ABSTRACT This paper proposes methods for comparing the accuracy of two competing sets of functional forecasts. This is increasingly relevant as many economic and financial forecasters are interested in predicting variables which are observed as functional data objects. However, to date there has been very little research on formal statistical tests to evaluate the relative accuracy of competing functional forecasts. We propose a suite of novel tests, building on the classic Diebold‐Mariano test, to provide formal statistical guidance on forecast accuracy in the case of functional data. We derive the asymptotic properties of the tests, including their self‐normalised versions, and demonstrate the validity of analytic or bootstrap‐based critical values. We investigate the finite sample performance of the tests using Monte Carlo simulations. We illustrate the use of our method by comparing forecasts of the U.S. yield curve based on forward rates with a random walk benchmark.
This article proposes a unified nonparametric framework for testing the stability of distributions and moments in time series models. We focus on the joint distribution of a response variable and its covariates and develop a generalized Cramér–von Mises test that compares a time‐invariant empirical joint distribution with a locally smoothed time‐varying counterpart. The test detects both smooth structural changes and abrupt breaks with unknown timing and is asymptotically pivotal under weak dependence. When the marginal distribution of the covariates is time‐invariant, stability of the joint distribution is equivalent to stability of the conditional distribution of the response given the covariates. Marginal stability itself corresponds to joint distributional stability in a lower‐dimensional setting and can therefore be examined within the same framework. We further develop moment‐specific tests based on moment‐weighted joint distributions. Under marginal stability of the covariates, these tests are equivalent to tests of stability of the corresponding conditional moments. They may be used either as targeted tests when a particular conditional moment is of primary interest or as diagnostic tools following rejection of the omnibus distributional test. Monte Carlo simulations show that the proposed procedures have accurate size and good power against both abrupt and smooth forms of instability. An application to economic policy uncertainty illustrates the empirical relevance of the framework.
ABSTRACT We establish the asymptotic normality, at the general order , of the quasi‐maximum likelihood estimator of random coefficient autoregressive (RCA) models evaluated by the Kalman filter. Because the lagged states are observed, the filter is exact, and the criterion is the conditional Gaussian quasi‐likelihood; this closes the gap between the non‐constructive general‐order maximum‐likelihood theory and the first‐order quasi‐likelihood results, with an effectively computable estimator. The score is a square‐integrable martingale difference, so the limit theorem is the martingale central limit theorem. Because the score is self‐normalised by the conditional variance, no moment of beyond the second—the second‐order stationarity already assumed—is required, and only finite fourth moments of the innovations enter; this is strictly weaker than the classical . The information matrices are in closed form and proved positive definite, and the covariance is of sandwich form; Gaussianity serves only to identify it with the inverse Fisher information, hence efficiency, and a distribution‐free version is our main robustness result. Each evaluation costs , and a Monte Carlo study confirms the rate, including a heavy‐tailed design with .
ABSTRACT Least squares (LS) estimation of canonical cointegrating regression (CCR) is explored allowing for potential misspecification from singularity in the system and multicointegration among the variables. The following properties are established: (i) superconsistent estimation with increased rates of convergence in degenerate directions; and (ii) limit theory that is nuisance parameter dependent in degenerate directions. Importantly for practitioners, inference for testing coefficient restrictions is shown to remain valid under certain conditions in CCR despite potential misspecification from singularity in the system.
ABSTRACT We study Bayesian inference for structural breaks in vector autoregressive (VAR) models, where both contemporaneous and dynamic interdependencies complicate regime identification. Within this framework, we extend the asymptotic analysis of Shimizu to multiple dependent series and establish conditions under which the posterior distribution of the change‐point location under conjugate priors asymptotically concentrates on the true break. Consistency is further demonstrated for non‐conjugate priors, and the results are extended to multiple change‐points under controlled overlaps between estimated and true regimes. A data‐driven hyperparameter specification is suggested to automate prior selection. Simulation studies confirm the accurate recovery of structural breaks, and empirical applications to U.S. macroeconomics data illustrate the method's advantages over the existing approaches.
