of durable products thus usually proceeds along other lines; seeLilien et al. (2000) for a detailed account. First, one considers the sales patterns of products that are similar, think of sales data for the PlayStation3 to predict the patterns of PlayStation4. For these similar data one obtains initial values of p and q. The expected value of the total sales is usually set by the manager. What is then helpful are cross-country comparison studies, where the innovation and imitation characteristics of countries are summarized, see for exampleChandrasekaran and Tellis(2008),Talukdar et al.(2002) andTellis et al.(2003). One can make the characteristics of the diffusion process a function of characteristics of countries.
To make point forecasts for sales one can use one of the above expressions, like in (4.35), (4.36) or (4.37). The model in (4.37) seems easiest to construct forecasts. When t = n is the forecasting origin, and h is the horizon, one can simply use
Xn+h = ˆm[Fn+h( ˆp, ˆq) − Fn+k−1( ˆp, ˆq)] + εt (4.39) When the error term is an ARMA type process, straightforward modifications of (4.39) can be made.