On Affine and Conjugate Nonparametric Regression

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.