Resumen de Goodness-of-fit in multivariate time series
Huong Nguyen Thu
Goodness-of-fit is an important task in time series analysis. In this thesis, we propose a new family of statistics and a new goodness-of-fit process for the wellknown multivariate autoregressive moving average VARMA(p,q) model. Some preliminary results are studied first for an initial goodness-of-fit method. Since the residuals of the fit play an important role in identification and diagnostic checking, relations between least squares residuals and true errors are studied. An explicit representation of the information matrix as a limit is also obtained. Second, we generalize a univariate goodness-of-fit process studied in Ubierna and Velilla (2007). An explicit form of the limit covariance function is presented, as well as a characterization of its limit properties in terms of a parametric Gaussian process. This motivates the introduction of a new goodness-of-fit process based on a transformed correlation matrix sequence. The construction and properties of the associated transformation matrices are investigated. We also prove the convergence of this new process to the Brownian bridge. Thus, statistics defined as functionals of our process use a null distribution that is free of unknown parameters. Finally, simulations, comparisons, and examples of application are presented to illustrate our theoretical findings and contributions. Our proposed goodness-of-fit statistics are shown to be quite sensitive for detecting lack of fit. They also seem to be relatively independent of the choice of a particular lag.
