Model Specification Tests of Heterogenous Agent Models with Aggregate Shocks under Partial Information

For a heterogeneous agent model with aggregate shocks, the seminal paper by Krusell and Smith (1998) provides an equilibrium framework depending only on the (conditional) mean wealth rather than the wealth distribution of all agents, which is referred to as approximate aggregation for their prototype model. Their result can be obtained through the analysis of a forward-backward system consisting of the Hamilton-Jacobi-Bellman equation, the Fokker-Planck equation, and some constraint. Different from the existing literature, this paper proposes a statistical method to verify whether a heterogeneous agent model features approximate aggregation in the scenario that only one agent's wealth together with the aggregate shocks is observable over time. Our main approach lies in studying a model specification testing problem for the evolution of the wealth (i.e. the Fokker-Planck equation) in some appropriate parametric family featuring approximate aggregation. The key challenge stems from the partially observed information where the wealth distribution of all agents is infeasible. To overcome this difficulty, first, a novel two-step estimate is proposed for estimating the parameter in the parametric family. Then, several testing statistics are constructed, and their asymptotic properties are established, which in turn provides several testing rules. Finally, some Monte Carlo simulations are conducted to illustrate the finite sample performance of the proposed tests.