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This paper expands the estimation theory for both quasi-maximum likelihood estimates (QMLEs) and Least Squares estimates (LSEs) for potentially misspecified constrained VAR(p) models. Our main result is a linear formula for the QMLE of a constrained VAR(p), which generalizes the Yule-Walker formula for the unconstrained case. We make connections with the known LSE formula and the determinant of the forecast mean square error matrix, showing that the QMLEs for a constrained VAR(p) minimize this determinant; however, the QMLEs need not minimize the component entries of the mean square forecast error matrix, in contrast to the unconstrained case. An application to computing mean square forecast errors from misspecified models is discussed, and numerical comparisons of the different methods are presented and explored.
Tucker McElroy and David Findley. (2013). Fitting Constrained Vector Autoregression Models. Center for Statistical Research & Methodology Research Report Series (Statistics #2013-06). U.S. Census Bureau. Available online at <http://www.census.gov/srd/papers/pdf/rrs2013-06.pdf>.
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