Ïðîìûøëåííûé ëèçèíã  Ìåòîäè÷êè  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 [ 42 ] 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 wbilWp = 0 (6.6.4)/èã è (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 Ê-Im tor model. The ve< lor ol weights isw/( = ËÃàÃÃÃà= (.ÅàÃÅà, (6.6.5)where the (superscript indicates the generalized inverse. The usefulness of this portfolio ,<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 Ë/, 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 (ÊÈÌ). 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 ~