Economists use hidden‑variable models to study how individual earnings change over time. These models are powerful but hard to estimate because they require integrating over many unobserved histories. Variational inference (VI) is a machine‑learning tool that makes this task fast by turning integration into an optimization problem. But VI can give biased answers if the chosen approximation to the hidden parts is too simple.

The authors first test VI in a sequence of earnings models of increasing complexity. These models allow for nonlinear persistence (how past earnings affect future earnings in a nonlinear way), non‑Gaussian and serially correlated short‑run shocks, and differences across people. They try different variational posterior families, including a common mean‑field choice that forces independence over time and more flexible Gaussian families that allow richer correlations. They find that the choice of variational family matters a lot: mean‑field approximations often produce substantial bias, while more flexible families do much better.

To fix the remaining bias, the paper introduces indirect variational inference (IVI). IVI treats VI as an auxiliary estimator and then corrects its bias using an indirect‑inference step. The correction keeps the main practical advantage of VI: it does not require computing the likelihood, so it stays scalable and compatible with automatic differentiation. The authors propose two concrete implementations of IVI, based on gradient descent and on fixed‑point iteration. They also show that, under their setup, IVI is consistent and asymptotically normal for the true parameters, whereas plain VI targets a so‑called pseudo‑true value when the variational family is misspecified.

In simulations the methods perform as expected. When the true model is linear and Gaussian, VI works well because the true posterior is Gaussian and falls inside the variational family. In more realistic nonlinear or non‑Gaussian cases, flexible Gaussian variational families recover the main features of the persistent component of earnings, such as nonlinear conditional means and state‑dependent variances. But VI still gives biased estimates for other quantities, such as higher moments of transitory shocks, latent heterogeneity, or serial correlation in the transitory component. IVI substantially reduces these biases and produces estimates closer to the true values, especially in longer panels where scalability matters.