# Transforming regression coefficients into variance terms

$$begin{equation} bar{w} Delta bar{x} = Cov(w_i, x_i) end{equation}$$

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).