Determining the Rank of the Beta Matrix in Linear Asset Pricing Models *

Transcription

1 Determining the Rank of the Beta Matrix in Linear Asset Pricing Moels * Seung C. Ahn a Arizona State University an Sogang University Alex R. Horenstein b IAM School of Business Na Wang c Arizona State University February 1, 011 Abstract his paper proposes a new metho to estimate the rank of the beta matrix in a factor moel. We consier the case in which possible factor variables, which we call factor-caniate variables, are observe. he iiosyncratic error terms are allowe to be correlate both over ifferent cross section units an over ifferent time perios. For the factor moel, estimating the rank of the beta matrix is equivalent to estimating the number of the relevant factors among the factor-caniate variables. he estimator we propose is easy to use because it is compute with the eigenvalues of the inner prouct of an estimate beta matrix. Simulation results show that the propose metho works well even in small samples. Our analysis of US iniviual stock returns is consistent with the notion that the three factors of Fama an French (FF, 1993) capture three ifferent risk sources. he five factors of Chen, Roll, an Ross (CRR, 1985) are correlate with one aitional factor that is not relate to the Fama- French factors. he momentum an reversal factors capture a further source of commovement that is not capture by the FF an CRR factors. In aition, the two factors propose by Chen, Novy-Marx an Zhang (CNZ, 010) capture a factor misse by all the previous ones. hese results suggest that there are six common risk factors in US iniviual stock returns among the thirteen factor caniate use. Furthermore, our analysis of portfolio returns reveals that the estimate number of common factors changes epening on how the portfolios are constructe. he number of risk sources foun from the analysis of portfolio returns is generally smaller than the number foun in iniviual stock returns. Key Wors: factor moels, beta matrix, rank, eigenvalues. JEL Classification: C01, C3, C31, G1 *Formerly calle Determining the Rank of the Beta Matrix in a Factor Moel with Factor-Caniate Regressors. We are grateful to Richar Roll, o Pronno, seminar participants at the 010 MFA annual meeting, the 010 EFA annual meeting, the 010 FMA annual meeting, the 010 Korean Econometric Society meeting, the Econometric Society 009 Far Eastern an South Asia meeting, the 009 Japan Association of Applie Economics meeting, Columbia University, Instituto ecnológico Autónomo e México, Universia Autónoma e Nuevo León, Sogang University an Arizona State University. a Department of Economics, W.P. Carey School of Business, Arizona State University, empe, AZ, miniahn@asu.eu. b IAM School of Business, Instituto ecnológico Autónomo e México, México, D.F , México. alex.horenstein@itam.mx. Alex Horenstein thanks the Asociación Mexicana e Cultura A.C. for its support. c Department of Economics, W.P. Carey School of Business, Arizona State University, empe, AZ, nwang10@asu.eu.

2 1. Introuction Jack reynor (196), William Sharpe (1964), John Lintner (1965) an Jan Mossin (1966) evelope the Capital Asset Pricing Moel (CAPM). he moel lai out the founations of moern asset pricing theory. Since the avent of the CAPM, it has become an important question whether a small number of economic or financial variables can capture the sources of non-iversifiable risk. If the answer is affirmative, then the variables shoul be price an the information containe in them is crucial for the agents portfolio strategies. Determining whether a factor is price or not became more important with the evelopment of multifactor asset pricing moels, like Merton s Intertemporal CAPM (197) an the Arbitrage Price heory (AP) of Ross (1976). hese multifactor moels tell us that if there exist multiple (r) factors etermining non-iversifiable sources of risks, then the factors shoul properly price the risky assets. However, these moels o not tell us what the factors are. In the empirical asset pricing literature many time-series variables have been propose as possible risk factors (see Chapter 6 of Campbell, Lo an MacKinlay (1997), Chen, Roll, an Ross (1986), an Fama an French (199)), which we call factor-caniate variables. Several important questions arise with respect to these factor caniates. Which ones shoul be inclue in the pricing equation? Are they capturing ifferent risk sources? By estimating the rank of the beta matrix, we can answer these questions. If we a one factor which oes not explain asset returns, we a a column of zero to the corresponing beta matrix, an the rank will not increase. If we a one factor which captures the same risk as the existing factors, we a a column of betas that can be spanne by the existing betas, an the rank will not increase. Hence, by choosing factors that increase the rank of the beta we will fin the ones that capture ifferent risk sources.

3 Estimating the rank of beta matrix is also a necessary conition for the two-pass (P) risk premium estimation. he two-pass estimation evelope by Fama an MacBeth (1973) has been wiely use to estimate the risk premium of each factor-caniate variable. Using this metho, the betas of caniate variables are first estimate using asset-by-asset timeseries regressions, an then the risk premiums relate to the variables are estimate by the cross sectional regression of the mean asset returns on the estimate betas. Whether a factorcaniate variable is price or not is etermine by the significance of the estimate risk premium. An important conition for the consistency of the P estimator is that the matrix of the true beta values has full columns. However, there are two cases in which the beta matrix may fail to have full columns. he first case is the true betas relate to a factor are all zeros. Kan an Zhang (1999) name such a factor useless factor. For a one-factor moel in which the factor is useless, Kan an Zhang (1999) have investigate the asymptotic properties of the P estimator. he useless factor cannot be price; that is, the premium of the useless factor shoul be unefine. However, Kan an Zhang show that the estimate coefficient of an unefine risk premium is asymptotically significant when using the P estimator. his happens because the estimate betas are not zeros although the true betas are. he secon case is when relevant factors are not the factor-caniate variables themselves, but rather a few linear combinations of them. For such cases, the true beta matrix is not full column, but the estimate matrix may appear to be of full column. Accoringly, some P premium estimates coul falsely appear to be statistically significant, although the corresponing premiums are in fact unefine. hus, when using the two-pass estimation metho researchers nee to check the rank of the beta matrix before continuing the secon pass cross sectional regression. 3

4 his paper proposes a new estimation metho, calle the hreshol estimation for the rank of the beta matrix in an approximate factor moel. We allowe the iiosyncratic error terms for iniviual observations to be both auto an cross-sectional correlate. Specifically, we estimate the rank using the eigenvalues of the inner prouct of the estimate beta matrix. he hreshol metho prouces consistent estimations as the time series imension goes to infinity. For the number of cross sectional units (N) the only requirement is to be greater or equal than the number of factor caniates use. A few papers in the literature have also consiere the estimation methos for the rank of a matrix. Zhou (1995) proposes a Wal test in samples with small N to test the hypothesis of a given rank. Cragg an Donal (1997) provie the tests for the rank of a matrix base on a minimum chi-square criterion. Robin an Smith (000) consier the tests base on certain estimate characteristic roots, an show that the limiting istributions of the test statistics are a weighte sum of inepenent chi-square variables. Kleibergen an Paap (006) propose a rank statistic using a consistent estimator of the unrestricte matrix, an the propose rank statistic has a stanar χ limiting istribution. However, all these methos are applicable only to ata with small N. When N is large, too many parameters nee to be estimate. his is very restrictive for asset prcing applications in which the number of crosssectional observations, N, is usually large. A metho closely relate to our metho is propose by Connor an Korajzcyk (1993). heir metho is esigne to be appropriate for the analysis of the ata with large N an relatively small observations. Autocorrelation is not allowe for the iiosyncratic components of stock returns. For such ata, the number of relevant factors is estimate by evaluating whether aing one more factor results in a significant ecrease in the sum of the squares of estimate error terms. o use this sequential metho, one nees to etermine the orer of the factor variables to be teste in an arbitrary matter. In contrast, the hreshol 4

5 metho we propose requires looser restrictions in ata. In aition, no orering of the factors is necessary. Estimating the rank of the beta matrix is also relate to estimating the number of factors. hey are relate in the sense that the number of the common factors in return ata equals to the rank of the beta matrix corresponing to the factors. Bai an Ng (00), Onatski (010), an Ahn an Horenstein (009) have evelope formal statistical proceures to estimate the number of the true factors in approximate factor moels. Our approach is ifferent from their approaches in one important aspect. Our hreshol metho is for the case in which the factor-caniate variables are available, while their methos are esigne for the cases in which factor-caniate variables are not observe. Our interest is not to estimate the number of all common factors in asset return ata, but to estimate the number of relevant factors containe in observe factor-caniate variables. For this purpose, we estimate the number of relevant factors using the estimate betas corresponing to the caniate variables. he hreshol estimator we propose possesses several goo properties. First, its consistency oes not require any particular restriction on the relation between N an. Its consistency only requires ata with large. Secon, the hreshol estimator allows iiosyncratic error terms to have weak time-series an cross-sectional epenence. hir, it has power to etect the weak factors which have only limite explanatory power. Fourth, it can be applie to the zero factor case. Finally, our simulation exercises inicate that the hreshol estimator has goo finite sample properties. Application of the hreshol estimation is conucte first on the US iniviual stock returns. We confirm that all of the Fama-French (1993) three factors have explanatory power. In contrast, only one or two among the five factors of Chen, Roll, an Ross (1986) have explanatory power. When we combine the three factors of Fama-French (FF) together with 5

6 the five factors of Chen, Roll, an Ross (CRR) we fin that a factor not capture by FF is capture by CRR. Furthermore, we fin that momentum an reversal factors (MOM) capture a source of risk not capture by either FF or CRR. Similarly, the two factors propose by Chen, Novy-Marx, an Zhang (010, CNZ) capture an aitional source misse by all the other factors. We fin evience for six factors in US iniviual stock returns among the thirteen factor caniates use. When we use Inustrial Portfolio returns, results remain the same. However, when we use portfolios that are better iversifie such as the ones sorte on characteristics like Size an Book to Market, the FF factors seem to be enough to capture all the common sources of risk among the thirteen factor caniates, except for the 100 Size an Book to Market portfolios in which an extra factor appears when aing the CNZ factors. Overall, our analysis of portfolio returns reveals that the estimate number of common factors changes epening on how the portfolios are constructe. he rank of the beta matrix foun from the analysis of portfolio returns is generally smaller than the one foun in iniviual stock returns, except for the inustry portfolios. his result suggest that some inustry specific factors isappear when well iversifie portfolios are use. he rank estimation propose in the paper has two implications for the asset pricing literature. First, it emphasizes the over-ientification problem, where all the available factors may be simply throw into the asset pricing moels. he rank estimation prouces the number of inepenent sources of commovement that we shoul inclue from all the factor caniates when searching for price risk premiums. he estimator works very well even when some important factors are not inclue in the set of factor caniates since we allow for a factor structure in the resiuals. Another implication is that the rank estimation metho is free of the ebate whether or not firm characteristics are price risk factors. Since we use the ouble emeane ata set, we exclue the effect of firm characteristics. If price, the risk sources capture by estimating the rank of the beta matrix can only be systematic risk. 6

7 he rest of this paper is presente as follows. Section introuces the factor moel we investigate an the assumptions impose on it. Section 3 erives the asymptotic properties of the hreshol estimator. Simulation results are reporte in section 4. Section 5 shows the application to the Fama-French three factors, the five factors of Chen, Roll, an Ross (1986), three factors that capture momentum profits an the IA an ROA factors from Chen, Novy-Marx, an Zhang (010). Concluing remarks follow in section 6. All of the proofs are given in the appenix.. Moel an Assumptions We begin by efining an approximate factor moel as the one consiere by Chamberlain an Rothschil (1983) an Bai an Ng (00). Let x it be the response variable for the th i cross-section unit at time t, where i 1,,, = L N, an t = 1,, L,. Explicitly, x it can be the (excess) return on asset i at time t. he response variables x it epen on the iniviual effect α i, the time effect δ t an the k factor-caniate variables in ft = ( f1 t, ft,..., f kt ). hat is, x = α + δ + β f + ε, (1) it i t i t it where βi = ( βi 1, βi, L, β ik ) is the beta vector for cross section unit i. he prouct β i ft is the common component of x it, an the ε it are iiosyncratic components or iiosyncratic risks. 1 Our interest for moel (1) is to estimate rank of the beta matrix Β, where Β = ( β, β,..., β N ). However, because of the presence of the time effects δ t, we are unable 1 to estimate β i. Instea we can estimate the emeane betas, i i & β = β β, where 1 In this moel, we consier only the case of time invariant betas. Our metho can be easily extene to the case of time-varying betas since the rank estimation is base on the estimate beta matrix. 7

8 β β 1 N = N Σ i= 1 i. Use of the emeane beta estimate instea of the raw beta estimates oes not cause any technical problem. As long as any β ij is varying over ifferent cross-section units, rank( Β ) = rank( Β ), where Β = ( & β1, & β,..., & β N ). In aition, the rank of Β matters more than the rank of Β for the two-pass regression, because the risk premiums corresponing to the factors in f t are estimate by the cross-section regression of the 1 iniviual mean of x it ( xi = Σ t= 1xit ) on one an β i. If any beta in β i is constant over i, the risk premiums are unefine. he premiums are ientifie only if the emeane beta matrix Β has full column. moel, he emeane betas can be estimate by estimating the following ouble emeane && x = & β f& + && ε, () it i t it && x = x x x + x, f & t = ft f, && ε = ε ε ε + ε, it it t i where it it t i x = N Σ x, 1 N t i= 1 it x = Σ x, 1 i t= 1 it = Σ Σ, f = ( Σ t= 1 ft ) /, an ε t, ε i, an ε are similarly efine. For each 1 N x ( N ) i= 1 t= 1xit time perio t, moel () can be written as where && xi = (&& xi1, && xi,,&& x i ) L we have, εi && x = i F& & β + && i εi, ( 1) ( k) ( k 1) ( 1) && is similarly efine, an F & = ( f & 1, f &, L, f & ). For all ata, X&& = F& Β + Ε&&, ( N ) ( k) ( k N ) ( N ) where X && = ( && x1, && x, L,&& x N ),, an Ε && = ( && ε1, && ε, L,&& ε N ). hen, the emeane beta matrix 1 can be estimate by the OLS estimator ˆ Β = X && F & ( F & F & ). Β 8

9 In what follows, we use λ ( A) to enote the j th j largest eigenvalue of a matrix A, an the norm of A is enote by A 1/ = [ tr( A A)]. We efine c as a generic positive constant. With this notation, we make the following assumptions: Assumption A (factors): F & F & / = Σ t= 1( ft f )( ft f ) / p Σ, an f f µ, where p f Σ f is finite an positive efinite matrix an µ f is a finite vector. Assumption B (betas): (i) βi c for all i = 1,, L, N. (ii) Β Β / N is positive semi- efinite an rank( ) rank( Β = Β Β ) = r k for all N r. (iii) If N, Β Β / N Σ, where Σ β is finite. Assumption C (iiosyncratic errors): E( ε it ) = 0 an 4 it E ε c for all i an t, an β N 1 1 N 1 εit = ( εitε is ) N i 1 = t= 1 N i= 1 t= 1 s= 1. E E c Assumption D (weak epenence between factors an iiosyncratic errors): N N tεit εitε is t s i= 1 t= 1 i= 1 t= 1 s= 1. E( Ε F ) = E( f ) = E( f f ) c N N N he four assumptions are a subset of the assumptions use in Bai an Ng (00) an Ahn an Horenstein (009). Assumption A implies that the factors shoul be stationary. Assumption B(i) ensures that each factor loaing oes not exploe. Assumption B(ii) allows that the rank of Β to be smaller than the number of the variables in f t. Assumption B (iii) implies that for the cases where N is large, Β Β / N is asymptotically finite. hat is, the explanatory power of each factor increases at the rate of N. he estimators we propose below o not require large N. Uner Assumption B (iii), the estimators are consistent regarless of the size of N. Uner Assumption B, we treat the betas as fixe constants. We can easily relax this assumption, but at the cost of more notation. 9

10 Assumption C allows weak time-series correlations an oes not impose any restrictions on the cross-sectional correlation among the error terms ε it. Our asymptotic results obtaine below epen not on the covariance among the errors, but on the epenence between the errors an factors. Assumption C implies that Σ 1 ε / is a boune ranom t= it variable for all i. his assumption is weaker than Assumption C of Bai an Ng (00): 1 N N Σi= 1Σ j= 1Σt = 1Σ s= 1 E( εitε js) < c. N Assumption D implies that the ranom vectors Σ 1 ε f / are boune. his t= it t assumption is require for the consistency of the orinary least squares (OLS) estimator of Β. Assumption D is essentially the same assumption as Assumption D of Bai an Ng (00). Furthermore, Assumption D allows the errors ε it to have a factor structure. o see why, consier a simple case in which the ε it have an one-factor structure: εit = ξigt where E( g t ) = 0, E( gt f t ) = 0, 4 E( gt ) 1 < c, an Σt 1Σ s 1 E( gsgt ft fs ) < c for all t, an ξ i < c for = = all i. For this case, the ranom variable Σ 1 g f / is boune. hus, we can easily show that Assumption C hols. In aition, t= t t Σt= 1Σ s 1E( εitεis f = t fs ) = ξi Σt= 1Σ s 1E( gtgs f = t fs ) < c. hus, Assumption D hols. Given that the ε it can have a factor structure, estimating the rank of Β is not equivalent to estimating the number of all of the common factors in response variables. he rank of Β is the maximum number of the common components in response variables among the factor caniate variables f t. Hence, the rank estimation metho works well even when the factor caniates o not inclue all the common unerlying factors. he missing information is capture in the error terms with a factor structure. 10

11 3. Rank Estimation using Eigenvalues he hreshol estimator we propose below uses the eigenvalues of ˆ ˆ Β Β / N. So, we begin this section by stuying the asymptotic properties of the eigenvalues. Below, we use the notation ˆ ˆ ˆ µ N, j = λ j ( Β Β / N) where j inicates that ˆ µ N, j is the j th largest eigenvalue of the matrix ˆ ˆ Β Β / N. he following theorem presents the asymptotic properties of the eigenvalues. heorem 1: Uner assumption A D, (i) p lim ˆ µ N, j > 0 for 0 < j r ; an (ii) % µ = O, for 0 r < j k. 1 N, j p ( ) heorem 1 shows that the first r > 0 largest eigenvalues of ˆ ˆ Β Β / N have the same convergence rates, which are ifferent from those of the other eigenvalues. his ifference in convergence rate is use to ientify the rank of the matrix Β, r. Notice that the asymptotic properties of the eigenvalues o not require N. heorem 1 hols for any fixe number N. herefore, the estimator we propose below oes not require large N. estimator. he following theorem efines the consistent estimator that we call hreshol heorem (hreshol Estimator): For a given threshol function g( ) > 0 such that g( ) 0 an g( ) as, efine rˆ #{1 : ˆ H = j k µ N, j > g( )}, where #{ } is the carinality of a set. hen, uner Assumptions A D, lim Pr( ˆ rh = r) = 1. he result of heorem is quite intuitive. Observe that g( ) converges to zero at a lower rate than the last ( k r) eigenvalues of ˆ ˆ Β Β / N o. he first r eigenvalues converge to positive numbers. Accoringly, for sufficiently large, the value of g( ) is most likely to be smaller than the first r eigenvalues an larger than the rest of the eigenvalues. he 11

12 threshol estimation proceure is similar to the methos suggeste by Bai an Ng (00) to estimate the number of unobservable common factors in an approximate factor moel with a large number of response variables. Note that we can also use the hreshol estimator propose in heorem for the cases in which (i) the ata is not generate by a factor moel an/or (ii) all the factor caniates are useless. We will call this situation no-factor case. For such a case, r = 0. A possible pitfall of the threshol estimator is that there are many possible choices for g( ). Whenever a function is an appropriate choice for g( ), so is a finite multiple of the function. If is large, the estimation results woul be insensitive to the choice of g( ). However, for the ata with relatively small, the estimation result coul change epening on the choice of g( ). he optimal choice of the threshol function g( ) may epen on the ata generating processes. In the following paragraph we propose a specific function for g( ) which provies reliable estimates for many ifferent ata generating processes we have consiere in our Monte Carlo experiments. Let 1 N ˆ σ = [( N 1)( 1)] Σi= 1Σ && t= 1eit, where the e&& it are the OLS resiuals from the regression of the ouble emeane moel (). he estimator var( ε it ). Also, let N N i= 1 t= 1 it i= 1 t= 1 it ˆ σ is a consistent estimator of R = 1 [ Σ Σ e&& ] / [ Σ Σ && x ] be the R-square from the OLS regression of moel (). hen, the threshol function we suggest to use for the hreshol estimator is given by: ˆ σ g(, ) =, (3) where R = 1 for R, = 0.3 for R <, an = 0.8 for R >

13 he function g(, ) is esigne to be a non-ecreasing function of R for sufficiently large. Specifically, for > 8, g(,) is a monotonically ecreasing function of. Because is a non-increasing function of R, g(,) is an increasing (specifically, nonecreasing) function of R. he use of g(,) is motivate by our finings from Monte Carlo simulations: when the ata are generate by weak factors (that have low explanatory power), smaller threshol values are neee to better estimate the rank of Β. Since g(,) shoul satisfy the two conitions given in heorem, we limit the range of to be [0.3, 0.8]. he choice of the range is somewhat arbitrary. However, this range is the best choice we have foun from simulations. he property of g(,) coul be state from Graph 1. When R is low, factors have low explanatory power, we nee a small value of g(,) to etect the weak factors. When R is high, factors are stronger an a relative larger threshol function is neee. When increases, the last ( k r) eigenvalues of ˆ ˆ Β Β / N converges to zero faster, an a smaller g(,) is neee. 4. Simulations Our simulation ata are rawn by the same moel use in Bai an Ng (00) an Ahn an Horenstein (009): x = α + δ + Σ β f + u ; k it i t j= 1 ij jt it u it = v it 1 ρ, 1+ Jδ where i 1 min( i J, N ) vit = ρvi, t 1 ξ it h max( i J,1) δ v Σ = ht + Σ h= i+ 1 δ vht, an the ξ it (1 i N) an the factor caniate variables f jt are ranomly rawn from N (0,1). In this setup, the variance of uit is roughly equal to one. For simplicity, we set α i = 0 for all i, an δ t = 0 for all t. he beta matrix Β is rawn by the following way. We raw a N r ranom matrix A, each entry of which is 13

14 N (0,1). We also raw a ranom k k positive efinite matrix, compute the first r orthonormalize eigenvectors of the matrix, an set a k r matrix C using the eigenvectors. hen, we set 1/ Β = AΛ C, where Λ = iag( λ1,..., λ r ). his setup is equivalent to the case in which the true factors are f * = Λ C f with Var( f ) = Λ an the beta matrix corresponing * 1/ t t t to f = ( f,..., f ) is A. * * * t 1t rt he parameter Λ controls the signal to noise ratio of each of the true factors (SNR, ratio of the variances of a factor an the iiosyncratic error, u it ). When the j th true factor, has the variance of λ = 1/ r, its SNR equals 1/ r, where r 1. In case of r = 0, we present j the table separately. For benchmark simulations, we use λ = 1/ r, for 1 j r. In other simulations we try ifferent λ j s. For the error terms, we consier four cases: (i) the cases with i.i.. errors ( ρ = J = δ = 0 ), (ii) with both cross-sectional an auto-correlate errors ( δ = 0., ρ = 0.5, J = 8 ), (iii) with only cross-sectional correlate errors ( ρ = 0 ), an (iv) with only autocorrelate errors ( J = δ = 0 ). For each case, we try 5 ifferent combinations of N an, where N, {50, 100, 00, 500, 1000}. 1,000 samples are rawn for each combination of N an. ables 1 3 report the results from our benchmark simulations ( λ 1 =... = λ r = 1/ r ). able 1 shows the estimation results from the cases with i.i.. iiosyncratic errors an both cross-sectional an auto-correlate errors. Specifically, for the correlate error cases, we set ρ = 0.5, δ = 0., J = 8. All factors have the SNRs of 1/ r, where r 1. he hreshol estimator performs very well, even in the case of small sample size (e.g., = 50 ). For every case, the mean of the rank estimates is almost equal to the true rank. Also, only for a few j * f tj, We first generate a N k matrix M whose entries are rawn from N(0,1), an then compute the r eigenvectors of M M. 14

15 cases, the stanar eviation of the estimates is larger than zero. he results with correlate errors are not noticeably ifferent from those with i.i.. errors. ables shows the results from the cases of cross-sectional correlation only ( ρ = 0, δ = 0., J = 8 ) an auto-correlation only ( ρ = 0.5, δ = 0. ). he factors are generate with λ = 1/ r, for 1 j r. For all cases, the hreshol estimator performs very well even if is small. j able 3 shows the results for the cases in which all factors are weak with the same SNRs. he left part of the table reports the results from the cases with i.i.. errors, while the right part presents the results from the cases with both cross-sectionally an auto-correlate errors. For small ( = 60), the hreshol estimator oes not perform well when the SNRs of the factors are as low as But it works well in the cases with the SNRs larger than For the case in which = 100, the hreshol estimator performs very well even in the cases with the SNRs of he estimation results from the ata simulate with i.i.. errors are more reliable than those from the ata with correlate errors, especially when is small an factors are weak. In fact, we can a one more imension of the SNR to the threshol function. If the weak factors efine as important factors nee SNRs at least larger than 1/5, we can ajust the threshol function with the simulate ata to make our estimation capturing al the factors with SNRs larger than 1/5. able 4 is esigne to investigate the performances of the hreshol estimator when both weak an strong factors coexist. As in table 3, the left part of the table reports the results from the cases with i.i.. errors, while the other part presents the results from the cases with cross-sectionally an auto-correlate errors. We conuct the test with two ifferent factor-caniates moels, both of them with k = 3. In each of these moels we stuy three ifferent possible SNRs for the weak factor. In one moel we construct a factor structure with r =, where the first true factor is strong with λ 1 fixe at one an the secon true factor 15

16 is weak with three ifferent λ values: λ = 0.1, 0., an 0.3. In the other moel we stuy the case with r = 3 where the first two true factors are strong with λ1 = λ = 1, the last one is weak with three ifferent λ 3 values: λ 3 = 0.1, 0., an 0.3. From the table, we can see that the hreshol estimator performs very well in small samples even if the weak factor s SNR is ten times smaller than the SNRs of the strong ones ( λ = 0.1, in the first moel an λ 3 = 0.1, in the secon moel). he structure of the error terms oes not show significant ifference in the results. able 5 is esigne to investigate the performances of the hreshol estimator for the ata generate without true factors. hat is, all of the factor caniate factors use for able 5 are useless. We consier the cases with ifferent numbers of useless factors. able 5 shows that the hreshol estimator correctly etects the cases in which all factors are useless, if the number of factor caniate variables is small (e.g., k = 1), or is large, or errors are only weakly correlate. When the errors are highly correlate, the estimator has relatively low power to etect useless factors unless is sufficiently large. Our simulation results can be summarize as follows. First the hreshol estimator provies quite reliable inferences on the rank of the beta matrix even if the sample size is small. he SNR of each factor, the egrees of correlations among the errors, an the number of cross section units o not substantially influence the performances of the estimators. Secon, the hreshol estimator can be use to check the possibility of all factor caniates being useless. he hreshol estimator is relatively less precise, if the number of the factor caniates analyze is too large, or if the errors are highly correlate. However, it performs reasonably well even uner such cases if the number of the time series observations is sufficiently large. 5. Application 16

17 In this section we estimate the rank of the beta matrix using ifferent factor-caniates as regressors. More specifically, we use the three factors propose in the moel of Fama an French (199, FF), the five factors of Chen, Roll, an Ross (1986, CRR), 3 the momentum an reversal factors (MOM) available on Kenneth French webpage: momentum, short-term reversal an long-term reversal, an the two new factors evelope in Chen, Novy-Marx, an Zhang (CNZ, 010): Investment to Asset (IA) an Return on Asset (ROA). 4 As response variables we use the US monthly iniviual stock returns an portfolio returns. 5 Returns are calculate in excess over the risk free rate. he iniviual stock returns are ownloae from CRSP. he returns inclue iviens. he risk free rate is the onemonth reasury bill rate, which is available from Kenneth French s webpage. For the iniviual stock returns, we exclue REIs (Real Estate Investment rusts), ADRs (American Depositary Receipts) an the stocks that o not have information for every month over a sample perio. We also exclue stocks that show more than 300% excess returns in a given month since we are trying to capture common variation. Excessively high or low returns are most likely to be iiosyncratic risks. US Stock portfolio returns are ownloae from Kenneth French s webpage. he portfolios use are 100 portfolios base on Size an Book to Market, 5 portfolios base on Size an Book to Market, 5 portfolios base on Size an Momentum, 49 Inustrial portfolios an 30 Inustrial portfolios. We use monthly returns in every ata set. 3 While the FF moel may be more relate to the AP, the CRR moel is more relate to Merton s (197) Intertemporal CAPM, in the sense that they try to fin the macroeconomic (state) variables that may influence future investment opportunities. he factors propose by CRR are inustrial prouction (MP), unexpecte inflation (UI), change in expecte inflation (DEI), the term premium (US), an the efault premium (UPR). Each of these factors is available from Laura Xiaolei Liu s webpage from January 1960 to December 004 ( For etaile information on how these factors have been constructe, see Liu an Zhang (008). he FF factors are the proxy for the market risk premium, SMB an HML. 4 We thank Long Chen for proviing us the latest version of their factors. 5 We o not use the aily returns since the ata of some factor caniates are only available at monthly frequency. 17

18 Response variables are always ouble-emeane as suggeste in the Equation (). We also use stanarize factors for the following reason. he beta values corresponing to each factor change epening on the scale of the factor. For example, if we rescale a factor by multiplying 10, the (absolute) beta values corresponing to the factor are scale own by the orer of 0.1. In this case, even if the factor has a high explanatory power, the estimate betas obtaine with the rescale factor woul not reflect the factor s true explanatory power. 5.1 Rank of beta matrices using iniviual US stock returns as response variables he time span inclue in the analysis is from 197 to 004. We ivie the iniviual stock returns into three samples: the entire time span ( ), two subsamples ( an ) an three subsamples ( , , an ). Uner both subivisions, we coul fit a polynomial tren to the value weighte market portfolio to estimate the up an own cycles. We o so to examine how the estimation results may change epening on time intervals. We keep the time span at aroun 100 or more since the simulation exercises show that the estimators are very accurate in this case. he number of cross-sectional observations N changes as changes in orer to maintain a balance panel. he value of N epens on the available observations with complete ata on CRSP for each sample perio after the ata has been cleane. he results from the estimation of the rank of the beta matrix for iniviual stock returns are shown in able 6. Each line of the table represents a ifferent estimate moel. For each moel we report the number of factor caniates use (k), the estimate number of factors among the factor caniates ( ˆr ) an the average R of the regressing the response variables on the factor-caniates. he first line of table 6 shows that the hreshol estimator preicts that the rank of the beta matrix equals three when using the three FF factors in ifferent sample perios. 18

19 he secon line of table 6 shows the results from the estimation of the five CRR factors. For any perio, the estimate rank oes not excee two. his means that only one or two common sources of comovement in iniviual stock returns are explaine by the CRR factors. his result provies strong evience that the risk premiums of some factors in the CRR moel are unefine. Given that the CRR factors can ientify one or two common factors in iniviual stock returns, a question we wish to answer is whether the CRR factors capture some sources of comovement that the FF factors fail to o. If the CRR factors capture ifferent information from what the FF factors o, we coul expect that the rank of the beta matrix from the joint moel of CRR an FF woul be equal to the sum of the ranks from the CRR an FF moels separately. Inee, the hreshol estimation results are consistent with this expectation in the entire sample an every subsample. In the thir line of result in table 6 the hreshol estimation suggests that the risks capture by the CRR an FF factors are ifferent. Since the five CRR factors capture a common source of comovement that is not capture by the FF factors, an interesting question is which of the CRR factors contain the information misse by the FF factors. For this purpose we a to the FF factors each CRR factor iniviually in orer to estimate the rank of the beta matrix of at most four. In unreporte results we fin that no iniviual CRR factor increases the rank of the beta matrix when combine with the FF factors. hen we use every possible combination of two CRR factors together ae to the three FF factors. In this case we foun that aing UI (unexpecte inflation) an DEI (changes in expecte inflation) increases the rank of the beta matrix to four. Results are shown in the 4 th line of table 6. his shows that a factor relate to inflation is misse by the FF factors. Furthermore, we analyze if momentum factors (as constructe by Kenneth French) capture a ifferent source of risk than the Fama-French factors. Results of estimating the 19

20 rank of the beta matrix of the three momentum factors an the FF factors are presente in the 5 th row of the table. he hreshol estimator fins strong evience for an extra factor containe in the three momentum factors in most samples. However, if we a any one or any two possible combinations of the momentum factors to FF three factors, unreporte results show that in most cases we fin the rank equals three. We conclue that there is evience for a momentum factor among the three momentum factors uring the perio uner analysis that is not capture by the FF factors when using iniviual stock returns. In the 6 th row of table 6 we test the rank of the beta matrix when using the three FF factors an the two new factors of CNZ an fin four factors in almost every subsample. 6 his is evience that the CNZ factors capture one imension misse by the FF factors. Finally, the last row of the table show the results of using the ten factor-caniates that seem to contain ifferent information together: the three FF factors, UI an DEI from CRR, the three momentum factors an the two CNZ factors. he table shows that there is evience for at least six factors among the 10 factor caniates. However, an open an important question is whether we nee to use iniviual stock returns or portfolio returns to estimate the beta matrix in orer to perform asset pricing tests 7. For example, imagine a hypothetical situation in which half of the sample of the iniviual stock returns have betas of 0.5 with respect to a factor an the other half have betas of In this case the factor will a a imension to the rank of the beta matrix when using iniviual stock returns, but this factor will isappear in properly iversifie portfolios (because the beta of the iversifie portfolio with respect to the factor will be zero). In the next section we estimate the rank of the beta matrix using the same factor caniates as before but using portfolio returns as response variables. 6 Most of the time aing ROA to the FF factors is sufficient to get a rank equal to four while aing only IA never increases the estimate rank of the beta matrix. For this reason, we can conclue that ROA posses most of the information not capture by the FF factors. 7 See Ang, Liu, an Schwarz (008). 0

21 5. Rank of beta matrices using US stock portfolio returns as response variables In this section we use five sets of portfolios ownloae from Kenneth French website as response variables. Since the number of portfolios is fixe in each ifferent set, we use for every estimation the full time span from January 197 to December 004 (=396). he cross-sectional imension N equals to the number of portfolios in each set. In table 7 we report the same statistics for portfolio returns as those in the previous table for iniviual stock returns. When using the FF factors we fin all the time an estimate rank of three except for the 5 Size an Book to Market portfolio set where we fin a rank of two. When we use the five CRR factors we fin the rank equals to one or two as in the case with iniviual stocks. When we test together the FF factors an the CRR factors (k=8), we o not fin evience of an extra factor except for the cases of the 49 an 30 Inustrial Portfolios. A common pattern observe in table 7 is that when testing the number of factors in Inustrial Portfolios the results are similar to those obtaine using iniviual stock returns. However, once we use portfolios base on Book to Market an Size or Size an Momentum, the rank of the beta matrix is at most four. he maximum rank we fin for 100 Size an Book to Market portfolios is four, an for 5 Size an Book to Market portfolios an 5 Size an Momentum portfolios is three. his is evience that the portfolios sorte base on these characteristics are better iversifie (these portfolios also show less resiual variance since their R is higher than the one of the Inustrial portfolios). A possible explanation is the existence of inustry specific factors that are iversifie away when constructing portfolios base on characteristics like Size an Book to Market. his is a useful result that can clarify the iscussion of whether to use portfolios or iniviual stock returns when testing factors an also the iscussion about which type of portfolios shoul be use. It is known that 1

22 inustry portfolios ten to have positive abnormal excess returns (intercepts are significantly larger than zero). Accoring to our result this is because the existence of inustry specific factors that isappear when well iversifie portfolios are use. In other wors, the positive α that appears in many of the Inustry Portfolios shoul not be consiere a moels mispricing since it is exposure to a source of iversifiable risk. Our empirical results can be summarize as follows. When using iniviual stock returns we fin evience for the existence of six common factors among the thirteen factor caniates use. hese factors are the three FF factors, a factor relate to inflation from the CRR factors, a Momentum factor an a factor capture by the new CNZ factors. When we use Inustrial Portfolio returns, results remain the same. However, when we use portfolios that are better iversifie such as the ones sorte on characteristics like Size an Book to Market, the FF factors seem to be enough to capture all the common sources of risk among the thirteen factor caniates, except for the 100 Size an Book to Market portfolios in which an extra factor appears when aing the CNZ factors. 6. Conclusions In this paper, we have propose a new rank estimator, calle hreshol estimator, for the beta matrix from a factor moel with observe factor-caniate variables. esting whether the beta matrix has full rank is important for the two-pass estimation of the risk premiums in empirical asset pricing moels. he (emeane) beta matrix nees to have full rank. Otherwise, risk premiums are unefine. he hreshol estimator is compute easily with the eigenvalues of the inner prouct of an estimate beta matrix. Our simulation exercises provie promising evience that the hreshol estimator has goo finite-sample properties. Different from the existing methos, this propose metho can be use to analyze the ata with a large number of cross-section units.

23 In our empirical investigation we fin that all of the Fama-French (1993) three factors have explanatory power when using US iniviual stock returns as response variables, In contrast, only one or two among the five factors of Chen, Roll, an Ross (1986) have explanatory power. When we combine the three factors of Fama-French (FF) together with the five factors of Chen, Roll, an Ross (CRR) we fin that a factor not capture by FF is capture by CRR. Furthermore, we fin that momentum an reversal factors capture a source of risk not capture by either FF or CRR. Similarly, the two factors propose by Chen, Novy- Marx, an Zhang (010, CNZ) capture a source of risk misse by all the other factors. We fin evience for six factors in US iniviual stock returns among the thirteen factor caniates use. When we use Inustrial Portfolio returns, results remain the same. However, when we use portfolios that are better iversifie such as the ones sorte on characteristics like Size an Book to Market, the FF factors seem to be enough to capture all the common sources of risk among the thirteen factor caniates, except for the 100 Size an Book to Market portfolios in which an extra factor appears when aing the CNZ factors. Bai an Ng (00), Onatski (006), an Ahn an Horenstein (009) have evelope the estimators for the number of factors without using factor caniate variables. heir stuies have foun one or two factors from US iniviual stock return ata. In contrast, our results provie evience of at least six factors in iniviual stock returns. All of the estimation methos propose by the above three stuies are base on the analysis of principal components of response variables. Ahn an Horenstein (009) foun that principal components provie poor estimates of the true factors when the true factors are weak an the iiosyncratic errors are cross sectional correlate. From their results, we can conjecture that the analysis of principal components might have limite power to etect weak factors. In contrast, the hreshol estimator propose in this paper utilizes observe factor caniate variables. Factors nee not be estimate. hus, we can expect that the new estimator woul 3

24 have a higher power to etect the weak factors hien among the factor-caniate variables. Our estimation results are consistent with this expectation. 4

25 Appenix he following two Lemmas are use to prove heorem 1. Lemma 1: Uner Assumption B an D, for any k p ( p k ) matrices A an G such that A = O p (1), an G = O p (1), we have two conclusions: 1 N && & ; 1/ (i) ( tr A Β Ε F G) = Op ( ) (ii) 1 N tr A F& &&&& FA & O. 1 ( ΕΕ ) = p( ) Proof: Assumption B implies Β = Β 1N1 N Β / N Β + 1N1 N / N Β N N N N = Β = N N βi c N i= 1 ; From Assumptions B D, we obtain 1 Ε && F& N = Ε && F& = Ε Ε 11 1N1 N Ε + 1N1 N Ε 11 F 11 F N N N N = Ε F Ε 11 F 1N1 N Ε F + 1N1 N Ε 11 F N N N = Ε F + Ε 11 F + 1N1 N Ε F + 1N1 N Ε 11 F N N N N N N 1 Ε F + Ε 11 F N N 1 N 1 Op(1) + Ε 1 1 F = Op(1) + εit f = Op(1), N N i= 1 t= 1 where 1 is a 1 vector of ones. hus, 1/ N O p Β = (1) an N && F& O p. 1/ ( ) Ε = (1) hen, we have (i), because 5

26 1 1 Β Ε && F& 1 1 tr( A Β Ε&& FG & ) A G = Op Op(1) Op N =. N N We obtain (ii), because tr( A F F A) AA F F A F O p & ΕΕ &&&& & ΕΕ &&&& & Ε && & = N N N. Lemma : Suppose that two matrices A an B are symmetric of orer p. hen, ψ ( ) ( ) ( ) j+ k 1 A + B ψ j A + ψ k B, j + k p + 1. Proof: See Onatski (006) or Rao (1973, p. 68). Proof of heorem 1: Observe that 1 1 F& XX &&&& F& = F& ( F& Β + Ε&& )( Β F& + Ε&& ) F& N N F& F& Β Β F& F& F& F& Β Ε&& F& F& ΕΒ && F& F& F& ΕΕ &&&& F& = N N N N hus, we have ˆ ˆ Β Β Β Β Β Ε&& F& F& ΕΒ && F& ΕΕ &&&& F& = + A + A + A A, N N N N N where A 1 = ( F & F & / ) an A Op (1) = by Assumption A. Now, let Ξ ˆ l be the matrix of the eigenvectors corresponing to the first l( k) largest eigenvalues ˆ µ ˆ ˆ N,1 µ N, L µ N, l of ˆ ˆ Β Β / N. Similarly, efine l Ξ for the matrix Β Β / N. For any l r, we have 6

27 l 1 ˆ ˆ l ˆ ˆ ˆ l Σ j= 1 µ N, j = tr Ξ Β Β Ξ N 1 ˆ l ˆ l 1 ˆ l ˆ l = tr Ξ Β Β Ξ + tr Ξ Β Ε&& FA & Ξ N N 1 ˆ l ˆ l + tr Ξ A F& ΕΕ &&&& F& A Ξ N 1 l l 1 1 tr Ξ Β Β Ξ + Op Op N + l Β Β 1 1 = Σ j= 1λ j + Op N Op, + by Lemma 1, because ˆ l Ξ = (1) an ˆ l ˆ l A Ξ A Ξ = O (1). In aition, O p p l 1 ˆ l ˆ ˆ ˆ l 1 l ˆ ˆ ˆ l Σ j= 1 µ N, j = tr Ξ Β Β Ξ tr Ξ Β Β Ξ N N 1 l l 1 l l = tr Ξ Β Β Ξ + tr Ξ Β Ε&& FA & Ξ N N, 1 l l + tr Ξ A F& ΕΕ &&&& FA & Ξ N l Β Β 1 1 = Σ j= 1λ j + Op + Op N Since these two results hol for any l r, we have hus, for 1 j r, we have 1 1 ˆ µ N, j λ Β Β = j + Op + Op N. Β Β p lim ˆ µ N, j = λ j > 0. N Next, since we have rank( Β ) = r, we can rewrite Β =, where A an C are N r AC an k r matrices, respectively, an rank( A) = rank( C) = r. Let =, an Q( A ) 1 P( A) A( A A) A = 1 P( A). Using the fact that P( A) Β = Β an Q( A) Β = 0, we can easily show 7

28 ˆ ˆ ˆ [ ( ) ( )] ˆ ˆ ( ) ˆ Β Β Β P A + Q A Β Β P A Β F& Ε&& Q( A) Ε&& F& = = + A A, N N N N hus, for j = 1, L, k r, we have ˆ ˆ ˆ ( ) ˆ Β Β Β P A Β F& Ε&& Q( A) Ε&& F& λr + j λr λ j A A N N N F& ΕΕ &&&& F& 1 + & ΕΕ &&&& = N N 1 0 λ j A ( ) ( ), A tr A F FA Op where the first inequality is ue to Lemma. hus, for any 1 r + 1 j k, N, j ( ) ˆ µ = O 1 /. Notice that the secon part hols even for r = 0, which is the no-factor case. p Proof of heorem : For 1 j r, p lim ˆ µ N, j > 0, because rank( Β Β / N ) = r. Since g( ) 0, lim ˆ Pr[ µ N, j > g( ) j r] = 1. For 0 r < j k, p lim ˆ µ N, j <. hus, lim ˆ Pr( µ N, j < g( ) 0 r < j k) = lim ˆ Pr( µ N, j < g( ) 0 r < j k) = 1, because g( ) an µ, = O (1). ˆN j p 8

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