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Mathematical Problems in Engineering
Volume 2013, Article ID 848120, 13 pages
Research Article

Maximum Likelihood Estimation of the VAR(1) Model Parameters with Missing Observations

Departamento de Estatística e Investigação Operacional, Faculdade de Ciências, Universidade de Lisboa, Edifício C6, Piso 4, Campo Grande, 1749-016 Lisboa, Portugal

Received 4 January 2013; Revised 29 March 2013; Accepted 8 April 2013

Academic Editor: Xuejun Xie

Copyright © 2013 Helena Mouriño and Maria Isabel Barão. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Missing-data problems are extremely common in practice. To achieve reliable inferential results, we need to take into account this feature of the data. Suppose that the univariate data set under analysis has missing observations. This paper examines the impact of selecting an auxiliary complete data set—whose underlying stochastic process is to some extent interdependent with the former—to improve the efficiency of the estimators for the relevant parameters of the model. The Vector AutoRegressive (VAR) Model has revealed to be an extremely useful tool in capturing the dynamics of bivariate time series. We propose maximum likelihood estimators for the parameters of the VAR(1) Model based on monotone missing data pattern. Estimators’ precision is also derived. Afterwards, we compare the bivariate modelling scheme with its univariate counterpart. More precisely, the univariate data set with missing observations will be modelled by an AutoRegressive Moving Average (ARMA(2,1)) Model. We will also analyse the behaviour of the AutoRegressive Model of order one, AR(1), due to its practical importance. We focus on the mean value of the main stochastic process. By simulation studies, we conclude that the estimator based on the VAR(1) Model is preferable to those derived from the univariate context.