observed factors. Such factors appear as time series whose values are common to all stocks at a particular point in time. In this section we consider models where the values of factors are unobserved. Two examples of such factors are fundamental and industry factors. Fundamental, Technical, and Industry (Sector) Factors When factor returns are unobserved, we need to estimate their values using information on their exposures and stock returns. This estimation is done using either a cross section of returns or their time series. In the case where factors are unobserved and they are defined in terms of fundamental, technical, or industry designations, a popular factor model is a linear cross-sectional model. R(t) = Bit - l)F(t) + u{t) (20.12) where R{t) = N-vector of one-period asset (stock) returns B{t -1) =Nx K matrix of asset exposures to factors as of time t-1 F(t) = K-vector of one-period factor returns u(t) = N-vector of one-period specific returns The columns of B{t - 1) represent exposures to a particular factor. The values of F(t) are estimated, typically, by a cross-sectional regression of time t returns on time t-1 exposures. We explain the linear cross-sectional factor model in more detail later in this chapter. Principal Components Principal component analysis (PCA) is often used to extract a number of unobserved factors from a set of returns. It is important to review principal component methods for two reasons. First, some commercially available risk systems use principal component analysis as part of their risk models. Second, for many practitioners principal components are what often come to mind when thinking about factors and factor models. We begin by reviewing the standard principal component method to estimate factors, and then discuss an alternative method to estimate principal components. This alternative method is known as the asymptotic principal component (APC) method. A typical application of PCA to factor models11 begins with the factor model (20.12) for t = 1, . . . , T. We assume that the factor returns are orthogonal and specific returns are uncorrelated so that the variance of R(t) is V{t)=B(t-l)B(t-l)T+A(t) (20.13) where A(t) is diagonal. We can relax the assumption that security-specific returns are uncorrelated and allow for nonzero off-diagonal elements of A(t), in which case nSee, for example, Johnson & Wichern (1982). In this section we explain a very simple method to extract factors. There are other approaches that involve, for example, maximum likelihood estimation. For an application of maximum likelihood to estimate factors see Lit-terman, Knez, and Scheinkman (1994).