Table 5 Summary of the models to be evaluated

PM3

\(\mathrm{log}\left({\mu }_{ij}\right)=\mathrm{log}\left({S}_{ij}\right)+{\alpha }_{0}+{\beta }_{ep1}\cdot \mathrm{log}\left({y}_{i,j-1}\right)+{u}_{i}+{v}_{i}+{\beta }_{ep2}\cdot \mathrm{log}(\sum\limits_{{\delta }_{i}}{y}_{{\delta }_{i},j-1})\)  

NBM3RS

\(\mathrm{log}\left({\mu }_{ij}\right)=\mathrm{log}\left({S}_{ij}\right)+{\alpha }_{0}+{\beta }_{ep1}\cdot \mathrm{log}\left({y}_{i,j-1}\right)+{u}_{i}+{v}_{i}+{\beta }_{ep2}\cdot \mathrm{log}(\sum\limits_{{\delta }_{i}}{y}_{{\delta }_{i},j-1})\)  

\(\mathrm{log}\left({r}_{i}\right)= {r}_{0}+{u}_{i}+{v}_{i}\)

NBM3RST

\(\mathrm{log}\left({\mu }_{ij}\right)=\mathrm{log}\left({S}_{ij}\right)+{\alpha }_{0}+{\beta }_{ep1}\mathrm{log}\left({y}_{i,j-1}\right)+{u}_{i}+{v}_{i}+{\beta }_{ep2}\cdot \mathrm{log}(\sum\limits_{{\delta }_{i}}{y}_{{\delta }_{i},j-1})\)  

\(\mathrm{log}\left({r}_{ij}\right)= {r}_{0}+{u}_{i}+{v}_{i}+{\beta }_{r}\cdot \mathrm{log}(\sum\limits_{{\delta }_{i}}{y}_{{\delta }_{i},j-1})\)