(>. Mitttipiitoi Pricing Models

where Var)}> is Ir..... (6.2.27). The Ian (hat {iK and y are independent

has been ulili/ed to set the covariance term in (6.3.4) to zero.

In the case where the (actors ate not traded portfolios, an estimator of the vet lor ol fat lor risk premia Ад is the sum of (he estimator of ihe mean ol the factor realizations and die estimator of y\,

V = /V+TV (6.3.Г>)

An estimator of the variance of Ад is

\тАд] = у, Пд-+Vai l7,], (6.3.6)

where \ai [->, is from (6.2.-15). Because fiJK and 7, are independent the covariance term in (6.3.6) is zero.

fhe loin th case, where the factor portfolios span the mean-variance frontier, is the same as the fust case except that real returns are substituted lor excess returns. I lerc Ад is the vector of factor portfolio sample means and An is zero.

Tor any asset the expected return can be estimated by substituting the estimates of B, A , and Ад into (6.1.H). Since (6.1.8) is nonlinear in the parameters, calculating a standard ei i or requires using a linear approximation and estimates of the covariances of the parameter estimates.

It is also of interest lo ask if the factors are jointly priced, (liven the vector ol risk premia estimates and its covariance matrix, tests of the null hypothesis that the factors are jointly not priced can be conducted using the following test statistic:

./ = АдАдГАд-. (b.3.7)

Asymptotically, under the null hypothesis that Ад- = 0, /, has an / distribution with К and Г-К degrees of freedom. This distributional result is an application of the llotelling /- statistic and will be exact in finite samples for the cases where die estimator of Ад is based only on the sample means of the hit tors. Wc tan also test the significance of any individual factor using

/, = -~ ~ Л/ >. П. (6.3.8)

where А,д is the /lb element of Ад- and ult is the (/, j)ih element о\агАд . Testing if individual factors are priced is sensible for cases where the factors have been theoretically specified. With empirically derived factors, such tests are not uselul because, as wc explain in Section 6/1.1, factors are identified only up to an orthogonal ir.uisloi iiialiou; hence individual fat tors do not have clear-cut economic interpretations.

6.4 Selection of Factors

The estimation and testing results in Section 6.2 assume that the identity of the factors is. known. In this section wc address the issue of specifying the factors. The approaches fall into two basic categories, statistical and theoretical. The statistical approaches, largely motivated by the APT, involve building factors from a comprehensive set of asset returns (usually much larger than the set of returns used to estimate and test the model). Sample data on these returns are used to construct portfolios that represent factors. The theoretical approaches involve specifying factors based on arguments that the factors capture economy-wide systematic risks.

6.4.1 Statistical Approaches

Our starting point for the statistical construction of factors is the lineajr factor model. We present the analysis in terms of real returns. The same analysis will apply to excess returns in cases with a riskfree asset. Recall that

6.4. Selection of Factors

Shanken (1992b) shows that factor risk premia can also be estimated using a two-pass cross-sectional regression approach. In the first passjhe factor sensitivities are estimated asset-by-asset using OLS. These estimators represent a measure of the factor loading matrix В which we denote B. This estimator of В will be identical to the unconstrained maximum likelihood estimators previously presented for jointly normal and IID residuals.

Using this estimator of В and the (N x 1) vector of asset returns for each time period, the ex post factor risk premia can be estimated time-period-by-time-period in the second pass. The second-pass regression is

Z, = 1Х0, + ЪХК1 + г),. (6.3.9)

The regression can be consistently estimated using OLS; however, GLS can also be used. The output of the regression is a time series of ex post risk premia, Ад- / = 1,..., T, and an ex post measure of the zero-beta portfolio return, А J = 1,.... 7.

Common practice is then to conduct inferences about the risk premia using the means and standard deviations of these ex post series. While this approach is a reasonable approximation, Shanken (1992b) shows that the calculated standard errors of the means will understate the true standard errors because they do not account for the estimation error in B. Shanken tierives an adjustment which gives consistent standard errors. No adjustment is needed when a maximum likelihood approach is used, because the maximum likelihood estimators already incorporate the adjustment.

for the linear mode! we have

R, = a + Bf, + €, (G.4.1)

Ele,e; I Г,] = E. (0.4.2)

where R, is the (Л/xl) vector of asset returns for time period (, f, is die (Axl) vector of factor realizations for time period /, and e, is the (Лх 1) vector of model disturbances for lime period (. The number of assets, N, is now very large and usually much larger than the number of time periods, /. There are two primary statistical approaches, factor analysis and principal components.

/Factor Analysis

(Estimation using factor analysis involves a two-step procedure. First the factor sensitivity matrix В and the disturbance covariance matrix £ are estimated ami then these estimates are used to construct measures of the factor realizations. For standard factor analysis it is assumed that there is a slrirl factor structure. With this structure К factors account for all the cross covari-nnce of asset returns and hence E is diagonal. (Ross imposes this structure n his original development of the APT.)

Given a strict factor structure and К factors, we can express the (NxN) covariance matrix of asset returns as the sum of two components, the variation from the factors plus the residual variation,

П = Bflk-B + D, (0.4.3)

where /iff, f,] = Пк a d E = D to indicate it is diagonal. With the factors unknown, a rotational indeterminacy exists and В is identified only up to <; nonsingular transformation. This rotational indeterminacy can be eliminated by restricting the factors to be orthogonal to each other and to have unit variance. In this case we have fiA = I and В is unique up to an orthogonal transformation. All transforms BG are equivalent for any (К x K) orthogonal transformation matrix G, i.e., GG = I. With these restrictions in place we can express the return covariance matrix as

П = BB + D. Mi. I. I)

With the structure in (0.4.4) and the assumption that asset returns are jointly normal and temporally III), estimators of В and D can be formulated using maximum likelihood factor analysis. Because ihe (irsl-ordcr conditions for maximum likelihood are highly nonlinear in the parameters, solving for the estimators with the usual iterative procedure can be slow and convergence difficult. Alternative algorithms have been developed by Joreskog (H)b7) and Rubin and Thayer (1982) which facilitate quick convergence lo the maximum likelihood estimators.

One interpretation of the maximum likelihood estimator of В given the maximum likelihood estimator of D is that D~ limes the estimator of В has the eigenvectors of D f2 associated with Ihe К largest eigenvalues as ils columns. For details of the estimation the interested reader can see these papers, or Morrison (1990, chapter 9) and references therein.

The second step in the estimation procedure is to estimate the factors given В and E. Since the factors are derived from the covariance structure, the means are not specified in (0.4.1). Without loss of generality, we can restrict the factors to have zero means and express die factor model in terms of deviations about the means,

(R, -/t) = Bf, + (.,. (0.4.5)

Given (0.4.5), a candidate to proxy for the factor realizations for time period / is the cross-sectional generalized least squares (GI.S) regression estimator. Using the maximum likelihood estimators of В and 1) we have lor each (

f, = (BD-BrBD-lR, - (i). (0.4.0)

Here we arc estimating f, by regressing (R, - ft) onto B. The factor realization series, f / = 1.....7, can be employed lo lest the model using the

approach in Section 0.2.3.

Since the factors are linear combinations of returns we can construct portfolios which are perfectly correlated with the factors. Denoting RA, as the {Kx 1) vector of factor portfolio returns for time period /, we have

kK, = AWR,. (0.4.7)

where

W = (BDB) BD1,

and A is defined as a diagonal matrix with 1 / IV) as the ;tli diagonal element, where is the /th element of Wi.

The lactor portfolio weights obtained for the /lb factor from this procedure are equivalent to the weights that would result fioiii solving the following optimization problem and then normalizing the weights to sum to one:

Miiiu Dw, (0.4.H)

subject to

urb* = 0 V/f ф j (0.4.9)

w(b* = I V/, = /. (0.4.10)

0. Multifactor Pricing Models

That is, the lactor portfolio weights minimize the residual variance subject to the constraints that each factor portfolio has a unit loading on its own factor and zero loadings on other factors. The resulting factor portfolio returns can be used in all the approaches discussed in Section 6.2.

If В and D are known, then the factor estimators based on GI.S with the population values of В and D will have the maximum correlation with the population factors. This follows from the minimum-variance unbiased estimator property of generalized least squares given the assumed normality of the disturbance vector. But in practice the factors in (6.4.6) and (6.4.7) need not have the maximum correlation with the population common factors since they arc based on estimates of В and D. I.ehmann and Modest (1988) present an alternative to GI.S. In the presence of measurement error, they find this alternative can produce factor portfolios with a higher population correlation with the common factors. They suggest for the jth factor to use u/R, where the (Nx 1) vector u>; is the solution to the following problem:

MjnojjDojy (6.4.11)

subject lo

cb, = 0 V/t ф j (6.4.12)

uy =1. (6.4.13)

This approach finds the portfolio which has the minimum residual variance of all portfolios orthogonal lo the other {K-\) factors. Unlike the G1.S procedure, this procedure ignores the information in the factor loadings of the )th factor. It is possible that this is beneficial because of the measurement error in the loadings. Indeed, I.ehmann and Modest find that this method of forming factor portfolios results in factors with less extreme weightings on the assets and a resulting higher correlation with the underlying common factors.

Principal Components

factor analysis represents only one statistical method of forming factor portfolios. An alternative approach is principal components analysis. Principal components is a technique lo reduce ihe number of variables being studied without losing too much information in the covariance matrix. In the present application, ihe objective is lo reduce the dimension from Л/ asset returns lo К fat tors. The principal components serve as the factors. The first principal cuinpoiicui is the (normalized) linear combination of asset returns with maximum variance. Ihe second principal component is the (normalized) linear combination of asset returns with maximum variance of all coin bin,11 ions orthogonal lo the first principal component. And so on.

6. /. Selection of Factors 237

The first sample principal component is x R, where the (N x 1) ve :tor xj is the solution to the following problem:

Maxx.ftxi (6.4.14)

subject lo !

x,x, = 1. (6.41.15)

Cl is the sample covariance matrix of returns. The solution x* is the eigenvector associated with the largest eigenvalue of fi. To facilitate the portfjolio interpretation of the factors we can define the first factor as w,R where Ы1 is x* scaled by the reciprocal of txj so that its elements sum to 4ne. The second sample principal component solves the above problem for X2 in the place of xj with the additional restriction x*x2 = 0. The solution x*2 is the eigenvector associated with the second largest eigenvalue of ft. xj can be scaled by the reciprocal of txj giving w2, and then the second factor portfolio will be uR,. In general the jth factor will be wR, where u>j is the rescaled eigenvector associated with the jth largest eigenvalue of Ct. The factor portfolios derived from the first К principal components analysis can then be employed as factors for all the tests outlined in Section 6.2.

Another principal components approach has been developed by Connor and Korajczyk (1986, 1988).4 They propose using the eigenvectors associated with the К largest eigenvalues of the (Tx T) centered returns cross-product matrix rather than the standard approach which uses the principal components of the (JVxA/) sample covariance matrix. They show that as the cross section becomes large the (KxT) matrix with the rows consisting of the К eigenvectors of the cross-product matrix will converge to the matrix of factor realizations (up to a nonsingular linear transformation reflecting the rotational indeterminancy of factor models). The potential advantages of this approach are that it allows for time-varying factor risk premia and that it is computationally convenient. Because it is typical to have a cross section of assets much larger than the number of time-series observations, analyzing a (ТхГ) matrix can be less burdensome than working with an (Д/хЛ/) sample covariance matrix.

Factor Analysis or Principal Components!

We have discussed two statistical primary approaches for constructing the model factors-factor analysis and principal components. Within each approach there are possible variations in the process of estimating the factors. A question arises as to which technique is optimal in the sense of providing the most precise measures of the population factors given a fixed sample of returns. Unfortunately the answer in finite samples is not clear although all procedures can he justified in large samples.

Set- also Mfi (ИШ).