4.2 (Optional) CEF Multivariate

Theorem 4.1 (Characterization of CEF) If  \({\mathbb{E}\left[ Y^2 \right]}<\infty\) and \({\boldsymbol{X}}\) is a random vector such that \(Y=m({\boldsymbol{X}})+e\), then the following statements are equivalent:
1. \(m({\boldsymbol{X}})={\mathbb{E}\left[ Y|{\boldsymbol{X}} \right]}\), the CEF of \(Y\) given \({\boldsymbol{X}}\)
2. \({\mathbb{E}\left[ e|{\boldsymbol{X}} \right]}=0\)