In this section we use the small-sample critical values presented in Table 1 (Section VII) to 'test' the three asset-pricing models described in Section IV using U.S. consumption and stock and bond return data from 1947:1-1997:4, treating both assets as risky. For the time separable model with logarithmic preferences the Z2 statistic is -13.131, suggesting a rejection of the model.
Significantly larger values for the curvature parameter do not lead to acceptance of the model. For example, a curvature parameter of 50 leads to a Z2 statistic of -12.861, leading to a rejection of the model again. (This accords with results for quarterly data obtained by Hansen and Jagannathan (1991).)
The Epstein-Zin model fares better than the time-separable model for some parameter values. With ρ = 1.3 (the inverse of intertemporal substitution) and σ = 5 (risk aversion), the Z2 statistic is -12.309, indicating a model rejection. However, holding intertemporal substitution
constant and increasing risk aversion to 14, the Z2 statistic is -0.1105, so we would fail to reject this model. Of course, the HJ bound is only a necessary condition and we would want to study the model more carefully before accepting it.
The results for the habit model are mixed. With δ = -0.712 and σ = 7.12, values that are consistent with most of the literature on habit formation, the value of the Z1 statistic is –1.4184, so we would not reject this model at conventional significance levels. For a more moderate amount of habit, say δ = -0.2, and σ = 10, the Z1 statistic is –14.087, which would call for a rejection of the model. These mixed results have a number of implications. First, because the model is soundly rejected with the latter set of parameters, we would not want to go through the effort to more carefully analyze that version of the model. Second, since the habit model with the first set of parameters is not rejected, we would want to subject this model to closer scrutiny.
Again, this is because the HJ bound is only a necessary condition.10
X. Conclusion
We have investigated the usefulness of the Hansen-Jagannathan (1991) bound for assessing the empirical plausibility of asset-pricing models. For completeness, we provided a derivation of the bound, the asymptotic distribution theory for a distance measure based on the bound, and a description of three sources of IMRSs—time separable preferences, state non-separable preferences, and habit-formation preferences. We studied how the bound can be used to assess the three asset-pricing models with data samples of the size typically encountered in applied work.
10 In Otrok, Ravikumar and Whiteman (2001) we develop a spectral approach to evaluate and understand
For the model with time-separable preferences, we showed that the Hansen-Jagannathan distance measure test is non-pivotal in finite samples. Under the null hypothesis that the model is true, the distributions of the test statistics depend in a nontrivial way on the curvature parameter of the utility function. We then showed that inference based on critical values taken from the almost linear utility (risk neutral) case yields type I errors no larger than intended across any positive values for the curvature parameter; i.e., our critical value calculations are designed to guarantee 'small' type I errors. We also documented that our critical values for the risk-neutral case are valid for the two non-separable preferences as well.
Next, we provided finite-sample critical values for the distance measure using a high-dimension Markov process for consumption growth. We studied the power of the procedure to detect departures from the null hypothesis, and conclude that the HJ bound is in fact a useful device for evaluating asset-pricing models. Finally, we investigated the three asset-pricing models using U.S. quarterly data and our small-sample critical values. We found evidence against the time-separable model, and mixed evidence for the time and the state non-separable preferences.
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Appendix: Asset Prices
A.1 Time Separable Preferences. Mehra and Prescott (1985) obtain closed-form
solutions for asset prices as follows. From (25) the price of the risky asset is:
∑
= −σTo solve for the unknown prices, Mehra and Prescott use the fact that prices are homogenous of degree one in consumption and write prices as: p(c,i)=wic. Substituting in (A1) yields a set of two equations with two unknowns:
, i = 1, 2. (A2)
The formula for the price of the riskless asset in state i is:
∑
The gross return on equity is
p
A.2 State Non-Separable Preferences. The w's now solve:
σ
The price of the riskless asset in state i is:
The unknown w’s are found using a nonlinear equation solver in Gauss.
A.3 Habit Formation Preferences. Following the same procedure as in Section IV.1, the price of the risky asset in state 1 can be written as:
( )
two equations and two unknowns, which can be solved for the prices in each state, as in the case of time-separable preferences. The price of the riskless assset in the habit case is calculated by substituting a 1 for (p(λic,i)+cλi) in equation (A6).Figure 1: PDFs of the Z1 Statistic for 1 Riskless Asset and 1 Risky Asset a: σ = 0.005 b: σ = 2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
-6.54 -3.31 -2.85 -2.39 -1.93 -1.46 -1.00 -0.54 -0.08
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
-4.38 -2.82 -2.37 -1.92 -1.47 -1.02 -0.56 -0.11 0.34
c: σ= 50 d: σ = 100
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.24 1.17 1.99 2.82 3.64 4.46 5.29 6.11 7.19
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
0.24 1.81 3.40 4.99 6.59 8.18 9.77 11.36 12.96
Figure 2: CDFs of the Z1 Statistic
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -1.70 0.30 2.30 4.30 6.30 8.30 10.30 12.30 14.30
σ = 0.005 σ = 2 σ = 50 σ=100 Normal
Figure 3: PDFs of Z2 Statistic for 1 Risky Asset a: σ = 0.005 b: σ = 2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
-4.32 -3.79 -3.26 -2.73 -2.20 -1.67 -1.15 -0.62 -0.09
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
-3.54 -3.09 -2.64 -2.20 -1.75 -1.30 -0.85 -0.41 0.04
c: σ = 50 d: σ = 100
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
-2.70 -2.23 -1.76 -1.29 -0.82 -0.35 0.12 0.59 1.06
0.00 0.02 0.04 0.06 0.08 0.10 0.12
-1.50 -1.05 -0.60 -0.14 0.31 0.76 1.21 1.67 2.12
Figure 4: CDFs of the Z2 Statistic
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
σ = 0.005 σ = 2 σ = 50 σ=100
Normal
Figure 5: CDFs of Z1 and Z2 for Fixed Intertemporal Substitution a: Z1 (1 Riskless Asset and 1 Risky Asset) b: Z2 (1 Risky Asset)
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
σ = 0.005 σ = 2 σ = 50 σ = 100
Normal
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.7 -2.7 -1.7 -0.7 0.3 1.3 2.3 3.3
σ = 0.005 σ = 2 σ = 50 σ = 100 Normal
Figure 6: CDFs of Z1 and Z2 for Fixed Risk Aversion
a: Z1 (1 Riskless Asset and 1 Risky Asset) b: Z2 (1 Risky Asset)
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
ρ = 0.005 ρ = 2 ρ = 50 ρ = 100 Normal
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
ρ = 0.005 ρ = 2 ρ = 50 ρ = 100 Normal
Figure 7: CDFs of Z1 and Z2 for Fixed Habit Persistence
a: Z1 (1 Riskless Asset and 1 Risky Asset) b: Z2 (1 Risky Asset)
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
σ = 0.005 σ = 2 σ = 50 Normal
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
σ = 0.005 σ = 2 σ = 50 Normal
Figure 8: CDFs of Z1 and Z2 for Fixed σ
a: Z1 (1 Riskless Asset and 1 Risky Asset) b: Z2 (1 Risky Asset)
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
δ = −0.35 δ = −0.50 δ = −0.65
Normal
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-3.70 -2.70 -1.70 -0.70 0.30 1.30 2.30 3.30
δ = −0.35 δ = −0.50 δ = −0.65
Normal
Figure 9: Power of Z2 Test Against Time-Separable Alternatives
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
σ
rejection probability
Figure 10: Population Bound and IMRS for Alternative Economies
a: Population Bound, σσσσ = 2 b: Population bound, σσσσ = 0.001, 2, 5
0 0.02 0.04 0.06 0.08 0.1 0.12
0.94 0.95 0.96 0.97 0.98 0.99 1 mean
std dev
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.93 0.95 0.97 0.99
mean
std dev
σ = 0.001 σ = 2 σ = 5
Figure 11: Power Against Habit Alternatives
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
σ
rejection probability
Figure 12: Power Against Epstein-Zin Alternatives
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
ρ
rejection probability