wbilWp = 0 (6.6.4)

/иг и (Kx\) viitinu> where V , is Ihe (NxK) matrix of asset weights for the factor portfolios, u /, is Ihe (N x I) vector of asset weights for the optimal orthogonal portfolio, aiitlw,! isthe(Nx I) vector of asset weights for the langemy portfolio. If one considers a model without any factor portfolios (K - 0) then Ihe optimal orthogonal portfolio will lie the langemy portfolio.

I lie weights ol porllolio /, can be expressed in terms ol the parameters of the К-Im tor model. The ve< lor ol weights is

w/( = ЛГаГГГа

= (.ЕаГЕа, (6.6.5)

where the (superscript indicates the generalized inverse. The usefulness of this portfolio <dines from the fact that when added lo (6.6. 1) the intercept will vanish and the fat lor loading matrix В will not be altered. The opiiinality restriction in (6.6.3) leads to die intercept vanishing, and the orthogonality condition in (6,6.4) leads to В being unchanged. Adding in /. ,:

/., = BZA( -rfli,/.! + u, (li.(i.ti)

lKI = <) (6.6.7)

Eu,ii, = Ф (6.6.8)

ЩУ-ы] = lit,. ЩУ-hi -Hi,)2l = ((,.0.9)

CovZA u, = 0 (6.6.10)

Oovz, ,ii,j = 0. (6.6.11)

We can relate the optimal orthogonal portfolio parameters lo the factor model deviations by comparing (li.li.l) and (6.6.6). Taking the unconditional expet unions of both sides,

= ft i,I4,. (6.6.12)

and by equating the variance of c, with the variance of ft /. , + u

s = = аа + Ф. (6.6.13)

Ihe key link between the model deviations and the residual variances and covariances emerges from (6.6.13). The intuition for the link is straightforward. Deviations from the model must be accompanied bv a common

component in the residual variance to prevent the formation of a portfolio with a positive deviation and a residual variance that decreases to zero as the number of securities in the portfolio grows, that is, an asymptotic arbitrage opportunity.

6.6.2 Squared Sharpe Ratios

The squared Sharpe ratio is a useful construct for interpreting much of live ensuing analysis. The tangency portfolio q has the maximum squared Sharpe measure of all portfolios. The squared Sharpe ratio of q, s, is

$ = ГГУ (6.6.14)

Given that the К factor portfolios and the optimal orthogonal portfolio Л can be combined to form the tangency portfolio, the maximum squared Sharpe ratio of these K+\ portfolios will be s. Since h is orthogonal to the portfolios K, MacKinlay (1995) shows that one can express as the sum of the squared Sharpe ratio of the orthogonal portfolio and the squared maximum Sharpe ratio of the factor portfolios,

s] = st + sl (6.6.15)

where s\ = ixj/crf and s\ = цкОГкх цк?

Empirical tests of multifactor models employ subsets of the N assets. The factor portfolios need not be linear combinations of the subset of assets. Results similar to those above will hold within a subset of N assets. For subset analysis when considering the tangency portfolio (of the subset), the maximum squared Sharpe ratio of the assets and factor portfolios, and th.e optimal orthogonal portfolio for the subset, it is necessary to augment the N assets with the factor portfolios K. Defining Z* as the {N+Kxl) vector [Z( ZKI] with mean u* and covariance matrix fl*, for the tangency portfolio of these N+K assets we have

si = ktcV;. (6.6.

The subscript s indicates that a subset of the assets is being considered. If tiny of the factor portfolios is a linear combination of the N assets, it will be necessary to use the generalized inverse in (6.6.16).

litis result is related to the work <ilC.il)li<ms, Ross, and Sliauken (1989).

6. Multifactor hiring Models

The analysis (with a subset of assets) involves the quadratic aS~a com-nitcd using the parameters for the N assets. Gibbons, Ross, and Slianken ; 1989) and Lehmann (1987, 1992) provide interpretations of this quadratic erm using Sharpe ratios. Assuming £ is of full rank, they show

aXa, = si - 4 (6.6.17)

Consistent with (6.6.15), for the subset of assets aE 1 a is the squared Sharpe ratio of the subsets optimal orthogonal portfolio A.,. Therefore, for a given subset оГ assets:

*s\ = aa, (6.6.18)

4 = 4 + 4- ( )

Note that the squared Sharpe ratio of the subsets optimal orthogonal portfolio is less than or equal to that of the population optimal orthogonal portfolio, that is,

s\ < sl (6.6.20)

Next we use the optimal orthogonal portfolio and the Sharpe ratios results together with the model deviation residual variance link to develop implications for distinguishing among asset pricing models. Hereafter the s subscript is suppressed. No ambiguity will result since, in the subsequent analysis, wc will be working only with subsets of the assets.

6.6.3 Implications for Separating Alternative Theories

If a given factor model is rejected a common interpretation is that more (or different) risk factors are required to explain the risk-return relation. This interpretation suggests that one should include additional factors so that the null hypothesis will be accepted. A shortcoming of this popular approach is that there arc multiple potential interpretations of why the hypothesis is accepted. One view is that genuine progress in terms of identifying the right asset pricing model has been made. But it could also be the case that inc apparent success in identifying a better model has come from finding d good within-sample fit through data-snooping. The likelihood of this possibility is increased by the fact that the additional factors lack theoretical njiotivation.

I This section attempts to discriminate between the two interpretations. Tp do this, wc compare the distribution of the test statistic under the null hypothesis with the distribution under each of the alternatives.

We reconsider the zero-intercept /-test of the null hypothesis thai the intercept vector a from (0.0.1) is 0. Let Ho be the null hypothesis and H.i

6.6. Interpreting Divinlions from lixacl initio hiring

be the allernalive:

a = 0

H : а ф 0.

Ho can be tested using the lest statistic j{ from (0.2.12):

VT-N-K) , ., - -i . , - - i j\ = -д-И + МЛ Ilk\ Л a. (0.0.21)

where 7is the number of lime-scries observations, N is the number of assets or portfolios of assets included, and К is ihe number of laclor portfolios. The hat superscripts indicate the maximum likelihood estimators. Under the null hypothesis, / is unconditionally distributed centra! / with N degrees of freedom in the numerator and {T - N - K) degrees of freedom in the denominator.

To interpret deviations from the null hypothesis, we require a general representation for the distribution of j\. Conditional on the laclor portfolio returns die distribution of Ji is

j\ ~ .y.y-n-кОЛ. (0.6.22)

<5 = T[l + fiKU~ii-Y. \x. (6.6.23)

where <5 is the noncenlrality parameter of the / distribution. If К = 0 then

the term 1 + fiKlK Ал] vvl1 ot appear in (0.0.21) or in (6.6.23), and Ji will be unconditionally distributed non-central /.

We consider the distribution of j\ under two different alternatives, which are separated by their implications for die maximum value of the squared Sharpe ratio. With the risk-based multifactor alternative there will be an upper bound on the squared Sharpe ratio, whereas with the mmi isk-based alternatives the maximum squared Sharpe ratio is unbounded as the number of assets increases.

first consider the distribution of jt under the alternative hypothesis that deviations are due lo missing factors. Drawing on the results for the squared Sharpe ratios, the noncenlrality parameter of the / distribution is

S= 7 [ 1 + /УА.П~ Ал-rS;. (0.0.24)

From (6.6.20), the third term in (6.6.24) is bounded above by .sjf and positive. The second term is bounded between zero and one. Thus there is an tipper bouud for o>,

<5 < Tsi < Ts;r (0.0.25)

The second inequality follows from the fact thai the langency portfolio ij has the maximum Sharpe ratio of any asset or portfolio.

(liven a iii.i\iiiniin value lor llie squared Shar)e ratio, ihe upper hound on ihe nonrcniiality parameter can he important. With this hound, independent ol how one arranges the assets to he included as dependent variables in the pricing model regression and for any value of Л/, there is a limit on the distance between the null distribution and the distribution of the test statistic under the missing-factor alternative. All the assets can be mispi iced and yet the bound will still apply.

ln contrast, when the alternative one has in mind is that the source ol deviations is iioiu isk-basccl, such as data snooping, market frictions, or market irrationalities, the notion of a maximum squared Sharpe ratio is not useful. Ihe squared Sharpe ratio (and the noncenlrality parameter) are in principle unbounded because the theory linking the deviations and the residual variances and covariances does not apply. When comparing alternatives with the intercepts of about the same magnitude, in general, one would expect lo see larger test statistics in this nonrisk-based case.

We examine the informativencss of ihe above analysis by considering alternatives with realistic parameter values. We consider the distribution of the test statistic for three cases: ihe* null hypothesis, the missing risk factors alternative, and the nonrisk-based alternative. For the risk-based alternative, the framework is designed lo be similar lo that in Fama and French (КИМ). For the nonrisk-based alternative we use a setup that is consistent with the analysis of l.o and MacKinlay (1990b) and the work of 1 .akonishok, Shleiler, and Vislniy (1991).

Consider a one-factor asset pricing model using a time series of the excess returns for .12 portfolios as die dependent variable. The one factor (independent variable) is the excess return of the market so that the zem-inlercept null hypothesis is the CAIM. The length of the lime series is 312 months. This setup corresponds to that of Fama and French (1993, Table I, regression (ii)). The null distribution of the test statistic /[ is

./i ~ <V:toi(0). (6.6.26)

To define the distribution of j\ under the alternatives of interest one needs lo specify the parameters necessary lo calculate the noncenlralitv parameter. For the risk-based alternative, given a value for the squared Sharpe ratio ol the optimal orthogonal portfolio, the distribution corresponding to the upper bound ol the noncenlrality parameter from (6.6.25) can be considered. The Sharpe ratio of the optimal orthogonal portfolio can be obtained using (6.6.15) given the squared Sharpe ratios of (he tangency portfolio and ol the hit hided lac lor porllolio.

lit pi.ti lii г wlu-ii using tlu- /-test ii will ik- necessary tor л to ik- u-ss than f- К so that XI will lie ol lull rank.

Willi tlata-sni loping the distribution of Ji is not exactly a noncentral F (see txiand MacKinlay I I.l.lOb]). However, for die purposes of this analysis, ihe noncentral F will be a good approximation.

MacKinlay (1995) argues that in a perfect capital markets setting, a reasonable value for the Sharpe ratio squared of the tangency portfolio for an observation interval of one month is 0.031 (or approximately 0.6 for the Sharpe ratio on an annualized basis). This value, for example, corresponds to a portfolio with an annual expected excess return of 10% and a standard deviation of 16%. If the maximum squared Sharpe ratio of the included factor portfolios is the exposl squared Sharpe ratio of the CRSP value-weighted index, the implied maximum squared Sharpe ratio for the optimal orthogonal portfolio is 0.021. This monthly value of 0.021 would be consistent with a portfolio which has an annualized mean excess return j of 8% and annualized standard deviation of 16%. We work through the j analysis using this value. I

Using this squared Sharpe ratio for the optimal orthogonal portfolio to } calculate <5, the distribution of J\ from equation (6.2.1) is i

ii ~ Osteal). (6.6.27)

This distribution will be used to characterize the risk-based alternative. One can specify the distribution for two nonrisk-based alternatives by

specifying values of a, E, z.nAjlKSlK p.K,and thencalculating<5 from (6.6.23). To specify the intercepts we assume that the elements of a are normally distributed with a mean of zero. We consider two values for the standard deviation, 0.0007 and 0.001. When the standard deviation of the elements of a is 0.001 about 95% of deviations will lie between -0.002 and +0.002, an annualized spread of about 4.8%. A standard deviation of0.0007 for the deviations would correspond to an annual spread of about 3.4%. These spreads are consistent with spreads that could arise from data-snooping.9 They are plausible and even somewhat conservative given the contrarian strategy returns presented in papers such as Lakonishok, Shleifer, and Vishny (1993). For E we use a sample estimate based on portfolios sorted by market capitalization for the Fama and French (1993) sample period 1963 to 1991. The effect of -i - -i .

fiKClK цк on S will typically be small, so it is set to zero. To get an idea of a reasonable value for the noncentrality parameter given this alternative, the expected value of tS given the distributional assumption for the elements of a conditional upon £ = Ё is considered. The expected value of the noncentrality parameter is 39.4 for a standard deviation of 0.0007 and 80.3 for a standard deviation of 0.001. Using these values for the noncentrality parameter, the distribution of J\ is

J\ ~ %soa(39.4) (6.6.28)