Testing Stationarity and Introducing a New Replace Age Distribution for Modelling Aggregate Sales of Consumer Durables

We conducted a survey of 8,077 households collecting data for the timing of their first and replacement purchases of six electronics products. Our analysis of this data makes two contributions. First, we introduce a new replacement distribution motivated by the deficiency of existing distributions when tested against our data. Our data revealed empirical replacement rates that decrease substantially for larger ages for all six products. As none of the existing replacement distributions can accommodate this shape, we develop a new distribution based on a modified Gamma distribution. We compare this distribution to earlier ones, both for the replacements distribution and the aggregate sales model. Our second contribution is to investigate whether replacement distributions are non- stationary (vary over time). The diffusion literature suggests that the average replacement age may change over time (e.g. Bayus 1988). Almost all models assume a constant replacement age over time. The only exception is Steffens (2001). His study demonstrates a time-varying model fit aggregate sales data for automobiles better, but this notion has not yet been empirically tested using disaggregate-level replacement data. We are able to test this proposition for TVs for which we have a longer history of replacements. Methods Many earlier studies of repeat purchases based on disaggregate (household) data have limited generalisability as they study only one product and / or low statistical power due to sample size restrictions. To overcome these limitations we decided to conduct an on-line survey to enable both a large sample size, and data collection for several products. We collected data for the year of first and replacement purchases. We had 8,077 useable responses. Three types of analyses were performed. Replacement Distribution Analysis. First, replacement distributions (or life tables, see Ruffin and Tippet 1975) were derived by considering the time until a replacement for each purchase of every household. The aggregate proportion of products in service (at any time) of age a that were replaced at that age was calculated. Each of the six replacement distributions were fitted to these data using non-linear regression (SPSS NLR routine). For the parameter estimation, the data (percentage replaced for each age) were weighted by the total number of observations. The distributions were compared both in terms of their fit to the data and to a hold-out sample.