The Forward Premium is Still a Puzzle Appendix

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1 The Forward Premium is Still a Puzzle Appendix Craig Burnside June 211 Duke University and NBER.

2 1 Results Alluded to in the Main Text 1.1 First-Pass Estimates o Betas Table 1 provides rst-pass estimates o the betas or the base case discussed in Section 2 o the main text. The rst pass is a time series regression o each portolio s excess return on the vector o risk actors: Rit e = a i + t i + it, t = 1; : : : ; T, or each i = 1; : : : ; n: (1) Here i represents the ith row in. The system o equations represented by (1) is estimated using equation-by-equation OLS. In Section 2 I also mention that tests o the SDF covariances lead to the same conclusions as tests o the SDF betas. The p-values associated with the the null hypothesis that cov(rit; e m t ) = c or all i are :94, :91, and :89 or SDF models (i), (ii) and (iii), respectively. The p-values associated with the the null hypothesis that cov(rit; e m t ) = or all i are :81, :7, and :7 or SDF models (i), (ii) and (iii), respectively. Table 2 provides rst-pass estimates o the betas or the case discussed in Section 4.3 o the main text. 1.2 Weak Identi cation In this section I argue that tests o the pricing errors ail to reject LV s model at low levels o signi cance due to an identi cation problem. In the second-pass regression with the constant, the parameters and are identi ed under the assumption that + = ( ) has ull column rank, where is an n 1 vector o ones. In the second-pass regression with no constant, is identi ed i has ull column rank. The same conditions must hold or the GMM procedure used to estimate b. When the rank conditions ail, conventional inerence drawn rom second pass regressions and GMM is unreliable because standard asymptotic theory does not apply. As Burnside (21) discusses, t-statistics or ^ and ^b have non-standard distributions, and, most importantly, pricing-error tests cannot reliably detect model misspeci cation. It is straightorward to test whether LV s model is identi ed using a rank test rom Jonathan H. Wright (23). Table 3 presents tests o the null hypothesis that has reduced rank. Since there are three risk actors in the model, has reduced rank i rank() < 3. As Table 3 indicates, it is not possible to reject that rank() = 1 at conventional signi cance 1

3 levels. In act, it is only when VARHAC standard errors are used that we can reject the null hypothesis that rank() =, which is equivalent to the null that every element o is zero. Similar results are obtained when I test the rank o +. A standard tool or conducting inerence under weak identi cation is to construct con - dence sets or weakly identi ed parameters using the objective unction corresponding to the continuously updated (CU) GMM estimator. 1 To put this method into practice, I construct an m t series using the ollowing values o b: (i) the vector corresponding to LV s two pass estimates o : b c = 21, b d = 13 and b r = 4:5, (ii) the vector corresponding to LV s GMM estimates o b: b c = 37, b d = 75 and b r = 4:7, (iii) the vector corresponding to the calibrated model discussed in section I.E o LV s paper: b c = 6:7, b d = 23 and b r = :31. I then treat m t as a risk actor, and construct a robust con dence set or m, the price o SDF risk, using the CU-GMM objective unction corresponding to the two-pass regression method (with no constant included in the regression). The reason or treating m t as a risk actor is that this reduces the dimensionality o the con dence set to one parameter. The objective unction is evaluated at each possible value o m, and the corresponding p-value is calculated. Those values o m or which the p-value exceeds.5 lie in the 95 percent con dence set or m. The di erence between this approach and standard GMM is that the weighting matrix is recalculated at each m, and no degrees o reedom are lost because m is not estimated. The resulting con dence sets are the entire real line in cases (i) and (iii). In case (ii) the con dence set is ( 1; :75) [ (:98; +1). 2 While the rank tests are the de nitive indication that the data are uninormative about, the act that the con dence sets are so vast helps to illustrate the problem. When con dence sets or parameters are constructed using the CU-GMM objective unction, lack o inormation about the parameters goes hand-in-hand with inability to reject the model. Given the con dence sets I have constructed, o course, we cannot ormally reject the model. It is easy to nd parameter values or which the test o the over-identiying restrictions ails to reject. But this hardly seems like a signal o the model s success. Table 4 presents results o rank tests or the cases where the six Fama-French equity portolios and the ve Fama bond portolios are added to the set o test assets. It also presents results o rank tests or the cases where LV s di erenced currency portolios are used 1 See James H. Stock and Wright (2), Stock, Wright and Yogo (22) and Yogo (24) or details. 2 The test or m = is equivalent to a test that the vector E(R e ) is equal to zero, i the GMM errors are demeaned beore the long-run covariance matrix is computed. I do not demean the GMM errors, but doing so does not change the act that the con dence region or m encompasses most o the real line. 2

4 as the test assets. The results o the tests suggest that the model remains underidenti ed in these cases. 1.3 Estimates o the Models with Additional Test Assets Table 5 provides GMM estimates o the model with no constant or the case where the currency portolios and six Fama-French equity portolios are used as test assets. Table 6 provides GMM estimates o the model with no constant or the case where the currency portolios, six Fama-French equity portolios, and ve Fama bond portolios are used as test assets. 2 Calculation o Standard Errors 2.1 Standard Errors or the Two-Pass Procedure This section discusses the computation o standard errors or the two-pass regression procedure. Lustig and Verdelhan compute standard errors under the assumption that the betas are known. I rst, consider this case, and then consider the case where the betas are treated as generated regressors. The derivations here are reproduced rom or based on Cochrane (25) and Shanken (1992) Betas are Known Equation (1) can be rewritten as R e t = a + t + t where a is an n 1 vector ormed rom the individual a i, and t is an n 1 vector ormed rom the individual it. Traditionally the actors and errors are assumed to be i.i.d. over time, with var( t ) = and var( t ) =, but these assumptions can be generalized. Taking averages over time: R e = a + + ; (2) where R e, and are the sample means o R e t, t and t. Without a Constant When the betas are known and the second stage excludes a constant ^ = ( ) 1 R e. This implies that ^ = R e ^ = I ( ) 1 R e = M R e : 3

5 Given that the beta representation implies that E( R e ) =, it ollows that plim ^ = M E( R e ) = M = : Also, the asymptotic covariance matrix o p T ^ is ^ = M RM where R is the asymptotic covariance matrix o p T ( R e ER e ). Given (2) and the assumptions made above: R = + hence ^ = M ( + ) M = M M : Since ^ has rank n k, C = T ^ 1 ^ ^ must be computed using a generalized inverse, and C is distributed 2 with n o p T (^ ) is k degrees o reedom. Also, the asymptotic covariance matrix ^ = ( ) 1 R( ) 1 = + ( ) 1 ( ) 1 : With a Constant When a constant is included in the second stage we have ^ = (X X) 1 X R e where = ( ), X = ( ) and is an n 1 vector o ones. Thereore, ^u = R e X^ = I X(X X) 1 X R e = M X R e : The beta representation states that E(R e ) = + = X where =. So E(^u) = M X E( R e ) = M X X = : Also, the asymptotic covariance matrix o p T ^u is ^u = M X RM X : The term can be written as X~ X with ~ = : 4

6 Thereore we can rewrite R as X~ X + so that: ^u = M X (X~ X + )M X = M X M X : Since ^u has rank n k 1, C = T ^u 1 ^u ^u must be computed using a generalized inverse, and C is distributed 2 with n k 1 degrees o reedom. Also, the asymptotic covariance matrix o p T (^ ) is ^ = (X X) 1 X RX (X X) 1 = (X X) 1 X (X~ X + )X (X X) 1 = ~ + (X X) 1 X X (X X) 1 : As suggested in the text, the constant should really be considered part o the pricing error. As such, its signi cance could be tested alone, as it is the rst element o ^. Alternatively one might also consider a reormulated 2 test based on ^ = R e ^ = ^u + ^: Letting we have P = I k ^ = R e XP^ = I XP(X X) 1 X R e = H R e : Thereore the asymptotic covariance matrix o p T ^ is ^ = H(X~ X + )H = HH : As in the other cases, this means that a test statistic can be ormed as C = T ^ 1 ^ ^. It will be distributed 2 n k since ^ is o rank n k and must be computed using a generalized inverse Shanken Corrections (Betas are Estimated) When the betas are unknown the rst stage estimates, ^ i, are given by ^ i = (^ ^) 1^ R e i where R e i is a T 1 vector with elements Rit e and ^ is a T k matrix with rows equal to ( t ). Given the model, R e i = a i + i + i, where is an T k matrix with rows equal to t and i is a T 1 vector with elements it. Hence 5

7 ^ i = (^ ^) 1^ (a i + i + i ) = i + (^ ^) 1^ i : Assuming that t and t are independent, the asymptotic covariance between p T (^ i i ) and p T (^ j j ) is given by ij 1 rearranged into a nk 1 stacked vector, where ij is the covariance between it and jt. I ^ is ^ v = ^ 1 ^ 2. ^ n 1 C A ; the asymptotic covariance matrix o p T (^ v v ) is 1. Without a Constant When the second stage excludes a constant ^ = ^A R e, where ^A = (^ ^) 1^. To work out the asymptotics we proceed as ollows. De ne = + : (3) The model implies that ER e = a+ =. Hence we can write a = ( this result into (2) we get R e = ( + ) + : ). Substituting Using (3) we have R e = + = ^ + (^ ): (4) Premultiplying (4) by ^A we get ^ = + ^A[ (^ ) ]; so that ^ = ^A[ (^ ) ]: (5) Now ^ = (^ ) + ( ) = ^A[ (^ ) ] + ( ): 6

8 The term is uncorrelated with (^ ) ollowing arguments in Shanken s (1992) Lemma 1. Also we can rewrite the term in brackets as (I n )(^ v v ). Since plim =, and plim ^A = A = ( ) 1 this means that the asymptotic variance o p T (^ ^ = A[ + (I n ) 1 (In )]A + : Using the rules or Kronecker products this reduces to The pricing errors are ^ = (1 + 1 )AA + : h ^ = R e ^^ = I ^(^^) i 1^ R e = M^ R e i h^ + (^ ) = M^ = M^ h i (^ ) Hence the asymptotic covariance matrix o p T ^ is ^ = (1 + 1 )M M : ) is Since ^ has rank n k, C = T ^ 1 ^ ^ must be computed using a generalized inverse, and C is distributed 2 with n k degrees o reedom. With a Constant unknown, we have When a constant is included in the second stage, but the betas are ^ = ( ^X ^X) 1 ^X R e where = ( ), ^X = ( n ^ into a n(k + 1) 1 stacked vector, ) and n is an n 1 vector o ones. I ^X is rearranged ^X v = 1 ^ 1 1 ^ 2. 1 ^ n 1 ; C A the asymptotic covariance matrix o p T ( ^X v X v ) is where = 1 : 7

9 We have ^ = ^A R e, where ^A = ( ^X ^X) 1 ^X. The model implies that ER e = a + = + = (since, under the null, = ). Hence we can write a = ( ). Substituting this result into (2) we get R e = ( + ) + : De ning ( ) we can then write this as R e = X + = ^X + ( ^X X) : (6) Premultiplying (6) by ^A we get ^ = + ^A[ ( ^X X) ]; so that ^ = ^A[ ( ^X X) ]: Now ^ = (^ ) + ( ) = ^A[ ( ^X X) ] + : The term in uncorrelated with the ( ^X X) term ollowing arguments in Shanken s (1992) Lemma 1. And we can rewrite the term in brackets as (I n )( ^X v X v ). Since plim = and plim ^A = A = (X X) 1 X this means that the asymptotic variance o p T (^ ) is ^ = A[ + (I n ) ( ) (I n )]A + ~ : Using the rules or Kronecker products this reduces to ^ = (1 + )AA + ~ ; but because o the orm o it can also be written as ^ = (1 + 1 )AA + ~ : The pricing errors are h i ^u = R e ^X^ = I ^X( ^X ^X) 1 ^X R e = M ^X R e h = M ^X ^X + ( ^X X) i h i = M ^X ( ^X X) 8

10 Hence the asymptotic covariance matrix o p T ^u is ^u = (1 + 1 )M X M X : Since ^u has rank n k 1, C = T ^u 1 ^u ^u must be computed using a generalized inverse, and C is distributed 2 with n k 1 degrees o reedom. As suggested in the text, the constant should really be considered part o the pricing error. As such, its signi cance could be tested alone, as it is the rst element o ^. Alternatively one might also consider a reormulated 2 test based on ^ = R e ^^ = ^u + ^: Letting we have P = I k h i ^ = R e ^XP^ = I ^XP( ^X ^X) 1 ^X R e = ^H R e h = ^H ^X + ( ^X X) i h = ^H ( ^X X) i : Thereore the asymptotic covariance matrix o p T ^ is ^ = (1 + 1 )HH : As in the other cases, this means that a test statistic can be ormed as C = T ^ 1 ^ ^. It will be distributed 2 n k since ^ is o rank n k and the covariance matrix must be inverted using a generalized inverse GMM Standard Errors (Betas are Estimated) Without a Constant The model is estimated by exploiting the moment restrictions E(R e it a i i t ) =, E[(R e it a i i t ) t] =, and E(R e it i) =, i = 1, : : :, n. Let ~ t = ( 1 t ), ~ i = ( a i i ) and = 9 ~ 1 ~ 2. ~ n 1 : C A

11 De ne the n(k + 2) 1 vector u t () = ~ t (R1t e ~ t 1 ~ ) ~ t (R2t e ~ t 2 ~ ) ~ t (Rnt e ~ t ~ n ) R e t 1 C A ; the n(k + 2) 1 vector g T () = 1 T P T t=1 u t(), and the [n(k + 1) + k] [n(k + 2) 1] matrix In(k+1) a T = ^OLS The GMM estimator that sets a T g T = reproduces the two-pass estimates o a,, and. De ne d T T = I n M ~ I n ^ : n(k+1)k ^ OLS! where M ~ = 1 T P T t=1 ~ t ~ t. Let a = plim a T and d = plim d T. The covariance matrix o p T (^ ) is and the covariance matrix o p T g T (^) is V = (ad) 1 asa (ad) 1 V g = [I d(ad) 1 a]s[i d(ad) 1 a] where S is the asymptotic covariance matrix o p T g T (). These results ollow rom the acts that p T (^ ) d! (ad) 1 a p T g T () and p T g T (^) d! [I d(ad) 1 a] p T g T (). The test statistic or the pricing errors is just T g T (^) Vg 1 g T (^), where the inverse is generalized. Since S = P 1 j= 1 E(u tu t j), I use a variant o a VARHAC estimator or S: due to limited sample size I only allow lags o an error to enter into the VAR equation or that error. With a Constant The model is estimated by exploiting the moment restrictions E(R e it a i i t ) =, E[(Rit e a i i t )t] =, and E(Rit e i) =, i = 1, : : :, n. Now let 1 ~ 1 ~ 2 =. : B ~ n A 1

12 De ne the n(k + 2) 1 vector u t () = ~ t (R1t e ~ t (R2t e ~ t 1 ~ ) ~ t 2 ~ ) ~ t (Rnt e ~ t n ~ ) R e t 1 C A ; P the n(k + 2) 1 vector g T () = 1 T T t=1 u t();and the (n + 1)(k + 1) [n(k + 2) 1] matrix In(k+1) a T = ^X : where ^X = ( n1 ^OLS ). The GMM estimator that sets a T g T = reproduces the twopass estimates o a,, and. De ne d T T = I n M ~ I n ^ n(k+1)(k+1) ^X! : Let a = plim a T and d = plim d T. The covariance matrix o p T (^ ) is and the covariance matrix o p T g T (^) is V = (ad) 1 asa (ad) 1 V g = [I d(ad) 1 a]s[i d(ad) 1 a] where S is the asymptotic covariance matrix o p T g T (). These results ollow rom the acts that p T (^ ) d! (ad) 1 a p T g T () and p T g T (^) d! [I d(ad) 1 a] p T g T (). The test statistic or the pricing errors is just T g T (^) Vg 1 g T (^), where the inverse is generalized. A test o the pricing errors inclusive o the contest can be derived rom the joint distribution o p T (^ ) and p T g T (^). Since S = P 1 j= 1 E(u tu t j), I use a variant o a VARHAC estimator or S. Due to limited sample size I only allow lags o an error to enter into the VAR equation or that error. 2.2 GMM Estimation o the Model Model without a Constant Asympotic Theory Let u 1t (b; ) = R e t[1 ( t ) b] (7) TX g 1T (b; ) = T 1 u 1t (b; ) = R e (1 + b) D T b: (8) t=1 11

13 where D T = T 1 P T t=1 Re t t and R e = T 1 P T t=1 Re t. Also de ne Finally, de ne the stacked vectors u1t (b; ) u t (b; ) = u 2t () and the matrix u 2t () = t (9) TX g 2T () = T 1 u 2t () = : (1) S = E[ 1X j= 1 t=1 g T (b; ) = u t (b ; )u t j (b ; ) ]: g1t (b; ) g 2T () The parameters b and are estimated by setting a T g T =, where (DT R a T = e ) W T ; I k and W T is some weighting matrix. Given the de nition o g T this means the GMM estimator is the solution to (DT R e ) W T implying that I k R e (1 + b) D T b = (11) ^ = (12) ^b = (d T W T d T ) 1 d T W T R e ; (13) where d T = D T R e. In the rst stage the weighting matrix is W T Verdelhan ollow Cochrane (25) and set W T = " T 1 = I n. In the second stage, Lustig and # 1 TX u 1t (^b; ^)u 1t (^b; ^) (14) t=1 where ^b = (d T d T ) 1 d T R e and ^ = are the rst stage estimates o the parameters. In this case plim W T = W = S 1 11 where S 11 = E[u 1t (b ; )u 1t (b ; ) ]: Given (13), plim ^b = b. This ollows rom the act that plim d T = d E[R e t( t ) ] and that plim R e = E(R e ). We then get plim ^b = (d Wd) 1 d WE(R e ). The model implies 12

14 that E(R e ) = db. Hence plim ^b = b. So the rst and second stage estimates o b are obviously consistent. The derivation o the asymptotic distribution o (^b; ^) relies on deriving the distance between g T (^b; ^) and g T (b ; ). Using (8) and the consistency o ^b and ^ we can argue that there is a pair ( b;) between (b ; ) and (^b; ^) such that g 1T (^b; ^) = g 1T (b ; ) + R e D T (^b b ) + R e b (^ ): (15) From (1) we also have g 2T (^) = g 2T ( ) (^ ): (16) Premultiplying (15) by d T W T one obtains = d T W T g 1T (^b; ^) = d T W T [g 1T (b ; ) + R e D T (^b b ) + R e b (^ )] (17) We can rewrite (16) and (17) together as = ^a T g T (b ; ) T ^b b ^ : (18) where d ^a T = T W T I k DT R T = e R e b I k We have plim ^a T = a and plim T = where d a = W d db b = I k I k ; : and I have used the act that plim R e = E(R e ) = db. Hence p ^b b d (d T! Wd) 1 b b d W ^ I k I k pt gt (b ; ): Thus we have p T (^b b ) p T (^ ) d! (d Wd) 1 d W b b d! I k p T gt (b ; ) = p T g 2T (b ; ) p T gt (b ; ) = B p T g T (b ; ) and the asymptotic covariance matrix o p T (^b b ) is V b = BSB : (19) 13

15 The act that is estimated a ects V b. I was known the covariance matrix would reduce to (d Wd) 1 d WS 11 Wd (d Wd) 1. To get a test o the pricing errors, Cochrane (25) ollows Hansen (1982) in showing that the asymptotic distribution o p T g T (^b; ^) is normal with covariance matrix [I (a) 1 a]s[i a (a) 1 ]: Some algebra shows that this implies that p T g 1T (^b; ^) is normal with covariance matrix V = [I d (d Wd) 1 d W]S 11 [I Wd (d Wd) 1 d ]: This is the same expression as one obtains when is known. Since the test o the pricing errors is obtained as T g 1T (^b; ^) V 1 T g 1T (^b; ^), where the inverse is generalized, and V T is a consistent estimate o V, the act that is estimated has no e ect on the statistic as compared to the case where is treated as known. Factor Risk Premia The GMM estimator produces estimates o b and. To obtain an estimate o we can use the expression = b. This requires estimation o. This can be done by adding moment conditions that identiy the unique elements o : E[( it i ) jt j ;ij ], i = 1; : : : ; k; j = i; : : : ; k: (2) The estimate ^ then corresponds to the sample covariance matrix o t. O course, standard errors or ^ should take into account estimation o. Equivalence Between the First Stage o GMM and the Two-Pass Procedure rst stage estimate o b based on W = I n is ^b = (d T d T ) 1 d T R e. The matrix d T The is the sample covariance between R t and t. Hence ^ = d T ^ 1, where ^ is the sample covariance o the risk actors. Thereore d T = ^ ^ and ^b = (^ ^^ ^ ) 1 ^ ^ R e = ^ 1 ^. Since ^ GMM ^ ^b, ^ GMM = ^ rom the two-pass procedure. VARHAC Spectral Density Matrix Since S = P 1 j= 1 E(u tu t j), I estimate it as ollows. De ne u 1t and u 2t as in (7) and (9). I use a VARHAC estimator or S, imposing the restriction that Eu 1t u t j = or j 1. This means that the VARHAC estimator or S 11, the sub-block o S equal to P 1 j= 1 E(u 1tu 1t j), is the same as the HAC estimator or S 11. But this is not true or the S 12, S 21 and S 22 sub-blocks. In practice, the VARHAC 14

16 procedure typically nds persistence in some elements o u 2t because these are the GMM errors corresponding to t ^. Since some o the risk actors are persistent it is important to allow or this possibility, which is not ruled out by theory. Equivalence o the Pricing Error Test at the First and Second Stages o GMM At the rst stage o GMM we have ^b 1 = (d T d T ) 1 d T R e so the pricing errors are ^ 1 = R e d T ^b1 = M d R e where M d = I d T (d T d T ) 1 d T. The estimated covariance matrix o ^ 1 is V T = M d^s 11 M d, so the test statistic is T R e Md (M d^s 11 M d ) 1 M d R e where the inverse is generalized. At the second stage o GMM we have ^b 2 = (d T W T d T ) 1 d T W T R e so the pricing errors are ^ 2 = R e d T ^b2 = M W R e where M W = [I d T (d T W T d T ) 1 d T W T ] R e. The estimated covariance matrix o ^ 2 is V T = M W^S 11 M W T R e M W (M W^S 11 M W) 1 M W R e : so the test statistic is Because M d (M d^s 11 M d ) 1 M d = M W (M W^S 11 M W ) 1 M W, when W = ^S 1 11, the two statistics are the same Model with a Constant Asymptotic Theory Let u 1t (b; ; ) = R e [1 ( t ) b] (21) TX g 1T (b; ; ) = T 1 u 1t (b; ; ) = R e (1 + b) D T b: (22) t=1 where b = ( b ), = ( ) and D T = ( n1 D T ). De ne u 2t and g 2T as in (9) and (1). De ne the stacked vectors u1t (b; ; ) g1t (b; ; ) u t (b; ; ) = g u 2t () T (b; ; ) = g 2T () 15

17 and the matrix S = E[ 1X j= 1 u t (b ; ; )u t j (b ; ; ) ]: The parameters b and are estimated by setting a T g T =, where ( D a T = T R e ) W T ; I k and W T is some weighting matrix. Given the de nition o g T this means the GMM estimator is the solution to ( D T R e ) W T implying that I k R e (1 + b) D T b = (23) ^ = (24) bb = d T W T dt 1 d T W T R e ; (25) where d T = D T R e b = ( n1 D T ) R e ( ) = ( n1 d T ). The rst and second stage estimates are calculated as in the case with the constant. In the rst stage W T = I n. In the second stage, W T is the inverse o a consistent estimator or S 11 = E[u 1t (b ; ; )u 1t (b ; ; ) ]. Equivalence Between the First Stage o GMM and the Two-Pass Procedure The rst stage estimate o b based on W = I n is b b = d T d T 1 d T R e. The matrix d T = ( n1 d T ), which can be rewritten as d T = ( n1 ^ ^ ). Hence bb = = =! =1 R e ^ ^ R e ^ ^ ^ ^ ^ ^^ ^ " 1 ^! # 1 1 ^ ^ ^^ ^ 1 ^! 1 R e ^ 1 ^ ^^ ^ : R e R e ^ ^ R e The two-step estimator o and is ^^ = ^ ^ ^^! 1 R e ^ R e 16

18 Hence bb = 1 ^ 1 ^^ = So the GMM estimator o is identical to the two-step estimator o. Also ^ GMM ^ ^b, ^ GMM = ^ rom the two-pass procedure. ^ ^ ^ 1 : 3 Replication o Lustig and Verdelhan s Results Some results in my paper are directly comparable to results presented in LV s original article. In some cases di erences arise, and I try to explain these di erences in this section. Their Table 5 (column EZ-DCAPM) and Table 14 (panel B, column C) are directly comparable to my Table 3. The point estimates o, OLS standard errors, and R 2 are identical. The p-values or the pricing error test are di erent. They report a p-value o :628, while I report a p-value o :483. The value o my test statistic is 3:4666. When a constant is included in the model, the covariance matrix o the error vector has rank n k 1 = 4. The p-value or a statistic o 3:4666, with 4 degrees o reedom is :483. I one incorrectly uses the n k = 5 as the degrees o reedom or the test, one obtains LV s p-value, :628. The Shanken standard errors are di erent. For consumption growth, durables growth and the market return, I report standard errors o 2:11, 2:42 and 18:8. They report slightly larger standard errors: 2:15, 2:52 and 19:8. I believe that this may be due to them using the ormula [1 + ( 1 )](AA + ~ ) in computing the standard errors instead o [1 + ( 1 )]AA + ~ (the meaning o these expressions is explained in a previous section o the appendix). When I use the incorrect ormula, I reproduce their standard errors to within one decimal place. We report di erent types o GMM standard errors (they use HAC, while I used VARHAC), so they are not directly comparable. The reported mean absolute pricing errors also di er. I cannot account or this di erence. LV s Table 6 reports individual actor betas, not partial actor betas obtained in the rst pass regressions, so it is not comparable to Table 1 in this appendix. Table 8 o LV s paper appears to repeat a typo contained in Yogo (26). The structural parameters o the Yogo model map to the bs according to b 1 = (1= + (1= 1=)) (26) b 2 = (1= 1=) (27) b 3 = 1 (28) 17

19 where = (1 )=(1 1=). This corresponds to equation (19) is Yogo (26). Given estimates o the bs, Yogo s approach is to set a value or and then solve the three equations above or, and. The solutions Yogo (26) states in his paper near the bottom o page 557 are The expression or (29) is wrong and should be: = 1 b 3 b 1 + b 3 (29) = b 1 + b 2 + b 3 (3) = b 2 b 1 + b 2 + (b 3 1) = : (31) = 1 b 3 b 1 + b 2 (32) With b c = 21:, b d = 13 and b r = 4:46 I obtain = :32 using (32). Using the incorrect ormula in (29) gives = :21, as in Lustig and Verdelhan s paper. This error does not a ect values o structural parameters given in Yogo (26), as the error appears to only be in the text, not in calculations. LV s Table 14 (panel A, column C) is directly comparable to my Tables 7 and 8. Our reported point estimates are identical. They appear to report the MAE or the rst stage o GMM rather than the second stage. We report di erent types o GMM standard errors, so they are not directly comparable. However, I believe their reported HAC standard errors to be incorrect. My code can be used or the HAC case, and I do not replicate their results. I believe their code ignores the sampling uncertainty induced by being estimated. When is known, the expression in (19) is simpler, and reduces to V k = (d Wd) 1 d WS 11 Wd (d Wd) 1 : (33) In the second stage o GMM W = S 1 11 so (33) reduces to V k = d S 1 11 d 1 : (34) I believe that LV base their GMM standard errors on (34). However, this is inappropriate when must be estimated. This is because V b, given in (19), does not reduce to V k unless b = or is known. This problem does not bias the standard errors sharply in a consistent direction, and the di erences it induces are small. Finally, my Tables 5(b) and 6(b) in this appendix are not directly comparable to LV s Table 14 (panel A, columns E/C and E/B/C). The reason is that LV do not provide the 18

20 equity and bond portolio data in their archive. Consequently, di erences between our data series or these test assets probably account or any di erencs in results. Appendix Reerences not Included in the Main Article Stock, James H. and Jonathan H. Wright (2) GMM with Weak Identi cation, Econometrica 68, Stock, James H., Jonathan H. Wright and Motohiro Yogo (22) A Survey o Weak Instruments and Weak Identi cation in Generalized Method o Moments Journal o Business and Economic Statistics 2: Yogo, Motohiro (24) Estimating the Elasticity o Intertemporal Substitution When Instruments Are Weak, Review o Economics and Statistics 86:

21 TABLE 1 (Part 1): First-Pass Estimates o the Betas, Portolios P1 P4 Portolio Factor ( j ) Test o (Pi) c d r W ij = 8j P1 :21 :28 :68 (:852) (:612) (:55) (:6) [:643] [:563] [:49] [:492] h 1:4; 2:i h 1:4; 1:i h :19; :3i P2 :74 :91 :34 (:889) (:638) (:58) (:579) [:62] [:47] [:62] [:164] h 1:; 2:1i h :8; 1:6i h :16; :13i P3 :639 :962 :19 (:882) (:633) (:57) (:464) [1:26] [:814] [:58] [:645] h 2:6; 1:5i h :8; 1:6i h :13; :11i P4 :546 :982 :89 (1:95) (:786) (:71) (:156) [1:75] [:749] [:69] [:65] h 2:6; 2:3i h 1:3; 1:7i h :25; :4i Notes: Annual data, The regression equation is Rit e = a i + t i + it, where Rit e is the excess return o portolio i at time t, t = ( c t d t r W t ), c is real per household consumption (nondurables & services) growth, d is real per household durable consumption growth, and r W is the value weighted US stock market return. The portolios are equally-weighted groups o short-term oreign-currency denominated money market securities sorted according to their interest di erential with the United States, where P1 and P8 are the portolios with, respectively, the smallest and largest interest di erentials. The table reports ^ ij and p-values or tests o the hypotheses that ij = 8j or each portolio i. For estimates o betas, OLS standard errors are in parentheses, and GMM-VARHAC standard errors are in square brackets. Bootstrapped 95 percent con dence regions appear in angled brackets. For test statistics, corresponding p-values are presented. 2

22 TABLE 1 (Part 2): First-Pass Estimates o the Betas, Portolios P5 P8 Portolio Factor ( j ) Test o (Pi) c d r W ij = 8j P5 :18 :485 :9 (1:6) (:722) (:65) (:74) [:754] [:714] [:65] [:746] h 1:5; 2:4i h 1:4; 1:5i h :15; :14i P6 :755 1:79 :23 (1:89) (:781) (:71) (:556) [:958] [:833] [:68] [:595] h 2:2; 2:5i h 1:6; 1:6i h :14; :15i P7 :36 1:234 :27 (1:44) (:749) (:68) (:11) [:797] [:73] [:63] [:126] h 1:2; 2:8i h 1:1; 1:7i h :21; :9i P8 1:342 1:426 :79 (1:674) (1:21) (:18) (:684) [1:646] [1:225] [:114] [:7] h 6:; 1:9i h 1:2; 2:8i h :18; :34i Notes: Annual data, The regression equation is Rit e = a i + t i + it, where Rit e is the excess return o portolio i at time t, t = ( c t d t r W t ), c is real per household consumption (nondurables & services) growth, d is real per household durable consumption growth, and r W is the value weighted US stock market return. The portolios are equally-weighted groups o short-term oreign-currency denominated money market securities sorted according to their interest di erential with the United States, where P1 and P8 are the portolios with, respectively, the smallest and largest interest di erentials. The table reports ^ ij and p-values or tests o the hypotheses that ij = 8j or each portolio i. For estimates o betas, OLS standard errors are in parentheses, and GMM-VARHAC standard errors are in square brackets. Bootstrapped 95 percent con dence regions appear in angled brackets. For test statistics, corresponding p-values are presented. 21

23 TABLE 2: First-Pass Estimates o the Betas, Portolios DQ2 DQ6 Portolio Factor (j) Test o (i) c d r W ij = 8j DQ2 :72 :522 :81 (:722) (:69) (:34) (:73) [:75] [:579] [:6] [:165] DQ3 :264 :147 :82 (:77) (:65) (:37) (:132) [:939] [:695] [:54] [:5] DQ4 :745 :6 :91 (:978) (:825) (:47) (:249) [1:98] [:833] [:55] [:298] DQ5 :43 :333 :156 (:948) (:799) (:45) (:5) [1:54] [:76] [:62] [:3] DQ6 :399 :162 :161 (1:87) (:917) (:52) (:21) [1:186] [:862] [:72] [:163] Notes: Quarterly data, 1976Q2 21Q1. The regression equation is R e it = a i + t i + it, where R e it is the excess return o portolio DQi at time t, t = ( c t d t r W t ), c is real per household consumption (nondurables & services) growth, d is real per household durable consumption growth, and r W is the value weighted US stock market return. The excess return to portolio DQi is the di erence between the quarterly excess returns o portolios Qi and Q1. At the monthly requency the excess returns o the Q-portolios are the payo s to holding long equally-weighted one-month orward positions in oreign currencies sorted by orward discount versus the US dollar. Q1 and Q6 are the portolios with, respectively, the smallest and largest orward discounts. The table reports ^ ij and p-values or tests o the hypotheses that ij = 8j or each portolio i. For estimates o betas, OLS standard errors are in parentheses, and GMM-VARHAC standard errors are in square brackets. For test statistics, corresponding p-values are presented. 22

24 TABLE 3: Tests o the Rank o the Factor Beta Matrix Test o H : rank() = r Test o H : rank( + ) = r + 1 r p-value r + 1 p-value 2 (:615) 3 (:529) [:657] [:699] 1 (:643) 2 (:618) [:444] [:326] (:639) 1 (:562) [:] [:3] Notes: Annual data, The matrix is obtained by running the regressions described in Table 1. The matrix must have ull column rank (3) or to be identi ed in the two-pass procedure without the constant. The matrix + = ( ) must have ull column rank (4) or and to be identi ed in the two-pass procedure with the constant. The table presents tests o the null hypothesis that these rank conditions ail. The p-values or the tests are presented in parentheses (OLS standard errors) and square brackets (GMM-VARHAC standard errors). 23

25 TABLE 4: Tests o the Rank o the Factor Beta Matrix with Dierent Test Assets Tests o H : rank() = r r Currencies & Equities Currencies, Equities & Bonds Di erenced Currencies (:422) (:732) (:529) [:76] 1 (:415) (:16) (:618) [:134] (:) (:) (:562) [:1] r + 1 Tests o H : rank( + ) = r + 1 Currencies & Equities Currencies, Equities & Bonds Di erenced Currencies (:449) (:728) (:46) [:626] 2 (:497) (:182) (:612) [:339] 1 (:) (:) (:63) [:5] Notes: See the note to Table 3. P-values or rank tests are presented in parentheses (OLS standard errors) and square brackets (GMM-VARHAC standard errors). Dashes indicate cases where the GMM-VARHAC standard errors cannot be calculated because the number o parameters in the matrix plus the number o equations (to account or the constants in the rst pass regressions) exceeds the sample size. 24

26 TABLE 5: GMM Estimates o the Model with no Constant Currency and Equity Portolios as Test Assets (a) 1st Stage (b) 2nd Stage (c) Iterated GMM Factor ^b ^ ^b ^ ^b ^ c 113:9 2:38 84:6 2:57 6:7 2:53 (82:1) (1:23) (55:4) (:62) (51:9) (:98) d 4:89 1:8 33:2 2:68 6:7 3:5 (65:8) (2:3) (27:9) (:88) (45:6) (1:59) r W 2:11 11:2 4:19 13:7 2:6 5:8 (3:37) (3:87) (1:71) (6:2) (2:14) (5:6) R 2 or currencies :3 :6 :76 MAE or currencies 1:44 1:37 1:88 Notes: Annual data, The table reports GMM estimates o b and obtained by exploiting the moment restrictions ER e t [1 ( t ) b]g =, E( t ) = and E[( t )( t ) ] =, where R e t is a vector o excess returns that includes the currency portolios described in Table 1, as well as Fama and French s (1993) six equity portolios created by sorting stocks on the basis o size and value, t = ( c t d t r W t ), c is real per household consumption (nondurables & services) growth, d is real per household durable consumption growth, r W is the value weighted US stock market return. GMM-VARHAC standard errors are in parentheses. The R 2 statistic and mean absolute pricing error (MAE) are presented or currency portolios only, and are comparable to the statistics in Tables 7 and 8 in the main text. 25

27 TABLE 6: GMM Estimates o the Model with no Constant Currency, Equity and Bond Portolios as Test Assets (a) 1st Stage (b) 2nd Stage (c) Iterated GMM Factor ^b ^ ^b ^ ^b ^ c 89:3 1:67 91:1 1:8 17:1 :4 (61:1) (1:7) (33:5) (:46) (23:2) (:8) d 14:7 :94 12:2 1: 28:8 1:33 (52:5) (1:73) (2:7) (:67) (22:8) (:74) r W 1:92 1:4 3:4 13:7 6:9 21:6 (2:64) (3:24) (1:24) (5:5) (1:55) (1:8) R 2 or currencies :3 :5 1:35 MAE or currencies 1:4 1:42 1:64 Notes: Annual data, The table reports GMM estimates o b and obtained by exploiting the moment restrictions ER e t [1 ( t ) b]g =, E( t ) = and E[( t )( t ) ] =, where R e t is a vector o excess returns that includes the currency portolios described in Table 1, the equity portolios described in Table 5, and ve Fama bonds portolios sorted by maturity (rom the Center or Research in Securities Prices, 27), t = ( c t d t r W t ), c is real per household consumption (nondurables & services) growth, d is real per household durable consumption growth, r W is the value weighted US stock market return. GMM-VARHAC standard errors are in parentheses. The R 2 statistic and mean absolute pricing error (MAE) are presented or currency portolios only, and are comparable to the statistics in Tables 7 and 8 in the main text. 26