TABLE 3.

Average realized and predicted discrepancy measures for the autoregressive multinomial and Dirichlet‐multinomial models and posterior predictive p‐values for Italian data

Autoregressive model
Multinomial	$\widehat{\text{Dist}}$	$\widetilde{\text{Dist}}$	p‐value
Model 1 (2nd order, without constraints)	1658.011	124.670	0.000
Model 2 (2nd order, with constraints)	2347.274	68.474	0.000
Model 3 (3rd order, without constraints)	1565.587	122.793	0.000
Model 4 (3rd order, with constraints)	2203.832	70.512	0.000
Dirichlet‐multinomial	$\widehat{\mathbf{\text{Dist}}}$	$\widetilde{\mathbf{\text{Dist}}}$	p‐value
Model 5 (2nd order, without constraints)	2608.502	3060.236	0.679
Model 6 (2nd order, with constraints)	2992.213	3629.419	0.750
Model 7 (3rd order, without constraints)	2414.970	2811.524	0.536
Model 8 (3rd order, with constraints)	2915.772	3344.208	0.661