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Multivariate Least-Squares Linear Regression

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  • Bruce C. Dieffenbach

    (Independent author)

Abstract

We analyze multivariate least-squares linear regression in moment space. For the multivariate dependent variable Y, choose fitted values X to minimize $$\left \langle {\textit {{\sf { {Y}}}}}-{\textit {{\sf { {X}}}}}, {\textit {{\sf { {Y}}}}}-{\textit {{\sf { {X}}}}}\right \rangle$$ . One expresses the minimizing choice via the cross-moment transformation between the subspaces of the dependent variables and the independent variables. The multivariate least-squares linear regression is the best linear regression, in that any other choice X makes larger the matrix $$\left ( {\textit {{\sf { {Y}}}}} -{\textit {{\sf { {X}}}}}\right ) ^{\top }\left ( {\textit {{\sf { {Y}}}}}-{\textit {{\sf { {X}}}}}\right )$$ of the second moments of the residual Y−X. We extend the analysis to reduced-rank least-squares linear regression, in which the fitted values X are constrained to having reduced rank l. The primal embodies the reduced-rank requirement. We present primal and dual solutions such that the primal and dual values are the same. A solution X is the truncation of the singular value decomposition of A⊤Y at l terms.

Suggested Citation

  • Bruce C. Dieffenbach, 2026. "Multivariate Least-Squares Linear Regression," Contributions to Economics,, Springer.
    • Handle: RePEc:spr:conchp:978-3-032-21396-9_54
    DOI: 10.1007/978-3-032-21396-9_54
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