Generalized AKM: Flexible Controls and Interactions in Wage Decompositions

How much wage dispersion is attributed to workers, firms, and their sorting depends on how wages are adjusted for observed characteristics. Standard AKM decompositions impose a known linear adjustment. We develop Generalized AKM, a framework that permits an unknown smooth covariate function and group-specific nonlinear interactions while preserving the original variance components. We prove consistency and asymptotic normality with heteroskedastic errors and many fixed effects, and characterize the stronger smoothness required for quadratic forms. In Portuguese employer-employee data, adding worker and firm-input controls lowers the bias-corrected worker-effect variance from $0.551$ to $0.474$ of total wage variance, firm-effect variance from $0.144$ to $0.121$, and sorting from $0.080$ to $0.047$. Across three group-specific nonlinear bases, firm-effect variance remains between $0.114$ and $0.117$ and sorting between $0.041$ and $0.042$, while worker-effect variance ranges from $0.474$to $0.491$. Which controls enter matters more for firm variance and sorting than how flexibly they enter; worker variance remains more sensitive to the basis.