Least-Squares Linear Regression

Bruce C. Dieffenbach ()
Additional contact information
Bruce C. Dieffenbach: Independent author

Chapter 53 in Conjugate Duality in Economic Analysis, 2026, pp 407-413 from Springer

Abstract: Abstract Let y denote the dependent random variable, and M is the subspace of independent variables. The least-squares linear regression of y on M is the x∈M such that the quadratic $$\left \langle \boldsymbol {y}-\boldsymbol {x},\boldsymbol {y}-\boldsymbol {x}\right \rangle$$ is minimized:$$\inf _{\boldsymbol {x}}\left [ \frac {1}{2}\left \langle \boldsymbol {y}-\boldsymbol {x},\boldsymbol {y}-\boldsymbol {x}\right \rangle + \delta _{M}\left ( \boldsymbol {x}\right ) \right ] \!.$$ The following three conditions are necessary and sufficient for solutions to the primal and the dual:$$\begin {array}{c} \boldsymbol {y}^{\ast }=\boldsymbol {y}-\boldsymbol {x}\\ \boldsymbol {x} \in M\\ \boldsymbol {y}^{\ast } \in M^{\bot }. \end {array}$$ These conditions are symmetric between the fitted valuex and the residualy*. Taking the dual of a transformation of the Fenchel dual obtains another fundamental result: the fitted values maximize the correlation with the dependent variable. We interpret least-squares linear regression as a fractional program.

Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_53

Ordering information: This item can be ordered from
http://www.springer.com/9783032213969

DOI: 10.1007/978-3-032-21396-9_53

Access Statistics for this chapter

More chapters in Contributions to Economics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().