Improvement in Finite Sample Properties
of the Hansen-Jagannathan Distance Test
Yu Ren
Department of Economics
Queen’s University
Katsumi Shimotsu
Department of Economics
Queen’s University
May 22, 2006
0-0
Motivation
• Hansen and Jagannathan (1997) propose the HJ-distance to
evaluate stochastic discount factor. It is the quadratic form of
the pricing error weighted by the second moment of the
portfolios returns.
• Jagannthan and Wang (1996) develop a specification test of the
HJ-distance based on linear asset pricing models .
• Ahn and Gadarowski (2004) find that the the HJ-distance test
overrejects correct models too severely in commonly used sample
size.
1
Table1.Rejection frequencies of the HJ-distance test
Number of Observations
T =160
T =330
T =700
1%
5.8
3.3
1.1
5%
15.1
10.6
7.1
10%
23.9
18.9
12.8
1%
99.8
53.7
13.8
5%
100.0
76.0
30.8
10%
100.0
84.3
44.6
Fama-French Model
25 Portfolios
100 Portfolios
2
Basic Setting
• N portfolios with T observations
• Rt : t-th period returns, N × 1
• Yt : K × 1 factor vector including a 1
• mt = Yt0 δ linear combination of factors
• mt : a stochastic discount factor proxy
3
Hansen-Jagannathan Distance
• Define wt (δ) = Rt Yt0 δ − 1N , G = E[Rt Rt0 ]
p
• HJ(δ) = E[wt (δ)]0 G−1 E[wt (δ)]
• This distance is equal to the least squares distances between the
stochastic discount factors proxies and the family of stochastic
discount factors that price correctly.
4
Sample Estimation of the HJ-Distance
• Jagannathan and Wang (1996) use sample moments to estimate
the HJ-distance
q
HJT (δ) = wT (δ)0 G−1
T wT (δ),
• wT (δ) and GT are sample analogue of E(wt (δ)) and G
PT
−1
0
• DT = T
R
Y
t
t
t=1
PT
−1
• wT (δ) = T
t=1 wt (δ) = DT δ − 1N
PT
−1
0
• GT = T
R
R
t
t
t=1
5
Estimation of SDF and null Hypothesis
q
• δT = arg min wT (δ)0 G−1
T wT (δ)
• We can test the hypothesis that the SDF, Yt0 δT , prices the
portfolios returns correctly.
• H0 : E[Rt (Yt0 δT )] = 1N
6
Asymptotical Distribution of the HJ-Distance
Asymptotic distribution of the HJ-distance is:
d
T [HJT (δT )]2 →
NX
−K
λj υj
j=1
where υ1 ,. . . υN −K are independent χ2 (1) random variables, and
λ1 ,. . . λN −K are nonzero eigenvalues of the following matrix:
ψ = S 1/2 G−1/2 [IN − (G−1/2 )0 D(D0 G−1 D)−1 D0 G−1/2 ](G−1/2 )0 (S 1/2 )0
7
p-value of the HJ-distance
PN −K
• The distribution of j=1 λj υj depends on ψ.
PN −K
• We simulate the distribution of j=1 λj υj by randomly
drawing N − K random variables from χ2 (1) distribution M
times.
• Obtain an empirical p-value by
PM
PN −K
−1
2)
p=M
I(
λ
v
≥
T
[HJ
(δ
)]
i
ij
T
T
j=1
i=1
8
Fama-French Three-Factor Model
Rit = α + X1t β1i + X2t β2i + X3t β3i + eit
• Fama and French (1993)
• Rit : Fama-French 25 (or 100)portfolio returns.
• Xit are Fama-French factors.
9
Monte Carlo Simulation
• Collect time series of monthly returns for Fama-French portfolios
and Fama-French factors.
• Simulate the factors from normal distribution with the mean
and the covariance equal to the sample mean and the sample
covariance matrix derived from the actual data of Fama-French
factors.
• Calibrate Fama-French portfolio returns according to
Fama-French three factor model.
• Do the HJ-distance test based on the simulated factors and
calibrated returns.
• Repeat the above process for 1000 times.
10
Table1.Rejection frequencies of the HJ-distance test
Number of Observations
T =160
T =330
T =700
1%
5.8
3.3
1.1
5%
15.1
10.6
7.1
10%
23.9
18.9
12.8
1%
99.8
53.7
13.8
5%
100.0
76.0
30.8
10%
100.0
84.3
44.6
Fama-French Model
25 Portfolios
100 Portfolios
11
Possible Reason for the Over-rejection
GT = T −1
T
X
d t ) + E(R
b t )0 E(R
b t)
Rt Rt0 = Cov(R
t=1
• E(Rt ) can be estimated accurately by the sample mean for the
sample size of our interest.
• The sample covariance matrix can be a very inaccurate estimate
of Cov(Rt ) when N/T is not negligible.
• Inverting the sample moment as the weighting matrix can
amplify the inaccuracy.
12
Table3. Rejection frequencies with
the exact weighting matrix G
Number of Observations
T =160
T =330
T =700
1%
1.0
1.2
0.7
5%
4.9
5.7
5.0
10%
9.9
11.8
11.9
1%
1.1
1.6
1.3
5%
7.4
7.4
5.8
10%
16.8
14.7
13.6
Fama-French Model
25 Portfolios
100 Portfolios
13
Covariance Matrix based on a Structure Model
• True covariance matrix of returns: Σ
• Asset pricing model (Structure model): R = Xβ + ²
• Covariance matrix of returns implied by the structure model:
Φ = β 0 Cov(Xt )β + ∆
• Estimated covariance matrix based on the structure model:
\ +D
F = b0 Cov(X)b
• Sample covariance: S
14
Improve Estimation of G by Shrinkage
• Shrinkage Estimation: combine sample covariance and
structure model covariance matrix with an optimal weight.
• Sample covariance S has small bias, but it has a large estimation
error.
• Covariance matrix F based on a structure model has a large
bias, but small estimation error.
• Shrinkage estimator balances these two with an optimal weight α
15
Shrinkage Estimation
• Loss function: L(α) = kαF + (1 − α)S − Σk2
• k · k is Frobenius norm: for any N × N matrix Z,
PN PN 2
2
2
kzk = T race(Z ) = i=1 j=1 zij
• Minimize the expected loss function: minα E[L(α)]
16
Optimal α
√
√
√
PN PN
PN PN
i=1
j=1 V ar( T sij ) −
i=1
j=1 Cov( T fij , T sij )
∗
α = PN PN
√
√
PN PN
2
i=1
j=1 V ar( T fij − T sij ) + T
i=1
j=1 (φij − σij )
• sij is the typical element of sample covariance S
• fij is the typical element of estimated structure based covariance
matrix F
• φij is the typical element of structure based covariance matrix Φ
• σij is the typical element of the true covariance matrix Σ
17
Consistent Estimators of the Optimal Weight
Assumption 1. Individual stock returns are independent and
identically distributed over time.
Assumption 2. Stock returns have finite fourth moments.
Use consistent estimators to replace the numerator and the
denominator in the formula of α∗
18
Shrinkage Estimator
Theorem 1. Define an estimate of the optimal shrinkage constant as
PN PN
PN PN
i=1
j=1 pij −
i=1
j=1 rij
α
ˆ=
PN PN
i=1
j=1 (gij + T cij )
ˆ is: Σ
ˆ =α
The shrinkage covariance matrix Σ
ˆ F + (1 − α
ˆ )S.
19
Table 5. Rejection frequencies with
shrinkage estimation of G
Number of Observations
T =160
T =330
T =700
1%
0.9
1.1
0.5
5%
5.3
6.0
4.8
10%
10.7
11.6
10.7
1%
4.9
2.2
1.7
5%
23.9
11.5
7.3
10%
39.9
22.1
15.0
Fama-French Model
25 Portfolios
100 Portfolios
20
Compare Table 1. with Table 5.
Number of Observations
T =160
T =330
T =700
Fama-French Model: 25 Portfolios
Table 1.
1%
5.8
3.3
1.1
5%
15.1
10.6
7.1
10%
23.9
18.9
12.8
1%
0.9
1.1
0.5
5%
5.3
6.0
4.8
10%
10.7
11.6
10.7
Table 5.(with Shrinkage)
21
Compare Table 1. with Table 5. (continued)
Number of Observations
T =160
T =330
T =700
Fama-French Model: 100 Portfolios
100 Portfolios
1%
99.8
53.7
13.8
5%
100.0
76.0
30.8
10%
100.0
84.3
44.6
1%
4.9
2.2
1.7
5%
23.9
11.5
7.3
10%
39.9
22.1
15.0
100 Portfolios
22
The Density Functions for α
ˆ
in Fama-French 25 Portfolios Model
20
density
T=160
15
10
5
0
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.8
0.85
0.9
0.95
1
0.8
0.85
0.9
0.95
1
20
density
T=330
15
10
5
0
0.65
0.7
0.75
20
density
T=700
15
10
5
0
0.65
0.7
0.75
23
The Density Functions for α
ˆ
in Fama-French 100 Portfolios Model
15
density
T=160
10
5
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.7
0.75
0.8
0.85
0.9
0.95
1
0.7
0.75
0.8
0.85
0.9
0.95
1
15
density
T=330
10
5
0
0.6
0.65
15
density
T=700
10
5
0
0.6
0.65
24
Premium-Labor Three-Factor Model
Rit = α + X1t β1i + X2t β2i + X3t β3i + eit
• Jagannathan and Wang (1996)
• Rit is constructed in the same way with Fama-French model, but
ONLY includes the stocks of nonfinanical firms listed in NYSE,
AMEX.
• Xit are Premium-Labor factors.
25
Compare Table 1. with Table 5.
Number of Observations
T =160
T =330
T =700
Premium-Labor Model: 25 Portfolios
Table 1.
1%
14.9
11.3
9.2
5%
31.9
26.0
19.5
10%
42.7
34.4
29.0
1%
4.7
5.9
6.4
5%
16.4
17.0
16.2
10%
26.3
26.9
23.8
Table 5.(with Shrinkage)
26
Compare Table 1. with Table 5. (continued)
Number of Observations
T =160
T =330
T =700
Premium-Labor Model: 100 Portfolios
100 Portfolios
1%
99.7
79.1
36.8
5%
99.9
90.1
59.1
10%
99.9
94.3
69.6
1%
16.6
14.9
11.1
5%
41.5
35.2
27.0
10%
58.3
50.4
38.9
100 Portfolios
27
The Density Functions for α
ˆ
in Premium-Labor 25 Portfolios Model
15
density
T=160
10
5
0
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.8
0.85
0.9
0.95
1
0.8
0.85
0.9
0.95
1
15
density
T=330
10
5
0
0.65
0.7
0.75
15
density
T=700
10
5
0
0.65
0.7
0.75
28
The Density Functions for α
ˆ
in Premium-Labor 100 Portfolios Model
20
density
T=160
15
10
5
0
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.75
0.8
0.85
0.9
0.95
1
0.75
0.8
0.85
0.9
0.95
1
25
T=330
density
20
15
10
5
0
0.65
0.7
25
T=700
density
20
15
10
5
0
0.65
0.7
29