I have a question regarding the model reparametrization of an ARDL model.
Consider the following ARDL$(p,q,q,\ldots,q)$ model:
\begin{equation} y_{it} = \alpha_i + \sum_{j=1}^{p} \lambda_{ij} y_{i,t-j} + \sum_{j=0}^{q} \delta'_{ij} x_{i,t-j} + \nu_{it} \end{equation}
where $\alpha_i,\lambda_{ij}$ , $y_{i,t-j}$, and $\nu_{it}$ are scalar. Futher, $\delta_{ij}$ and $x_{i,t-j}$ are column vectors of size $K$.
The book proposes the following ECM reparameterization:
\begin{equation} \Delta y_{it} = \alpha_i + \phi_i y_{i,t-1} + \beta'_i x_{it} + \sum_{j=1}^{p-1} \lambda^{\ast}_{ij} \Delta y_{i,t-j} + \sum_{j=1}^{q-1} \delta^{\ast '}_{ij} \Delta x_{i,t-j} + \nu_{it} \end{equation}
where,
$\phi_i = - (1- \sum_{j=1}^{p} \lambda_{ij})$, and $\beta_i = \sum_{j=0}^{q} \delta_{ij}$
$\lambda^{\ast}_{ij} = - \sum_{m=j+1}^{p} \lambda_{im}$, $j =1,2,\ldots,p-1$,
$\delta^{\ast}_{ij} = - \sum_{m=j+1}^{q} \delta_{im}$, $j =1,2,\ldots,q-1$
How can I derive the reparameterization given in equation 2?
I attempted the following approach:
First, $y_{it} = \Delta y_{it} + y_{i,t-1}$, with $\Delta y_{it} = y_{it} - y_{i,t-1}$. Similarly for $x_{it} = \Delta x_{it} + x_{i,t-1}$.
Second, $y_{i,t-j} = y_{i,t-1} - (\Delta y_{i,t-1} + \Delta y_{i,t-2}+ \ldots + \Delta y_{i,t-j+1})$,
and
$x_{i,t-j} = x_{i,t-1} - (\Delta x_{i,t-1} + \Delta x_{i,t-2}+ \ldots + \Delta x_{i,t-j+1})$.
Substituting these results in the first equation:
\begin{equation} \Delta y_{it} = \alpha_i - y_{i,t-1} + \sum_{j=1}^{p} \lambda_{ij} [y_{i,t-1} - (\Delta y_{i,t-1} + \Delta y_{i,t-2}+ \ldots + \Delta y_{i,t-j+1})] + \delta'_{i0} (\Delta x_{it} + x_{i,t-1}) +\sum_{j=0}^{q} \delta'_{ij} [x_{i,t-1} - (\Delta x_{i,t-1} + \Delta x_{i,t-2}+ \ldots + \Delta x_{i,t-j+1})] + \nu_{it} \end{equation} And this yields, \begin{equation} \Delta y_{it} = \alpha_i - \left(1 - \sum_{j=1}^{p} \lambda_{ij} \right)y_{i,t-1} - \sum_{j=1}^{p} \lambda_{ij} \left[ \sum_{m=1}^{j-1}\Delta y_{i,t-m} \right] + \delta'_{i0} (\Delta x_{it} + x_{i,t-1}) + \sum_{j=0}^{q} \delta'_{ij} [x_{i,t-1} - (\Delta x_{i,t-1} + \Delta x_{i,t-2}+ \ldots + \Delta x_{i,t-j+1})] + \nu_{it} \end{equation}
Which reads \begin{equation} \Delta y_{it} = \alpha_i - \left(1 - \sum_{j=1}^{p} \lambda_{ij} \right)y_{i,t-1} - \sum_{j=1}^{p} \lambda_{ij} \left[ \sum_{m=1}^{j-1}\Delta y_{i,t-m} \right] + \delta'_{i0} (\Delta x_{it} + x_{i,t-1}) + \sum_{j=0}^{q} \delta'_{ij} \{ x_{i,t-1} - \left[ \sum_{m=1}^{j-1}\Delta x_{i,t-m} \right] \} + \nu_{it} \end{equation}
which does not read as equation 2. Any suggestion about it?
