What is the quantile covariance? Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:
\begin{equation}
\text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])^2\}
\end{equation}
And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:
\begin{equation}
\text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\}
\end{equation}
where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).
I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:


*

*How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

*Where is this definition as an optimization problem for the covariance coming from?

*Any insights that help understanding the quantile covariance.
References:


*

*Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

*Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.
 A: Setup
We'll assume that the feature covariance $\mathrm{var}(X) \in \mathbb{R}^{p \times p}$ is full rank. As you have in the original post, define $$(\hat{\alpha}, \hat{\beta}) = \arg\min_{\alpha, \beta} \mathbb{E} \left[ \left(Y - \left\{ \alpha + \beta^T \Sigma_{XX}^{-1} (X - \mu_X) \right\} \right)^2 \right]$$ and $$(\tilde{\alpha}, \tilde{\beta}) = \arg\min_{\alpha, \beta} \mathbb{E} \left[ \rho_\tau \left(Y - \left\{ \alpha + \beta^T \Sigma_{XX}^{-1} (X - \mu_X) \right\} \right) \right],$$ where e.g. $\mu_Y = \mathbb{E}[Y]$, $\Sigma_{XX} = \mathrm{cov}(X, X) = \mathrm{var}(X)$, and $\Sigma_{YX} = \mathrm{cov}(Y,X)$. Recall that the covariance $\Sigma_{XY} = \left(\Sigma_{YX}\right)^T$ and that the covariance $\Sigma_{XY}$ is itself just a vector whose $j^{\mathrm{th}}$ entry is the (scalar) covariance between $X_j$ and $Y$.
Review of OLS
Write that $$(\hat{u}, \hat{v}) = \arg\min_{u, v} \mathbb{E} \left[ \left(Y - \left\{ u + v^T (X - \mu_X) \right\} \right)^2 \right],$$ where $\hat{u} \in \mathbb{R}$ is the population-level OLS intercept and $\hat{v} \in \mathbb{R}^p$ is the population-level OLS slope, and so $\hat{u} + \hat{v}^T (X - \mu_X)$ is the population-level OLS line-of-best-fit. 
Connection between OLS and covariance
Using matrix algebra, we could derive that $\hat{u} = \mu_Y$ and $$\hat{v}^T = \Sigma_{YX}\Sigma_{XX}^{-1}. \tag{*}$$ In terms of your first displayed equation, we see that $\hat{u} = \hat{\alpha}$ and $\hat{v} = \Sigma_{XX}^{-1}\,\hat{\beta}$; that is, $$\hat{\beta} = \Sigma_{XX} \, \hat{v} = \Sigma_{XX} \left( \Sigma_{XX}^{-1}\,\Sigma_{XY} \right) = \Sigma_{XY},$$ the covariance.
Quantile covariance
In order to define the "quantile covariance" $\Sigma_{YX}^{\mathrm{QUANTILE}}$, the authors continue to treat $(*)$ as true while switching away from a squared error loss function. Specifically, they could consider $$(\tilde{u}, \tilde{v}) = \arg\min_{u, v} \mathbb{E} \left[ \rho_\tau \left(Y - \left\{ u + v^T (X - \mathbb{E}[X]) \right\} \right) \right],$$ which is analogous to the population-level OLS fit earlier except now with quantile loss instead of squared loss. At this point, the goal is to connect the population-level quantile slope $\tilde{v}$ to a "quantile covariance". To do this, we could define that $$\tilde{v}^T =: \Sigma_{YX}^{\mathrm{QUANTILE}}\,\Sigma_{XX}^{-1}, \tag{**}$$ so that $\Sigma_{YX}^{\mathrm{QUANTILE}}$ is recovered through $\beta$ in your second displayed equation, i.e. $\tilde{\beta} = \Sigma_{XY}^{\mathrm{QUANTILE}}$.
