One way to think about pairwise model comparison is to ask whether two competing beta pricing models have the same population cross-sectional R2. Kan et al.
(2010) show that the asymptotic distribution of the difference between the sample cross-sectionalR2s of two models depends on whether the models are nested or non-nested and whether the models are correctly specified or not. In this section, we focus on theR2 of the CSR with known weighting matrixW and on theR2of the GLS CSR that usesWO D OV221 as the weighting matrix. Since the weighting matrix of the WLS CSR is model dependent, it is not meaningful to compare the WLS cross-sectionalR2s of two or more models. Therefore, we do not consider the WLS cross-sectionalR2in the remainder of the article. Our analysis in this section is based on the earlier work ofVuong(1989),Rivers and Vuong(2002), andGolden (2003).
Consider two competing beta pricing models. Letf1t,f2t, andf3t be three sets of distinct factors at timet, wherefit is of dimensionKi 1,i D1; 2; 3. Assume that model 1 usesf1t andf2t, while Model 2 usesf1t andf3t as factors. Therefore, model 1 requires that the expected returns on the test assets are linear in the betas or covariances with respect tof1tandf2t, i.e.,
2 D1N1;0CCovŒRt; f1t01;1CCovŒRt; f2t01;2DC11; (9.85)
whereC1DŒ1N; CovŒRt; f1t0; CovŒRt; f2t0and1DŒ1;0; 01;1; 01;20. Model 2 requires that expected returns are linear in the betas or covariances with respect to f1t andf3t, i.e.,
2D1N2;0CCovŒRt; f1t02;1CCovŒRt; f3t02;3DC22; (9.86) whereC2DŒ1N; CovŒRt; f1t0; CovŒRt; f3t0and2DŒ2;0; 02;1; 02;30.
In general, both models can be misspecified. Thei that maximizes the 2 of modeliis given by
i D.Ci0W Ci/1Ci0W2; (9.87) whereCiis assumed to have full column rank,i D1; 2. For each model, the pricing- error vector ei, the aggregate pricing-error measure Qi, and the corresponding goodness-of-fit measure i2are all defined as in Sects.9.4and9.5.
WhenK2D0, model 2 nests model 1 as a special case. Similarly, whenK3 D0, model 1 nests model 2. When bothK2 > 0andK3 > 0, the two models are non- nested.
We study the nested models case next and deal with non-nested models later in the section. Without loss of generality, we assumeK3 D 0, so that model 1 nests model 2. Since 21 D 22 if and only if1;2 D 0K2 (this result is applicable even when the models are misspecified), testing whether the models have the same 2is equivalent to testingH0W1;2D0K2. Under the null hypothesis,
TO01;2V .O O1;2/1O1;2 A
2K2; (9.88)
whereV .O O1;2/is a consistent estimator of the asymptotic covariance ofp
T .O1;2 1;2/given in Sect.9.6. This statistic can be used to testH0W 21 D 22. It is important to note that, in general, we cannot conduct this test using the usual standard error of O, which assumes that model 1 is correctly specified. Instead, we need to rely on the misspecification-robust standard error ofOgiven in Sect.9.6.
Alternatively, one can derive the asymptotic distribution of O12 O22and use this statistic to testH0W 21D 22. PartitionHQ1D.C10W C1/1as
HQ1 D
"HQ1;11HQ1;12
HQ1;21HQ1;22
#
; (9.89)
whereHQ1;22isK2K2. Under the null hypothesisH0W 12D 22,
T .O12 O22/A
K2
X
iD1
i
Q0xi; (9.90)
where thexi’s are independent21random variables and thei’s are the eigenvalues ofHQ1;221V .O1;2/. Once again, it is worth emphasizing that the misspecification-robust version ofV .O1;2/should be used to testH0W 21 D 22. Model misspecification tends to create additional sampling variation in O12 O22. Without taking this into account, one might mistakenly reject the null hypothesis when it is true. In actual testing, we replacei with its sample counterpartOi, where theOi’s are the eigenvalues of HOQ11;22V .O O1;2/, andHOQ1;22andV .O O1;2/are consistent estimators ofHQ1;22andV .O1;2/, respectively.
The test of H0 W 12 D 22 is more complicated for non-nested models. The reason is that underH0, there are three possible asymptotic distributions for O12 O22, depending on why the two models have the same cross-sectionalR2. To see this, first let us define the normalized stochastic discount factors at timet for models 1 and 2 as
y1t D1.f1tEŒf1t/01;1.f2tEŒf2t/01;2; (9.91) y2t D1.f1tEŒf1t/02;1.f3t EŒf3t/02;3: (9.92) Kan et al.(2010) show that y1t D y2t implies that the two models have the same pricing errors and hence 21 D 22. Ify1t ¤ y2t, there are additional cases in which 12D 22. A second possibility is that both models are correctly specified (i.e.,
21 D 22 D 1). This occurs, for example, if model 1 is correctly specified and the factorsf3t in model 2 are given byf3t D f2tCt, wheret is pure “noise” – a vector of measurement errors with mean zero, independent of returns. In this case, we haveC1DC2and both models produce zero pricing errors. A third possibility is that the two models produce different pricing errors but the same overall goodness of fit. Intuitively, one model might do a good job of pricing some assets that the other prices poorly and vice versa, such that the aggregation of pricing errors is the same in each case ( 21 D 22 < 1). As it turns out, each of these three scenarios results in a different asymptotic distribution for O21 O22.
For non-nested models, Kan et al. (2010) show that y1t D y2t if and only if 1;2 D 0K2 and2;3 D 0K3. This result, which is applicable even when the models are misspecified, implies that we can test H0 W y1t D y2t by testing the joint hypothesis H0 W 1;2 D 0K2; 2;3 D 0K3. Let D Œ01;2; 02;30 and O D ŒO01;2; O02;30. Under H0 W y1t D y2t, the asymptotic distribution of O is
given by p
T .O /A N.0K2CK3; V . //;O (9.93) where
V . /O D X1 jD1
EŒqQtqQ0tCj; (9.94) andqQtis aK2CK3vector obtained by stacking up the lastK2andK3elements of hQt for models 1 and 2, respectively, wherehQt is given in Sect.9.6.
LetV .O /O be a consistent estimator ofV . /O . Then, under the null hypothesis H0 W D0K2CK3,
T O0V .O /O 1O A 2K2CK3; (9.95) and this statistic can be used to testH0 W y1t D y2t. As in the nested models case, it is important to conduct this test using the misspecification-robust standard error of .O
Alternatively, one can derive the asymptotic distribution of O21 O22givenH0 W y1t Dy2t. LetHQ1D.C10W C1/1andHQ2D.C20W C2/1, and partition them as
HQ1D
"HQ1;11HQ1;12
HQ1;21HQ1;22
#
; HQ2D
"HQ2;11HQ2;13
HQ2;31HQ2;33
#
; (9.96)
whereHQ1;11andHQ2;11 are.K1C1/.K1C1/. Under the null hypothesisH0 W y1t Dy2t,
T .O21 O22/A
KX2CK3 iD1
i
Q0xi; (9.97)
where the xi’s are independent 21 random variables and the i’s are the eigenvalues of " HQ1;221 0K2K3
0K3K2 QH2;331
#
V . /:O (9.98)
Note that we can think of the earlier nested models scenario as a special case of testing H0 W y1t D y2t with K3 D 0. The only difference is that the i’s in (9.90) are all positive whereas some of the i’s in (9.97) are negative. As a result, we need to perform a two-sided test based on O12 O22 in the non-nested models case.
If we fail to rejectH0 W y1 D y2, we are finished since equality of 12 and 22 is implied by this hypothesis. Otherwise, we need to consider the casey1t ¤ y2t. As noted earlier, wheny1t ¤ y2t, the asymptotic distribution of O21 O22 given H0 W 21 D 22 depends on whether the models are correctly specified or not. A simple chi-squared statistic can be used for testing whether models 1 and 2 are both correctly specified. As this joint specification test focuses on the pricing errors, it can be viewed as a generalization of the CSRT ofShanken(1985), which tests the validity of the expected return relation for a single pricing model.
Letn1 D N K1K21andn2 D N K1K31. Also letP1 be an Nn1orthonormal matrix with columns orthogonal toW 12C1andP2be anNn2
orthonormal matrix with columns orthogonal toW12C2. Define gt./D
"
g1t.1/ g2t.2/
# D
"
1ty1t
2ty2t
#
; (9.99)
where1t and2tare the residuals of models 1 and 2, respectively, D.01; 02/0, and
S
"
S11 S12
S21 S22
# D X1
jD1
EŒgt./gtCj./0: (9.100) Ify1t ¤y2tand the null hypothesisH0W 21D 22D1holds, then
T
"
PO10WO 12eO1
PO20WO 12eO2
#0"
PO10WO 12SO11WO 12PO1PO10WO 12SO12WO 12PO2
PO20WO 12SO21WO 12PO1PO20WO 12SO22WO 12PO2
#1"
PO10WO 12eO1
PO20WO 12eO2
#
A2n1Cn2; (9.101) whereeO1andeO2are the sample pricing errors of models 1 and 2, andPO1,PO2, andSO are consistent estimators ofP1,P2, andS, respectively.
An alternative specification test makes use of the cross-sectionalR2s. Ify1t ¤ y2tand the null hypothesisH0W 21D 22D1holds, then
T .O12 O22/A
nX1Cn2 iD1
i
Q0xi; (9.102)
where the xi’s are independent 21 random variables and the i’s are the eigenvalues of "
P10W12S11W12P1 P10W12S12W 12P2
P20W12S21W12P1 P20W12S22W12P2
#
: (9.103)
Note that thei’s are not all positive because O21 O22can be negative. Thus, again, we need to perform a two-sided test ofH0 W 12D 22.
If the hypothesis that both models are correctly specified is not rejected, we are finished, as the data are consistent withH0 W 12 D 22 D 1. Otherwise, we need to determine whether 21 D 22 for some value less than one. As in the earlier analysis for O2, the asymptotic distribution of O21 O22 changes when the models are misspecified. Supposey1t ¤y2tand0 < 12D 22 < 1. Then,
pT .O12 O22/A N 0
@0;
X1
jD1
EŒdtdtCj 1
A: (9.104)
When the weighting matrixW is known,
dt D2Q10 .u2ty2tu1ty1t/; (9.105) where u1t D e10W .Rt 2/and u2t D e20W .Rt 2/. With the GLS weighting matrixWO D OV221,
dt DQ10 .u21t 2u1ty1tu22tC2u2ty2t/: (9.106)
Note that ify1t Dy2t, then 21 D 22, u1t D u2t, and hencedt D0. Or, ify1t ¤ y2t, but both models are correctly specified (i.e., u1t Du2t D0and 12D 22D1), then againdt D0. Thus, the normal test cannot be used in these cases.
Given the three distinct cases encountered in testingH0 W 21 D 22 for non- nested models, the approach we have described above entails a sequential test, as suggested byVuong(1989). In our context, this involves first testingH0Wy1t Dy2t
using (9.95) or (9.97). If we rejectH0 W y1t D y2t, then we use (9.101) or (9.102) to testH0 W 12 D 22 D 1. Finally, if this hypothesis is also rejected, we use the normal test in (9.104) to testH0 W 0 < 21 D 22 < 1. Let˛1,˛2, and˛3 be the significance levels employed in these three tests. Then the sequential test has an asymptotic significance level that is bounded above by maxŒ˛1; ˛2; ˛3.
Another approach is to simply perform the normal test in (9.104). This amounts to assuming that y1t ¤ y2t and that both models are misspecified. The first assumption rules out the unlikely scenario that the additional factors are completely irrelevant for explaining cross-sectional variation in expected returns. The second assumption is sensible because asset pricing models are approximations of reality and we do not expect them to be perfectly specified.