Abstract
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 but not the component entries of the mean square forecast error matrix, as opposed 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.
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Notes
- 1.
For any vector a, we have \(a^{{\prime}}\hat{\varOmega }(\xi )a = (2\pi n)^{-1}\int _{-\pi }^{\pi }\vert a^{{\prime}}\,\varPsi ^{-1}(e^{-i\lambda })\,\sum _{t=1}^{n}\mathbf{W}_{t}e^{-i\lambda t}\vert ^{2}\,d\lambda\), so that the expression equals zero iff \(a^{{\prime}}\,\varPsi ^{-1}(e^{-i\lambda }) \cdot \sum _{t=1}^{n}\mathbf{W}_{t}e^{-i\lambda t} = 0\) almost everywhere with respect to λ; because both terms in this product are polynomials in e −i λ, the condition is equivalent to one or the other of them being zero. In the one case that \(a^{{\prime}}\,\varPsi ^{-1}(e^{-i\lambda }) = 0\), we at once deduce that a is the zero vector; in the other case, we have that the discrete Fourier Transform \(\sum _{t=1}^{n}\mathbf{W}_{t}e^{-i\lambda t} = 0\) for almost every λ, which can only be true if the data is zero-valued.
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McElroy, T., Findley, D. (2015). Fitting Constrained Vector Autoregression Models. In: Beran, J., Feng, Y., Hebbel, H. (eds) Empirical Economic and Financial Research. Advanced Studies in Theoretical and Applied Econometrics, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-319-03122-4_28
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DOI: https://doi.org/10.1007/978-3-319-03122-4_28
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