Estimator’s Properties of Specific Time-Dependent Multivariate Time Series

There is now a vast body of literature on ARMA and VARMA models with time-dependent or time-varying coefficients. A large part of it is based on local stationary processes using time rescaling and assumptions of regularity with respect to time. A recent paper has presented an alternative asymptotic theory for the parameter estimators based on several distinct assumptions that seem difficult to verify at first look, especially for time-dependent VARMA or tdVARMA models. The purpose of the present paper is to detail several examples that illustrate the verification of the assumptions in that theory. These assumptions bear on the moments of the errors, the existence of the information matrix, but also how the coefficients of the pure moving average representation of the derivatives of the residuals (with respect to the parameters and evaluated at their true value) behave. We will do that analytically for two bivariate first-order models, an autoregressive model, and a moving average model, before sketching a generalization to higher-order models. We also show simulation results for these two models illustrating the analytical results. As a consequence, not only the assumptions can be checked but the simulations show how well the small sample behavior of the estimators agrees with the theory.