Seasonality and non-linear price effects in scanner-data-based market-response models
Introduction
This paper deals with the econometric aspects of market-response models, when these are calibrated for weekly scanner data, typically for fast moving consumer goods (FMCGs). Market-response models usually seek to correlate sales or market shares with marketing-mix instruments such as price, promotions like feature and display, and advertising, see Hanssens et al. (2001) and Leeflang et al. (2000). Due to in-store scanner techniques, the data that are typically available to estimate model parameters are weekly data for 4–8 years. The amount of years is due to the fact that the life cycle of products and, sometimes also of brands, does not often extend beyond that time frame. The weekly data are usually provided by a particular retail chain, and they concern the most important brands in several categories across the outlets within that chain. It is common to stack the information on brands across categories, and to consider models for, say, N brands, where these brands thus cover a variety of FMCG categories like margarine, tissues, ketchup and so on, see Pauwels and Srinivasan (2004), Nijs et al. (2001) and Fok et al. (2006). In sum, the relevant market-response models are calibrated for panels of time series, where N ranges from, say, 50 to 300, and where time T covers 52 weeks for 4–8 years. Note that this type of data is not always suited to analyze the market response across all supermarkets in a particular area as some retailers, usually hard discounters like Wall-Mart, do not supply data. There are some recent developments to overcome this problem, such as the use of consumer hand-scan panels, see, for example, Fox et al. (2004) and van Heerde et al. (2005). In this paper we however consider a single retailer.
Given the availability of large N and large T data, one could simply want to consider N different models, or different models per product category. However, in marketing research it is common practice to search for, so-called, empirical generalizations, that is here, common features across the N models. In market-response models such common features could concern the effects of price changes or of promotions. These effects could partly be idiosyncratic, and partly be the same for similar brands in similar categories, for example. This usually means that a useful market-response model has a second layer in which the parameters in the N models are correlated with characteristics of brands and categories which are constant over time, see Fok et al. (2006) among others. In the present paper, we also propose such a two-level Hierarchical Bayes model and we use a Markov Chain Monte Carlo (MCMC) approach to obtain posterior results.
The first advantage of Hierarchical Bayes models for panels of time series is that it often amounts to a plausible reduction of the number of parameters. Hence, there is an increase in the degrees of freedom. This is particularly useful when the first-level parameters are less easy to estimate due to a lack of degrees of freedom. For example, as we will discuss below, the inclusion of 52 weekly dummies to capture seasonality in weekly market-response models amounts to a serious loss of degrees of freedom, particularly when there would be only 4 years of data. Hence, a plausible strategy here is to introduce a second-level model where seasonality is captured by a sinusoid regressor and an error term.
The model that we will propose below further allows for the possibility that past prices have an effect on the current short-run price elasticity. Such an effect is often documented in the marketing literature, see Pauwels et al. (2006) and the references cited therein, and it means that the difference between the current price and the previously observed price has an impact on the current price effect. Such a non-linear effect can occur in many N equations, but perhaps not in all. Hence, a second advantage of a two-level model is that by including all N equations, the parameters in each of these get shrunk towards a common value in the second level, see Blattberg and George (1991), while of course the error term allows for brand-specific variation.
In sum, in this paper we put forward a two-level Hierarchial Bayes model for a panel of weekly time series on sales and marketing-mix instruments, where we use the second level to effectively reduce the number of parameters to capture seasonality and to shrink (potentially difficult to estimate) non-linear effects towards interpretable parameters. In the second level, the latter parameters are correlated with brand-specific and category-specific characteristics. In Section 2, we describe the representation of the model. In Section 3 we propose an MCMC sampler to obtain posterior results. In Section 4, we apply our model to data on 96 brands for close to 8 years of weekly data. We demonstrate that the model yields plausible and reliable estimates. In Section 5 we conclude with some remarks.
Section snippets
Representation and interpretation
When modeling weekly sales of FMCGs, a typical model that relates log sales to log prices and promotion variables, amongst other marketing instruments, iswhere denotes the sales of brand i for at time t, for , and denotes the price of brand i at time t, and where , see Wittink et al. (1988) and many others. The vector captures promotion activities for brand i at time t. In recent years it has been recognized that
Bayes analysis
The total model is given bywhere is given in (3), is given in (5) together with (8) and for and . The likelihood function belonging to this model iswhere is the density function of a normal distribution with mean m and covariance matrix and
An illustration
To illustrate the usefulness of our model, we consider weekly sales volumes for 96 brands of fast moving consumer goods in 24 distinct categories. These data are obtained from the database of the US supermarket chain Dominick's Finer Foods. The data cover the period September 1989 to May 1997 in the Chicago area. The same data are used in Srinivasan et al. (2004). We take the top four brands of each product category. Next to the actual price we include in the typical promotion
Conclusion
In this paper we have put forward a hierarchical Bayes model for a panel of time series on sales and marketing activities, where we allowed for weekly seasonality and for non-linear asymmetric price effects. As sales data can be available at a weekly basis, the standard dummy variable approach to model seasonality involves too many parameters. Instead, we have proposed a combination of a deterministic cycle and of random effects to capture seasonality. In the empirical section we showed that
Acknowledgments
We thank Herman van Dijk and two anonymous reviewers for their comments. Over the years, members of the Econometric Institute showed an interest in modeling data that are not necessarily of a macroeconomic nature, and that show seasonality and non-linearity. Also, panel data have been considered. We combine all these research interests into a single model for marketing data in the present paper, which has been prepared for the commemorative special issue of the Journal of Econometrics,
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