Conditional Moment Kernels: Joint Modeling of Expected Returns and Covariances in Unbalanced Panels

We propose the conditional moment kernel (CMK), a joint model of the conditional mean and covariance matrix of asset returns given their characteristics. A single kernel function of the characteristics gives the means and second moments of all returns. The implied covariance matrices are positive semidefinite for any number and composition of assets, so the model applies directly to unbalanced panels. Its estimation is a convex program whose size does not depend on the number of assets. A floor at the scale of the idiosyncratic noise keeps the estimated systematic covariance away from zero where the data cannot resolve it, and a trace penalty selects the number of factors. A closed form, the CMK estimator, approximates the solution. For fixed features and hyper-parameters, we prove identification, consistency and finite-sample bounds. In out-of-sample tests on US stocks from 1972 to 2021, the CMK estimator fits the conditional moments significantly better than instrumented principal component analysis (IPCA) with the same characteristics. Almost all of the gain is in the covariance matrix, and nonlinear functions of the characteristics add to it. The implied unconditionally mean--variance efficient portfolios earn annualized Sharpe ratios of 4.0 to 5.1 before costs, close to those of IPCA and declining over time.