Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models

Linear bootstrap methods for vector autoregressive moving-average models

“…However, it is noteworthy that echelon form VARMA models should not be confounded with structural VARMA processes since the innovations u t are not, although 0 is a lower triangular matrix. Now, a brief description of the linear procedures developed in [29] for estimating echelon form VARMA models follows.…”

“…The next theorem provides the theoretical justification of the asymptotic validity of bootstrapping stationary invertible VARMA models using the multistep linear estimation procedures of Dufour and Jouini. [29] Theorem 3.16 Let {y t : t ∈ Z} satisfy the stationary invertible echelon form VARMA model given by Equations (1)- (4). Also, assume that n T is function of T such that n T = o(T 1/8 ) as n T , T → ∞.…”

“…The finite-sample performance of our bootstrap proposals for echelon form VARMA models compared with standard methods [29] is evaluated with MC simulations. Although the sample sizes 100, 200, 300, 400, 500 and 1000 are considered in the simulations, results associated with T = 200 only are reported to save space (more results are available by the author upon request).…”

“…Bootstrapping provides powerful tools as alternative to conventional asymptotics for building reliable small-sample inference and forecasting even in the absence of closed-form solutions of the variances of the estimators of possibly high dynamics such as impulse responses, error variance decompositions, predictability measures or long-term forecasts. Our results rely on the estimation procedures developed in [29] for echelon form VARMA models, but also apply to other model set-ups ensuring parameter identifiability. In particular, we establish the asymptotic validity of bootstrapping their two-and three-step linear estimators using residual-based methods under parametric and nonparametric assumptions.…”

“…Section 2 presents VARMA models under the parsimonious echelon form parametrization ensuring model parameter identifiability, then briefly describes the estimation procedure studied in [29]. Section 3 provides the resampling algorithms then establishes the asymptotic validity of bootstrapping (parametrically and nonparametrically) stationary invertible VARMA models.…”