  Suppose the nonparametric regression function of a response variable $Y$ on
covariates $X$ and $Z$ is an affine function of $X$ such that the slope $\beta$
and the intercept $\alpha$ are real valued measurable functions on the range of
the completely arbitrary random element $Z$. Assume that $X$ has a finite
moment of order greater than or equal to $2$, $Y$ has a finite moment of
conjugate order, and $\alpha\left(Z\right)$ and $\alpha\left(Z\right)X$ have
finite first moments. Then, the nonparametric regression function equals the
least squares linear regression function of $Y$ on $X$ with all the moments
that appear in the expression of the linear regression function calculated
conditional on $Z$. Consequently, conditional mean independence implies zero
conditional covariance and a degenerate version of the aforesaid affine form
for the nonparametric regression function, whereas the aforesaid affine form
and zero conditional covariance imply conditional mean independence. Further,
it turns out that the nonparametric regression function has the aforesaid
affine form if $X$ is Bernoulli, and since $1$ is the conjugate exponent of
$\infty$, the least squares linear regression formula for the nonparametric
regression function holds when $Y$ has only a finite first moment and $Z$ is
completely arbitrary.
