$z$ has mean $\mu \in {{\mathbb R}}^d$ and positive definite variance/covariance matrix $\Sigma$. Our model assumptions are further discussed in @Ste18a [Remark 7.1]. No additional restrictions will be placed on the regression coefficients $\vartheta$ and $\theta$, on the moments $\mu$ and $\Sigma$, or on the error distribution $\mathcal L(\epsilon)$.

We do place some assumptions on the distribution of the explanatory variables. First, we assume that $z$ can be written as an affine transformation of independent random variables. With this, we can represent the $d$-vector $z$ as $$\label{z}
z \quad=\quad \mu + \Sigma^{1/2} R \tilde{z}$$ for a $d$-vector $\tilde{z}$ with independent (but not necessarily identically distributed) components so that ${{\mathbb E}}[\tilde{z}]=0$ and ${{\mathbb E}}[\tilde{z}\tilde{z}'] = I_d$, where $\Sigma^{1/2}$ is the positive definite and symmetric square root of $\Sigma$, and where $R$ is an orthogonal (non-random) matrix. Second, we assume that $\tilde{z}$ has a Lebesgue density, which we denote by $f_{\tilde{z}}$, with bounded marginal densities and finite marginal moments of sufficiently high order. In particular, we will assume that $f_{\tilde{z}}$ belongs to one of the classes ${\mathcal F}_{d,k}(D,E)$ that are defined in the next paragraph, for appropriate constants $k$, $D$ and $E$. Our assumptions on $z$ are similar to those maintained by 