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A Unifying Approach to the Empirical Evaluation of Asset Pricing Models Francisco Peñaranda and Enrique Sentana CenFIS Working Paper 10-03 December 2010
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A Unifying Approach to the Empirical Evaluation of Asset Pricing Models Francisco Pearanda and Enrique Sentana CenFIS Working Paper 10-03 December 2010 Theauthors thankAbhay Abhyankar, ManuelArellano, Antonio DezdelosRos, Lars Hansen,Raymond Kan, Craig MacKinlay, Cesare Robotti, Rosa Rodrguez, Amir Yaron, participants at the XVIII Finance Forum (Elche) andtheXXXVSimposiodelaAsociacinEspaoladeEconoma(Madrid),andaudiencesattheAtlantaFed, Princeton University, Universitat Pompeu Fabra, and the Wharton School for helpful comments, suggestions, and discussions.FinancialsupportfromtheSpanishMinistryofScienceandInnovationthroughgrantsECO2008-03066 (Pearanda) and ECO 2008-00280 (Sentana) is gratefully acknowledged. Pearanda also acknowledges the financialsupportoftheBarcelonaGSEandtheGovernmentofCataloniaandthehospitalityoftheCenterfor FinancialInnovationandStabilityattheAtlantaFed.Theviewsexpressedherearetheauthorsandnot necessarilythoseoftheFederalReserveBankofAtlantaortheFederalReserveSystem.Anyremainingerrors are the authors responsibility. PleaseaddressquestionsregardingcontenttoFranciscoPearanda,AssistantProfessor,Departmentof EconomicsandBusiness,UniversitatPompeuFabra,RamonTriasFargas25-27,08005Barcelona,Spain,(+34) 935422638,fax(+34)935421746,francisco.penaranda@upf.edu,orEnriqueSentana,CEMFI,Casadodel Alisal 5, 28014 Madrid, Spain, +34 914290551, fax +34 914291056, sentana@cemfi.es. CenFIS Working Papers from the Federal Reserve Bank of Atlanta are available online at frbatlanta.org/frba/cenfis/. Subscribe online to receive e-mail notifications about new papers. A Unifying Approach to the EmpiricalEvaluation of Asset Pricing Models Francisco Pearanda and Enrique Sentana CenFIS Working Paper 10-03 December 2010 Abstract: Two main approaches are commonly used to empirically evaluate linear factor pricing models:regressionandstochasticdiscountfactor(SDF)methods,withcenteredanduncentered versions of the latter. We show that unlike standard two-step or iterated gerneralized method of moments(GMM)procedures,single-stepestimatorssuchascontinuouslyupdatedGMMsyield numerically identical values for prices of risk, pricing errors, Jensens alphas, and overidentifying restrictions tests irrespective of the model validity. Therefore, there is arguably a single approach regardlessofthefactorsbeingtradedortheuseofexcessorgrossreturns.Weillustrateour results with the currency returns constructed by Lustig and Verdelhan (2007). JEL classification: G11, G12, C12, C13 Key words: CU-GMM, factor pricing models, forward premium puzzle, generalized empirical likelihood, stochastic discount factor 1 IntroductionAsset pricing theories are concerned with determining the expected returns of assets whosepayos are risky. Specically, these models analyse the relationship between risk and expectedreturns, and address the crucial question of how to value risk. The most popular empiricallyoriented asset pricing models eectively assume the existence of a common stochastic discountfactor (SDF) that is a linear function of some risk factors, which discounts uncertain payosdierently across dierent states of the world. Those factors can be either the excess returns onsome traded securities, as in the traditional CAPM of Sharpe (1964), Lintner (1965) and Mossin(1966) or the so-called Fama and French (1993) model, non-traded economy wide sources ofuncertainty related to macroeconomic variables, like in the Consumption CAPM (CCAPM) ofBreeden (1979), Lucas (1978) or Rubinstein (1976), or a combination of the two, as in the exactversion of Ross (1976) APT.There are two main approaches to formally evaluate linear factor pricing models from anempirical point of view. The traditional method relies on regressions of excess returns on factors,and exploits the fact that an asset pricing model imposes certain testable constraints on theintercepts. More recent methods rely on the SDF representation of the model instead, andexploit the fact that the corresponding pricing errors should be zero. There are in fact twovariants of the SDF method, one that demeans the factors and another one that does not.Although the initial asset pricing tests tended to make the assumption that asset returns andfactors were independently and identically distributed as a multivariate normal vector, nowadaysthese approaches are often implemented by means of the generalized method of moments (GMM)of Hansen (1982), which has the advantage of yielding asymptotically valid inferences even if theassumptions of serial independence, conditional homoskedasticity or normality are not totallyrealistic in practice (see Campbell, Lo and MacKinlay (1996) or Cochrane (2001a) for advancedtextbook treatments).Unfortunately, though, each approach typically yields dierent estimates of prices of risk,pricing errors and Jensens alphas, and dierent values for the overidentifying restrictions test.This begs the question of which approach is best, and there is some controversy surrounding theanswer. For example, Kan and Zhou (2000) advocated the use of the regression method over theuncentred SDF method because the former provides more reliable risk premia estimators andmore powerful specication tests than the latter. However, Cochrane (2001b) and Jagannathanand Wang (2002) criticised their conclusions on the grounds that they did not consider theestimation of factor means and variances. In this respect, Jagannathan and Wang (2002) showedthat if excess returns and factors are independently and identically distributed as a multivariate1normal random vector, in which case the regression approach is optimal, the (uncentred) SDFapproach is asymptotically equivalent under the null. Kan and Zhou (2002) acknowledged thisequivalence result, and extended it to compatible sequences of local alternatives under weakerdistributional assumptions.More recently, Burnside (2007) and Kan and Robotti (2008) have also pointed out thatin certain cases there may be dramatic dierences between the results obtained by applyingstandard two-step or iterated GMM procedures to the centred and uncentred versions of theSDF approach. At the same time, Kan and Robotti (2008, footnote 3) eectively exploit theinvariance to coecient normalisations of the continuously updated GMM estimator (CU-GMM)of Hansen, Heaton and Yaron (1996) to prove the numerical equivalence of the overidenticationtests associated to the centred and uncentred versions of the SDF approach. As is well known,CU-GMM is a single-step method that integrates the heteroskedasticity and autocorrelationconsistent (HAC) estimator of the long-run covariance matrix as part of the objective function.In this context, the main contribution of our paper is to show the more subtle result that theapplication to both the regression and SDF approaches of single-step GMM methods, includingCU-GMM, gives rise to numerically identical estimates of prices of risk, pricing errors, Jensensalphas and overidentifying restrictions tests irrespective of the validity of the asset pricing model.Therefore, one could argue that there is eectively a single method to empirically evalu-ate asset-pricing models. Although the rationale for our results is the well-known functionalinvariance of maximum likelihood estimators, our results do not depend on any distributionalassumption, the number of assets, the specic combination of traded and non-traded factors, orthe sample size, and remain true regardless of whether or not the researcher works with excessreturns or gross returns, or the asset pricing restrictions hold. To ease the exposition, we cen-tre most of our discussion on models with a single priced factor. Nevertheless, our numericalequivalence results do not depend in any way on this simplication.In fact, the proofs of ourmain results explicitly consider the general multifactor case. Similarly, our empirical applicationincludes both single and multifactor models.Importantly, our results apply to optimal GMM inference procedures. In particular, we donot consider sequential GMM methods that x the factor means to their sample counterparts. Wedo not consider either procedures that use alternative weighting matrices such as the uncentredsecond moment of returns. Although the choice in Hansen and Jagannathan (1997) is reasonablein SDF contexts for the purposes of comparing several misspecied models, it does not have anatural regression counterpart. For analogous reasons, we do not consider the popular two-pass regressions, which do not have a natural SDF counterpart. In any case, those generally2suboptimal GMM estimators fall outside the realm of single-step methods such as CU-GMM,and therefore they would typically give rise to numerically dierent statistics.While single-step methods are not widespread in empirical nance applications, this situationis likely to change in the future because they do not require to choose the number of iterations,and more importantly, they often yield more reliable inferences in nite samples than two stepor iterated methods (see Hansen, Heaton and Yaron (1996)). Such Monte Carlo evidence isconrmed by Newey and Smith (2004), who highlight the nite sample advantages of CU andother generalised empirical likelihood estimators over two-step GMM by going beyond the usualrst-order asymptotic equivalence results.In fact, the recent papers by Julliard and Gosh (2008), Almeida and Garcia (2009) or Camp-bell, Gilgio and Polk (2010) attest the increasing popularity of these modern GMM variants.However, the CU-GMM estimator and other single-step methods such as empirical likelihoodor exponentially-tilted methods are often more dicult to compute than two-step estimators,particularly in linear models, and they may sometimes give rise to multiple local minima andextreme results.Although we explain in Pearanda and Sentana (2010) how to compute CU-GMM estimators by means of a sequence of OLS regressions, here we derive simple, intuitiveconsistent parameter estimators that can be used to obtain good initial values, and which willbe ecient for elliptically distributed returns and factors. Interestingly, we can also show thatthese consistent estimators coincide with the GMM estimators recommended by Hansen andJagannathan (1997), which use the second moment of returns as weighting matrix. In addition,we suggest the imposition of good deal restrictions (see Cochrane and Saa-Requejo (2000)) thatrule out implausible results.We illustrate our results with the currency portfolios constructed by Lustig and Verdelhan(2007). We consider three popular linear factor pricing models: the CAPM, as well as linearlisedversions of the Consumption CAPM and the Epstein and Zin (1989) model. Our ndingsconrm that the conict among criteria for testing asset pricing models that we have previouslymentioned is not only a theoretical possibility, but a hard reality. Nevertheless, such a conictdisappears when one uses single-step methods. A dierent issue, though, is the interpretation ofthe restrictions that are eectively tested.In this sense, our results conrm Burnsides (2007)ndings that US consumption growth seems to be poorly correlated to currency returns. Thisfact could explain the discrepancies between the dierent two-step and iterated procedures thatwe nd because non-traded factors that are uncorrelated to excess returns will automaticallyprice those returns with a SDF whose mean is 0. Obviously, such a SDF is not satisfactory froman economic point of view, but strictly speaking, the vector of risk premia and the covariances3between excess returns and factors will be proportional. On the other hand, lack of correlationbetween factors and returns is not an issue when all the factors are traded, as long as they arepart of the set of returns to be priced. In this sense, our empirical results indicate that therejection of the CAPM that we nd disappears when we do not attempt to price the market.The rest of the paper is organised as follows. We study the case of traded factors fromthe theoretical and empirical perspectives in section 2, while in sections 3 and 4 we analysenon-traded factors, and a mixture of traded and non-traded factors, respectively.In section 5we show that our results hold not only for excess returns but also for gross returns. Finally,we summarise our conclusions in section 6.A brief description of CU-GMM as an example ofsingle-step methods, multifactor models, and formal proofs are gathered in appendices A, B andC, respectively. We also include a supplemental appendix with additional results.2 Traded factorsLet r be an :1 vector of excess returns.Standard arguments such as lack of arbitrageopportunities or the rst order conditions of a representative investor imply that1 (:r) = 0for some random variable : called SDF, which discounts uncertain payos in such a way thattheir expected discounted value equals their cost.The standard approach in empirical nance is to model : as an ane transformation ofsome risk factors, even though this ignores that : must be positive with probability 1 to avoidarbitrage opportunities (see Hansen and Jagannathan (1991)).With a single factor ), we canexpress the pricing equation as1 [(a + /)) r] = 0 (1)for some real numbers (a, /).Although r only contains assets with 0 cost, which leaves the scale and sign of : undeter-mined, we would like our candidate SDF to price other assets with positive prices.Therefore,we require a scale normalisation to rule out the trivial solution (a, /) = (0, 0) (see Cochrane(2001a, pp. 256-258)). For example, we could choose the popular asymmetric normalisationsa = 1 or 1(:) = a + /j = 1, where j = 1()).Alternatively, we could choose the symmetricnormalisation a2+/2= 1, together with a sign restriction on one of the nonzero coecients. Aswe shall see below, this seemingly innocuous issue may have important empirical consequences.In this section we assume that the pricing factor ) is itself the excess return on another4asset, such as the market portfolio in the CAPM.1As forcefully argued by Farnsworth et al.(2002) and Lewellen, Nagel and Shanken (2009) among others, the pricing model applies to )too, which means that1 [(a + /)) )] = 0. (2)Following Chamberlain (1983b) we also know that a +/) will constitute an admissible SDFif and only if )lies on the mean-variance frontier generated by )and r. Then, the well-known properties of mean-variance frontiers imply that the least squares projection of r ontoa constant and ) should be proportional to ).As result, we can equivalently write the abovepricing equations as1__(r d))(r d)))__ =__ 00__, (3)where the parameter vector d represents the slopes of the projection.2.1 Existing approachesEquations (1) and (2) are particularly amenable to GMM estimation once we choose anormalisation for (a, /). As we mentioned before, there are two widespread asymmetric choicesin empirical nance: a = 1 and a +/j = 1, with the corresponding SDFs typically expressed as: = 1 )` and : = 1 () j) t, respectively. Kan and Zhou (2000), Cochrane (2001b) andJagannathan and Wang (2002) only study the rst variant, but the second one is also widelyused in the literature (see e.g. Parker and Julliard (2005) or Yogo (2006)). Burnside (2007)refers to the rst approach as the a-normalisation and to the second one as the -normalisation.We will refer to them instead as the uncentred and centred SDF parametrisations since theydier in their use of either 1 (r)) or Co (r, )) in explaining the cross-section of risk premia.Specically, the uncentred SDF test relies on the overidentied, : + 1 linear moment condi-tions1__ r (1 )`)) (1 )`)__ = 0, (4)where the only unknown parameter is `. Given that the last moment condition implies that1 ()) = 1_)2_`, we will have that` = j, (5)where is the second moment of the factor, which allows us to interpret ` as a price of riskfor the factor. Under standard regularity conditions, the overidentifying restrictions (J) test1It is important to mention that our assets could include managed portfolios.Similary, the factor could alsobe a scaled version of a primitive excess return to accommodate conditioning information; see the discussion inchapter 8 of Cochrane (2001a).5will follow an asymptotic chi-square distribution with : degrees of freedom (2n) when (4) iscorrectly specied.In contrast, the centred SDF test relies on the overidentied, :+2 non-linear in parametersmoment conditions1__r (1 () j) t)) (1 () j) t)) j__ = 0, (6)where the two unknown parameters are (j, t), with the additional moment condition allowingfor the estimation of j. Once again, we can use the middle moment condition to show that:t =jo2, (7)where o2= j2denotes the variance of ), which means that t also has a price of riskinterpretation. Not surprisingly, the corresponding J test also converges in distribution to a 2nunder correct specication.The regression (or beta) representation of the pricing model is also amenable to GMM esti-mation.In particular, we can follow MacKinlay and Richardson (1991) in regarding (3) as 2:overidentied, linear moment conditions, where the : unknown parameters are the slope coe-cients d, which under the null coincide with both 1(r)),1()2) and Co(r,)),\ ()). Therefore,the J test will be asymptotically distributed as 2n under the null.The regression method identies j with the expected excess return of a portfolio whosebeta is equal to 1. Hence, this parameter may also be interpreted as an alternative price ofrisk. To estimate it, we can add ) j to (3) as in (6), and simultaneously obtain d and j.The overidentication tests are regularly complemented by three standard evaluation mea-sures, which correspond to the value of the dierent moment conditions when the linear factorpricing model is incorrect. In this way, we dene Jensens alphas:o = 1 (r) d1 ()) (8)for the regression method, the pricing errors obtained from the uncentred SDF representation:r = 1 (r) 1 (r)) `,or from the centred SDF representation:) = 1 (r) 1 (r () j)) t.Under the null hypothesis these three measures should be simultaneously 0, but otherwise theirvalues will be dierent.62.2 Numerical equivalence resultsAs we mentioned in the introduction, Kan and Zhou (2000, 2002), Cochrane (2001b), Ja-gannathan and Wang (2002), Burnside (2007) and Kan and Robotti (2008) compare some ofthe aforementioned approaches when researchers rely on traditional, two-step or iterated GMMprocedures. In contrast, our main result is that all the methods coincide if one uses insteadsingle-step procedures such as CU-GMM. More formally:Proposition 1 If we apply single-step procedures to the uncentred SDF method based on themoment conditions (4), the centred SDF method based on the moment conditions (6), and theregression method based on the moment conditions (3), then for a common specication of thecharacteristics of the HAC weighting matrix the following numerical equivalences hold:1) The three overidentication restrictions (J) tests.2) The direct estimate of the price of risk ` from (4), the indirect estimate` = o2 t,from (6) extended to include , and the indirect estimate` = jfrom (3) extended to include (j, ). Analogous results apply to t and j.3) The estimates of Jensens alphas in (8) obtained by replacing 1 () by an unrestricted sampleaverage and d by their direct estimates obtained from the regression method, or the indirectestimates obtained from SDF methods extended to include d. Analogous results apply to thealternative pricing errors r and ).Importantly, these numerical equivalence results do not depend in any sense on the number ofassets, the sample size, or indeed the number of factors, and remain true regardless of the validityof the asset pricing restrictions. In order to provide some intuition, imagine that for estimationpurposes we assumed that the joint distribution of r and ) is i.i.d. multivariate normal. Inthat context, we could test the mean-variance eciency of ) by means of a likelihood ratio(LR) test. We could then factorise the joint log-likelihood function of r and ) as the marginallog-likelihood of ), whose parameters j and o2would be unrestricted, and the conditional log-likelihood of r given ). As a result, the LR version of the original Gibbons, Ross and Shanken(1989) test would be numerically identical to the LR test in the joint system irrespective ofthe chosen parametrisation. The CU-GMM overidentication test, which implicitly uses theGaussian scores as inuence functions, inherits the invariance of the LR test.The advantage,though, is that we can make it robust to departures from normality, serial independence orconditional homoskedasticity.Alternatively, we could understand the CU-GMM procedure as being based on the originalmoment conditions (1) and (2), which are valid for any normalisation of the SDF scale of the7form a2+/2,= 0. In this light, the equivalence between the two SDF approaches is a direct con-sequence of the fact that single-step procedures are numerical invariant to normalisation, whilethe additional, less immediate results relating the regression and SDF approaches in Proposition1 follow from the fact that those GMM procedures are also invariant to parameter dependentlinear transformations of the moments and reparametrisations (see appendix A).One drawback of CU-GMM and other GEL procedures, though, is that they induce a non-linearity in the GMM objective function of the uncentred SDF and regression approaches, whichmay induce multiple local minima. In this sense, the uncentred SDF method has a non-trivialcomputational advantage because it only contains a single unknown parameter.2At the sametime, one can also exploit the numerical equivalence of the three approaches to check that aglobal minimum has been reached. A much weaker convergence test is given by the fact that thevalue of the criterion function at the CU-GMM estimators cannot be larger than at the iteratedGMM estimators (see Hansen, Heaton and Yaron (1996)).In any case, it is convenient to have consistent initial parameter values. For that reason,we propose a computationally simple intuitive estimator that is always consistent, but whichwould become ecient for i.i.d. elliptical returns, a popular assumption in nance because itguarantees the compatibility of mean-variance preferences with expected utility maximisationregardless of investors preferences (see Chamberlain (1983a) and Owen and Rabinovitch (1983)).In particular, if we derive the optimal moment condition for the uncentred SDF model (seeHansen (1982)), then we can immediately show that:Lemma 1 If (rt, )t) is an i.i.d. elliptical random vector with bounded fourth moments, and thenull hypothesis of linear factor pricing holds, then the most ecient estimator of ` obtainedfrom (4) will be given by_`T =

Tt=1 )t

Tt=1 )2t. (9)Intuitively, this means that under those circumstances (2), which is the moment involving), exactly identies the parameter `, while (1), which are the moments corresponding to r,provide the : overidentication restrictions to test. Although the elliptical family is ratherbroad (see e.g. Fang, Kotz and Ng (1990)), and includes the multivariate normal and Studentt distribution as special cases, it is important to stress that _`T will remain consistent underlinear factor pricing even if the assumptions of serial independence and ellipticity are not totallyrealistic in practice.32This advantage becomes more relevant as the number of factors I increases because the centred SDF methodrequires the additional estimation of I factor means, while the regression method the estimation of nI factorloadings.3We can also prove that we obtain an estimator of ` that is asymptotically equivalent to (9) if we followSpanos (1991) in assuming that the so-called Haavelmo distribution, which is the joint distribution of the T(n+1)observed random vector (r1, )1, . . . , rt, )t, . . . , rT, )T), is an ane transformation of a scale mixture of normals,8A rather dierent justication for the estimator (9) is that it coincides with the GMMestimator of ` that we would obtain from (4) if we used as weighting matrix the second momentof the vector of excess returns (r, )), as recommended by Hansen and Jagannathan (1997).Hansen, Heaton and Yaron (1996) also indicate that CU-GMM occasionally generates ex-treme estimators that lead to large pricing errors with even larger variances.In those circum-stances, we would suggest the imposition of good deal restrictions (see Cochrane and Saa-Requejo(2000)) to rule out implausible results.42.3 Empirical applicationOver the last thirty years most empirical studies have rejected the hypothesis of uncoveredinterest parity, which in its basic form implies that the expected return to speculation in theforward foreign exchange market conditioned on available information should be zero.Speci-cally, many of those studies nd support for the so-called the forward premium puzzle, whichimplies that, contrary to the theory, high domestic interest rates relative to those in the for-eign country predict a future appreciation of the home currency. In fact, the so-called carrytrade, which involves borrowing low-interest-rate currencies and investing in high-interest-rateones, constitutes a very popular currency speculation strategy developed by nancial marketpractitioners to exploit this anomaly (see Burnside et al. (2006)).One of the most popular explanations among economists is that such a seemingly anomalouspattern might reect a reward to the exposure of foreign currency positions to certain systematicrisk factors. To study this possibility, Lustig and Verdelhan (2007) constructed eight portfoliosof currencies sorted at the end of the previous year by their nominal interest rate dierentialto the US dollar, creating in this way annual excess returns (in real terms) on foreign T-Billinvestments for a US investor over the period 1953-2002. Then they used two-pass regressions totest if some popular empirical asset pricing models that rely on certain domestic US risk factorswere able to explain the cross-section of risk premia.Table 1 reports some simple descriptive statistics for those portfolios. Interestingly, thebroadly monotonic relationship between the level of interest rates dierentials and risk premiaprovides informal evidence on the failure of uncovered interest rate parity.and therefore elliptical. Intuitively, the reason is that a single sample realisation of such a Haavelmo distributionis indistinguishible from a realisation of size T of an i.i.d. multivariate normal distribution for (rt, ft).4Specically, given that we know from Hansen and Jagannathan (1991) thatcv2 12(i)\ (i) = 12,where cv is the maximum attainable Sharpe ratio of any portfolio of the assets under consideration, and 12is thecoecient of determination in the (theoretical) regression of ) on a constant and the tradeable assets, one couldestimate the linear factor pricing model subject to implicit restrictions that guarantee that the values of cv or thecoecient of variation of m computed under the null should remain within some loose but empirically plausiblebounds. In the case of traded factors both bounds should coincide because 12= 1.9(TABLE 1)Given that for pedagogical reasons we have only considered a single traded factor in ourtheoretical analysis, we focus on the CAPM. Following Lustig and Verdelhan (2007), we take thepricing factor to be the US market portfolio (MK), which we also identify with the CRSP value-weighted excess return. Table 2 contains the results of applying the dierent inference procedurespreviously discussed to this model. In all cases, we estimate the asymptotic covariance matrixof the relevant inuence functions by means of its sample counterpart, as in Hansen, Heatonand Yaron (1996), except for the rst-step estimators, for which we use the identity matrix asinitial weighting matrix.5The rst thing to note is that the value of CU-GMM J statistic (=21.375, 0.6% p-value)is the same across the three methods, as stated in point 1 of Proposition 1. In contrast, thereare marked numerical dierences between the three two-step versions of the overidenticationrestrictions test. In particular, the centred SDF yields a much higher J statistic. These numericaldierences are substantially reduced but not eliminated if we use those two-step GMM estimatesto compute a new weighting matrix, which we then use to obtain three-step parameter estimates,and so on and so forth.6On the other hand, while the two-step, iterated and CU-GMM estimatesof ` and j are fairly similar, the CU-GMM estimate of t is higher than its two-step counterpart,although the t-ratio is lower. In any case, all tests reject the null hypothesis of linear factorpricing. Interestingly, these rejections do not seem to be due to poor nite sample propertiesof the J statistics in this context since the 1 version of the Gibbons, Ross and Shanken (1989)regression test, which remains asymptotically valid in the case of conditional homoskedasticity,yields a p-value of 0.3%.(TABLE 2)The J tests reported in Table 2 can also be interpreted as distance metric tests of the nullhypothesis of zero pricing errors in the eight currency returns only. The rationale is as follows.A distance metric test, which is the GMM analogue to a likelihood ratio statistic, is given by thedierence between the criterion function under the null and the alternative. But if we saturate(1) by adding : pricing errors, then the joint system of moment conditions becomes exactly5We have also considered another two-step GMM procedure that as rst-step estimator uses (9), whose nu-merical value is 0.0186, but the results are qualitatively similar.6As Hansen, Heaton and Yaron (1996) show, though, such iterated GMM estimators do not generally coincidewith the CU ones.10identied, which in turn implies that the optimal criterion function under the alternative willbe zero.Similarly, we can also consider the distance metric test of the null hypothesis of zero pricingerror for the traded factor. Once again, the criterion function under the null takes the valuereported in Table 2. Under the alternative, though, we need to conduct a new estimation.Specically, if we saturate the moment condition (2) corresponding to the traded factor byadding a single pricing error, then the exact identiability of this modied moment conditionmeans that the joint system of moment conditions eectively becomes equivalent to anothersystem that relies on (1) only. Such a model delivers a borderline signicant t ratio for `, anda CU-GMM J test of 7.601, whose p-value is 0.42. As a result, the CAPM restrictions arenot rejected when we do not force this model to price the market. In contrast, the distancemetric test of zero pricing error for the traded factor, which is equal to the dierence betweenthis J statistic and the one reported in Table 2, is 14.314, with a tiny p-value.Therefore, thefailure of the CAPM to price the US stock market portfolio provides the clearest source of modelrejection, thereby conrming the relevance of the recommendation in Farnsworth et al. (2002)and Lewellen, Nagel and Shanken (2009).Importantly, these distance metric tests avoid the problems that result from the degeneratenature of the joint asymptotic distribution of the pricing error estimates recently highlighted byGospodinov, Kan and Robotti (2010), and which would be particularly relevant in the ellipticalcase in view of Lemma 1.Table E1 in the supplemental appendix also conrms the numerical equality of the CU-GMM estimators of prices of risk (`, t and j) and pricing errors (o, r and )) regardless of theapproach used to estimate them, as expected from points 2 and 3 of Proposition 1. In contrast,two-step and iterated GMM yield dierent results, which explains the three dierent columnsrequired for each of them. Finally, Figure 1, which plots the CU-GMM criterion as a functionof `, conrms that we have obtained a global minimum.3 Non-traded factorsLet us now consider a situation in which ) is a scalar non-traded factor, such as the growthrate of per capita consumption.The main dierence with the analysis in section 2 is that thefactor may not satisfy the pricing equation (2), so that the SDF is simply dened by (1). As inthe case of a traded factor, we can equivalently write this pricing condition as a restriction onthe least squares projection of r onto a constant and ). Specically, if we dene = 1(r) dj11andd = Co (r,)) ,o2(10)as the vectors of intercepts and slopes in that projection, respectively, then (1) is equivalent to1 [(a + /)) ( +d))] = c + dd = 0, (11)where c and d are two scalars not simultaneously equal to 0, so that the projection and the SDFshould be orthogonal. Intuitively, (1) implies that we can nd a non-trivial linear combination of1(r) and 1(r)) (or Co(r, ))) that is zero, which in turn implies that we can nd a non-triviallinear combination of and d that is zero too.7In practice, we can easily impose the implicit constraint (11) by writing the moment condi-tions that dene the projection as1__(r (d c)))(r (d c))))__ =__ 00__, (12)where is some : 1 vector such that = d and d = c. This moment condition closelyresembles (3), except for the fact that when ) is traded, the additional condition (2) impliesthat = 0 and d = 0. More generally, we need to solve the scale indeterminacy of c and d bychoosing either the popular asymmetric normalisation c = 1, so that d = , or the symmetricnormalisation c2+d2= 1. As we shall see below, this seemingly innocuous issue may once againhave important empirical consequences.3.1 Existing approachesAs we mentioned in the case of traded factors, some normalisation is required to identify(a, /) from (1). Specically, the uncentred SDF approach implicitly sets a = 1, and relies on the: overidentied, linear moment conditions1_ r (1 )`) _ = 0, (13)where the only parameter to estimate is `. This parameter still has a price of risk interpreta-tion, as in (5), but this time in terms of factor mimicking portfolios. In particular` = 1(r+)1(r+2), (14)wherer+= 1()r0)11(rr0)r (15)7Note that c = o + /j is the mean of the SDF, while d = oj + / would be the price of ) if it was traded.12is the uncentred least squares projection of ) on r. As expected, the asymptotic distributionof the J test will be 2n1 under the null. For that reason, in what follows we assume that thenumber of assets exceeds the number of factors to ensure that the linear factor pricing modelimposes testable restrictions on asset returns.In contrast, the centred SDF test implicitly sets a +/j = 1, and relies on the overidentied,: + 1 non-linear in parameters moment conditions1__ r (1 () j) t)) j__ = 0, (16)where the two parameters to estimate are (j, t). Once again, t has a price of risk interpretationas in (7), but in terms of factor mimicking portfolios. Specicallyt = 1(r++)\ (r++), (17)wherer++= co(), r0)\ 1(r)ris the centred least squares projection of ) on r. The link to the uncentred price of riskparameter is simplyt =`1 `j,which eectively divides ` by the SDF mean since the centred SDF approach normalises thatmean to 1. As before, the J test will asymptotically converge to a 2n1 when the asset pricingmodel is correct.On the other hand, if we normalise c = 1 then we can write (12) as the following 2:overidentied, non-linear in parameters moment conditions:1__(r d({ + )))(r d({ + ))))__ = 0, (18)where the : + 1 parameters to estimate are ({, d), with { = c j, so that we can interpret cas the expected excess return of a portfolio whose beta is equal to 1.8Given that the riskpremium of r++is t\ (r++) and its beta is \ (r++),o2, there is again a simple connection tothe price or risk of the centred SDF. Specically,c = o2t, (19)8Jagannathan and Wang (2002) use c j instead of {, and add the inuence functions ) j and () j)2o2to estimate j and o2too. The addition of these moments is irrelevant for the estimation of { and the J testbecause they exactly identify j and o2. Consequently, we will ignore them for the time being to simplify theexposition, although we will use them in our proofs to link the regression and SDF approaches.13which simply re-scales t by the factor variance. This is the usual regression (or beta) test ofthe pricing model, which implicitly exploits the restrictions on the regression intercepts (seeCampbell, Lo and MacKinlay (1996, chap. 5)).Finally, the centred and uncentred pricing errors ) and r are dened as in section 2, whileJensens alpha is now dened aso = 1 (r) dc. (20)Unfortunately, the existing approaches may run into diculties in those cases in which theirimplicit normalisations are invalid. For instance, if 1 (r)) = 0 so that the true SDF wouldbe proportional to ) and r+= 0, then the normalisation of (1) with a = 1 in the uncentredSDF approach (13) is not well dened in population terms. Similarly, if Co (r,)) = 0 so thatthe true SDF would be proportional to () j) and r++= 0, then neither the normalisationof (1) with a + /j = 1 in the centred SDF approach (16), nor the normalisation of (12) withc = 1 in the centred regression approach (18) are properly dened. In contrast, the symmetricnormalisations a2+ /2= 1 and c2+ d2= 1 continue to be well dened in those circumstances.We shall return to this issue in the empirical application.3.2 Numerical equivalence resultsAs in the case of traded factors, we can show that all the approaches discussed in theprevious subsection coincide if one uses single-step methods such as CU-GMM. More formally9Proposition 2 If we apply single-step procedures to the uncentred SDF method based on themoment conditions (13), the centred SDF method based on the moment conditions (16), and theregression method based on the moment conditions (18), then for a common specication of thecharacteristics of the HAC weighting matrix the following numerical equivalences hold:1) The three overidentication restrictions (J) tests.2) The direct estimate of the price of risk ` from (13), the indirect estimate` =t1 + tjfrom (16), and the indirect estimate of` =co2 + cj9We could also consider a nonlinear SDF such as i = )

, with 0 unknown, so that the moments would become1(r)

) = 0.In this context, we can easily show that the CU-GMM overidentifying restrictions test would be numericallyequivalent to the one obtained from the regression-based moment conditions12664(r m()

mjm))(r m()

mjm))))

)

jm)2m3775 = 0,whose unkown parameters are (0, m, jm, m).14from (18) parametrised in terms of c and extended to include , provided that 1 + tj=_o2+ cj_,o2,= 0. Analogous results apply to t and c.3) The estimates of Jensens alphas in (20) obtained by replacing 1 () by an unrestricted sampleaverage and d and c by their direct estimates obtained from the regression method or the indirectestimates obtained from SDF methods extended to include d, and j. Analogous results applyto the pricing errors r and ).It is important to distinguish this proposition from the results in Jagannathan and Wang(2002) and Kan and Zhou (2002). These authors showed that the centred regression and uncen-tred SDF approaches lead to asymptotically equivalent inferences under the null and compatiblesequences of local alternatives in single factor models. In contrast, Proposition 2 shows thatin fact both SDF approaches and the regression method yield numerically identical conclusionsif we work with one-step GMM procedures such as CU-GMM. Since our equivalence result isnumerical, it holds regardless of the validity of the pricing model, and does not depend on :, Tor the number of factors.10The numerical equivalence of the three approaches gives once more a non-trivial computa-tional advantage to the uncentred SDF method, which only contains the single unknown para-meter `.11At the same time, one can also exploit the fact that the three approaches coincideto check that a global minimum has been obtained.Still, it is convenient to have consistent initial values. For that reason, we propose a compu-tationally simple intuitive estimator that is always consistent, but which would become ecientwhen the returns and factors are i.i.d. elliptical, which nests the joint Gaussian assumption inJagannathan and Wang (2002). In particular, if we derive the optimal moment condition forthe uncentred SDF model in this context, then we can immediately show that:Lemma 2 If (rt, )t) is an i.i.d. elliptical random vector with bounded fourth moments, and thenull hypothesis of linear factor pricing holds, then the most ecient estimator of ` obtainedfrom (13) will be given by

`T =

Tt=1 r+t

Tt=1 r+2t(21)where r+tis the uncentred factor mimicking portfolio dened in (15), whose sample counterpartwould be~ r+t=_T

s=1)sr0s__T

s=1rsr0s_1rt.10Kan and Robotti (2008) also show that CU-GMM versions of the SDF approach are numerically invariant toane transformations of the factors with known coecients, which is not necessarily true of two-step or iteratedGMM methods. Not surprisingly, it is easy to adapt the proof of Proposition 2 to show that the regressionapproach is also numerically invariant to such transformations.11This advantage becomes more relevant as the number of factors I increases because the centred SDF methodrequires the additional estimation of I factor means, while the regression method the estimation of nI factorloadings if we use the parametrisation in terms of {, and even more elements if we use the parametrisation interms of .15Once again, it is important to stress that the feasible version of `T will remain consistentunder linear factor pricing even if the assumptions of serial independence and a multivariateelliptical distribution are not totally realistic in practice.Similarly, (21) also coincides with the GMM estimator of ` that we would obtain from (13)if we used as weighting matrix the second moment of the excess returns in r, as recommendedby Hansen and Jagannathan (1997).3.3 Empirical applicationWe look again at the eight currency portfolios in Lustig and Verdelhan (2007), but this timewe focus on linear factor pricing models with non-traded factors. Given that for pedagogicalreasons we have only considered single factor models in our theoretical analysis, we considera linearised version of the CCAPM, which denes the US per capita consumption growth ofnondurables as the only pricing factor.Table 3 displays the results from the application of the dierent inference procedures previ-ously discussed to this data set for the purposes of testing the CCAPM. The computation of theweighting matrix in two-step and CU-GMM is the same as we explained for the case of tradedfactors.12As stated in point 1 of Proposition 2, the value of CU-GMM J statistic (=5.663,58% p-value) is the same across the three methods. Therefore, in this case this J test doesnot reject the null hypothesis implicit in (1) and (12), which is in agreement with the empiricalresults in Lustig and Verdelhan (2007).This result is conrmed by a p-value of 83.9% for thetest of the same null hypothesis computed from the regression using the expressions in Beatty,LaFrance and Yang (2005). Their 1-type test is asymptotically valid in the case of conditionalhomoskedasticity, and may lead to more reliable inferences in nite samples.In contrast, there are again numerical dierences between the standard two-step GMM imple-mentation of the three existing approaches. Unlike what happened in the case of traded factors,though, in this case the numerical dierences lead to dierent conclusions at conventional sig-nicance levels. Specically, while the centred SDF approach rejects the null hypothesis, theuncentred SDF and the regression approach do not. These numerical dierences are attenuatedwhen we use iterated GMM procedures, but the conicting conclusions remain. On the otherhand, while the two-step, iterated and CU-GMM estimates of ` and c are fairly close, the CU-GMM estimate of t is higher than its two-step and iterated counterparts, although the t-ratiois much lower.(TABLE 3)12We have also considered another two-step GMM procedure that as rst-step estimator uses (21), whosenumerical value is 0.492, but the results are qualitatively similar.16Table E2 in the supplemental appendix also conrms the numerical equality of the CU-GMMestimators of prices of risk (`, t and j) and pricing errors (c, and c) regardless of the approachused to estimate them, as expected from points 2 and 3 of Proposition 2. In contrast, two-stepand iterated GMM yield dierent results, which explains the three dierent columns required foreach of them. Finally, Figure 2, which plots the CU-GMM criterion as a function of `, conrmsthat we have obtained a global minimum.Importantly, Burnside (2007) argues that the usual two-step implementation of the uncen-tred SDF approach has no power against potentially misspecied SDFs when the populationcovariance of the pricing factors with the excess returns on the assets is 0. Similarly, one couldeasily modify his arguments to say that the usual two-step implementation of the centred SDFapproach would have no power if the cross moment between pricing factors and excess returnswere 0, and the same would apply to the centred regression approach.Given the numerical equivalence of the single-step implementation of the three approachesin Proposition 2, in our view the focus should not be on the statistical properties of the dierentestimators and tests, but rather, on the interpretation of the restrictions that are eectivelytested in those two special cases.For the sake of clarity, let us study these issues with a single factor. Specically, whenCo(r, )) = 0 but 1(r) ,= 0, (12) will be satised with c = 0 and d = 1(r), while the momentconditions (1) will be satised by any ane transformation of ) such that a+/j = 0. Therefore,the value of ` given by (14) will trivially satisfy (13), as Burnside (2007) shows. In contrast, onecannot nd any nite value of t that will satisfy (16) because the centred mimicking representingportfolio r++will be 0. Likewise, one cannot nd any nite value of { that will satisfy (18)because d = 0. Therefore, the lack of correlation between excess returns and consumptiongrowth could explain the striking numerical dierences between the empirical results obtainedwith the centred and uncentred SDF moment conditions using two-step and iterated GMM.Similarly, when 1(r)) = 0 but 1(r) ,= 0, (12) will be satised with dj c = 0 andd_o2,_ = 1(r), while the moment conditions (1) will be satised by any SDF which isexactly proportional to ) (so that a = 0). Therefore, the value of t given by (17) will triviallysatisfy (16), and the same applies to (18) with (10) and { = ,j. In contrast, one cannot ndany nite value of ` that will satisfy (13) because the uncentred mimicking portfolio r+will be0.From an economic point of view those solutions are clearly unsatisfactory, but strictly speak-ing the corresponding SDFs correctly price the vector of excess returns at hand. In our view,the best way to solve these problems would be to add assets whose cost is not 0, which would17implicitly x the scale of the SDF by xing its mean; see e.g.Hodrick and Zhang (2001) andFarnsworth et al. (2002). For that reason, we devote the section 5 to this case. Unfortunately,most empirical studies, including Lustig and Verdelhan (2008) only include zero cost assets.From an econometric point of view, though, the truly problematic case arises when (1) and(12) hold but 1(r) = 0, in which case both Co(r, )) and 1(r)) must be 0 too. In this situation,the SDF parameters a and / and the projection parameters c and d are underidentied evenafter normalisation, which renders standard GMM inferences invalid.13To investigate whether these theoretical situations are empirically relevant,we performsingle-step GMM overidentication tests of the following null hypotheses:(a) the mean excess return is 0, i.e.1(r) = 0,(b) the cross moment between excess returns and factor is 0, i.e.1(r)) = 0,and (c) the covariance between excess returns and factor is 0, i.e.1__r (r ) )__ =__ 00__,where is a vector of parameters to estimate.Hypothesis (a) is clearly rejected with a J statistic of 39.97, whose p-value is essentially 0.Therefore, there are statistically signicant risk premia in search of pricing factors to explainthem. The next step is to investigate if US consumption growth can play such a role.Hypothesis (b) is also unambiguously rejected with a statistic of 34.85 and a p-value close to0, but there is not much evidence against hypothesis (c) for non-durable consumption, with atest statistic of 8.39 and p-value of almost 40%. Thus, we cannot reject that this factor is jointlyuncorrelated with the currency portfolios, which indicates that the seemingly positive evaluationof the consumption based asset pricing model in Table 3 must be interpreted with some care.In this sense, note that the joint test of (c) is eectively testing that any portfolio formedfrom the original eight currency portfolios in Lustig and Verdelhan (2008) is uncorrelated to USconsumption growth. In this sense, it is worth noting that once again CU-GMM proves usefulto unify the empirical results because the zero covariance test above is numerically equivalentto a test that all the betas are 0. For analogous reasons, we obtain the same J test whether weregress r on ) or ) on r. This lack of correlation does not seem to be due to excessive reliance13See Kan and Zhang (1999) or Burnside (2007) for the implications that identication failures have for two-stepGMM procedures.18on asymptotic distributions, because it is corroborated by a p-value of 81.7% for the 1 testof the latter regression, which like the corresponding LR test, is also invariant to exchangingregressand and regressors. Obviously, if we computed t-tests between every conceivable portfolioand consumption growth, a non-negligible fraction of them will be statistically signicant, so theusual trade o between power and size applies (see Lustig and Verdelhan (2008) and Burnside(2009) for further discussion of this point). In any case, the number of portfolios must be strictlylarger than the number of pricing factors for (1) to have testable implications.Finally, the joint hypothesis (a)+(b), or equivalently (a)+(c), is also rejected with a statisticof 53.039 and a negligible p-value, which conrms that the parameters appearing in (1) and(12) are point identied in a single factor model. Interestingly, this test coincides with a simpleversion of the underidentication test of Arellano, Hansen and Sentana (2009) adapted to linearfactor pricing models by Manresa (2008).In summary, our empirical results with a non-traded factor indicate that although we cannotreject the overidentication restrictions implicit in (1) and (12), this may be due to the factthat US consumption growth is uncorrelated to the eight portfolios of currencies in Lustig andVerdelhan (2007). In this sense, the CCAPM results are very similar to the ones obtained withthe CAPM when we treat the market portfolio as non-traded, as we mentioned at the end ofsection 2, which is not very surprising given that the correlations between the eight currencyportfolios and the excess returns on the US market portfolio and consumption growth are ofsimilar order.4 Mixed factorsLet us now consider a model with two pricing factors in which )1 is a scalar traded factor,such as the market portfolio, and )2 is another scalar non-traded factor, such as growth rate ofper capita consumption. An important example of such a mixed linear factor pricing model willbe the linearised CCAPM with Epstein and Zin (1991) preferences. Apart from the multifactornature of this model, the main dierence with the analysis in the previous sections is that while)1 must satisfy the pricing equation (2), )2 does not. As a result, if we dene I0 = ()1, )2), thenthe SDF will be implicitly dened by1__(a +L0I ) r(a +L0I ) )1__ = 0, (22)subject to some scale normalisation of the vector (a, L).19In this context, the uncentred test relies on the overidentied, :+1 linear moment conditions1__r (1 I0X))1 (1 I0X)__ = 0,where the two unknown parameters are contained in X. In contrast, the centred SDF test relieson the : + 3 overidentied, nonlinear in parameters moment conditions1__r_1 (I )0r_)1_1 (I )0r_I __ = 0,whose four unknown parameters are in and r. Finally, the regression version can be writtenin terms of the following 3: overidentied, nonlinear in parameters moment conditions:1__r d1)1d2 ({2 + )2)(r d1)1d2 ({2 + )2)) )1(r d1)1d2 ({2 + )2)) )2__ = 0,where the 2: + 1 parameters to estimate are ({2, d1, d2), with (d1[d2) = co(r, I )\ 1(I ) and{2 = c2 j2, so that we can interpret c2 as the expected excess return of a portfolio whosebetas on )1 and )2 are equal to 0 and 1, respectively.Not surprisingly, it is straightforward to combine the proofs of Propositions 1 and 2 to showthat all these three approaches numerically coincide if one uses single-step methods such asCU-GMM.Table 4 contains the results of estimating a linearised version of the CCAPM with Epsteinand Zin (1991) preferences with the dataset that we have been considering. This amounts toidentifying )1 with the US market portfolio and )2 with US per capita consumption growth.Therefore, this model nests both the CAPM and the CCAPM studied in previous sections.(TABLE 4)As can be seen from Table 4, the common CU-GMM J statistic is 4.785, with a p-valueof 68.6%, so like the CCAPM considered in the previous section, the combined model is notrejected even though we attempt to price the market portfolio as in the CAPM. In addition, theestimates of X show that consumption growth and not the market portfolio is the driving forcebehind this model.In addition, Figure 3, which plots the CU-GMM criterion as a function ofthe vector X, conrms that we have obtained a global minimum.20Once again, the two-step, iterated and CU GMM implementations of the uncentred SDFapproach provide similar results, with slightly higher dierences for the regression method. Butthe results of the two-step implementation of the centred SDF approach clearly diverge fromthe CU-GMM results, with negative estimates of the prices of risk and a very high J statistic.Moreover, in this case the iterated GMM fails to converge, cycling over three dierent solutions,and for that reason we do not report them.Once again, the cause of the wedge between the results obtained with the centred anduncentred SDF moment conditions using two-step GMM seems to be the lack of correlationbetween consumption growth and asset returns. Specically, if Co(r, )2) = 0 and Co()1, )2) =0 then (22) will be satised with /1 = 0 and a + /2j2 = 0 even though 1(r) is not proportionalto Co(r, )1). This is another example of an economically unattractive SDF with 1(:) = 0,which nevertheless satises the moment restrictions, as the CU GMM estimates conrm. Inthis context, the uncentred SDF equations will trivially hold, while the centred SDF equationscannot hold because they implicitly normalise with 1(:) = 1. Not surprisingly, exactly theopposite situation would arise if 1(r)2) = 0 and 1()1)2) = 0.As we mentioned in section 3, the most satisfactory solution to these normalisation-inducedproblems would be to add assets with non-zero cost, which are unfortunately not readily availablein the Lustig and Verdelhan (2008) dataset.Still, it is worth extending our theoretical resultsto this case.5 Extension to gross returnsLet us dene H as an : 1 vector of gross returns, which are such that their cost is givenby the :1 vector of ones /n. We focus again on the case of a single factor ) to simplify theexposition, and relegate the general multifactor case to the supplemental appendix.In this context, the analogue to the SDF pricing equation (1) is1 ((a + /)) H) = /n(23)for some real numbers (a, /).As we mentioned in the previous section, there is no longer anyneed to normalise (a, /), unlike in the case of excess returns.Not surprisingly, we can equivalently write this pricing condition as a restriction on the leastsquares projection of H onto a constant and ). Specically, if we dene and d as the vectorsof intercepts and slopes in that projection, respectively, then (23) is equivalent toc + dd = /n, (24)21where c is the SDF mean and d is the shadow price of the factor, or its actual price when istraded.When the factor is a gross return, it will also satisfy1 ((a + /)) )) = 1. (25)Therefore, the moments of the uncentred SDF method will be (23) and (25). Since the onlyparameters are (a, /), the J test will have : 1 degrees of freedom under correct specication.We can also reparametrise the SDF in terms of c instead of a and rely on the momentconditions1__H(c () j) t) /n) (c () j) t) 1) j__ = 0,whose parameters are (c, t, j), which are analogous to (6).Regarding the regression approach, the fact that the factor is traded implies that d = 1,which simplies the least squares constraint to = i(/nd) , i = 1,c,when c ,= 0.14The parameter i is usually referred to as the zero-beta return since it correspondsto the expectation of returns uncorrelated with ).Therefore, the moments associated to the regression method will be1__((Hi/n) d () i))((Hi/n) d () i)) )__ = 0,with parameters (d, i).We show in the supplemental appendix that there is a direct counterpart to Proposition 1for gross returns too, so that all the approaches are numerically equivalent when implementedby single-step GMM methods.Let us now consider the case of a non-traded factor, in which case the moments of the SDFmethod are simply (23) with parameters (a, /).Consequently, the J test has : 2 degrees offreedom under correct specication.14The case c = 0 is such that ) is equal to the minimum variance portfolio, whose return 1

satisesCo (R, 1

) = \ ov (1

) /n.It is well known that this asset is the only element of the mean-variance frontier that cannot be used as abenchmark in beta pricing.22Once again, we can also reparametrise the SDF in terms of c instead of a and rely on themoment conditions1__ H(c () j) t) /n) j__ = 0with parameters (c, t, j), which are analogous to (16).On the other hand, when c ,= 0 we can re-express the least squares constraint as = i/n +d{, i = 1,c, { = d,c.Therefore, the moments of the regression method are1__((Hi/n) d () +{))((Hi/n) d () +{)) )__ = 0,with parameters (d,i, {).We show in the supplemental appendix that there is a direct counterpart to Proposition 2 forgross returns too, so that all the approaches are numerically equivalent when implemented bysingle-step GMM methods. More generally, it is straightforward to show that analogous resultshold for the mixed case discussed in section 4.Finally, it is worth explaining the advantages of working with gross returns in the situationdiscussed in section 3.3 in which the non-traded factor is uncorrelated with excess returns.For simplicity, let us assume that the reference asset used in computing r is a portfolio whosegross return has zero variance, so that Co(H, )) = 0 if and only if Co(r, )) = 0. As wementioned in that section, the moment condition (1) will hold with a = /j regardless of 1 (r),which corresponds to a valid but economically unsatisfactory SDF. In contrast, (23) cannotbe satised unless 1 (H) is proportional to /n. If this risk neutral condition does not hold,then the J test based on H will reject with probability 1 in large samples, while the rejectionprobability of the J test based on r will coincide with its size.On the other hand, an unsatisfactory SDF that will satisfy (23) would be one in which thefactor is correlated with returns but cannot discriminate across assets, so that Co(H, )) = //nfor some scalar constant / ,= 0. Similar arguments apply to 1(H)).6 ConclusionsThere are two main approaches in empirical nance to evaluate linear factor pricing models.The oldest method relies on regressions of excess returns on factors, while the other more recentmethods rely instead on the SDF representation of the model. There are two variants of theSDF approach, one that subtracts the mean of the factors and another one which does not.23Given that these dierent procedures may lead to dierent empirical conclusions, it is perhapsnot surprising that there has been some controversy about which approach is most adequate.In this context, our paper shows that if we use single-step methods such as CU-GMM insteadof standard two-step or iterated GMM procedures, then all these procedures provide the sameestimates of prices of risk, overidentifying restrictions tests, pricing errors, and Jensens alphasirrespective of the validity of the model, and regardless of : and T. In this way, we eliminate thepossibility that dierent researches report potentially contradictory results with the same dataset.We prove our numerical equivalence results hold for any combination of traded and non-traded factors, and also for excess returns and gross returns. Thus, we would argue that ineect the regression and SDF approaches are dierent representations of a single method toempirically evaluate asset-pricing models. Nevertheless, the uncentred SDF method has a non-trivial computational advantage because it contains fewer unknown parameters. At the sametime, one can also exploit the numerical equivalence of the three approaches to check that aglobal minimum has been reached.For the benet of practitioners, we also develop simple, intuitive consistent parameter es-timators that can be used to obtain good initial conditions for CU-GMM, and which will beecient for elliptically distributed returns and factors. Interestingly, these consistent estimatorscoincide with the GMM estimators recommended by Hansen and Jagannathan (1997), whichuse as weighting matrix the second moment of returns.We illustrate our results with the currency portfolios constructed by Lustig and Verdelhan(2007). We consider three popular linear factor pricing models: the CAPM, as well as linearlisedversions of the Consumption CAPM and the Epstein and Zin (1989) model. Our ndings clearlypoint out that the conict among criteria for testing asset pricing models that we have previouslymentioned is not only a theoretical possibility, but a hard reality. Nevertheless, such a conictdisappears when one uses single-step methods.A dierent issue, though, is the interpretation of the restrictions that are eectively tested.In this sense, our results conrm Burnsides (2007) ndings that the discrepancies betweentraditional estimators are due to the fact that US consumption growth seems to be uncorrelatedto currency returns. We also nd that if we force the CAPM to price the market portfolio, thenwe reject the asset pricing restrictions.Given that our results are not specic to CU-GMM, it might be worth comparing the prop-erties in nite samples of dierent members of the generalised empirical likelihood family ofestimators. In addition, it would be interesting to compare them in our empirical application.24An alternative application of our numerical equivalence results would be the performanceevaluation of mutual and hedge funds. This literature can also be divided between papersthat rely on regression methods, such as Kosowski et al. (2006), and papers that rely on SDFmethods, such as Dahlquist and Soderlind (1999) and Farnsworth et al. (2002). Undoubtedly,these topics constitute interesting avenues for further research.25ReferencesAlmeida, C. and R. Garcia (2009): Assessing misspecied asset pricing models with empir-ical likelihood estimators, mimeo, Fundao Getulio Vargas.Antoine, A., H. Bonnal and E. 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(1991): A parametric model to dynamic heteroskedasticity: the Students t andrelated models, mimeo, Virginia Polytechnic Institute and State University.28Yogo, M. (2006): A consumption-based explanation of expected stock returns, Journal ofFinance 61, 539-580.29AppendicesA Single-step methods: Continuously Updated GMMLet xtTt=1 denote a strictly stationary and ergodic stochastic process, and dene L(xt; 0)as a vector of known functions of xt, where 0 is a vector of unknown parameters. The trueparameter value, 00, which we assume belongs to the interior of the compact set O _ Rdim(),is implicitly dened by the (population) moment conditions:1[L(xt; 00)] = 0,where the expectation is taken with respect to the stationary distribution of xt. In our context ofasset pricing models, xt = (r0t, I0t)0 represents data on excess returns and factors, and 0 representsthe parameters of the specic model under evaluation.GMM estimators minimise a specic norm L0T(0)XTLT(0) of the sample moments LT(0) =T1

Tt=1L(xt; 0) dened by some weighting matrix XT.In overidentied cases such as ours,Hansen (1982) showed that if the long-run covariance matrix of the moment conditions S(00) =aar[_TLT(00)] has full rank, then S1(00) will be the optimal weighting matrix, in the sensethat the dierence between the asymptotic covariance matrix of the resulting GMM estimatorand a GMM estimator based on any other norm of the same moment conditions is positivesemidenite. Therefore, the optimal GMM estimator of 0 will be`0T = arg min2JT(0),whereJT(0) = L0T(0)S1_00_LT(0).This optimal estimator is infeasible unless we know S(00), but under additional regularityconditions, we can dene an asymptotically equivalent but feasible two-step optimal GMM esti-mator by replacing S(00) with an estimator ST (0) evaluated at some initial consistent estimatorof 00, `0T say. There is an extensive literature on heteroskedasticity and autocorrelation consis-tent (HAC) estimators of long-run covariance matrices (see for example DeJong and Davidson(2000) and the references therein). In practice, we can repeat this two-step procedure manytimes to obtain iterated GMM estimators, although there is no guarantee that such a procedurewill converge, and in fact it may cycle around several values instead.An alternative way to make the optimal GMM estimator feasible is by explicitly taking intoaccount in the criterion function the dependence of the long-run variance on the parametervalues, as in the single-step CU-GMM estimator of Hansen, Heaton and Yaron (1996), which is30dened as0T = arg min2 ~JT(0),where~JT(0) = L0T(0)S1T (0)LT(0).Pearanda and Sentana (2010) discuss how to express the CU-GMM criterion in terms ofOLS output, which facilitates its optimisation.Although this estimator is often more dicultto compute than two-step and iterated estimators, particularly in linear models, an importantadvantage is that it is numerically invariant to normalisation, bijective reparametrisations andparameter-dependent linear transformations of the moment conditions, which will again proveuseful in our context. In contrast, these properties do not necessarily hold for two-step or iteratedGMM.Newey and Smith (2004) highlight other important advantages of CU- over two-step GMM bygoing beyond the usual rst-order asymptotic equivalence results. They also discuss alternativegeneralised empirical likelihood (GEL) estimators, such as empirical likelihood or exponentially-tilted methods. In fact, Antoine, Bonnal and Renault (2006) study the Euclidean empiricallikelihood estimator, which is numerically equivalent to CU-GMM as far as 0 is concerned. Im-portantly, it is straightforward to show that these GEL methods share the numerical invarianceproperties of CU-GMM.Our empirical application will consider two-step, iterated and CU-GMM. Under standardregularity conditions (see Hansen (1982)),_T(`0T00) and_T(0T00) will be asymptoticallydistributed up to rst-order as the same normal random vector with zero mean and variance_D0(00)S1_00_D(00)_1,where D(00) denotes the probability limit of the Jacobian of LT(0) evaluated at 00. In ourempirical application, we replace D(00) by 0LT(`0T),000 in the case of two-step and iteratedGMM estimators. In contrast, for the CU-GMM estimator0T we compute a consistent estimatorof D(00) that takes into account that the weighting matrix S1T (0) is not xed in the criterionfunction. Specically, we estimate the asymptotic variance of 0T as_T0T(0T)S1T (0T)TT(0T)_1,whereTT(0T) =0LT(0T)000 12_L0T(0T)S1T (0T) Idim(h)_ 0cc_ST(0T)_000.Finally, TJT(`0T) and T ~JT(0T) will be asymptotically distributed as the same chi-squarewith dim(L) dim(0) degrees of freedom if 1[L(x; 0)] = 0 holds, so that we can use thosestatistics to compute overidentifying restrictions (J) tests.31B Multifactor modelsIn what follows we represent a set of / factors by the vector I , and their mean vector, secondmoment and covariance matrix by , I and X = I 0, respectively. In this multifactorcontext, the connection between the SDF and regression approaches is given by1(r)a + 1(rI0)L = 1(r)(a +L0) + Co(r, I )L = (a +L0) +H(a +L0) +HXL= (a +L0) +H(a +IL) = c +Hd = 0, (B1)where c is the mean of the SDF, d is the shadow price of I (or its actual price if it is a vectorof traded payos) andH = _ d1dk_is the :/ matrix of regression slopes. We can interpret condition (B1) as reecting theorthogonality between the SDF and the projection of r onto a constant and I because c+Hd =1 [(a +L0I ) ( +HI )] = 0.Traded factorsIf the factors are excess returns themselves then condition (B1) is equivalent to = 0because c = a + L0 ,= 0 and the price of I must satisfy d = a + IL = 0. Therefore, we canevaluate the corresponding asset pricing model by means of the SDF inuence functionsLS (r, I ; a, L) =__ r (a +I0L)I (a +I0L)__, (B2)or the regression inuence functionsLR (r, I ; H) =__r HIcc ((r HI ) I0)__. (B3)The SDF functions require some normalisation in their implementation. A symmetricallynormalised version of the SDF approach would use the normalisation a2+L0L = 1 but asymmetricnormalisations are more common in empirical work. The uncentred SDF method imposes a = 1and relies on the inuence functions__ r (1 I0X)I (1 I0X)__with parameters X, while the inuence functions of the centred SDF method impose a+0L = 1,and become__r_1 (I )0r_I _1 (I )0r_I __32with parameters (, r). The link between both sets of parameters is = IX = Xr.In all these methods the degrees of freedom of the corresponding J tests are : regardless ofthe number of factors /. The Jensens alphas and pricing errors of excess returns r are denedbyo = 1 (r) H1 (I ) ,r = 1 (r) 1_rI0_X,) = 1(r) 1_r (I )0_r.Non-traded factorsIf the factors are not traded payos then we need :/ so that there are some overidenti-fying restrictions to test. As in the case of traded factors, the condition (B1) holds for a validasset pricing model. Such a constraint is equivalent to both and H belonging to the span ofsome : / matrix that we can denote asI = _ 1k_.Assuming that H has full column rank, we can impose this implicit constraint on the interceptsand slopes of the regression of r on a constant and I as follows: = Id = 1d1 + ... +kdk,H = _ d1dk_ = _ c1ck_ = cI.Therefore, we can evaluate the corresponding asset pricing model by means of the SDFinuence functionsgS (r, I ; a, L) = _r_a +I0L_, (B4)or the regression inuence functionsgR (r, I ; I, c, d) =__r I(d cI )cc ((r I(d cI )) I0)__. (B5)The SDF functions require some normalisation in their implementation. A symmetricallynormalised version of the SDF approach would use the normalisation a2+ L0L = 1, while theregression would rely on the normalisation c2+ d0d = 1, but asymmetric normalisations aremore common in empirical work. The uncentred SDF method imposes a = 1 and relies on theinuence functions_r_1 I0X_33with parameters X. The inuence functions of the centred SDF method impose a+0L = 1 andthey are__ r_1 (I )0r_I __with parameters (, r). The usual regression imposes c = 1 and relies on the inuence func-tions__r H(I +{)cc ((r H(I +{)) I0)__,with parameters ({, H). Alternatively we can dene the vector of factor risk premia as1 (r) = H, so that = { +, and add the estimation of . This yields__r H(I +)cc ((r H(I +)) I0)I __.Assuming that H has full column rank, the link between these parameters is = _X+0X = Xr.In any case, the degrees of freedom of the J test will be : /.Regarding Jensens alphasand pricing errors, they are dened byo = 1 (r) H,r = 1 (r) 1_rI0_X,) = 1(r) 1_r (I )0_r.C ProofsAll proofs consider the multifactor context of appendix B instead of the simplifying singlefactor set up in the main text. In addition, the proofs of Proposition 1 and 2 do not rely onany particular normalisation since they are irrelevant for single-step methods, even though theproposition statements in the main text refer to the usual normalisations in empirical work.34Proposition 1:Let us dene an extended regression system that adds the estimation of (, I) to LR (r, I ; H)dened in (B3). SpecicallyhR (r, I ; H, ,cc/(I)) =__LR (r, I ; H)I cc/_0I___ =__r HIcc ((r HI ) I0)I cc/_0I___.Importantly, by adding the exactly identied parameters (,cc/(I)), hR (r, I ; H, ,cc/(I))will be numerically equivalent to LR (r, I ; H) in terms of both the estimates of the originalparameters H and the J test.If we choose (a, L) such thata +IL = 0,then we can carry out the following transformations of the system hR (r, I ; H, ,cc/(I))_ aInL0InaH (L0H) T ___r HIcc ((r HI ) I0)I cc/_0I___=_r_a +L0I_H[a +IL] = r_a +L0I_,where the matrix T denotes the appropriate duplication matrix, that is, the matrix such thatcc () = Tcc/() (see Magnus and Neudecker (1990)). Similarly_ 0 0 aIk(L0Ik) T _________r HIcc ((r HI ) I0)I cc/_0I_________=_I _a +I0L_[a +IL] = I _a +I0L_.As we mentioned before, single-step methods are numerically invariant to normalisation, bi-jective reparametrisations and parameter-dependent linear transformations of the moment con-ditions. Therefore, for a given choice of HAC weighting matrix CU-GMM renders the extended35regression system hR (r, I ; H, ,cc/(I)) and the systemhS (r, I ; a, L, H, cc/(I)) =__r (a +I0L)I (a +I0L)cc ((r HI ) I0)cc/_0I___numerically equivalent. In particular, the estimates of H and cc/(I) are the same, the implied = IL,a is the same, and the J test is the same.Given the denition of LS (r, I ; a, L) in (B2), the last system can also be expressed ashS (r, I ; a, L, H, cc/(I)) =__LS (r, I ; a, L)cc ((r HI ) I0)cc/_0I___,where the inuence functions added to LS (r, I ; a, L) are exactly identied for (H, cc/(I)). ThushS (r, I ; a, L, H, cc/(I)) is numerically equivalent to relying on the rst block LS (r, I ; a, L) interms of both the original parameters (a, L) and the J test. Therefore, single-step methodsrender the systems LR (r, I ; H) and LS (r, I ; a, L) numerically equivalent.Finally, it is worth noting that in the case of the extended systems the numerical equivalenceof the CU-GMM criterion function will hold not only at the optimum but also for any compatibleset of parameter values. Lemma 1:We assume that the vector y = (I0, r0)0 follows an elliptical distribution, and denote thecorresponding coecient of multivariate excess kurtosis as i, which is equal to i = 2,( 4)in the case of Student t with degrees of freedom, and i = 0 under normality (see Fang, Kotzand Ng (1990) and the references therein for further details).Let us order the estimating functions in (4) for a multifactor model asL(y; X) =__ I (1 I0X)r (1 I0X)__ =__L1(I ; X)L2(r, I ; X)__.Thus, we can dene the relevant Jacobian asD = 1_0L(y; X)0X0_ =__I1_rI0___ =__ D1D2__,and similarly we can decompose the relevant asymptotic covariance matrix asS = aar_1_TT

t=1L(yt; X)_ =__ S11S12S21S22__.36If we apply Lemma D1 in Pearanda and Sentana (2010), then we ndS11 = .1I + .20,.1 = (1 H) (1 + iH) , .2 = 2 (1 H)2+_3H25H+ 2_i,where H = 1 (y)0 11(yy0) 1 (y), and similarlyS21 = .11_rI0_+ .21 (r) 0.Thus, we only need to check that condition (C1) in Lemma C1 in Pearanda and Sentana(2010) holds, which in our context becomesD2D11S11 = S21.This restriction will be satised becauseD2D11S11 = 1_rI0_I1_.1I + .20 = .11_rI0_+ .21_rI0_I10and, since 1 (r) = 1_rI0_I1 under the null of tangency,.11_rI0_+ .21_rI0_I10 = .11_rI0_+ .21 (r) 0 = S21.Therefore, the linear combinations of the moment conditions in 1 (L(y; X)) = 0 that providethe most ecient estimators of X will be given by1(0X I ) = 0.

Proposition 2:Let us dene a extended regression system that adds the estimation of (, I) to the inuencefunctions gR (r, I ; I, c, d) dened in (B5),gR (r, I ; I, c, d, ,cc/(I)) =__gR (r, I ; I, c, d)I cc/_0I___ =__r I(d cI )cc ((r I(d cI )) I0)I cc/_0I___.We are adding exactly identied parameters (,cc/(I)), and hence gR (r, I ; I, c, d, ,cc/(I))is numerically equivalent to the inuence functions in gR (r, I ; I, c, d) in terms of both theoriginal parameter estimates and the J test.37If we choose (a, L) such that__ aL__ =__ 1 0 I__1__ cd__ =__ c_1 +0X1_0X1dX1(d c)__then we can compute the following : 1 transformation of gR (r, I ; I, c, d, ,cc/(I)) :_ aInL0InI_acIk +dL0_ c (L0I) T ___r I(d cI )cc ((r I(d cI )) I0)I cc/_0I___=_r_a +I0L_I_d_a +L0_c (a +IL) = r_a +I0L_,where the matrix T denotes the corresponding duplication matrix.Accordingly, we can also reparametrise (c, d) in terms of the other parameters in the secondblock of inuence functions and then construct the systemgS (r, I ; a, L, I, ,cc/(I)) =__r (a +I0L)cc ((r I(d cI )) I0)I cc/_0I___,where__ cd__ =__ 1 0 I____ aL__.Under CU-GMM, and a specic choice of HAC estimator, gS (r, I ; a, L, I, ,cc/(I)) pro-vides the same estimates and J test as gR (r, I ; I, c, d, ,cc/(I)) because single-step proce-dures are numerically invariant to normalisation, bijective reparametrisations and parameter-dependent linear transformations of the moment conditions. As a result, the estimator of (c, d)obtained from gR (r, I ; I, c, d, ,cc/(I)) and (a, L) from gS (r, I ; a, L, I, ,cc/(I)) coincidewith their implied counterparts in the other system.This last system can be related to the inuence function gS (r, I ; a, L) dened in (B4), wherethe inuence functions that are added are exactly identied for (I, ,cc/(I)) given (a, L).Thus gS (r, I ; a, L, I, ,cc/(I)) is numerically equivalent to relying on r (a +L0I ) in terms ofboth the estimates of the common parameters (a, L) and the J test. Therefore, single-stepmethods render the systems gR (r, I ; I, c, d) and gS (r, I ; a, L) numerically equivalent.Once again, note that in the case of the extended systems the numerical equivalence of theCU-GMM criterion function will hold not only at the optimum but also for any compatible setof parameter values. 38Lemma 2:We extend the results in appendix D in Pearanda and Sentana (2010) for elliptical dis-tributions to the case of non-traded factors. The optimal moments are given by the linearcombinations D0S1LT (0). The uncentred SDF method has the following long-run varianceunder the nullaar_1_TT

t=1gU (rt, It; X)_ = (1 + i) H1 + 1(H2 + 1)21_rr0_ iH1 + 2 (1 i)(H2 + 1)21 (r) 1 (r)0 ,where H1 = 0X1 and H2 = 0X1. This asymptotic variance represents a multifactor andelliptical extension of the Gaussian computations in Jagannathan and Wang (2002).Given that D = 1_rI0_ for the uncentred SDF method, the optimal moments are thenproportional to the linear transformation1_Ir0_ _1_rr0_.1 (r) 1 (r)01, . = iH1 + 2 (1 i)(1 + i) H1 + 1 .Computing the inverse, we obtain1_Ir0__11_rr0_+.1 .1 (r)0 11 (rr0) 1 (r)11_rr0_1 (r) 1 (r)0 11_rr0__and imposing the null hypothesis 1 (r) = 1_rI0_X, we get1_Ir0__11_rr0_+.1 .X01_Ir0_11 (rr0) 1_rI0_X11_rr0_1_rI0_XX01_Ir0_11_rr0__=_Ik +.1 .X01_Ir0_11 (rr0) 1_rI0_X1_Ir0_11_rr0_1_rI0_XX0_1_Ir0_11_rr0_.Since the / / matrix in brackets has full rank, we can conclude that the optimal estimatorof X solves the sample moments1TT

t=1_r+t_1 I0tXT_ = 0,withr+t= 1_Ir0_11_rr0_rt.Finally, note that to implement this optimal estimator in practice, we need a consistent estima-tion of 1_Ir0_ and 1 (rr0), which we can easily obtain from their sample counterparts. 39Table 1: Descriptive statistics in annual % terms, 1953-20021 2 3 4 5 6 7 8Mean -2.336 -0.873 -0.747 0.329 -0.151 -0.213 2.988 2.031SD 6.346 6.628 6.614 8.415 7.443 8.121 8.090 12.417Sharpe ratio -0.368 -0.132 -0.113 0.039 -0.020 -0.026 0.369 0.164Real excess returns on the 8 currency portfolios in Lustig and Verdelhan (2007) for a US investor. Portfolio1 contains currencies with the lowest interest rates, while portfolio 8 contains currencies with the highestinterest rates.40Table 2: Empirical evaluation of the CAPMCU Iterated 2SUncentred SDF (`)Market 0.043 0.039 0.038(0.005) (0.005) (0.004)J test 21.375 22.003 28.068(0.006) (0.005) (0.000)Centred SDF (t)Market 0.122 0.031 0.026(0.070) (0.010) (0.009)J test 21.375 43.259 46.046(0.006) (0.000) (0.000)Regression (j)Market 14.974 14.468 14.627(1.694) (1.939) (2.071)J test 21.375 24.920 28.202(0.006) (0.002) (0.000)This table displays estimates of prices of risk (`, t or j) with standard errors in parenthesis, and theJ test with p-values in parenthesis. We implement each method by CU, iterated and two-step GMM.41Table 3: Empirical evaluation of the (linearised) CCAPMCU Iterated 2SUncentred SDF (`)Nondurables 0.495 0.488 0.488(0.042) (0.043) (0.043)J test 5.663 5.691 5.711(0.580) (0.576) (0.574)Centred SDF (t)Nondurables 4.388 1.154 1.201(5.727) (0.477) (0.181)J test 5.663 16.925 91.624(0.580) (0.018) (0.000)Regression (c)Nondurables 5.566 5.611 5.308(2.056) (1.623) (1.483)J test 5.663 5.677 5.627(0.580) (0.578) (0.584)This table displays estimates of prices of risk (`, t or c) with standard errors in parenthesis, and theJ test with p-values in parenthesis. We implement each method by CU, iterated and two-step GMM.42Table 4: Empirical evaluation of the (linearised) Epstein-Zin modelCU Iterated 2SUncentred SDF (`)Market 0.007 0.006 0.006(0.004) (0.005) (0.005)Nondurables 0.476 0.479 0.474(0.052) (0.054) (0.051)J test 4.785 4.827 5.171(0.686) (0.681) (0.639)Centred SDF (t)Market 0.057 -0.002(0.053) (0.006)Nondurables 3.948 -0.706(3.659) (0.220)J test 4.785 69.285(0.686) (0.000)Regression (j, c)Market 7.861 7.351 7.284(2.347) (2.279) (2.312)Nondurables 5.746 5.582 4.748(2.130) (1.638) (1.271)J test 4.785 4.964 5.283(0.686) (0.664) (0.626)This table displays estimates of prices of risk (`, t or j and c) with standard errors in parenthesis,and the J test with p-values in parenthesis.We implement each method by CU, iterated and two-stepGMM.43Supplemental AppendixD Evaluation with gross returnsThe SDF pricing equation is1__a +L0I_H_ = /n,where H is an :1 vector of gross returns and /n is an :1 vector of ones. Therefore, we nd1 (H) a + 1_HI0_L = 1 (H) c + Co (H, I ) L = /n,where c = a +L0 is the SDF mean. We can also relate the pricing of H to a constraint on theleast squares intercepts and slopes H. Specically,1 (H) c + Co (H, I ) L = c +Hd = /n,where d = c +XL is the shadow price of the factors, or their actual prices if they are traded.When the factors are gross returns themselves, they also satisfy1__a +L0I_I_ = /k, (D6)where /k is a / 1 vector of ones, which implies thatd = a +IL = c +XL = /k.In that case, the least squares constraint simplies toc +H/k = /n,which can also be expressed as = i(/nH/k) , i = 1,c,when c ,= 0.The inuence functions of the SDF and regression methods with traded factors areLS (H, I ; a, L) =__ H(a +I0L) /nI (a +I0L) /k__,LR (H, I ; H,i) =__(Hi/n) H(I i/k)cc (((Hi/n) H(I i/k)) I0)__,and the J tests have : 1 degrees of freedom.44If the factors are not traded then we discard the moment condition (D6) that prices thefactors. At the same time, we can re-express the least squares constraint as = i/n +H{, i = 1,c, { = d,c.when c ,= 0.In this context, the inuence functions of the SDF and regression approaches with non-tradedfactors aregS (H, I