Habit, Long Run Risk, Prospect?

Transcription

1 Habit, Long Run Risk, Prospect? Eric Aldrich Duke University Department of Economics Durham NC USA A. Ronald Gallant Duke University Fuqua School of Business Durham NC USA Robert E. McCulloch University of Texas Graduate School of Business Austin TX USA

2 Methodology - 1 Gallant and McCulloch (2008), On the Determination General Statistical Models with Application to Asset Pricing, Journal of the American Statistical Association, forthcoming. A statistical model determines the likelihood. A prior is placed on the statistical model by imposing 1. a scientific model, which restricts its parameters to a lower dimensional manifold 2. a prior on the parameters of the the scientific model 3. a prior on functionals of the the scientific model Relaxed versions of the prior attract the parameters of the statistical model to the lower dimensional manifold.

3 Tinker Toy Example Scientific Model: p(y θ) = n(y θ, θ 2 ) Statistical Model: f(y η) = n(y η 1, η 2 ) Implied map: g : θ η = (η 1, η 2 ) = (θ, θ 2 ) The scientific model states that excess returns y t = r dt r ft are iid normal with a Sharpe ratio of one. The statistical model states that excess returns are iid normal.

4 Fig 1. The Scientific and Statistical Model eta eta eta1 eta1 eta eta eta1 eta1 The the dotted lines are contours of the likelihood of the statistical model f(y x, η) of the tinker toy example. The line is the prior on η determined by the implied map η = g(θ) from the parameters θ of scientific model p(y x, θ) to the parameters η of the statistical model. In the left panels the scientific model is true, in the right it is false. The thickness of the line is proportional to the posterior of η. The prior π(η) can be relaxed as indicated by the shading. The lower panels are more relaxed than the upper. The solid contours show the posterior under the relaxed prior. Relaxation causes the contours to enlarge in all cases. When the scientific model is false, the posterior shifts in search of the likelihood.

5 Determination of Model Adequacy Ranking Scientific Models Put equal prior probability on each model. Model comparision is by means of posterior probabilities.

6 Determination of Model Adequacy Scientific Model vs. Statistical Model Relax the prior determined by the scientific model. The parameters of the statistical model are now free to move off the lower dimensional manifold into their natural higher dimensional parameter space. But the prior retains enough influence to overcome data sparseness. Model adequacy is determined by observing if parameters or functionals of the statistical model that have scientific relevance change. Model comparision is by means of posterior probabilities.

7 Fig 2. Determination of Model Adequacy kappa=5, restriction true kappa=5, restriction false kappa=100, restriction true kappa=100, restriction false The posterior of the Sharpe ratio for the tinker toy example is the solid line; the dashed line is prior. In the left panels the scientific model is true, in the right it is false. The prior is more relaxed in the lower panels than it is in the upper panels. The panels correspond to those of Figure 1

8 Data Annual observations , 72 years, on Pdt a Dt a Ct a rdt a Q a t end-of-year per capita stock market value annual aggregate per capita dividend annual per capita consumption annual real geometric return annual quadratic variation Data are real, i.e. inflation adjusted. Source: Bansal, Gallant, and Tauchen (2007).

9 Data for Estimation Used here ( c a t c a t 1 ) r a dt Used by BGT d a t ca t c a t ca t 1 p a dt da t r a dt

10 Habit Persistence Asset Pricing Model Driving Processes Consumption: c t c t 1 = g + v t Dividends: d t d t 1 = g + w t Random Shocks: ( ) vt w t NID [( ) 0, 0 ( σ 2 ρσσ w ρσσ w σ 2 w )] The time increment is one month. Lower case denotes logarithms of upper case quantities; i.e. c t = log(c t ), d t = log(d t ). From Campbell and Cochrane (1999).

11 Habit Persistence Asset Pricing Model Utility function E 0 t=0 Habit persistence δ t (S tc t ) 1 γ 1 1 γ Surplus ratio: s t s = φ(s t 1 s) + λ(s t 1 )v t 1 { Sensitivity function: λ(s) = 1 S 1 2(s s) 1 s t s max 0 s t > s max, E t is conditional expectation with respect to S t, S t 1,.... Lower case denotes logarithms of upper case quantities: s t = log(s t ). S and s max can be computed from model parameters θ = (g, σ, ρ, σ w, φ, δ, γ) as S = σ γ/(1 φ), s max = s + (1 S 2 )/2. From Campbell and Cochrane (1999).

12 Habit Persistence Asset Pricing Model Return on dividends V (S t ) = E t r dt = log ( ) γ ( St+1 δ C t+1 Dt+1 S t C t D t ( )] Dt [ 1 + V (St ) V (S t 1 ) D t 1 ) [1 + V (St+1) ] V ( ) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. P dt /D t = V (S t ), where P dt is the price of the asset that pays the dividend stream. r dt is the logarithmic real return, i.e. r dt = log(p dt + D t ) log(p d,t 1 ), where P dt and D t are measured in real (inflation adjusted) dollars. Dividend error can be integrated out analytically. Consumption error integrated by quadrature. From Campbell and Cochrane (1999).

13 Habit Persistence Asset Pricing Model Solution Method Approximate the log policy function v(s t ) = log V (e s t ) by a piecewise linear function and use policy function iteration. Campbell and Cochrane used Gauss s intquad1 and set join points at s, s max, s max 0.01, s max 0.02, s max 0.03, s max 0.04, and log[is/(m + 1)] for i = 1,..., m = 10. We used Gauss- Hermite quadrature; we added the abscissae of the Gauss- Hermite quadrature formula at the maximum and minimum of the above join points; we deleted all points less than apart. Figure 3, next slide, plots the approximation at the Campbell and Cochrane parameter values.

14 Fig 3. Piecewise Linear Approximation p d x x xx Annualized Log Policy Function v(s) x x x x x x x x x x x x x P/D x xx x x x Annualized Policy Function V(S) x x x x x x x x x xxxxx x s mark Campbell and Cochrane join points; o s mark extra join points from the quadrature rule.

15 Habit Persistence Asset Pricing Model Risk Free Rate r ft = log E t δ ( St+1 C t+1 S t C t ) γ r ft is the logarithmic return on an asset that pays one real dollar one month hence with certainty. From Campbell and Cochrane (1999). Solution method is similar to the foregoing.

16 Habit Persistence Asset Pricing Model Large Model Output Given model parameters θ = (g, σ, ρ, σ w, φ, δ, γ) simulate monthly and aggregate to annual: C a t = 11 C 12t k k=0 c a t = log(ca t ) rdt a 11 = rft a 11 = r d,12t k k=0 r f,12t k k=0

17 Habit Persistence Asset Pricing Model Prior Distribution p(θ) = N [ r f 0.89, ( ) 2 ] p i=1 N θ i θ i, ( 0.1θ i 1.96 ) 2 where the θi are the calibrated values from Campbell and Cochrane (1999).

18 Table 1. Habit Model Prior Posterior Parameter Mode Std.Dev. Mode Std.Dev. g e e-05 σ ρ σ w φ δ γ r f r d r f σ rd Parameter values are for the monthly frequency. Returns are annualized. Mode is the mode of the multivariate density. It acutually ocurrs in the MCMC chain whereas other measures of central tendancy may not even satisfy support conditions. In the data, r d r f = = 5.13 and σ rd =

19 Fig 4. Posterior Returns bond returns Density equity premium Density stock returns Density sdev stock returns Density

20 Long Run Risk Asset Pricing Model Driving Processes Consumption: c t+1 c t = µ c + x t + σ t η t+1 Long Run Risk: x t+1 = ρx t + φ e σ t e t+1 Stochastic Volatility: σt+1 2 = σ2 + ν(σt 2 σ2 ) + σ w w t+1 Dividends: d t+1 d t = µ d + φ d x t + π d σ t η t+1 + φ u σ t u t+1 Random Shocks: η t e t w t u t NID , The time increment is one month. Lower case denotes logarithms of upper case quantities; i.e. c t = log(c t ), d t = log(d t ). From Bansal, Kiku, and Yaron (2007).

21 Long Run Risk Asset Pricing Model Epstein-Zin utility function U t = [ (1 δ)c 1 γ θ t + δ ( ] θ E t U 1 γ )1 1 γ θ t+1 where γ ψ is the coefficient of risk aversion is the elasticity of intertemporal substitution and θ = 1 γ 1 1/ψ E t is conditional expectation with respect to x t, σ t.

22 Long Run Risk Asset Pricing Model Return on consumption mrs t+1 = δ θ exp[ (θ/ψ)(c t+1 c t ) + (θ 1)r c,t+1 ] V C (x t, σ t ) = E t { r ct = log ( Ct+1 mrs t+1 C t [ 1 + VC (x t, σ t ) V C (x t 1, σ t 1 ) ) [1 + VC (x t+1, σ t+1) ] } ( )] Ct C t 1 V C ( ) is defined as the solution of the Euler condition above. It is the price consumption ratio; i.e. P ct /C t = V C (x t, σ t ), where P ct is the price of the asset that pays the consumption stream. r ct is the logarithmic real return, i.e. r ct = log(p ct +C t ) log(p c,t 1 ), where P ct and C t are measured in real (inflation adjusted) dollars.

23 Long Run Risk Asset Pricing Model Solution Method Use the log linear approximation r c,t+1. = κ 0 + κ 1 z t+1 + c t+1 z t κ 1 = [exp( z)]/[1 + exp( z)] k 0 = log[1 + exp( z)] κ 1 z where z t = log(p c,t /C t ) and z is its endogenous mean. To compute z, use the approximation z t. = A 0 ( z) + A 1 ( z) x t + A 2 ( z) σ 2 t A i ( z) = tedious expressions in model parameters and z and solve the fixed point problem z = A 0 ( z) + A 1 ( z) x t + A 2 ( z) σ 2 t

24 Long Run Risk Asset Pricing Model Return on dividends mrs t+1 = δ θ exp[ (θ/ψ)(c t+1 c t ) + (θ 1)r c,t+1 ] V D (x t, σ t ) = E t { r dt = log ( Dt+1 mrs t+1 D t [ 1 + VD (x t, σ t ) V D (x t 1, σ t 1 ) ) [1 + VD (x t+1, σ t+1) ] } ( )] Dt D t 1 V D ( ) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. P dt /D t = V D (x t, σ t ), where P ct is the price of the asset that pays the dividend stream. r dt is the logarithmic real return, i.e. r dt = log(p dt + D t ) log(p d,t 1 ), where P dt and D t are measured in real (inflation adjusted) dollars. Solution method is similar to the foregoing.

25 Long Run Risk Asset Pricing Model Risk Free Rate r ft = log E t { δ θ exp [ (θ/ψ)(c t+1 c t ) + (θ 1)r c,t+1 ]} r ft is the logarithmic return on an asset that pays one real dollar one month hence with certainty. Solution method is similar to the foregoing.

26 Long Run Risk Asset Pricing Model Large Model Output Given model parameters θ = (δ, γ, ψ, µ c, ρ, φ e, σ 2, η, σ w, µ d, φ u ) simulate monthly and aggregate to annual: C a t = 11 C 12t k k=0 c a t = log(ca t ) rdt a 11 = rft a 11 = r d,12t k k=0 r f,12t k k=0

27 Long Run Risk Asset Pricing Model Prior Distribution p(θ) = N [ r f 0.89, ( ) 2 ] p i=1 N θ i θ i, ( 0.1θ i 1.96 ) 2 where the θ i are the calibrated values from Kiku (2006).

28 Table 2. Long Run Risk Model Prior Posterior Parameter Mode Std.Dev. Mode Std.Dev. δ γ ψ µ c e e-05 ρ φ e σ e e e e-06 ν σ w e e e e-07 µ d e e-05 φ d π d φ u r f r d r f σ rd Parameter values are for the monthly frequency. Returns are annualized. Mode is the mode of the multivariate density. It acutually ocurrs in the MCMC chain whereas other measures of central tendancy may not even satisfy support conditions. In the data, r d r f = = 5.13 and σ rd =

29 Fig 5. Posterior Returns bond returns Density equity premium Density stock returns Density sdev stock returns Density

30 Prospect Theory Asset Pricing Model Driving Processes Aggregate Consumption: c t+1 c t = g C + σ C η t+1 Dividends: d t+1 d t = g D + σ D ǫ t+1 Random Shocks: ( ) ηt ǫ t NID [( ) ( )] 0 1 ω, 0 ω 1 C t is aggregate, per capita consumption which is exogenous to the agent. The time increment is one year. Lower case denotes logarithms of upper case quantities; i.e. c t = log( C t ), d t = log(d t ). All variables are real. From Barberis, Huang, Santos (2001).

31 Prospect Theory Asset Pricing Model Other Model Variables Gross Stock Return: R t Gross Risk Free Rate: R f = ρ 1 exp ( γg C γ 2 σ 2 C /2) Allocation to Risky Asset: S t Gain or Loss: X t+1 = S t (R t+1 R f ) Benchmark Level (State Variable): z t+1 = η ( z t R R t+1 )+(1 η) Choose R to make Median {z t } = 1 The Agent s Consumption: C t

32 Prospect Theory Asset Pricing Model Utility function E 0 t=0 ρ t C1 γ t 1 1 γ + b 0 C t γ ρ t+1 [ S tˆv(r t+1, z t ) ] Utility from Gains and Losses: [ S tˆv(r t+1, z t ) ] ˆv(R t+1, z t ) = R t+1 R f z t 1, R t+1 z t R f = (z t R f R f ) + λ(r t+1 z t R f ) z t 1, R t+1 < z t R f = R t+1 R f z t > 1, R t+1 R f = λ(z t )(R t+1 R f ) z t > 1, R t+1 < R f λ(z t ) = λ + k(z t 1)

33 Fig 6. Utility of Gains and Losses Utility Gain/Loss The dot-dash line represents the case where the investor has prior gains (z < 1), the dashed line the case of prior losses (z > 1), and the solid line the case where the investor has neither prior gains nor losses (z = 1).

34 Prospect Theory Asset Pricing Model Return on dividends 1 = ρexp ( g D γg C + γ 2 σ 2 C (1 ω2 )/2 ) r dt = log [ 1 + f(zt+1 ) E t exp[(σ D γωσ C )ǫ t+1 ] f(z t ) ( )] 1 + f(zt+1 ) + b 0 ρ E t [ˆv exp(g D + σ D ǫ t+1 ), z t f(z t ) [ ] 1 + f(zt ) f(z t 1 ) exp(g D + σ D ǫ t ) f( ) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. P dt /D t = f(z t ), where P ct is the price of the asset that pays the dividend stream. r dt is the logarithmic real return, i.e. r dt = log(p dt + D t ) log(p d,t 1 ), where P dt and D t are measured in real (inflation adjusted) dollars. ]

35 Prospect Theory Asset Pricing Model Self Referential Equations z t+1 = η ( R z t R t+1 ) + (1 η) R t+1 = 1 + f(z t+1) f(z t ) exp(g D + σ D ǫ t+1 ) 1 = Median{z t }

36 Prospect Theory Asset Pricing Model Solution Method Approximate f by a piecewise linear function f (0) (z). Approximate R by (1+f(1))exp(g D )/f(1), which is a departure from Barberis, Huang, and Santos (2001). Define h (0) such that z t+1 = h (0) (z t, ǫ t+1 ) solves the self referential equations that define z t+1 and R t+1 on previous slide. A root finding problem. We use Brent s method. Substitute h (0) (z t, ǫ t+1 ) into the Euler equation. Use Gauss- Hermite quadrature to integrate out ǫ t+1. Solve for f (1) (z). A root finding problem at each join point of f (1). Repeat h (i) f (i+1) until convergence.

37 Fig 7. Piecewise Linear Approximation P/D z

38 Prospect Theory Asset Pricing Model Risk Free Rate r f = log [ ρ 1 exp ( γg C γ 2 σ 2 C /2)] r ft is the logarithmic return on an asset that pays one real dollar one year hence with certainty.

39 Prospect Theory Asset Pricing Model Large Model Output Given model parameters θ = (g C, g D, σ C, σ D, ω, γ, ρ, λ, k, b 0, η) simulate annually and set c a t = log(c t) r a dt = r dt r a ft = r f

40 Prospect Asset Pricing Model Prior Distribution p(θ) = N [ r f 0.89, ( ) 2 ] p i=1 N θ i θ i, ( 0.1θ i 1.96 ) 2 where the θ i are the calibrated values from Barberis, Huang, Santos (2001).

41 Table 3. Prospect Model Prior Posterior Parameter Mode Std.Dev. Mode Std.Dev. g C g D σ C σ D ω γ ρ λ k b η r f r d r f σ rd Parameter values are for the annual frequency. Mode is the mode of the multivariate density. It acutually ocurrs in the MCMC chain whereas other measures of central tendancy may not even satisfy support conditions. In the data, r d r f = = 5.13 and σ rd =

42 Fig 8. Posterior Returns bond returns Density equity premium Density stock returns Density sdev stock returns Density

43 Model Assessment Posterior Probabilities for Three Models Long Run Risk Habit Persistence Prospect Theory

44 Model Assessment Posterior Probabilities for Long Run Risk κ = κ = κ =

45 Methodology - 2 Gallant and Hong (2007), A Statistical Inquiry into the Plausibility of Recursive Utility, Journal of Financial Econometrics 5, Pricing kernel: θ = (θ 1,..., θ n+1 ) Hierarchical likelihood: L(θ, η) = L(θ) n t=1 f(θ t+1 θ t,..., θ 1, η) L(θ) function of returns derived from the Euler equation 1=E t θ t+1 R it by a conditioning argument. f(, η) seminonparametric transition density on θ. Composite prior: p(θ, η) = p(η) p T (θ, η) p(η) prior on hyperparameter η : habit, long run risk, etc. p T (θ, η) imposes technical conditions Inference: MCMC posterior probabilities P(A) = I A (θ, η)p(dθ, dη)

46 Data: Two Data Sets First: 551 monthly observations from February 1959 through December, 2004 real returns including dividends on 24 Fama-French (1993) portfolios. real returns on U.S. Treasury debt of ten year, one year, and thirty day maturities real returns including dividends on the aggregate stock market real, per-capita, consumption expenditure and labor income growth Second: The second is 75 annual observations from 1930 through 2004 on the same variables except U.S. Treasury debt of ten and one year maturities. Exclusion of one Fama-French portfolio and debt due to missing values.

47 Findings Monthly data: P(Habit) = 0.51 P(Long Run Risk) = 0.49 Annual data: P(Habit) = 0.50 P(Long Run Risk) = 0.50

48 Remainder Use long run risk model to illustrate Law of motion of pricing kernel f(θ t θ t 1,..., θ 1, η) Prior p(η) on hyperparameter η Likelihood for observables L(θ) Implementation of L(θ) Technical prior p T (θ, η)

49 Consumption Endowment and Cash Flows C t consumption endowment P ct price of an asset that pays the consumption endowment R ct = (P ct + C t )/P c,t 1 gross return on consumption endowment D st cash flow S P st price of cash flow S R st = (P st + D st )/P s,t 1 gross return on cash flow S Prices are real

50 LRR Uses Recursive Utility Marginal rate of substitution ( ) (β/ψ) M t,t+1 = δ β Ct+1 (R c,t+1) (β 1), C t δ time preference parameter γ coefficient of risk aversion ψ elasticity of intertemporal substitution β = 1 γ 1 1/ψ

51 In General Euler equation 1 = E t ( θt+1 R t,t+1 ) Satisfied by any pricing kernel θ t+1 including M t,t+1 given by recursive utility

52 Law of motion of pricing kernel f(θ t θ t 1,..., θ 1, η) Fit SNP expansion to simulated data from a calibrated long run risk economy. Use BIC to determine the truncation point. The form of the fitted density of θ t = M t 1,t is f(θ t θ t 1,..., θ 1, η) = f(y t y t 1,..., y 1, η)/exp(y t ) y t = log(θ t 1,t ) f(y t y t 1,..., y 1, η) = P 2 (y t a) n(y t µ t 1, σ 2 t 1) η = (a 1, a 2, a 3, a 4, b 0, b 1, r 0, r 1, r 2 ) P(y t a) = 1 + a 1 y t + a 2 y 2 t + a 3y 3 t + a 4y 4 t µ t t = b 0 + b 1 y t 1 σ 2 t 1 = r r 2 1σ 2 t 2 + r 2 2(y t 1 µ t 2 ) 2

53 Likelihood Derivation 1 of 4 Consider a general set of moment conditions m n = 1 n n t=1 m t (y t, θ t ) m t R K that a scientific theory implies ought to have expectation zero where both y t and θ t are regarded as random To estimate the variance when the m t (y t, x t, θ o ) are uncorrelated, use W n = 1 n n t=1 When correlated, use HAC. [m t (y t, θ t ) m n ][m t (y t, θ t ) m n ] Rely on CLT for normality of z = Wn 1/2 n mn

54 Likelihood Derivation 2 of 4 View as a data summary: z = Z(Y, θ) Z : (Y, θ) nw 1/2n m n where Y = [y 1 y 2 y n ], θ = [θ 1 θ 2 θ n ] Y is all possible Y Information loss: Probability can only be assigned to sets A expressed in terms of the original data (Y, θ) that are preimages A = Z 1 (B) of a Borel set B in R K. Assignment formula: P(A) = P Z (B) = I B (z)(2π) K/2 exp( z z/2) dz 1 dz K

55 Likelihood Derivation 3 of 4 Let Z θ denote the map Y Z(Y, θ) for θ fixed. The sets C expressed in terms of Y to which we can assign conditional probability knowing that θ has occurred have the form C = Z 1 θ (B θ ) for some Borel set B θ R K. The principle of conditioning is that one assigns conditional probability proportionately to joint probability. The constant of proportionality is 1/P(R θ ), where R θ is the Borel set for which Y = Z 1 θ (R θ ). The conditional probability of C is computed as P(C θ) = P Z(B θ ) P Z (R θ )

56 Likelihood Derivation 4 of 4 Conditional probability is computed as P(C θ) = 1 P Z (R θ ) The data summary is In our application P Z (R θ ) = 1 I Bθ (z)(2π) K/2 exp( z z/2) dz 1 dz K z = [W n (θ)] 1/2 n m n (θ) Conclude that the likelihood is { L(θ) exp n 2 m n(θ) [W n (θ)] 1 m n (θ) } Same argument as Fisher (1930) used to define fiducial probability. Is Bayesian inference for continuously updated GMM.

57 Implementation of L(θ) (monthly data) s t : Fama-French gross returns S11 S54 b t : T-debt gross returns t30ret, b1ret, and b10ret. Parameters θ = θ 1,..., θ 551 Euler equation errors e t (θ) = e t (s t+1, b t+1, θ t+1 ) = 1 θ t+1 ( st+1 b t+1 Instruments: Z t = (s t 1, b t 1, log(cg t ) 1, log(lg t ) 1, 1) Moment conditions: t = 1,..., n = 550 m t (θ) = m t (s t, b t, s t+1, b t+1, θ t+1 ) = Z t e t (s t+1, b t+1, θ t+1 ) Length of m t (θ): K = 810 Number of overidentifying restrictions: 260. )

58 Computing the Weighting Matrix W Assume factor structure for the random part of (s t, b t ). Σ e : one error common for s t, one error common for b t, one idiosyncratic error for each of (s t, b t ). Strengthen zero correlation condition E[Z i,t e j,t (s t+1, b t+1, θt+1 o )] = 0 to Var[Z t e t (s t+1, b t+1, θ o t+1 )] = Σ z Σ e, Implies Σ z Σ e is diagonalized by U z U e where U z and U e are known, constant, orthogonal matrices: 1000 fold efficiency gain. Mean correction when estimating diagonal elements to avoid acceptance of an absurd MCMC proposal due to a large variance estimate ( ) 2 s i (θ) = 1 n v t,i (θ) 1 n v t,i (θ) n n t=1 t=1

59 Implementation of L(θ) (monthly data) The likelihood for the pricing kernel θ is, then, { L(θ) exp n 2 m n(θ) (U z U e )Sn 1 (θ)(u z U e ) m n (θ) m n (θ) = 1 n n t=1 Z t e t (s t+1, b t+1, θ t+1 ), } Full likelihood L(η, θ) L(θ) n t=1 f(θ t+1 θ t, θ t 1,..., θ 1, η). dim(η, θ) = 560 implies high correlation in MCMC chain. To reduce, sample 30,000 out of 30 million MCMC draws.

60 Technical Prior p(θ, η) Support condition: Impose mean reversion on law of motion f(θ t θ t 1,..., θ 1, η) Constrain the size of the Euler equation errors to be about the same Annual data: P( 0.5 < e i,n (θ) < 0.5) = 0.95 Monthly data: P( 0.05 < e i,n (θ) < 0.05) = 0.95 Require an approximation P B = n t=1 θ t to the average price of a risk-free, one period bond to be between 1% and 4% per annum, f(p B ) (1 + cos(α + βp B )) a < P B < b where β = 2π/(b a), α = π βb.

61 Summary Pricing kernel: θ = (θ 1,..., θ n+1 ) Hierarchical likelihood: L(θ, η) = L(θ) n t=1 f(θ t+1 θ t,..., θ 1, η) L(θ) is a function of returns implied by Euler equations f(, η) is the seminonparametric transition density of the pricing kernel. Composite prior: p(θ, η) = p(η)p T (θ, η) p(η) prior on hyperparameter η : tight, intermediate, loose p T (θ, η) imposes technical conditions Inference: MCMC Moments of posterior distribution P(θ, η) Posterior probabilities P(A) = I A (θ, η)p(dθ, dη) of events A that correspond to hypotheses

62 Figure 5. The Posterior Mean of the Monthly Pricing Kernel

63 Figure 6. The Posterior Mean of the Annual Pricing Kernel