# Dynamic panel data model with AR(2) process in the errors

I set up the following dynamic panel data model: $$y_{it}=\alpha y_{it-1}+x_{it}^T\beta+v_{it}$$ Additionally, I have the process in the errors: $$v_{it}=\rho_1u_{it-1}+\rho_2u_{it-2}+\epsilon_{it}$$ under the assumptions that $$\epsilon$$'s are i.i.d. with zero mean and constant variance. Also, $$u_i$$ is independent of $$x_{it}$$.

I've read the Arellano and Bond (1991), and I am intending to use systems GMM to estimate the model. I'm not entirely sure how I can retrieve the parameters $$\rho_1$$ and $$\rho_2$$. Any help would be appreciated!

• Welcome to CV, marco11. This looks like a straightforward mixed linear regression model. E.g., in Stata something like mixed y L1.y x, nocons || i: L1.u L2.u. You do not say how $p_{1}$ and $p_{2}$ are distributed: assuming zero covariance add , cov(independent) to the end of the previous command, or if allowed to covary add , cov(un) instead. Is something like this what you are after, or am I misunderstanding you? (Estimation details like reml or vce(robust) left to your discretion.) Dec 4, 2019 at 4:34
• Hi, that's for the input! $\rho$'s are just parameters, not random variables. I'm estimating robust standard errors to account for the autocorrelation in the errors, but how can I efficiently estimate the values of $\rho$'s? Dec 4, 2019 at 20:04
• I am not sure I follow: robust estimation does not "account" for autocorrelation (autocorrelation is a model feature). The fact that your data is hierarchical, with times of observation $t$ nested within $i$ implies you should be modeling random effects. Also, if $p_{1}$ and $p_{2}$ are simply fixed effects, then why isn't your model simply $y_{it} = \alpha y_{it-1} + \beta x^{T}_{it} + p_{1}u_{it-1} + p_{2}u_{it-2} + \varepsilon_{it}$? E.g., reg y L1.y x L1.u L2.u, nocons? Dec 4, 2019 at 20:27