Suppose we start with:

begin{equation}

bar{w} Delta bar{x} = Cov(w_i, x_i)

end{equation}

We also have a regression formula for $w_i$:

$w_i = beta_0 + beta_{w_ix_i cdot x_j} x_i + beta_{w_ix_j cdot x_i} x_j + epsilon$

where $beta_0$ and $epsilon$ are constants, $beta_{w_ix_i cdot x_j}$ is the partial regression coefficient of $w_i$ on $x_i$, holding $x_j$ constant, and $beta_{w_ix_j cdot x_i}$ is the partial regression coefficient of $w_i$ on $x_j$, holding $x_i$ constant.

Subbing the second formula into the first formula, we (somehow) derive:

$bar{w} Delta bar{x} = Cov(alpha, x_i) + beta_{w_ix_i}V(x) + beta_{w_ix_j}beta_{x_jx_i}V(x) + Cov(epsilon, x_i) = beta_{w_ix_i}V(x) + beta_{w_ix_j}beta_{x_jx_i}V(x)$

Two parts of this derivation are surprising to me: **(1) where do these new coefficients come from? And (2) how did we transform the second covariance into a variance term?**

Here are some relevant formulas: for random variables $X,Y,Z$ and constant $a$…

$Cov(aX,Y) = Cov(X,aY) = aCov(X,Y)$

$Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z)$

$beta_{XY} = frac{Cov(X,Y)}{V(Y)}$

Even employing these formulas, I’m having trouble answering questions (1) and (2).