Linear regression

Equation (37) follows also from predicting $m_i$ from $m_p$ by a linear regression model. Specifically, assume $n$ data pairs $(y_j,x_j)$. In the case of the CAPM these data will be pairs $(m_i(t_j),m_p(t_j))$, sampled for example at times $t_j$. Out aim is to find an optimal prediction of the $y_i$ based on the $x_i$ (i.e.,, for CAPM, prediction of the $m_i(t_j)$ based on the $m_p(t_j)$). To determine an optimal predicting function $f(x)$ we define a quadratic error measure

$$E=\sum_i \Big(f(x_i)-y_i\Big)^2 .$$ (40)

In the particular situation of linear regression the function $f$ is linear, $f(x)$ = $\alpha+\beta x$. (A more flexible $f$ would for example be a neural network.) The optimal parameters $\alpha$, $\beta$ are now found as usual by setting $\partial E/\partial \alpha$ = 0 and $\partial E/\partial \beta$ = 0. Thus,

$$0 = 2 \sum_i \Big(\alpha+\beta x_i-y_i\Big)$$ (41)

$$0 = 2 \sum_i x_i \Big(\alpha+\beta x_i-y_i\Big)$$ (42)

or

$$\alpha = <y> -\beta <x>,$$ (43)

$$0 = \alpha <x> + \beta <x^2>-<xy> ,$$ (44)

from which follows

$$\beta = \frac{<xy>-<x><y>}{<x^2> - <x>^2} = \frac{\sigma_{xy}}{\sigma_x^2} .$$ (45)

The definition of $\beta$ being thus the same, Eq. (43) can now be identified with Eq. (37) with $\alpha$ corresponding to $m_0$, $\sigma_{xy}$ to $C_{ip}$, and $\sigma_{x}^2$ to $D_{p}$.