ir paper:

> [*When fitted models are approximations, conditioning on the regressor is no longer permitted ... Two effects occur: (1) parameters become dependent on the regressor distribution; (2) the sampling variability of the parameter estimates no longer derives from the conditional distribution of the response alone. Additional sampling variability arises when the nonlinearity conspires with the randomness of the regressors to generate a $1/\sqrt{n}$ contribution to the standard errors.*]{}

Moreover, it is not possible to estimate the distribution of the conditional projection parameter estimate in the distribution free framework. To see that, note that the least squares estimator can be written as $\hat\beta(j) = \sum_{i=1}^n w_i Y_i$ for weights $w_i$ that depend on the design matrix. Then $\sqrt{n}(\hat\beta(j) - \beta(j)) = \sum_{i=1}^n w_i \epsilon_i$ where $\epsilon_i = Y_i - \mu_n(i)$. Thus, for each $j \in \{1,\ldots,d\}$ we have that $\sqrt{n}(\hat\beta(j) - \beta(j))$ is approximately $\approx N(0,\tau^2)$, where $\tau^2 = \sum_i w_i^2 \sigma_i^2$, with $\sigma_i^2 = {\rm Var}(\epsilon_i | X_1,\ldots, X_n)$. The problem is that there is no consistent estimator of $\tau^2$ under the nonparametric models we are considering. Even if we assume that $\sigma_i^2$ is constant (an assumption we avoid in this paper), we still have that $\tau^2 =\sigma^2 \sum_i w_i^2$ which cannot be consistently estimated without assuming that the linear model is correct. Again, we refer the reader to [@buja2015models] for more discussion. In contrast, the projection parameter $\beta = \Sigma^{-1}\alpha$ is a fixed functional of the data generating distribution $P$ and is estimable. For these reasons, we focus in this paper on the projection parameter rather than the conditional projection parameter.

Goals and Assumptions {#goals-and-assumptions .unnumbered}
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Our main goal is to provide statistical guarantees for each of the four random parameters of variable significance introduced above, under our 