  We develop a model-free theory of general types of parametric regression for
iid observations. The theory replaces the parameters of parametric models with
statistical functionals, to be called "regression functionals'', defined on
large non-parametric classes of joint $\xy$ distributions, without assuming a
correct model. Parametric models are reduced to heuristics to suggest plausible
objective functions. An example of a regression functional is the vector of
slopes of linear equations fitted by OLS to largely arbitrary $\xy$
distributions, without assuming a linear model (see Part~I). More generally,
regression functionals can be defined by minimizing objective functions or
solving estimating equations at joint $\xy$ distributions. In this framework it
is possible to achieve the following: (1)~define a notion of well-specification
for regression functionals that replaces the notion of correct specification of
models, (2)~propose a well-specification diagnostic for regression functionals
based on reweighting distributions and data, (3)~decompose sampling variability
of regression functionals into two sources, one due to the conditional response
distribution
  and another due to the regressor distribution interacting with
misspecification, both of order $N^{-1/2}$, (4)~exhibit plug-in/sandwich
estimators of standard error as limit cases of $\xy$ bootstrap estimators, and
(5)~provide theoretical heuristics to indicate that $\xy$ bootstrap standard
errors may generally be more stable than sandwich estimators.
