5.2 Results
Figure 3 compares the autocorrelation patterns generated by the model (the solid line) and the actual data (the dashed line) for sales per hour (Figure 3A), purchases per hour (Figure 3B) and labour productivity (Figure 3C). The patterns are very similar, especially for labour productivity, where the autocorrelations are relatively lower and decline more rapidly as the lag length increases than do those of sales per hour and purchases per hour.
The cross-autocorrelations generated by the model (the solid line) and the actual data (the dashed line) are compared in Figure 4. Figure 4A shows the cross-autocorrelations between sales per hour and purchases per hour. Again, the patterns are remarkably similar. The calibrated model captures the high contemporaneous correlation between the two variables, the abrupt drop between lag-0 and lag-1 and the gradual decline thereafter, described in stylised fact 2.
Figures 4B and 4C compare cross-autocorrelations between sales per hour and labour productivity, and purchases per hour and labour productivity generated by the model (the solid line) and the actual data (the dashed line). The calibrated model does not do quite as well at replicating stylised facts 3 and 4 as it did with stylised facts 1 and 2. The cross-autocorrelations generated by the model are somewhat higher than those from the actual data. However, the model does capture the relative difference between cross-autocorrelations, with those between purchases and labour productivity being much lower than those between sales and labour productivity.
It is also possible to calculate transition probabilities for the calibrated model, similar to those of Section 4.2. These are presented in the Appendix. While patterns in the various transition probabilities generated by the model and the actual data do share some similarities, it is fair to say that the model does not perform as well in this regard as it did in terms of replicating observed autocorrelation and cross-autocorrelation patterns. This is not particularly surprising however, as the model was calibrated to stylised facts 1 to 4. Even so, this may suggest that the choice of the normal distribution from which to draw each of the models various shocks may be inappropriate. A distribution with more weight in its tails and around its mean may perform better.
