I employ a panel dataset (4,191 participants; 20,618 person-year observations) to model income dynamics. My model is as follows,
log(y_{it}) =X\beta + \lambda_t*\alpha_i+\delta_t (u_{it} +v_{it}) \\ u_{it} = \rho * u_{it-1} +\epsilon_{it}, \\ \alpha_i \sim N(0, \sigma^2_{\alpha}), \epsilon_{it} \sim N(0,\sigma^2_{\epsilon}), v_{it} \sim N(0,\sigma^2_{v})\\
y_{it} is annual earnings. X include age and its square. Both \lambda_t and \delta_t are loading factors. \alpha_i is permanent component, and u_{it} +v_{it} is transitory component.
How should I perform sensitivity analysis? Can I follow the methodology on pages 159-161 of Bayesian Data Analysis and compare the posterior predictive distributions of four test statistics—max(y), min(y), mean(y), and sd(y)—with the observed values of the test statistics?
You can, but @andrewgelman advises against posterior predictive checks that directly represent parameters of the model as they almost pass. So if there’s an intercept in your data matrix X, maybe don’t check means. Or you can check them, but don’t be surprised if they’re always OK.
I’m not sure what to do with the multiple “error” terms in your expression for y (\epsilon_{i, t}, v_{i, t}), so I’d just go ahead and test the standard deviation, too, but that may be unnecessary.