The penetration of CDs in the sound recording market: issues in specification, model selection and forecasting

Abstract

Annual data on the market penetration of music CDs in 12 countries are used to consider two issues in a forecasting comparison of 12 model specifications and three sample sizes. Firstly, particular attention is paid to the impact of stochastic specification of the models on the estimation of the saturation levels and forecasts. Secondly, the issue of model complexity and sample size is addressed by including two four-parameter models with various specialisations, other three-parameter models, and three different sample sizes. There is reasonably strong support for one particular model which is a member of Bewley and Fiebig’s (Int. J. Forecasting 4 (1988) 177) flexible logistic class.

Introduction

Since the seminal contributions of Griliches (1957), who used the logistic model to characterise the diffusion of hybrid varieties of wheat, and Chow (1967), who based his analysis of the growth of the market for computers on a Gompertz specification, many alternative specifications of S-shaped growth curves have been introduced and compared to one another. Not only have fundamentally different specifications been proposed, such as the logistic and Gompertz functions, but, as Dixon (1980) argues when commenting on Griliches, there are also many alternative stochastic specifications for any given deterministic form. Since equations can be, and often are, also expressed as a differential, or difference equation, on the one hand, or as a model solution, or levels form, on the other, the number of possible specifications for tackling any one application is very large. As a result, theoretical and empirical surveys have attempted to unravel the properties of competing models; see, for example, Mahajan, Muller and Bass (1990) and Meade & Islam, 1995, Meade & Islam, 1998.

One problem faced when comparing these models in any applied setting is the extent to which the diffusion process embodied in the data is free from changes in economic and other external factors. Failure to account for such factors may influence the choice of functional form in any given application. For example, in Meade and Islam’s (1995) telecommunications demand study, price was falling with technological improvements and new competing services were being offered. These and other external influences possibly also played a major role in determining demand (Bewley, 1997) and, indeed, the indicative graph of demand in Meade and Islam (1995) did not, at least visually, exhibit any clear S-shaped growth pattern.

On the other hand, there are many situations, such as with the introduction of a new pharmaceutical product under patent, that lend themselves to a more purist version of a diffusion process. The empirical study reported in this paper is another such example. Music compact discs (CDs) were introduced by Phillips in the Netherlands in 1983 and, because of the multinational nature of the manufacturers of CDs and CD players, one might expect some similarity in the diffusion process across the 12 industrialised countries considered in this study. There were no other major innovations in the formats of sound recordings being supplied and the relative price of CDs to the other formats did not alter greatly during that period. As a result, one might expect each country’s data to exhibit classic S-shaped growth patterns and, in the absence of other factors, some similarity in CD penetration across countries. Thus, an empirical investigation with these data sets provides a relatively controlled but realistic setting in which to investigate certain general questions of model specification.

Two issues are confronted in this model comparison. Firstly, particular attention is paid to the impact of stochastic specification of the models on the estimation of the saturation levels and forecasts. In a related study, Bewley and Griffiths (2001) considered this issue for just one model, the logistic, from a Bayesian perspective for including prior information. Secondly, the issue of model complexity and sample size is addressed by including two four-parameter models with various specialisations, and other three-parameter models, and three different sample sizes.

Using annual data for 12 countries and three sample sizes, 12 alternative models are evaluated in terms of their abilities to provide a realistic estimated saturation level, goodness-of-fit and forecasting ability. In the smallest sample, 1984–1990, it is not clear that the point of inflection has been reached while in the largest, 1984–1996, it is a relatively simple matter to estimate the saturation level. The intermediate sample, 1984–1993, is included to replicate what is thought to be a typical modelling situation.

While there are visual differences in the curves for the 12 countries analysed in this study, it is found that there is reasonably strong support for one particular functional form. This support is also consistent over the three sample sizes. Moreover, it is shown that the width of bootstrapped forecast confidence intervals varies across countries and they appear to be informative in that they are reasonably narrow, but also sufficiently wide to accommodate a change in trend that occurred in CD penetration in one of the countries.

The 12 models under consideration are introduced, and their properties discussed in Section 2. The data and the comparative fits of the models are discussed in Section 3. Forecast comparisons are also reported and are linked back to model fit. This section also contains bootstrapped forecast confidence intervals of the preferred model, the Box–Cox variant of Bewley and Fiebig’s (1988) flexible logistic (FLOG) family of models, for two countries that can be regarded as extreme cases. The issues and broader conclusions are summarised in Section 4.