Guy Mélard

In the present contribution, we propose and exploit a link between vector threshold autoregressive moving average (TVARMA) and time-dependent VARMA (tdVARMA) models. We show that an adequate parametrization permits the TVARMA model to be included in the broad class of tdVARMA structures. It allows us to derive new results on the asymptotic properties of the estimators of TVARMA parameters obtained under weaker conditions than those given in the literature, at least when the threshold variable is exogenous. As a consequence, new tests are proposed. A simulation study gives evidence of the achieved results that are applied to study the behavior of the monthly CPI, US unemployment, and US federal funds effective rate.

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.

Several authors have attempted to compute the asymptotic Fisher information matrix for a univariate or multivariate time-series model to check for its identifiability. This has the form of a contour integral of a matrix of rational functions. A recent paper has proposed a short Wolfram Mathematica notebook for VARMAX models that makes use of symbolic integration. It cannot be used in open-source symbolic computation software like GNU Octave and GNU Maxima. It was based on symbolic integration but the integrand lacked symmetry characteristics in the appearance of polynomial roots smaller or greater than 1 in modulus. A more symmetric form of the integrand is proposed for VARMA models that first allows a simpler approach to symbolic integration. Second, the computation of the integral through Cauchy residues is also possible. Third, an old numerical algorithm by Söderström is used symbolically. These three approaches are investigated and compared on a pair of examples, not only for the Wolfram Language in Mathematica but also for GNU Octave and GNU Maxima. As a consequence, there are now sufficient conditions for exact model identifiability with fast procedures.

The article will focus on vector autoregressive‐moving average (VARMA) models with time‐dependent coefficients (td) to represent general nonstationary time series, not necessarily Gaussian. The coefficients depend on time, possibly on the length of the series , hence the name tdVARMA for the models, but not necessarily on the rescaled time . As a consequence of the dependency on of the model, we need to consider array processes instead of stochastic processes. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is consistent in probability and asymptotically normal. The theoretical results are illustrated using three examples of bivariate processes, the first two with marginal heteroscedasticity. The first example is a tdVAR(1) process while the second example is a tdVMA(1) process. In these two cases, the finite‐sample behavior is checked via a Monte Carlo simulation study. The results are compatible with the asymptotic properties even for small . A third example shows the application of the tdVARMA models for a real time series.

In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations.

About 25 years ago, effective methods for dealing with time series models that vary with time appeared in the statistical literature. Except in a few cases, they have never been used for economic statistics. In this chapter, we consider autoregressive integrated moving average (ARIMA) models with time-dependent coefficients (tdARIMA) applied to monthly industrial production series. We start with a small-size study with time-dependent integrated autoregressive (tdARI) models on Belgian series compared to standard ARI models with constant coefficients. Then, a second, bigger, illustration is given on 293 U.S. industrial production time series with tdARIMA models. We employ the software package Tramo to obtain linearized series and model specifications and build both the ARIMA models with constant coefficients (cARIMA) and the tdARIMA models, using specialized software. In these tdARIMA models, we use the simplest specification for each coefficient: a simple regression with respect to time. Surprisingly, for a large part of the series, there are statistically significant slopes, indicating that the tdARIMA models fit better the series than the cARIMA models.