Functional forms

Model_1

$\begin{array}{c}{w}_{it}={\alpha }_i+{\displaystyle \sum _{j=1}^n{\gamma }_{ij}\ln P_j}+{\beta }_i\ln \left(X_t/P_t\right)+{\lambda }_i{\left(\ln \left(X_t/P_t\right)\right)}^2/Q\\ +{\alpha }_i^c\cos \frac{2\pi t}{4}+{\alpha }_i^s\sin \frac{2\pi t}{4}+{\alpha }_i^t\cdot t\end{array}$

Model_2

$w_{it}={\alpha }_i+{\displaystyle \sum _{j=1}^n{\gamma }_{ij}\ln P_j}+{\beta }_i\ln \left(X_t/P_t\right)+{\alpha }_i^c\cos \frac{2\pi t}{4}+{\alpha }_i^s\sin \frac{2\pi t}{4}+{\alpha }_i^t\cdot t$

Model_3

$w_{it}={\alpha }_i+{\displaystyle \sum _{j=1}^n{\gamma }_{ij}\ln P_j}+{\beta }_i\ln \left(X_t/P_t\right)+\log \left(X_t\right)$

Model_4

$w_{it}={\alpha }_{it}+{\displaystyle \sum _{j=1}^n{\gamma }_{ij}\ln p_{jt}}+{\beta }_i\ln \left(X_t/P_t\right)$