Thcimaximum likelihood estimates can he obtained by iterating over (6.2.23) to (6.2.25). В from (6.2.20) and Ё from (6.2.21) can be used as starting values for В and E in (6.2.25).

Exact maximum likelihood estimators can also be calculated without iteration for this case. The methodology is a generalization of the approach outlined for the Black version of the CAPM in Chapter 5; it is presented by Shankcn (1985a). The estimator of yu is the solution of a quadratic equation. Given ya, the constrained maximum likelihood estimators of В and £ follow from (6.2.23) and (6.2.24).

The restrictions of the constrained model in (6.2.22) on the unconstrained model in (6.2.14) arc

a = (t-Bi.)y . (6.2.26)

These restrictions can be tested using the likelihood ratio statistic / in (6.2.1). Under the null hypothesis the degrees of freedom of the null distribution will be N-l. There is a reduction of one degree of freedom in comparison to the case with a riskfree asset. A degree of freedom is used up in estimating the zero-beta expected return.

For use in Section 6.3, we note that the asymptotic variance of y evaluated at the maximum likelihood estimators is

Var()>()] = \. [\ + {fiK - yUL)nK(ilK - p0t))

x [(t-B*(.)E l(i-B*t)]-. (6.2.27)

6.2.3 Macroeconomic Variables as Factors

Factors need not be traded portfolios of assets; in some cases proposed factors fpetude macroeconomic variables such as innovations in GNP, changes in bond yields, or unanticipated inflation. We now consider estimating and testing exact factor pricing models with such factors.

Again define R( as an (Л/xl) vector of real returns for N assets (or portfolios of assets). For the unconstrained model we have a A-factor linear modl:

t R, = a-fBfK1 + €, (6.2.28)

E[e,] = 0 (6.2.29)

E[€,£,] = E (6.2.30)

Cov[fKl,e,l = O.

(6.2.31) (6.2.32)

В is the {N x K) matrix of factor sensitivities, fKl is the (Лх 1) vector of factor realizations, and a and e, arc (Nxl) vectors of asset return intercepts and disturbances, respectively. О is a (Ax/V) matrix of zeroes.

For the unconstrained model in (6.2.14) the maximum likelihood estimators are

a = ix - B/i/A;

(6.2.33)

В =

E =

£(R,-А)(ГЛ-/-A;K)

£(л-, - А/л) Сл/ - Аул)

1=1 1 7

j, ]£(R, - а - BfA-,)(R, - а - ВГ*,).

(6.2.34) (6.2.35)

where

1 v-л 1 .

y-ER< aiul А/л =

The constrained model is most conveniently formulated by comparing the unconditional expectation of (6.2.28) with (6.1.8). The unconditional expectation of (6.2.28) is

ц = a + Bit

(6.2.36)

where fjt/K = E[fAi). Equating the right hand sides of (6.1.8) and (6.2.36) we have

a = iAo + B(AA- - nJK). (6.2.37)

Defining yu as the zero-beta parameter \ and defining 7, as (Ал - Мул) where AA- is the (Kxl) vector of factor risk premia, for die constrained model, we have

R, = £.у + В7, +HfKt + c,.

(6.2.38)

The constrained model estimators are

B* =

3jR,-t70)(fA-, + 7,) 1

4-7t)(f / + 7t)

(6.2.39)

t* = y,X]l(R(-Po)-B*(fA(+7i)]

x(R,-1-й )-B*(fK(+ 7,)] (6.2.40)

7 = IXEXI1 X£, 1(A-BAM)1. (6.2,11)

where in (0.2. I 1) X н= I,. В* 1 and 7=1)/ 7, ].

Tlu maximum likelihood estimates can be obtained by iterating over (6.2.39) to (6.2.41). В from (6.2.34) and £ from (6.2.35) can be used as starling values for В and S in ((i.2.l 1).

The restrictions of (li.2.38) on (ti.2.28) are

a = (./ + В7,. (6.2.42)

These restrictions can be tested using the likelihood ratio statistic / in (6.2.1). Under the null hypothesis the degrees of freedom of the null distribution is N - К - 1. There are /V restrictions but one degree of freedom is lost estimating yu, and A. degrees of freedom tire used estimating the К elements of Ад .

Ihe asymptotic variance of 7 follows from the maximum likelihood approach. The variance evaluated al the maximum likelihood estimators is

Va7l7l = 7 ( + <ТП + A/a) a<7i + A/a-)) [ХЁ* X]-. (6.2.43)

Applying the partitioned inverse rule to (6.2.43), for the variances of the components of 7 we have estimators

Varlul = f (l + (7, +£/a-> kI<7i +£/*))

US* \. - l£ lB*(B*S* 1B*)-B*S* lt]- (6.2.44)

Vl7il = f(l +(7,+A/a)l(7t+Аул))(В*Е ~b*)-

+ (B*E*~BMlB*E* \(Va7{y,)])

x /.Ё* 11Г(В,Ё* 1В*) 1. (6.2.45)

We will use these variance results for inferences concerning die factor risk picmia in Section 6.3.

6.2.-I Factor Iortjiilios Spanning the Mean-Variance Frontier

When factor portfolios span the mean-variance frontier, the intercept term of the exact pricing relation Xt) is /его without the need for a riskfree asset.

Thus this case retains the simplicity of the first case with the riskfree asset. In the context of the APT, spanning occurs when two well-diversified portfolios are on the minimum-variance boundary. Chamberlain (1983a) provides discussion of this case.

The unconstrained model will be a ЛГ-factor model expressed in rel returns. Define R, as an (A/xl) vector of real returns for N assets (dr portfolios of assets). Then for real returns we have a K-factor linear modelj:

R, = a + BR + c, (6.2.4f)

E[€,] = 0 (6.2.47)

E[€,e;i = S (6.2.48)

E[Ra-,1 = Мл:. E[(R*, - Ma) (R*/ - /**)] = а- (6.2.49)

Cov[Ra e,] = O. (6.2.50)

В is the (Л/х K) matrix offactor sensitivities, Ra-< is the (Afx 1) vector of factoj-portfolio real returns, and a and e, are (N x 1) vectors of asset return inter-cepts and disturbances, respectively. О is a (KxN) matrix of zeroes. The restrictions on (6.2.46) imposed by the included factor portfolios spanning the mean-variance frontier are:

a = 0 and Bt = 6. (6.2.51)

To understand the intuition behind these restrictions, we can return to die Black version of the CAPM from Chapter 5 and can construct a spanning example. The theory underlying the model differs but empirically the restrictions are the same as those on a two-factor APT model with spanning. The unconstrained Black model can be written as

R, = a + f3om R0, + f3m Rm, + e (6.2.52)

where R, , and R0, are the return on the market portfolio and the associated zero-beta portfolio, respectively. The restrictions on the Black model are a = 0and/3om+/3m = l as shown in Chapter 5. These restrictions correspond to those in (6.2.51).

For the unconstrained model in (6.2.46) the maximum likelihood estimators are

a = A-BAa-

(6.2.53)

6. Multifactor Pricing Models

В =

]T(R, - £)(RA/ - (iK) i

]T(RA-, - £A) (RA-, - /хл)

7-X>-

a-BRA-,)(R,-a-BRA-,),

(0.2.54) (0.2.55)

where

To estimate the constrained model, we consider the unconstrained model in (6.2.46) with the matrix В partitioned into an (Nx 1) column vector b and an (/Vx(A-l)) matrix B and the factor portfolio vector partitioned into the first row Rj, and the last (K-l) rows RA-.,. With this partitioning the constraint В t = t can be written b) + B)t = t. For the unconstrained model we have

R, = a + b,R + B,RA., + e,. (6.2.56)

Substituting а = 0 and b = t - Bt into (6.2.56) gives the constrained model,

R, - iRw = B, (RA-., - iR ) + e,. (6.2.57)

I Using (6.2.57) the maximum likelihood estimators arc

ь: =

]T(R, - iR ) (RA-, - iR )

£(RA-, - tRu) (RA-, - *R )

£ = - jjR, - BRA,)(R, - BRai).

(6.2.58)

(6.2.59) (6.2.60)

flhe null hypothesis a equals zero can be tested using die likelihood ratio statistic J in (6.2.1). Under the null hypothesis the degrees of freedom of the null distribution will be 2N since a = 0 is N restrictions and Bt = t is N additional restrictions.

We can also construct an exact test of the null hypothesis given the linearity of the restrictions in (6.2.51) and the multivariate normality assumption.

6.3. Estimation of I{isk Premia and Expected Returns

Defining }i as the test statistic we have

T-N-K) h ~ -N-

Under the null hypothesis, Ji is unconditionally distributed central F with 2N degrees of freedom in the numerator and 2(7- N - K) degrees of freedom in the denominator. Ilubcrman and Ivandcl (1987) present a derivation of this test.

(6.2.61)

6.3 Estimation of Risk Premia and Expected Returns

All the exact factor pricing models allow one to estimate the expected return on a given asset. Since the expected return relation is fi = (,A -f BAA, one needs measures of the factor sensitivity matrix B, the riskfree rate or the zero-beta expected return A.()l and the factor risk premia AA. Obtaining measures of В and the riskfree rate or the expected zero-beta return is straightforward. For the given case the constrained maximum likelihood estimator B* can be used for B. The observed riskfree rate is appropriate for the riskfree asset or, in the cases without a riskfree asset, the maximum likelihood estimator y can be used for the expected zero-beta return.

Further estimation is necessary to form estimates of the factor risk premia. The appropriate procedure varies across the four cases of exact factor pricing. In the case where the factors are the excess returns on traded portfolios, the risk premia can be estimated directly from the sample means of the excess returns on the portfolios. For this case we have

An estimator of the variance of AA- is - 1 1

Var[AA = -fiA- = - V(ZA, - (ZA, - fiK). (0.3.2)

In the case where portfolios are factors but there is no riskfree asset, the factor risk premia can be estimated using the difference between the sample mean ol the factor portfolios and the estimated zero-beta return:

AA = fiK ~ (0.3.3)

In this case, an estimator of the variance of AA- is

Va ilAA] = Па+УИЙ,]. (6.3.4)