Factor models aim to characterize variations in systematic risk by a set of factors.

$F$ captures the deviation of the common factor from its expected value. There are many ways to estimate different $F$

Single Factor Model

To simplify the exposition, we treat right now F has a single factor. A single-factor model would therefore postulate the following return dynamic for asset $i$: $r_i - E[r_i] = \beta_i * F + \epsilon_i$

where $\forall i ≠ j$

$E[\epsilon_i] = 0$

$E[\epsilon_i F] = 0$

$E[\epsilon_i\epsilon_j] = 0$

Two Factor Model

The next more general linear factor model is a two-factor model. As an example, consider we want to test whether business cycle risk and interest rate risk is part of the economy's systematic risk factor.

So, for example, we say that GDP and IR capture unexpected innovations to GDP growth and unexpected innovations to the one-month nominal interest rate. The respective 2-factor model for asset $i$ would read:

$r_i - E[r_i] = \beta_{i, GDP} \times GDP + \beta_{i, IR} \times IR + \epsilon_i$

Multi-Factor Models

$F$ is a $T \times K$ factor matrix, $\beta_i$ is a vector with factor c for $T$ time periods and $K$ factors coefficients

$r_{i,t}$ is a return of the asset $i$ for the time period $t$ $r_{i,t} = \beta_{0,i} + \beta_{1,i} * f_{1,t} + ... + \beta_{K,i} * f_{K,t} + \epsilon_{i,t}$