where T = number of time series observations on returns; 4; (2,,~?~.

Given the normality assumption, the null hypothesis in (3) can be tested using "Hotelling's T2 test," a multivariate generalization of the univariate t-test (e.g., see alinvaud (1980, page 230)). A brief derivation of the equivalent F test is included for completeness and as a means of introducing some notation that will be needed later. If we estimate the multivariate system of (1) using ordinary least squares for each individual equation, the estimated intercepts have a multivariate normal distribution, conditional on r,, (Vt = 1,..., T), with where T = number of time series observations on returns; 4; (2,,~?~. ~. 8, = ~,/s,; = sample mean of ;,?, and sj = sample varinance of $, wlthout an adjustment for degrees of freedom. Furthermore, 2, and H are independent with (T- 2)2 having a Wishart distribution with parameters (T- 2) and 2. These facts imply (see orrison (1976, page 131)) that (T(T- N - l)/n(t - 2))Wu has a noncentral F distribution with degrees of freedom N and (T- N - I), where and 2 = unbiased residual covariance matri~.~ (The corresponding statistic based on the maximum likelihood estimate of H will be denoted as W.) The noncentrality parameter, A, is given by Under the null hypothesis that a e uals zero, X = 0, and we have a central F 4 9. distribution. ore generally, the d~stnbution under the alternative provides a way to study the power of the test; more will be said about this in a later section. It is also interesting to note that under the null hypothesis the W, statistic has a central Fdistribution unconditionally, for the parameters of this central Fdo not depend on G, in any way. However, we do not know the unconditional distribution of 2, or W, under the alternate, for the conditional distribution depends on the sample values of i,, through 8;.

TABLE 1 A COPARlSON OF FOUR ASYPTOTICALLY EQUIVALENT TESTS OF EX ANTE EFFICIENCY OF A GIVEN PORTFOLIO. THE W STATISTIC IS DISTR~BUTEDAS A TRANSFOR OF A CENTRAL F DISTR~BUTIONIN FINITE SAPLES. THEWALDTEST, THE LIKELIHOOD RATIO TEST (LRT), AND THE LAGRANGEULTIPLIER TEST (LT) ARE ONOTONE TRANSFORS OF W, AND EACH IS DISTRIBUTED AS CHI-SQUARE WITH N DEGREES OF FREEDO AS T APPROACHES INFINITY. P-Values Using P-Value Using Asymptotic Approximations Exact Distribution of W Wald LRT LT Note: N is the number of assets used together with portfolio p to construct the ex post frontier, and T is the number of time series observations.

ARTICLE IN PRESS J. Lewellen, S. Nagel / Journal of Financial Economics 82 (2006) 289 314 293 2.2. Unconditional alphas and betas The conditional CAP says that stocks expected returns are proportional to their conditional betas: E t 1 [R it ] ¼ b t g t. Taking unconditional expectations implies that E[R it ] ¼ bg+cov(b t,g t ), as observed by Jagannathan and Wang (1996). An asset s unconditional alpha is defined as a u E[R it ] b u g, and substituting for E[R it ] yields: a u ¼ gðb b u Þþcovðb t ; g t Þ. (1) Under some conditions, discussed below, a stock s unconditional and expected conditional betas will be similar, in which case a u is approximately equal to the covariance between beta and the market risk premium. ore generally, Appendix A shows that b u ¼ b þ g s 2 covðb t ; g t Þþ 1 s 2 cov b t ; ðg t gþ 2 1 þ s 2 cov b t ; s 2 t. (2) This expression says that b u differs from the expected conditional beta if b t covaries with the market risk premium (second term), if it covaries with (g t g) 2 (third term), or if it covaries with the conditional volatility of the market (last term). Roughly speaking, movement in beta that is positively correlated with the market risk premium or with market volatility, g t or s 2 t, raises the unconditional covariance between R i and R (the other term is generally quite small, as we explain in a moment). Substituting (2) into (1), the stock s unconditional alpha is a u ¼ 1 g2 s 2 covðb t ; g t Þ g s 2 cov b t ; ðg t gþ 2 g s 2 cov b t ; s 2 t : (3)

2.3. agnitude Our goal is to understand whether a u in Eq. (3) might be large enough to explain observed anomalies. We begin with a few observations to simplify the general formula. Notice, first, that the market s squared Sharpe ratio, g 2 /s 2 in the first term, is very small for monthly returns: for example, using the Center for Research in Security Prices (CRSP) value-weighted index from 1964 to 2001, g ¼ 7% and s ¼ 4.5%, so the squared Sharpe ratio is 11. Further, the quadratic (g t g) 2, in the second term, is also quite small for plausible parameter values: if g equals % and g t varies between, say, % and %, the quadratic is at most 05 2 ¼ 00025. Plugging a variable this small into the second term would have a negligible effect on alpha. These observations suggest the following approximation for a u : a u ¼ covðb t ; g t Þ g s 2 cov b t ; s 2 t. (4) Eq. (4) says that, when the conditional CAP holds, a stock s unconditional alpha depends primarily on how b t covaries with the market risk premium and with market volatility 3. To explore the magnitude of Eq. (4), it is useful to consider the simplest case when b t covaries only with the market risk premium: a u Ecov(b t,g t ) ¼ rs b s g, where s denotes a standard deviation and r is the correlation between b t and g t. Table 1 3 The approximation becomes perfect as the return interval shrinks because g 2 and (g t g) 2 go to zero more quickly than the other terms in Eq. (3). We thank John Campbell for this observation.

Table 3 Average conditional alphas, 1964 2001 The table reports average conditional alphas for size, B/, and momentum portfolios (% monthly). Alphas are estimated quarterly using daily returns, semiannually using daily and weekly returns, and annually using monthly returns. The portfolios are formed from all NYSE and Amex stocks on CRSP/Compustat. We begin with 25 size- B/ portfolios (5 5 sort, breakpoints determined by NYSE quintiles) and ten return-sorted portfolios, all value weighted. Small is the average of the five low-market-cap portfolios, Big is the average of the five high-marketcap portfolios, and S B is their difference. Similarly, Grwth is the average of the five low-b/ portfolios, Value is the average of the five high-b/ portfolios, and V G is their difference. Return-sorted portfolios are formed based on past six-month returns. Losers is the bottom decile, Winrs is the top decile, and W L is their difference. Bold values denote estimates greater than two standard errors from zero. Size B/ omentum Small Big S B Grwth Value V G Losers Winrs W L Average conditional alpha (%) Quarterly 2 0 2 1 9 0 9 5 1.33 Semiannual 1 6 0 6 8 0 7 1 9 9 Semiannual 2 6 1 5 2 6 8 3 3 1.37 Annual 6 8 4 0 2 3 6 1 7 Standard error Quarterly 0 6 2 2 4 4 0 3 6 Semiannual 1 1 6 3 2 4 5 9 4 5 Semiannual 2 1 6 3 4 5 6 0 5 7 Annual 6 7 9 6 7 4 1 7 9 Quarterly and Semiannual 1 alphas are estimated from daily returns, Semiannual 2 alphas are estimated from weekly returns, and Annual alphas are estimated from monthly returns.

Figure 1. Population R 2 s for artificial factors. This figure explores how easy it is to find factors that explain, in population, the cross section of expected returns on Fama and French s 25 size-b/ portfolios. We randomly generate factors either factor loadings directly or zero-investment factor portfolios, as described in the figure and estimate the population R 2 when the size-b/ portfolios expected returns are regressed on their factor loadings. The average returns and covariance matrix of the portfolios, quarterly from 1963 2004, are treated as population parameters in the simulations. The plots are based on 5,000 draws of 1 to 5 factors. Panel A: Random draws of factor loadings. Loadings for the 25 size-b/ portfolios are drawn from a VN distribution with mean zero and covariance matrix proportional to the return covariance matrix. Cross-sectional R 2 Panel B: Random draws of factor portfolios. Zero-investment factors, formed from the size-b/ portfolios, are generated by randomly drawing a 25 1 vector of weights from a standard normal distribution. Cross-sectional R 2 Panel C: Random draws of zero-mean factor portfolios. Zero-investment factors, formed from the size-b/ portfolios, are generated by randomly drawing a 25 1 vector of weights from a standard normal distribution, but only factors with roughly zero expected returns are kept. Cross-sectional R 2

Figure 2: Sample distribution of the cross-sectional adj. R 2. This figure shows the sample distribution of the cross-sectional adj. R 2 (average returns regressed on estimated factor loadings) for Fama and French s 25 size-b/ portfolios from 1963 2004 (quarterly returns). The plots use one to five randomly generated factors that together have the true R 2 reported on the x-axis. In the left-hand panels, the factors are combinations of the size-b/ portfolios (the weights are randomly drawn to produce the given R 2, as described in the text). In the right-hand panels, noise is added to the factors equal to 3/4 of a factor s total variance, to simulate factors that are not perfectly spanned by returns. The plots are based on 40,000 bootstrap simulations (10 sets of random factors; 4,000 simulations with each). 1 factor 1 factor, w/ noise Sample adj. R 2 0 1 0 1 - - 3 factors 3 factors, w/ noise Sample adj. R 2 0 1 0 1 - - 5 factors 5 factors, w/ noise Sample adj. R 2 0 1 0 1 - - True R 2 True R 2

Figure 4. Population OLS and GLS R 2 s for artificial factors. This figure explores how easy it is to find factors that explain, in population, the cross section of expected returns on Fama and French s 25 size-b/ portfolios. We randomly generate factors either factor loadings directly or zero-investment factor portfolios, as noted and estimate the population OLS and GLS R 2 s when expected returns are regressed on factor loadings. The average returns and covariance matrix of the size-b/ portfolios, quarterly from 1963 2004, are treated as population parameters in the simulations. The plots are based on 5,000 draws of 1 to 5 factors. OLS R 2 GLS R 2 Panel A: Random draws of factor loadings. Loadings for the 25 size-b/ portfolios are drawn from a VN distribution with mean zero and covariance matrix proportional to the return covariance matrix. OLS R 2 GLS R 2 Panel B: Random draws of factor portfolios. Zero-investment factors, formed from the size-b/ portfolios, are generated by randomly drawing a 25 1 vector of weights from a standard normal distribution. OLS R 2 GLS R 2 Panel C: Random draws of zero-mean factor portfolios. Zero-investment factors, formed from the size-b/ portfolios, are generated by randomly drawing a 25 1 vector of weights from a standard normal distribution, but only factors with roughly zero expected returns are kept.

Prescription 4. If a proposed factor is a traded portfolio, include the factor as one of the test assets on the left-hand side of the cross-sectional regression. Prescription 4 builds on Prescription 2, in particular, the idea that the cross-sectional price of risk for a factor portfolio should be the factor s expected excess return. One simple way to build this restriction into a cross-sectional regression is to ask the factor to price itself; that is, to test whether the factor portfolio itself lies on the estimated cross-sectional regression line. Prescription 4 is most important when the cross-sectional regression is estimated with GLS rather than OLS. As mentioned above, when a factor portfolio is included as a left-hand side asset, GLS forces the regression to price the asset perfectly: the estimated slope on the factor s loading exactly equals the factor s average return in excess of the estimated zero-beta rate (in essence, the asset is given infinite weight in the regression). Thus, a GLS cross-sectional regression, when a traded factor is included as a test portfolio, is similar to the time-series approach of Black, Jensen, and Scholes (1972) and Gibbons, Ross, and Shanken (1989).

Table 1. Empirical tests of asset-pricing models, 1963 2004. The table reports slopes, Shanken (1992b) t-statistics (in parentheses), and adj. R 2 s from cross-sectional regressions of average excess returns on estimated factor loadings for eight models proposed in the literature. Returns are quarterly (%). The test assets are Fama and French s 25 size-b/ portfolios used alone or together with their 30 industry portfolios. The cross-sectional T 2 (asymptotic χ 2 ) statistic tests whether residuals in the cross-sectional regression are all zero, as described in the text, with simulated p-values in brackets. T 2 is proportional to the distance, q, that a model s true mimicking portfolios are from the minimum-variance boundary, measured as the difference between the maximum generalized Sharpe ratio and that attainable from the mimicking portfolios; the sample estimate of q is reported in the final column. 95% confidence intervals for the true R 2 and q are reported in brackets below the sample values. The models are estimated from 1963 2004 except Yogo s, for which we have factor data through 2001. odel and test assets Variables Adj. R 2 T 2 q Lettau & Ludvigson const. cay Δc cay Δc FF25 3.33-1 5 0 8 33.9 4 (2.28) (-1.25) (4) (2) [0, 0] [p=8] [0, 2] FF25 + 30 industry 2.50-8 9-0 0 193.8 1.31 (3.29) (-1.23) (3) (-1.10) [0, 5] [p=0] [8, 8] Lustig & V Nieuwerburgh const. my Δc my Δc FF25 3.58 4.23 2 0 7 2 5 (2.22) (6) (4) (1.57) [5, 0] [p=7] [0, 8] FF25 + 30 industry 2.78 7-2 3 9 157.1 1.32 (3.51) (3) (-9) (1.40) [0, 0] [p=4] [0, 6] Santos & Veronesi const. R s ω R FF25 2.45-2 -2 1 19.7 3 (1.39) (-7) (-2.04) [5, 0] [p=3] [0, 8] FF25 + 30 industry 2.29-7 -5 3 188.7 1.28 (2.75) (-6) (-1.51) [0, 0] [p=1] [6, 0] Li, Vassalou, & Xing const. ΔI HH ΔI Corp ΔΙ Ncorp FF25 2.47-0 -2.65-8.59 0 25.2 4 (2.13) (-9) (-3) (-1.96) [5, 0] [p=9] [0, 8] FF25 + 30 industry 2.22 0-3 -5.11 2 141.2 1.27 (3.14) (9) (-8) (-2.32) [0, 0] [p=1] [0, 4] Yogo const. Δc Ndur Δc Dur R FF25 1.98 8 7 8 8 24.9 6 (1.36) (0) (2.33) (9) [0, 0] [p=9] [0, 0] FF25 + 30 industry 1.95 8 9 2 2 159.3 1.24 (2.27) (1) (1.52) (1) [0, 0] [p=6] [0, 8] CAP const. R FF25 2.90-4 -3 77.5 6 (3.18) (-9) [0, 5] [p=0] [2, 8] FF25 + 30 industry 2.03 0-2 225.2 1.34 (2.57) (9) [0, 5] [p=0] [8, 6]

Table 1, continued. odel and test assets Variables Adj. R 2 T 2 q Consumption CAP const. Δc FF25 1.70 4 5 6 6 (2.47) (9) [0, 0] [p=1] [6, 6] FF25 + 30 industry 2.07 3-2 224.5 1.34 (3.51) (5) [0, 5] [p=0] [8, 2] Fama & French const. R SB HL FF25 2.99-1.42 0 1.44 8 56.1 7 (2.33) (-8) (1.70) (3.11) [0, 0] [p=0] [6, 2] FF25 + 30 industry 2.21-9 0 7 1 20 1.24 (2.14) (-1) (1.24) (1.80) [0, 0] [p=0] [2, 0] Variables: R = CRSP value-weighted excess return Δc = log consumption growth cay = Lettau and Ludvigson s (2001) consumption-to-wealth ratio my = Lustig and Van Nieuwerburgh s (2004) housing collateral ratio (based on mortgage data) s ω = labor income to consumption ratio ΔI HH, ΔI Corp, ΔI Ncorp = log investment growth for households, non-financial corporations, and the non-corporate sector Δc Ndur, Δc Dur = Yogo s (2005) log consumption growth for non-durables and durables SB, HL = Fama and French s (1993) size and B/ factors