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Habit, Long Run Risk, Prospect? Eric Aldrich Duke University Department of Economics Durham NC 27708-0097 USA A. Ronald Gallant Duke University Fuqua School of Business Durham NC 27708-0120 USA Robert E. McCulloch University of Texas Graduate School of Business Austin TX 78712-1175 USA 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. Tinker Toy Example Scientific Model: p(y | θ) = n(y | θ, θ2) Statistical Model: f (y | η) = n(y | η1, η2) Implied map: g : θ 7→ η = (η1, η2) = (θ, θ2) The scientific model states that excess returns yt = rdt − rf t are iid normal with a Sharpe ratio of one. The statistical model states that excess returns are iid normal. 1.0 1.5 2.0 2.5 3.0 2 4 6 8 10 eta2 2 4 6 8 10 eta2 Fig 1. The Scientific and Statistical Model 3.5 1.0 1.5 1.5 2.0 2.5 eta1 3.0 3.5 3.0 3.5 3.0 3.5 2 4 6 8 10 eta2 1.0 2.5 eta1 2 4 6 8 10 eta2 eta1 2.0 1.0 1.5 2.0 2.5 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. Determination of Model Adequacy Ranking Scientific Models • Put equal prior probability on each model. • Model comparision is by means of posterior probabilities. 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. 4 3 2 1 0 0 1 2 3 4 Fig 2. Determination of Model Adequacy 0.5 1.0 1.5 2.0 0.5 1.5 2.0 3 2 1 0 0 1 2 3 4 kappa=5, restriction false 4 kappa=5, restriction true 1.0 0.5 1.0 1.5 2.0 kappa=100, restriction true 0.5 1.0 1.5 2.0 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 Data Annual observations 1929–2001, 72 years, on a Pdt end-of-year per capita stock market value Dta annual aggregate per capita dividend Cta annual per capita consumption a rdt annual real geometric return Qat annual quadratic variation Data are real, i.e. inflation adjusted. Source: Bansal, Gallant, and Tauchen (2007). Data for Estimation Used here cat − cat−1 a rdt ! dat − cat cat − cat−1 padt − dat a rdt Used by BGT Habit Persistence Asset Pricing Model Driving Processes Consumption: ct − ct−1 = g + vt Dividends: dt − dt−1 = g + wt Random Shocks: vt wt ! ∼ NID " ! σ2 0 ρσσw , 2 0 ρσσw σw !# The time increment is one month. Lower case denotes logarithms of upper case quantities; i.e. ct = log(Ct), dt = log(Dt). From Campbell and Cochrane (1999). Habit Persistence Asset Pricing Model Utility function E0 ∞ X t=0 1−γ (StCt) −1 t δ , 1−γ Habit persistence Surplus ratio: st − s̄ = φ (st−1 − s̄) + λ(st−1)vt−1 Sensitivity function: λ(s) = ( 1 S̄ q 1 − 2(s − s̄) − 1 st ≤ smax 0 st > smax Et is conditional expectation with respect to St, St−1, ... . Lower case denotes logarithms of upper case quantities: st = log(St). S̄ and smax can be computed p from model parameters θ = (g, σ, ρ, σw , φ, δ, γ) as S̄ = σ γ/(1 − φ), smax = s̄ + (1 − S̄ 2)/2. From Campbell and Cochrane (1999). Habit Persistence Asset Pricing Model Return on dividends St+1Ct+1 V (St) = Et δ St C t " !−γ ! i Dt+1 h 1 + V (St+1) Dt Dt 1 + V (St) rdt = log V (St−1) Dt−1 !# V (·) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. Pdt /Dt = V (St), where Pdt is the price of the asset that pays the dividend stream. rdt is the logarithmic real return, i.e. rdt = log(Pdt + Dt) − log(Pd,t−1 ), where Pdt and Dt are measured in real (inflation adjusted) dollars. Dividend error can be integrated out analytically. Consumption error integrated by quadrature. From Campbell and Cochrane (1999). Habit Persistence Asset Pricing Model Solution Method Approximate the log policy function v(st) = log V (est ) by a piecewise linear function and use policy function iteration. Campbell and Cochrane used Gauss’s intquad1 and set join points at s̄, smax, smax −0.01, smax −0.02, smax −0.03, smax −0.04, and log[iS/(m + 1)] for i = 1, . . . , m = 10. We used GaussHermite quadrature; we added the abscissae of the GaussHermite quadrature formula at the maximum and minimum of the above join points; we deleted all points less than 0.001 apart. Figure 3, next slide, plots the approximation at the Campbell and Cochrane parameter values. Fig 3. Piecewise Linear Approximation x x x x x x xxxxx x x x 2.4 2.0 p−d 2.8 3.2 Annualized Log Policy Function v(s) x xx x −5.0 x x −4.5 x −4.0 −3.5 −3.0 −2.5 15 x 10 P/D 20 25 Annualized Policy Function V(S) x x x x x x x x xxxxxx x x x x xx 0.02 0.04 0.06 0.08 x’s mark Campbell and Cochrane join points; o’s mark extra join points from the quadrature rule. Habit Persistence Asset Pricing Model Risk Free Rate rf t = − log Et δ !−γ St+1Ct+1 St C t rf t 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. Habit Persistence Asset Pricing Model Large Model Output Given model parameters θ = (g, σ, ρ, σw , φ, δ, γ) simulate monthly and aggregate to annual: Cta = 11 X C12t−k k=0 cat = log(Cta) a = rdt 11 X rd,12t−k 11 X rf,12t−k k=0 rfat = k=0 Habit Persistence Asset Pricing Model Prior Distribution " p(θ) = N rf | 0.89, 2 # Y p 1 1.96 i=1 N θi | θi∗, !2 0.1θi∗ 1.96 where the θi∗ are the calibrated values from Campbell and Cochrane (1999). Table 1. Habit Model Prior Parameter g σ ρ σw φ δ γ rf rd − rf σr d Posterior Mode Std.Dev. Mode Std.Dev. 0.0015640 0.0042114 0.21045 0.033386 0.98691 0.98850 2.17383 0.87084 6.8610 20.560 8.8319e-05 0.00021870 0.010179 0.0016510 0.00033643 0.00062957 0.0861067 0.60467 0.38795 1.1170 0.00157928 0.00494385 0.18604 0.033142 0.98837 0.98934 2.12305 2.0611 5.4221 18.6145 7.2418e-05 0.00016592 0.010208 0.0017159 0.00029442 0.00033657 0.050678 0.55120 0.35970 0.73469 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, rd − rf = 6.02 − 0.89 = 5.13 and σrd = 19.29. Fig 4. Posterior Returns 0.4 0.0 0.2 Density 0.6 0.8 bond returns 1.0 1.5 2.0 2.5 3.0 equity premium 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Density 0.5 5.0 5.5 6.0 6.5 1.0 0.0 0.5 Density 1.5 2.0 stock returns 7.0 7.5 8.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Density sdev stock returns 18 19 20 21 Long Run Risk Asset Pricing Model Driving Processes Consumption: ct+1 − ct = µc + xt + σtηt+1 Long Run Risk: xt+1 = ρxt + φeσtet+1 2 = σ̄ 2 + ν(σt2 − σ̄ 2) + σw wt+1 Stochastic Volatility: σt+1 Dividends: dt+1 − dt = µd + φdxt + πdσtηt+1 + φuσtut+1 1 0 ηt 0 0 e t Random Shocks: ∼ NID , 0 0 wt 0 0 ut 0 1 0 0 0 0 1 0 0 0 0 1 The time increment is one month. Lower case denotes logarithms of upper case quantities; i.e. ct = log(Ct), dt = log(Dt). From Bansal, Kiku, and Yaron (2007). Long Run Risk Asset Pricing Model Epstein-Zin utility function " Ut = (1 − 1−γ δ)C θ t 1−γ θ + δ Et Ut+1 where γ ψ 1 is the coefficient of risk aversion is the elasticity of intertemporal substitution and 1−γ θ= 1 − 1/ψ Et is conditional expectation with respect to xt , σt. # θ 1−γ Long Run Risk Asset Pricing Model Return on consumption mrst+1 = δ θ exp[−(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1] ( Ct+1 VC (xt, σt) = Et mrst+1 Ct " ! h 1 + VC (xt+1, σt+1) 1 + VC (xt, σt) Ct rct = log VC (xt−1, σt−1) Ct−1 i ) !# VC (·) is defined as the solution of the Euler condition above. It is the price consumption ratio; i.e. Pct /Ct = VC (xt, σt ), where Pct is the price of the asset that pays the consumption stream. rct is the logarithmic real return, i.e. rct = log(Pct + Ct ) − log(Pc,t−1), where Pct and Ct are measured in real (inflation adjusted) dollars. Long Run Risk Asset Pricing Model Solution Method Use the log linear approximation . rc,t+1 = κ0 + κ1zt+1 + ∆ct+1 − zt κ1 = [exp(z̄)]/[1 + exp(z̄)] k0 = log[1 + exp(z̄)] − κ1z̄ where zt = log(Pc,t/Ct) and z̄ is its endogenous mean. To compute z̄, use the approximation . zt = A0(z̄) + A1(z̄) xt + A2(z̄) σt2 Ai(z̄) = tedious expressions in model parameters and z̄ and solve the fixed point problem z̄ = A0(z̄) + A1(z̄) xt + A2(z̄) σt2 Long Run Risk Asset Pricing Model Return on dividends mrst+1 = δ θ exp[−(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1] ( Dt+1 VD (xt, σt) = Et mrst+1 Dt " ! h i 1 + VD (xt+1, σt+1) Dt 1 + VD (xt, σt) rdt = log VD (xt−1, σt−1) Dt−1 ) !# VD (·) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. Pdt /Dt = VD (xt , σt), where Pct is the price of the asset that pays the dividend stream. rdt is the logarithmic real return, i.e. rdt = log(Pdt + Dt) − log(Pd,t−1), where Pdt and Dt are measured in real (inflation adjusted) dollars. Solution method is similar to the foregoing. Long Run Risk Asset Pricing Model Risk Free Rate h io θ rf t = − log Et δ exp −(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1 n rf t is the logarithmic return on an asset that pays one real dollar one month hence with certainty. Solution method is similar to the foregoing. Long Run Risk Asset Pricing Model Large Model Output Given model parameters θ = (δ, γ, ψ, µc, ρ, φe, σ̄ 2, η, σw , µd, φu) simulate monthly and aggregate to annual: Cta = 11 X C12t−k k=0 cat = log(Cta) a = rdt 11 X rd,12t−k 11 X rf,12t−k k=0 rfat = k=0 Long Run Risk Asset Pricing Model Prior Distribution " p(θ) = N rf | 0.89, 2 # Y p 1 1.96 i=1 N θi | θi∗, !2 0.1θi∗ 1.96 where the θi∗ are the calibrated values from Kiku (2006). Table 2. Long Run Risk Model Prior Parameter δ γ ψ µc ρ φe σ̄ 2 ν σw µd φd πd φu rf rd − rf σr d Mode Std.Dev. 0.99919 0.00012674 10.191 0.21061 1.4882 0.061835 0.0014458 8.011084e-05 0.98672 0.0024294 0.033661 0.0010018 4.2468e-05 1.3089e-06 0.99295044 0.0016819 1.7397e-06 4.8917e-07 0.0012093 7.2206e-05 2.78515 0.14961 3.9492 0.63146 6.0507 0.66856 1.17504 0.27676 6.24790 1.5816 19.076 1.8840 Posterior Mode 0.99949 11.269 1.5585 0.0015296 0.98812 0.032684 4.2528e-05 0.98648 1.7844e-06 0.0011482 2.6835 4.1523 5.9648 0.59225 7.1733 18.970 Std.Dev. 0.00010942 0.29135 0.065250 6.6133e-05 0.00088681 0.0010092 1.2106e-06 0.00093868 5.0889e-07 7.7365e-05 0.15192 0.62532 0.51864 0.20659 0.51500 1.1396 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, rd − rf = 6.02 − 0.89 = 5.13 and σrd = 19.29. Fig 5. Posterior Returns 0.0 0.5 1.0 1.5 2.0 Density bond returns 0.6 5.5 1.0 6.0 6.5 1.2 1.4 7.0 7.5 8.0 stock returns 0.0 0.2 0.4 0.6 0.8 1.0 Density 0.8 equity premium 0.0 0.2 0.4 0.6 0.8 Density 0.4 7.0 7.5 8.0 8.5 0.2 0.0 Density 0.4 sdev stock returns 15 16 17 18 19 Prospect Theory Asset Pricing Model Driving Processes Aggregate Consumption: c̄t+1 − c̄t = gC + σC ηt+1 Dividends: dt+1 − dt = gD + σD ǫt+1 Random Shocks: ηt ǫt ! ∼ NID " ! 0 1 ω , 0 ω 1 !# C̄t 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. c̄t = log(C̄t), dt = log(Dt). All variables are real. From Barberis, Huang, Santos (2001). Prospect Theory Asset Pricing Model Other Model Variables • Gross Stock Return: Rt −1 2 2 • Gross Risk Free Rate: Rf = ρ exp γgC − γ σC /2 • Allocation to Risky Asset: St • Gain or Loss: Xt+1 = St(Rt+1 − Rf ) • Benchmark Level (State Variable): zt+1 = η zt R R̄ +(1−η) t+1 • Choose R̄ to make Median {zt} = 1 • The Agent’s Consumption: Ct Prospect Theory Asset Pricing Model Utility function E0 ∞ X t=0 1−γ h i C −1 −γ t+1 ρt t + b0C̄t ρ St v̂(Rt+1, zt) 1−γ h Utility from Gains and Losses: St v̂(Rt+1, zt) v̂(Rt+1, zt) = Rt+1 − Rf = (ztRf − Rf ) + λ(Rt+1 − ztRf ) = Rt+1 − Rf = λ(zt)(Rt+1 − Rf ) λ(zt) = λ + k(zt − 1) i zt ≤ 1, Rt+1 ≥ ztRf zt ≤ 1, Rt+1 < ztRf zt > 1, Rt+1 ≥ Rf zt > 1, Rt+1 < Rf 0.0 −0.6 −0.4 −0.2 Utility 0.2 0.4 0.6 Fig 6. Utility of Gains and Losses −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 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). Prospect Theory Asset Pricing Model Return on dividends 2 2 2 1 = ρ exp gD − γgC + γ σC (1 − ω )/2 " # 1 + f (zt+1) × Et exp[(σD − γωσC )ǫt+1] f (zt) " !# 1 + f (zt+1) + b0ρ Et v̂ exp(gD + σD ǫt+1), zt f (zt) " 1 + f (zt) exp(gD + σD ǫt) rdt = log f (zt−1) # f (·) is defined as the solution of the Euler condition above. It is the price dividend ratio; i.e. Pdt/Dt = f (zt), where Pct is the price of the asset that pays the dividend stream. rdt is the logarithmic real return, i.e. rdt = log(Pdt + Dt) − log(Pd,t−1 ), where Pdt and Dt are measured in real (inflation adjusted) dollars. Prospect Theory Asset Pricing Model Self Referential Equations zt+1 = η zt R̄ Rt+1 ! + (1 − η) 1 + f (zt+1) exp(gD + σD ǫt+1) Rt+1 = f (zt) 1 = Median{zt} Prospect Theory Asset Pricing Model Solution Method Approximate f by a piecewise linear function f (0)(z). Approximate R̄ by (1 + f (1)) exp(gD )/f (1), which is a departure from Barberis, Huang, and Santos (2001). Define h(0) such that zt+1 = h(0)(zt, ǫt+1) solves the self referential equations that define zt+1 and Rt+1 on previous slide. A root finding problem. We use Brent’s method. Substitute h(0)(zt, ǫt+1) into the Euler equation. Use GaussHermite quadrature to integrate out ǫt+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. 20 15 10 P/D 25 30 Fig 7. Piecewise Linear Approximation 0 1 2 z 3 4 Prospect Theory Asset Pricing Model Risk Free Rate i −1 2 2 rf = log ρ exp γgC − γ σC /2 h rf t is the logarithmic return on an asset that pays one real dollar one year hence with certainty. Prospect Theory Asset Pricing Model Large Model Output Given model parameters θ = (gC , gD , σC , σD , ω, γ, ρ, λ, k, b0, η) simulate annually and set cat = log(Ct) a =r rdt dt rfat = rf Prospect Asset Pricing Model Prior Distribution " p(θ) = N rf | 0.89, 2 # Y p 1 1.96 i=1 N θi | θi∗, !2 0.1θi∗ 1.96 where the θi∗ are the calibrated values from Barberis, Huang, Santos (2001). Table 3. Prospect Model Prior Parameter gC gD σC σD ω γ ρ λ k b0 η rf rd − rf σr d Mode Std.Dev. 0.017212 0.00092156 0.018188 0.00089243 0.036621 0.0020330 0.12402 0.0060679 0.14941 0.0081401 0.99219 0.051017 0.99982 0.0010958 2.28125 0.11124 10.09375 0.52937 2.18018 0.033314 0.93909 0.014177 1.6604 0.166210 6.3091 0.385231 30.984 2.074061 Posterior Mode Std.Dev. 0.017700 0.00096645 0.019165 0.00087639 0.034668 0.0019947 0.11035 0.0051616 0.14746 0.0076743 1.00781 0.0498657 0.99823 0.0013179 2.28125 0.11559 10.28125 0.60546 1.96436 0.0572277 0.85754 0.0120773 1.8999 0.170305 4.5090 0.347421 21.3325 1.33432 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, rd − rf = 6.02 − 0.89 = 5.13 and σrd = 19.29. Fig 8. Posterior Returns 1.5 1.0 0.0 0.5 Density 2.0 bond returns 1.8 2.0 2.2 2.4 2.6 equity premium 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Density 1.6 4.0 4.5 5.0 5.5 0.0 0.2 0.4 0.6 0.8 1.0 Density stock returns 5.5 6.0 6.5 7.0 7.5 0.20 0.10 0.00 Density 0.30 sdev stock returns 20 22 24 26 Model Assessment Posterior Probabilities for Three Models Long Run Risk 0.34736 Habit Persistence 0.33278 Prospect Theory 0.31986 Model Assessment Posterior Probabilities for Long Run Risk κ = 0.01 0.32531 κ = 50.00 0.33692 κ = 75.00 0.33777 Methodology - 2 • Gallant and Hong (2007), “A Statistical Inquiry into the Plausibility of Recursive Utility,” Journal of Financial Econometrics 5, 523–590. • Pricing kernel: θ = (θ1, . . . , θn+1) Qn • Hierarchical likelihood: L(θ, η) = L(θ) t=1 f (θt+1|θt, . . . , θ1, η) – L(θ) function of returns derived from the Euler equation 1 = Etθt+1Rit by a conditioning argument. – f (·|·, η) seminonparametric transition density on θ. • Composite prior: p(θ, η) = p(η) pT (θ, η) – p(η) prior on hyperparameter η : habit, long run risk, etc. – pT (θ, η) imposes technical conditions • Inference: – MCMC posterior probabilities P (A) = RR IA(θ, η)P (dθ, dη) 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. 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 Remainder • Use long run risk model to illustrate – Law of motion of pricing kernel f (θt|θt−1, . . . , θ1, η) – Prior p(η) on hyperparameter η – Likelihood for observables L(θ) – Implementation of L(θ) – Technical prior pT (θ, η) Consumption Endowment and Cash Flows Ct consumption endowment Pct price of an asset that pays the consumption endowment Rct = (Pct + Ct)/Pc,t−1 gross return on consumption endowment Dst cash flow S Pst price of cash flow S Rst = (Pst + Dst)/Ps,t−1 gross return on cash flow S Prices are real LRR Uses Recursive Utility Marginal rate of substitution Mt,t+1 = δ β Ct+1 Ct !−(β/ψ) (Rc,t+1)(β−1), δ time preference parameter γ coefficient of risk aversion ψ elasticity of intertemporal substitution 1−γ β = 1−1/ψ In General Euler equation 1 = Et θt+1Rt,t+1 Satisfied by any pricing kernel θt+1 including Mt,t+1 given by recursive utility Law of motion of pricing kernel f (θt|θt−1, . . . , θ1, 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 θt = Mt−1,t is f (θt |θt−1, . . . , θ1 , η) yt f (yt |yt−1 , . . . , y1 , η) η P(yt|a) µt−t 2 σt−1 = = = = = = = f (yt |yt−1, . . . , y1 , η)/ exp(yt) log(θt−1,t ) 2 ) P 2(yt|a) n(yt | µt−1, σt−1 (a1, a2 , a3 , a4 , b0, b1, r0, r1, r2) 1 + a1 yt + a2 yt2 + a3 yt3 + a4 yt4 b0 + b1 yt−1 2 r02 + r12σt−2 + r22 (yt−1 − µt−2)2 Likelihood Derivation – 1 of 4 • Consider a general set of moment conditions n 1 X m̄n = mt(yt, θt) n t=1 mt ∈ ℜK that a scientific theory implies ought to have expectation zero where both yt and θt are regarded as random • To estimate the variance when the mt(yt, xt, θo) are uncorrelated, use n 1 X Wn = [mt(yt, θt) − m̄n] [mt(yt, θt) − m̄n]′ n t=1 • When correlated, use HAC. −1/2√ n m̄n • Rely on CLT for normality of z = Wn Likelihood Derivation – 2 of 4 • View as a data summary: z = Z(Y, θ) Z : (Y, θ) 7→ √ nW −1/2n m̄n where Y = [y1|y2| · · · |yn], θ = [θ1|θ2| · · · |θn] • 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 ℜK . • Assignment formula: P (A) = PZ (B) = Z ··· Z IB (z) (2π)−K/2 exp(−z ′z/2) dz1 · · · dzK Likelihood Derivation – 3 of 4 • Let Zθ denote the map Y 7→ 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 = Zθ−1(Bθ ) for some Borel set Bθ ∈ ℜK . • The principle of conditioning is that one assigns conditional probability proportionately to joint probability. • The constant of proportionality is 1/P (ℜθ ), where ℜθ is the Borel set for which Y = Zθ−1(ℜθ ). • The conditional probability of C is computed as P (C|θ) = PZ (Bθ ) PZ (ℜθ ) Likelihood Derivation – 4 of 4 • Conditional probability is computed as 1 P (C|θ) = PZ (ℜθ ) Z ··· Z IBθ (z) (2π)−K/2 exp(−z ′z/2) dz1 · · · dzK • The data summary is √ −1/2 n mn(θ) z = [Wn(θ)] • In our application PZ (ℜθ ) = 1 • Conclude that the likelihood is n L(θ) ∝ exp − mn(θ) ′[Wn(θ)]−1mn(θ) 2 • Same argument as Fisher (1930) used to define fiducial probability. Is Bayesian inference for continuously updated GMM. Implementation of L(θ) (monthly data) • st: Fama-French gross returns S11 − S54 bt: T-debt gross returns t30ret, b1ret, and b10ret. • Parameters θ = θ1, . . . , θ551 • Euler equation errors s et(θ) = et(st+1, bt+1, θt+1) = 1 − θt+1 t+1 bt+1 ! • Instruments: Zt = (st − 1, bt − 1, log(cgt) − 1, log(lgt) − 1, 1) • Moment conditions: t = 1, . . . , n = 550 mt(θ) = mt(st, bt, st+1, bt+1, θt+1) = Zt ⊗ et(st+1, bt+1, θt+1) • Length of mt(θ): K = 810 Number of overidentifying restrictions: 260. Computing the Weighting Matrix W • Assume factor structure for the random part of (st, bt). • Σe : one error common for st, one error common for bt, one idiosyncratic error for each of (st, bt). o )] = 0 to • Strengthen zero correlation condition E[Zi,t ej,t(st+1 , bt+1 , θt+1 o Var[Zt ⊗ et(st+1 , bt+1 , θt+1 )] = Σz ⊗ Σe , • Implies Σz ⊗ Σe is diagonalized by Uz ⊗ Ue where Uz and Ue 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 n n X X 1 1 si (θ) = vt,i(θ) vt,i(θ) − n t=1 n t=1 Implementation of L(θ) (monthly data) • The likelihood for the pricing kernel θ is, then, n L(θ) ∝ exp − mn(θ) ′(Uz ⊗ Ue)Sn−1(θ)(Uz ⊗ Ue) ′mn(θ) 2 n 1 X Zt ⊗ et(st+1, bt+1, θt+1), mn(θ) = n t=1 • Full likelihood L(η, θ) ∝ L(θ) × Qn t=1 f (θt+1 |θt, θt−1, . . . , θ1 , η). • dim(η, θ) = 560 implies high correlation in MCMC chain. • To reduce, sample 30, 000 out of 30 million MCMC draws. Technical Prior p(θ, η) • Support condition: – Impose mean reversion on law of motion f (θt|θt−1, . . . , θ1, η) • Constrain the size of the Euler equation errors to be about the same – Annual data: P (−0.5 < ei,n(θ) < 0.5) = 0.95 – Monthly data: P (−0.05 < ei,n(θ) < 0.05) = 0.95 • Require an approximation PB = n t=1 θt to the average price of a risk-free, one period bond to be between 1% and 4% per annum, P f (PB ) ∝ (1 + cos(α + βPB )) where β = 2π/(b − a), α = π − βb. a < PB < b Summary • Pricing kernel: θ = (θ1, . . . , θn+1) Q • Hierarchical likelihood: L(θ, η) = L(θ) n t=1 f (θt+1 |θt , . . . , θ1 , 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)pT (θ, η) – p(η) prior on hyperparameter η : tight, intermediate, loose – pT (θ, η) imposes technical conditions • Inference: MCMC – Moments of posterior distribution P (θ, η) RR – Posterior probabilities P (A) = IA(θ, η)P (dθ, dη) of events A that correspond to hypotheses 0.5 1.0 2.0 5.0 10.0 Figure 5. The Posterior Mean of the Monthly Pricing Kernel 1962 1964 1966 1968 1970 0.5 1.0 2.0 5.0 10.0 1960 1975 1980 1985 1990 0.5 1.0 2.0 5.0 10.0 1970 1990 1995 2000 2005 0.1 0.2 0.5 1.0 2.0 5.0 Figure 6. The Posterior Mean of the Annual Pricing Kernel 1940 1960 1980 2000