Robust Estimation of the VAR Model

Vector autoregressive model is a very popular tool in multiple time series analysis. Its parameters are usually estimated by the least squares procedure which is very sensitive to the presence of errors in data, e.g. outliers. If outliers were present, the estimation results would become unreliable. Therefore in the presented paper we will propose a new procedure for estimating multivariate regression model. This method is a multivariate generalization of the least weighted squares (LWS) of residuals and we will use it for estimating the coefficients of vector autoregressive model. Introduction In many situations one does not observe just a single time series, but several series, possibly interacting with each other. The aim of mutliple time series analysis is then statistically describe the data (including the relationship among variables), suggest a model best fitting our data and estimate a future development (forecasting). For these multiple time series the vector autoregressive model became very popular. The year 1980, when Christopher Sims in his article Sims, C. A. [1980] advocated vector autoregressive (VAR) model as an alternative to simultaneous equation model, used to be refered to as a milestone in their development. Because VAR models are linear models it is relatively easy to deal with them in the theoretical and practical way. The ease of computation and quality forecasts were the main reasons of their wide usage. Nowadays is VAR described in standard textbooks on time series (e.g. Lütkepohl [2005] and Hamilton, J. D. [1994]) and econometrics (Greene, W. H. [2002]). Let {Yt|t ∈ {−p + 1, . . . , T}} be a K−dimensional stationary time series. The object of interest is the vector autoregressive model of order p (VAR(p)), given by Yt = ν +B1Yt−1 +B2Yt−2 + . . .+BpYt−p + Ut, (1) where B1, . . . , Bp(Bp 6= 0) are (K × K) coefficient matrices, ν is (K × 1) vector of intercept term and the K−dimensional error terms {Ut, t ∈ Z} are supposed to be independently and identically distributed with a density of the form fUt(x) = g(x ′ Σ−1x) (det(Σ))1/2 , (2) where Σ and g are a positive definite matrix called the scatter matrix and a positive function. When the second moments of Ut exist, Σ will be proportional to the covariance matrix of the error term. Throughout this paper A ′ will stand for the transpose of a matrix A. This model is usually estimated by LS procedure (c.f. Lütkepohl [2005]) which is extremely sensitive to outliers, therefore it is important to investigate robust multivariate regression methods which could be used for estimating the VAR model. Suppose we have observations of time series Yt for t = −p+1, . . . , T . When we use following notation: for t = 1, . . . , T denote Xt = (1, Y ′ t−1, . . . , Y ′ t−p) ′ , B = (ν,B1, . . . , Bp) ′ , we can rewrite the equation (1) as the multivariate regression model Yt = B ′ Xt + Ut. (3) 143 WDS'09 Proceedings of Contributed Papers, Part I, 143–147, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS