I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {1,2,\ldots,N }$), and $t$ represent time, denoted as available data for country $i$ ($t = {1,2,\ldots,T_i }$). Let's consider the data-generating process(DGP) for $\Delta y_{it}$, which follows an AR(1) process: \begin{equation} \Delta y_{it} = \alpha_i + \rho_i y_{i,t-1} + u_{it} \end{equation}
Then, consider the OLS estimate of country-specific persistent parameter $\hat{\rho}_i$ for country $i$ equal to: \begin{equation} \hat{\rho}_i = \frac{ \sum_{t \in T_i}(\Delta y_{it} - \overline{\Delta y}_i) (y_{i,t-1} -\bar{y}_{i,-})}{\sum_{t \in T_i} (y_{i,t-1} - \bar{y}_{i,-})^2} \end{equation} where $\overline{\Delta y}_i= \frac{\sum_{t \in T_i} \Delta y_{it}}{T_i}$ and $\bar{y}_{i,-} = \frac{\sum_{t \in T_i} y_{i,t-1}}{T_i}$
Since the true DGP is given by the first equation, $\hat{\rho}_i$ is a consistent estimator for $\rho_i$, i.e., $plim(\hat{\rho}_i)= \rho_i$.
Now, let's consider the OLS estimate of $\rho$ from the following equation:
\begin{equation} \Delta y_{it} = \rho y_{i,t-1} + \delta_t + u_{it} \end{equation} Thus, $\hat{\rho}$ reads as
\begin{equation} \hat{\rho} = \frac{\sum_{i \in N} \sum_{t \in T_i}(\Delta y_{it} - \overline{\Delta y}) (y_{i,t-1} -\bar{y}_{-})}{\sum_{i \in N} \sum_{t \in T_i} (y_{i,t-1} - \bar{y}_{-})^2} \end{equation}
where $\overline{\Delta y}= \frac{\sum_{i \in N} \sum_{t \in T_i} \Delta y_{it}}{\sum_{i \in N} T_i}$ and $\bar{y}_{-} = \frac{\sum_{i \in N} \sum_{t \in T_i} y_{i,t-1}}{\sum_{i \in N} T_i}$
Focusing on the numerator of $\hat{\rho}$: \begin{equation} \begin{aligned} \sum_{i \in N} \sum_{t \in T_i}(\Delta y_{it} - \overline{\Delta y}) (y_{i,t-1} -\bar{y}_{-}) =& \sum_{i \in N} \sum_{t \in T_i}\Delta y_{it} (y_{i,t-1} -\bar{y}_{-}) \\ =& \sum_{i \in N} \sum_{t \in T_i}\Delta y_{it} (y_{i,t-1} -\bar{y}_{i,-}) + \sum_{i \in N} \sum_{t \in T_i} \overline{\Delta y}_i (\bar{y}_{i,-} - \bar{y}_{-}) \end{aligned} \end{equation}
The paper says the first equality occurs because $\sum_{t \in T_i} (y_{i,t-1} - \bar{y}_{-})=0$. Could you show me how to exploit such basic statistical result? I mean, I know that the sum of the deviations from the average is zero, but how can exploit it?
What about the second equality?
