Combining time series and cross sectional data for the analysis of dynamic marketing systems
Vector AutoRegressive (VAR) models have become popular in analyzing the behavior of competitive marketing systems. However, an important drawback of VAR models is that the number of parameters to be estimated can become very large. This may cause estimation problems, due to a lack of degrees of freedom. In this paper, we consider a solution to these problems. Instead of using a single time series, we develop pooled models that combine time series data for multiple units (e.g. stores). These approaches increase the number of available observations to a great extent and thereby the efciency of the parameter estimates. We present a small simulation study that demonstrates this gain in efficiency. An important issue in estimating pooled dynamic models is the heterogeneity among cross sections, since the mean parameter estimates that are obtained by pooling heterogenous cross sections may be biased. In order to avoid these biases, the model should accommodate a sufficient degree of heterogeneity. At the same time, a model that needlessly allows for heterogeneity requires the estimation of extra parameters and hence, reduces efciency of the parameter estimates. So, a thorough investigation of heterogeneity should precede the choice of the nal model. We discuss pooling approaches that accommodate for parameter heterogeneity in different ways and we introduce several tests for investigating cross-sectional heterogeneity that may facilitate this choice. We provide an empirical application using data of the Chicago market of the three largest national brands in the U.S. in the 6.5 oz. tuna sh product category. We determine the appropriate level of pooling and calibrate the pooled VAR model using these data.
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