Data Cloning in Latent-Variable Time-Series Models: Likelihood Theory and Estimability Diagnostics

This paper characterizes what data cloning estimates and how it behaves in large samples, in nonlinear latent-variable models. Our main result is a Bernstein--von Mises theorem for the cloned posterior at a fixed sample size: as the number of clones K grows, the posterior concentrates at the maximum likelihood estimator of the observed sample, and its covariance contracts at the canonical 1/K rate. Inference therefore becomes invariant to the prior, and the cloned posterior provides a computational route to observed-information standard errors. We also show that the data-cloning estimator inherits standard likelihood properties under joint (T,K) asymptotics, including consistency and asymptotic normality for the true parameter. The theory yields a simple estimability diagnostic: in estimable directions, the standardized eigenvalues of the cloned posterior covariance contract linearly in 1/K, while plateaus signal weak identification or nonidentification. We illustrate the results in stochastic volatility models with symmetric, linear-asymmetric, threshold, and time-varying downside-asymmetric features. In an application to U.S. broad-market and financial-sector equity returns, the diagnostic certifies full identification of the benchmark specifications and reveals that the leverage-state dynamics of the time-varying model are not identified, correctly attributing the failure to a likelihood-flat direction.