I have been struggling a lot with implementing Arellano-Bover estimator for the following panel model: $Y_{it} = \rho Y_{it-1} + \lambda_i + \varepsilon_{it}$, with $Y$ observed data, $\rho$ autoregressive parameter, $\lambda_i$ the time-invariant heterogeneous intercept. I have been struggling a lot on estimating $\rho$ with Arellano-Bover. To put it more concretely, the optimization problem I am struggling with is like following:
First, define $Y^*_{it} = Y_{it} - \frac{1}{T-t}\sum_{s = t+1}^{T}Y_{is}$ for $t =1,..,T-1$. Furthermore, let $$m_{it}(\rho) = (Y^*_{it} - \rho Y^*_{it-1})Y^{0:t-1}_i\textrm{,}$$ where $Y_i^{0:t-1} = (Y_{i0},...,Y_{it-1})$. Finally, let $$m_i(\rho) = (m_{i1}(\rho),...,m_{iT-1}(\rho))$$ Then, the optimization problem is $$\hat{\rho}_{GMM} = argmin_{\rho}\left(\sum_{i=1}^{N}m_i(\rho)\right)'\left(\sum_{i=1}^{N}m_i(\rho)m_i(\rho)'\right)^{-1}\left(\sum_{i=1}^{N}m_i(\rho)\right)\textrm{.}$$
I have no idea how to solve this numerically, or analytically. I have tried to gradient descent, but I am not able to compute the gradient, mainly because of the inverse term in the objective function.
Thanks in advance.