Goodness-of-fit in Multivariate Time Series
Abstract: Goodness-of-fit is an important task in time series analysis. In this thesis, wepropose a new family of statistics and a new goodness-of-fit process for the wellknownmultivariate 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 diagnosticchecking, relations between least squares residuals and true errors are studied. Anexplicit representation of the information matrix as a limit is also obtained.Second, we generalize a univariate goodness-of-fit process studied in Ubierna andVelilla (2007). An explicit form of the limit covariance function is presented, aswell as a characterization of its limit properties in terms of a parametric Gaussianprocess. This motivates the introduction of a new goodness-of-fit process based ona transformed correlation matrix sequence. The construction and properties of theassociated transformation matrices are investigated. We also prove the convergenceof 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 toillustrate our theoretical findings and contributions. Our proposed goodness-of-fitstatistics are shown to be quite sensitive for detecting lack of fit. They also seem tobe relatively independent of the choice of a particular lag.
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