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    Ludvigson methodslecture 1 Ludvigson methodslecture 1 Presentation Transcript

    • NBER Summer Institute Econometrics Methods Lecture: GMM and Consumption-Based Asset Pricing Sydney C. Ludvigson, NYU and NBER July 14, 2010 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... ...emphasize here: all models are misspecified, and macro variables often measured with error. Therefore: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... ...emphasize here: all models are misspecified, and macro variables often measured with error. Therefore: 1 move away from specification tests of perfect fit (given sampling error), Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... ...emphasize here: all models are misspecified, and macro variables often measured with error. Therefore: 1 move away from specification tests of perfect fit (given sampling error), 2 toward estimation and testing that recognize all models are misspecified, Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... ...emphasize here: all models are misspecified, and macro variables often measured with error. Therefore: 1 move away from specification tests of perfect fit (given sampling error), 2 toward estimation and testing that recognize all models are misspecified, 3 toward methods permit comparison of magnitude of misspecification among multiple, competing macro models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Themes Why care about consumption-based models? True systematic risk factors are macroeconomic in nature–derived from IMRS over consumption–asset prices are derived endogenously from these. Some cons-based models work better than others, but... ...emphasize here: all models are misspecified, and macro variables often measured with error. Therefore: 1 move away from specification tests of perfect fit (given sampling error), 2 toward estimation and testing that recognize all models are misspecified, 3 toward methods permit comparison of magnitude of misspecification among multiple, competing macro models. Themes are important in choosing which methods to use. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Incorporating conditioning information: scaled consumption-based models Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Incorporating conditioning information: scaled consumption-based models Generalizations of CRRA utility: recursive utility Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Incorporating conditioning information: scaled consumption-based models Generalizations of CRRA utility: recursive utility Semi-nonparametric minimum distance estimators: unrestricted LOM for data Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Incorporating conditioning information: scaled consumption-based models Generalizations of CRRA utility: recursive utility Semi-nonparametric minimum distance estimators: unrestricted LOM for data Simulation methods: restricted LOM Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM and Consumption-Based Models: Outline GMM estimation of classic representative agent, CRRA utility model Incorporating conditioning information: scaled consumption-based models Generalizations of CRRA utility: recursive utility Semi-nonparametric minimum distance estimators: unrestricted LOM for data Simulation methods: restricted LOM Consumption-based asset pricing: concluding thoughts Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Economic model implies set of r population moment restrictions E{h (θ, wt )} = 0 (r × 1 ) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Economic model implies set of r population moment restrictions E{h (θ, wt )} = 0 (r × 1 ) wt is an h × 1 vector of variables known at t Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Economic model implies set of r population moment restrictions E{h (θ, wt )} = 0 (r × 1 ) wt is an h × 1 vector of variables known at t θ is an a × 1 vector of coefficients Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Economic model implies set of r population moment restrictions E{h (θ, wt )} = 0 (r × 1 ) wt is an h × 1 vector of variables known at t θ is an a × 1 vector of coefficients Idea: choose θ to make the sample moment as close as possible to the population moment. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , t= 1 (r × 1 ) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , t= 1 (r × 1 ) ′ ′ ′ ′ yT ≡ wT , wT −1 , ...w1 is a T · h × 1 vector of observations. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , t= 1 (r × 1 ) ′ ′ ′ ′ yT ≡ wT , wT −1 , ...w1 is a T · h × 1 vector of observations. The GMM estimator θ minimizes the scalar ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1) ( 1× r ) (r×r) ( r × 1) ∞ {WT }T =1 a sequence of r × r positive definite matrices which may be a function of the data, yT . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , t= 1 (r × 1 ) ′ ′ ′ ′ yT ≡ wT , wT −1 , ...w1 is a T · h × 1 vector of observations. The GMM estimator θ minimizes the scalar ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1) ( 1× r ) (r×r) ( r × 1) ∞ {WT }T =1 a sequence of r × r positive definite matrices which may be a function of the data, yT . If r = a, θ estimated by setting each g(θ; yT ) to zero. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , t= 1 (r × 1 ) ′ ′ ′ ′ yT ≡ wT , wT −1 , ...w1 is a T · h × 1 vector of observations. The GMM estimator θ minimizes the scalar ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1) ( 1× r ) (r×r) ( r × 1) ∞ {WT }T =1 a sequence of r × r positive definite matrices which may be a function of the data, yT . If r = a, θ estimated by setting each g(θ; yT ) to zero. GMM refers to use of (1) to estimate θ when r > a. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Asym. properties (Hansen 1982): θ consistent, asym. normal. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Asym. properties (Hansen 1982): θ consistent, asym. normal. Optimal weighting WT = S−1   ∞     ′ S = r×r ∑ E  [h (θo , wt )] h θo , wt−j  . j=− ∞   r×1 1× r Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) Asym. properties (Hansen 1982): θ consistent, asym. normal. Optimal weighting WT = S−1   ∞     ′ S = r×r ∑ E  [h (θo , wt )] h θo , wt−j  . j=− ∞   r×1 1× r In many asset pricing applications, it is inappropriate to use WT = S−1 (see below). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) ST depends on θT which depends on ST . Employ an iterative procedure: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Review (Hansen, 1982) ST depends on θT which depends on ST . Employ an iterative procedure: (1) 1 Obtain an initial estimate of θ = θT , by minimizing Q (θ; yT ) subject to arbitrary weighting matrix, e.g., W = I. (1) (1) 2 Use θT to obtain initial estimate of S = ST . (1 ) 3 Re-minimize Q (θ; yT ) using initial estimate ST ; obtain (2) new estimate θT . 4 Continue iterating until convergence, or stop. (Estimators have same asym. dist. but finite sample properties differ.) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Classic Example: Hansen and Singleton (1982) Investors maximize utility ∞ max Et Ct ∑ β i u (C t + i ) i= 0 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Classic Example: Hansen and Singleton (1982) Investors maximize utility ∞ max Et Ct ∑ β i u (C t + i ) i= 0 Power (isoelastic) utility  1− γ  u (C ) =  Ct γ>0 t 1− γ (2)   u (Ct ) = ln(Ct ) γ = 1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Classic Example: Hansen and Singleton (1982) Investors maximize utility ∞ max Et Ct ∑ β i u (C t + i ) i= 0 Power (isoelastic) utility  1− γ  u (C ) =  Ct γ>0 t 1− γ (2)   u (Ct ) = ln(Ct ) γ = 1 N assets => N first-order conditions −γ −γ Ct = βEt (1 + ℜi,t+1 ) Ct+1 i = 1, ..., N. (3) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions −γ Ct + 1 0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4) Ct Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions −γ Ct + 1 0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4) Ct 2 params to estimate: β and γ, so θ = ( β, γ) ′ . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions −γ Ct + 1 0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4) Ct 2 params to estimate: β and γ, so θ = ( β, γ) ′ . ∗ xt denotes info set of investors −γ −γ ∗ 0=E 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt i = 1, ...N (5) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions −γ Ct + 1 0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4) Ct 2 params to estimate: β and γ, so θ = ( β, γ) ′ . ∗ xt denotes info set of investors −γ −γ ∗ 0=E 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt i = 1, ...N (5) ∗ . Conditional model (5) => unconditional model: xt ⊂ xt −γ Ct + 1 0=E 1− β (1 + ℜi,t+1 ) −γ xt i = 1, ...N Ct (6) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Let xt be M × 1. Then r = N · M and,  −γ  C +1  1 − β (1 + ℜ1,t+1 ) t−γ xt  Ct    −γ Ct+1   1 − β (1 + ℜ2,t+1 ) −γ xt   Ct     h (θ, wt ) =  ·  (7)  r×1  ·     ·     −γ C +1  1 − β (1 + ℜN,t+1 ) t−γ xt Ct Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Let xt be M × 1. Then r = N · M and,  −γ  C +1  1 − β (1 + ℜ1,t+1 ) t−γ xt  Ct    −γ Ct+1   1 − β (1 + ℜ2,t+1 ) −γ xt   Ct     h (θ, wt ) =  ·  (7)  r×1  ·     ·     −γ C +1  1 − β (1 + ℜN,t+1 ) t−γ xt Ct Model can be estimated, tested as long as r ≥ 2. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Let xt be M × 1. Then r = N · M and,  −γ  C +1  1 − β (1 + ℜ1,t+1 ) t−γ xt  Ct    −γ Ct+1   1 − β (1 + ℜ2,t+1 ) −γ xt   Ct     h (θ, wt ) =  ·  (7)  r×1  ·     ·     −γ C +1  1 − β (1 + ℜN,t+1 ) t−γ xt Ct Model can be estimated, tested as long as r ≥ 2. Take sample mean of (7) to get g(θ; yT ), minimize ′ min Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )] θ r×r 1× r r×1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) HS use lags of cons growth and returns in xt ; index and industry returns, NDS expenditures. Estimates of β ≈ .99, RRA low = .35 to .999. No equity premium puzzle! But.... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) HS use lags of cons growth and returns in xt ; index and industry returns, NDS expenditures. Estimates of β ≈ .99, RRA low = .35 to .999. No equity premium puzzle! But.... ...model is strongly rejected according to OID test. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) HS use lags of cons growth and returns in xt ; index and industry returns, NDS expenditures. Estimates of β ≈ .99, RRA low = .35 to .999. No equity premium puzzle! But.... ...model is strongly rejected according to OID test. Campbell, Lo, MacKinlay (1997): OID rejections stronger whenever stock returns and commercial paper are included as test returns. Why? Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) HS use lags of cons growth and returns in xt ; index and industry returns, NDS expenditures. Estimates of β ≈ .99, RRA low = .35 to .999. No equity premium puzzle! But.... ...model is strongly rejected according to OID test. Campbell, Lo, MacKinlay (1997): OID rejections stronger whenever stock returns and commercial paper are included as test returns. Why? Model cannot capture predictable variation in excess returns over commercial paper ⇒ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ θ; yT ∼ χ2 (r − a) HS use lags of cons growth and returns in xt ; index and industry returns, NDS expenditures. Estimates of β ≈ .99, RRA low = .35 to .999. No equity premium puzzle! But.... ...model is strongly rejected according to OID test. Campbell, Lo, MacKinlay (1997): OID rejections stronger whenever stock returns and commercial paper are included as test returns. Why? Model cannot capture predictable variation in excess returns over commercial paper ⇒ Researchers have turned to other models of preferences. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Results in HS use conditioning info xt –scaled returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Results in HS use conditioning info xt –scaled returns. Another limitation with classic CCAPM: large unconditional Euler equation (pricing) errors even when params freely chosen. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Results in HS use conditioning info xt –scaled returns. Another limitation with classic CCAPM: large unconditional Euler equation (pricing) errors even when params freely chosen. Let Mt+1 = β(Ct+1 /Ct )−γ . Define Euler equation errors: j j eR ≡ E[Mt+1 Rt+1 ] − 1 j j f eX ≡ E[Mt+1 (Rt+1 − Rt+1 )] Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Results in HS use conditioning info xt –scaled returns. Another limitation with classic CCAPM: large unconditional Euler equation (pricing) errors even when params freely chosen. Let Mt+1 = β(Ct+1 /Ct )−γ . Define Euler equation errors: j j eR ≡ E[Mt+1 Rt+1 ] − 1 j j f eX ≡ E[Mt+1 (Rt+1 − Rt+1 )] ′ Choose params: min β,γ gT WT gT where jth element of gT j gj,t (γ, β) = 1 T ∑T=1 eR,t t j gj,t (γ) = 1 T ∑T=1 eX,t t Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Unconditional Euler Equation Errors, Excess Returns j j f eX ≡ E[ β(Ct+1 /Ct )−γ (Rt+1 − Rt+1 )] j = 1, ..., N j j f RMSE = 1 N ∑N 1 [eX ]2 , j= RMSR = 1 N N ∑j=1 [E(Rt+1 − Rt+1 )]2 Source: Lettau and Ludvigson (2009). Rs is the excess return on CRSP-VW index over 3-Mo T-bill rate. Rs & 6 FF refers to this return plus 6 size and book-market sorted portfolios provided by Fama and French. For each value of γ, β is chosen to minimize the Euler equation error for the T-bill rate. U.S. quarterly data, 1954:1-2002:1. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Magnitude of errors large, even when parameters are freely chosen to minimize errors. Unlike the equity premium puzzle of Mehra and Prescott (1985), large Euler eq. errors cannot be resolved with high risk aversion. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Magnitude of errors large, even when parameters are freely chosen to minimize errors. Unlike the equity premium puzzle of Mehra and Prescott (1985), large Euler eq. errors cannot be resolved with high risk aversion. Lettau and Ludvigson (2009): Leading consumption-based asset pricing theories fail to explain the mispricing of classic CCAPM. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Magnitude of errors large, even when parameters are freely chosen to minimize errors. Unlike the equity premium puzzle of Mehra and Prescott (1985), large Euler eq. errors cannot be resolved with high risk aversion. Lettau and Ludvigson (2009): Leading consumption-based asset pricing theories fail to explain the mispricing of classic CCAPM. Anomaly is striking b/c early evidence (e.g., Hansen & Singleton) that the classic model’s Euler equations were violated provided the impetus for developing these newer models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More GMM results: Euler Equation Errors Magnitude of errors large, even when parameters are freely chosen to minimize errors. Unlike the equity premium puzzle of Mehra and Prescott (1985), large Euler eq. errors cannot be resolved with high risk aversion. Lettau and Ludvigson (2009): Leading consumption-based asset pricing theories fail to explain the mispricing of classic CCAPM. Anomaly is striking b/c early evidence (e.g., Hansen & Singleton) that the classic model’s Euler equations were violated provided the impetus for developing these newer models. Results imply data on consumption and asset returns not jointly lognormal! Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? One reason: assessing specification error, comparing models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? One reason: assessing specification error, comparing models. Consider two estimated models of SDF, e.g., (1) 1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected (2) 2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? One reason: assessing specification error, comparing models. Consider two estimated models of SDF, e.g., (1) 1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected (2) 2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected May we conclude Model 1 is superior? Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? One reason: assessing specification error, comparing models. Consider two estimated models of SDF, e.g., (1) 1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected (2) 2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected May we conclude Model 1 is superior? No. Hansen’s J-test of OID restricts depends on model specific S: J = gT S−1 gT . ′ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Asset pricing applications often require WT S−1 . Why? One reason: assessing specification error, comparing models. Consider two estimated models of SDF, e.g., (1) 1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected (2) 2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected May we conclude Model 1 is superior? No. Hansen’s J-test of OID restricts depends on model specific S: J = gT S−1 gT . ′ Model 1 can look better simply b/c the SDF and pricing errors gT are more volatile, not b/c pricing errors are lower. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 HJ: compare models Mt (θ) using distance metric: T 1 DistT (θ) = mingT (θ)′ GT 1 gT (θ), − GT ≡ ∑ Rt Rt′ θ T t= 1 N ×N T 1 gT (θ) ≡ T ∑ [Mt (θ)Rt − 1N ] t= 1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 HJ: compare models Mt (θ) using distance metric: T 1 DistT (θ) = mingT (θ)′ GT 1 gT (θ), − GT ≡ ∑ Rt Rt′ θ T t= 1 N ×N T 1 gT (θ) ≡ T ∑ [Mt (θ)Rt − 1N ] t= 1 DistT does not reward SDF volatility => suitable for model comparison. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 HJ: compare models Mt (θ) using distance metric: T 1 DistT (θ) = mingT (θ)′ GT 1 gT (θ), − GT ≡ ∑ Rt Rt′ θ T t= 1 N ×N T 1 gT (θ) ≡ T ∑ [Mt (θ)Rt − 1N ] t= 1 DistT does not reward SDF volatility => suitable for model comparison. DistT is a measure of model misspecification: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 HJ: compare models Mt (θ) using distance metric: T 1 DistT (θ) = mingT (θ)′ GT 1 gT (θ), − GT ≡ ∑ Rt Rt′ θ T t= 1 N ×N T 1 gT (θ) ≡ T ∑ [Mt (θ)Rt − 1N ] t= 1 DistT does not reward SDF volatility => suitable for model comparison. DistT is a measure of model misspecification: Gives distance between Mt (θ) and nearest point in space of all SDFs that price assets correctly. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 HJ: compare models Mt (θ) using distance metric: T 1 DistT (θ) = mingT (θ)′ GT 1 gT (θ), − GT ≡ ∑ Rt Rt′ θ T t= 1 N ×N T 1 gT (θ) ≡ T ∑ [Mt (θ)Rt − 1N ] t= 1 DistT does not reward SDF volatility => suitable for model comparison. DistT is a measure of model misspecification: Gives distance between Mt (θ) and nearest point in space of all SDFs that price assets correctly. Gives maximum pricing error of any portfolio formed from the N assets. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Appeal of HJ Distance metric: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Appeal of HJ Distance metric: Recognizes all models are misspecified. Provides method for comparing models by assessing which is least misspecified. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Appeal of HJ Distance metric: Recognizes all models are misspecified. Provides method for comparing models by assessing which is least misspecified. Important problem: how to compare HJ distances statistically? Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Comparing specification error: Hansen and Jagannathan, 1997 Appeal of HJ Distance metric: Recognizes all models are misspecified. Provides method for comparing models by assessing which is least misspecified. Important problem: how to compare HJ distances statistically? One possibility developed in Chen and Ludvigson (2009): White’s reality check method. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Chen and Ludvigson (2009) compare HJ distances among K competing models using White’s reality check method. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Chen and Ludvigson (2009) compare HJ distances among K competing models using White’s reality check method. 1 Take benchmark model, e.g., model with smallest squared distance d1,T ≡ min{d2 }K 1 . 2 j,T j= 2 Null: d2 − d2 ≤ 0, where d2 is competing model with 1,T 2,T 2,T the next smallest squared distance. √ 3 Test statistic T W = T (d2 − d2,T ). 1,T 2 4 If null is true, test statistic should not be unusually large, given sampling error. 5 Given distribution for T W , reject null if historical value T W is > 95th percentile. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Chen and Ludvigson (2009) compare HJ distances among K competing models using White’s reality check method. 1 Take benchmark model, e.g., model with smallest squared distance d1,T ≡ min{d2 }K 1 . 2 j,T j= 2 Null: d2 − d2 ≤ 0, where d2 is competing model with 1,T 2,T 2,T the next smallest squared distance. √ 3 Test statistic T W = T (d2 − d2,T ). 1,T 2 4 If null is true, test statistic should not be unusually large, given sampling error. 5 Given distribution for T W , reject null if historical value T W is > 95th percentile. Method applies generally to any stationary law of motion for data, multiple competing possibly nonlinear, SDF models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Distribution of T W is computed via block bootstrap. T W has complicated limiting distribution. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Distribution of T W is computed via block bootstrap. T W has complicated limiting distribution. Bootstrap works only if have a multivariate, joint, continuous, limiting distribution under null. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Distribution of T W is computed via block bootstrap. T W has complicated limiting distribution. Bootstrap works only if have a multivariate, joint, continuous, limiting distribution under null. Proof of limiting distributions exists for applications to most asset pricing models: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Statistical comparison of HJ distance: Chen and Ludvigson, 2009 Distribution of T W is computed via block bootstrap. T W has complicated limiting distribution. Bootstrap works only if have a multivariate, joint, continuous, limiting distribution under null. Proof of limiting distributions exists for applications to most asset pricing models: For parametric models (Hansen, Heaton, Luttmer ’95) For semiparametric models (Ai and Chen ’07). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons to use identity matrix: econometric problems − − Econometric problems: near singular ST 1 or GT 1 . Asset returns are highly correlated. We have large N and modest T. If T < N covariance matrix for N asset returns is singular. Unless T >> N, matrix can be near-singular. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons to use identity matrix: economically interesting portfolios Original test assets may have economically meaningful characteristics (e.g., size, value). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons to use identity matrix: economically interesting portfolios Original test assets may have economically meaningful characteristics (e.g., size, value). − − Using WT = ST 1 or GT 1 same as using WT = I and doing GMM on re-weighted portfolios of original test assets. Triangular factorization S−1 = (P′ P), P lower triangular min gT S−1 gT ⇔ (gT P′ )I(PgT ) ′ ′ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons to use identity matrix: economically interesting portfolios Original test assets may have economically meaningful characteristics (e.g., size, value). − − Using WT = ST 1 or GT 1 same as using WT = I and doing GMM on re-weighted portfolios of original test assets. Triangular factorization S−1 = (P′ P), P lower triangular min gT S−1 gT ⇔ (gT P′ )I(PgT ) ′ ′ Re-weighted portfolios may not provide large spread in average returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons to use identity matrix: economically interesting portfolios Original test assets may have economically meaningful characteristics (e.g., size, value). − − Using WT = ST 1 or GT 1 same as using WT = I and doing GMM on re-weighted portfolios of original test assets. Triangular factorization S−1 = (P′ P), P lower triangular min gT S−1 gT ⇔ (gT P′ )I(PgT ) ′ ′ Re-weighted portfolios may not provide large spread in average returns. May imply implausible long and short positions in test assets. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons not to use WT = I: objective function dependence on test asset choice Using WT = [ET (R′ R)]−1 , GMM objective function is invariant to initial choice of test assets. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons not to use WT = I: objective function dependence on test asset choice Using WT = [ET (R′ R)]−1 , GMM objective function is invariant to initial choice of test assets. Form a portfolio, AR from initial returns R. (Note, portfolio weights sum to 1 so A1N = 1N ). −1 [E (MR) − 1N ]′ E RR′ [E (MR − 1N )] ′ −1 = [E (MAR) − A1N ] E ARR′ A [E (MAR − A1N )] . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • GMM Asset Pricing With Non-Optimal Weighting Reasons not to use WT = I: objective function dependence on test asset choice Using WT = [ET (R′ R)]−1 , GMM objective function is invariant to initial choice of test assets. Form a portfolio, AR from initial returns R. (Note, portfolio weights sum to 1 so A1N = 1N ). −1 [E (MR) − 1N ]′ E RR′ [E (MR − 1N )] ′ −1 = [E (MAR) − A1N ] E ARR′ A [E (MAR − A1N )] . With WT = I or other fixed weighting, GMM objective depends on choice of test assets. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More Complex Preferences: Scaled Consumption-Based Models Consumption-based models may be approximated: Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 ) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More Complex Preferences: Scaled Consumption-Based Models Consumption-based models may be approximated: Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 ) Example: Classic CCAPM with CRRA utility 1− γ Ct u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ at = a0 bt = b0 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More Complex Preferences: Scaled Consumption-Based Models Consumption-based models may be approximated: Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 ) Example: Classic CCAPM with CRRA utility 1− γ Ct u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ at = a0 bt = b0 Model with habit and time-varying risk aversion: Campbell and Cochrane ’99, Menzly et. al ’04 ( Ct S t ) 1 − γ C t − Xt u ( Ct , S t ) = , St + 1 ≡ 1−γ Ct ⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1 at bt Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • More Complex Preferences: Scaled Consumption-Based Models Consumption-based models may be approximated: Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 ) Example: Classic CCAPM with CRRA utility 1− γ Ct u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ at = a0 bt = b0 Model with habit and time-varying risk aversion: Campbell and Cochrane ’99, Menzly et. al ’04 ( Ct S t ) 1 − γ C t − Xt u ( Ct , S t ) = , St + 1 ≡ 1−γ Ct ⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1 at bt Proxies for time-varying risk-premia should be good proxies for time-variation in at and bt . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Mt+1 ≈ at + bt ∆ct+1 Empirical specification: Lettau and Ludvigson (2001a, 2001b): at = a0 + a1 zt , bt = b0 + b1 zt zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual) cayt related to log consumption-(aggregate) wealth ratio. cayt strong predictor of excess stock market returns Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Mt+1 ≈ at + bt ∆ct+1 Empirical specification: Lettau and Ludvigson (2001a, 2001b): at = a0 + a1 zt , bt = b0 + b1 zt zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual) cayt related to log consumption-(aggregate) wealth ratio. cayt strong predictor of excess stock market returns Other examples: including housing consumption 1 1− σ ε Ct ε −1 ε −1 ε −1 U(Ct , Ht ) = 1 Ct = χCt ε + (1 − χ) Ht ε , 1− σ pC Ct t ⇒ ln Mt+1 ≈ at + bt ∆ ln Ct+1 + dt ∆ ln St+1 , St + 1 ≡ pC Ct t + pH Ht t Lustig and Van Nieuwerburgh ’05 (incomplete markets): at = a0 + a1 zt , bt = b0 + b1 zt , dt = d0 + d1 zt zt = housing collateral ratio (measures quantity of risk sharing) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Two kinds of conditioning are often confused. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Two kinds of conditioning are often confused. Euler equation: E Mt+1 Rt+1 |zt = 1 Unconditional version: E Mt+1 Rt+1 = 1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Two kinds of conditioning are often confused. Euler equation: E Mt+1 Rt+1 |zt = 1 Unconditional version: E Mt+1 Rt+1 = 1 Two forms of conditionality: 1 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 2 scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 : ′ Mt+1 = bt ft+1 with bt = b0 + b1 zt = b′ ft+1 ⊗ (1 zt )′ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Two kinds of conditioning are often confused. Euler equation: E Mt+1 Rt+1 |zt = 1 Unconditional version: E Mt+1 Rt+1 = 1 Two forms of conditionality: 1 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 2 scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 : ′ Mt+1 = bt ft+1 with bt = b0 + b1 zt = b′ ft+1 ⊗ (1 zt )′ Scaled consumption-based models are conditional in sense that Mt+1 is a state-dependent function of ∆ ln Ct+1 ⇒ scaled factors Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Two kinds of conditioning are often confused. Euler equation: E Mt+1 Rt+1 |zt = 1 Unconditional version: E Mt+1 Rt+1 = 1 Two forms of conditionality: 1 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 2 scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 : ′ Mt+1 = bt ft+1 with bt = b0 + b1 zt = b′ ft+1 ⊗ (1 zt )′ Scaled consumption-based models are conditional in sense that Mt+1 is a state-dependent function of ∆ ln Ct+1 ⇒ scaled factors Scaled consumption-based models have been tested on unconditional moments, E Mt+1 Rt+1 = 1 ⇒ NO scaled returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled CCAPM turns a single factor model with state-dependent weights into multi-factor model ft with constant weights: Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1 = a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 ) f1,t+1 f2,t+1 f3,t+1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled CCAPM turns a single factor model with state-dependent weights into multi-factor model ft with constant weights: Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1 = a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 ) f1,t+1 f2,t+1 f3,t+1 ′ Multiple risk factors ft ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled CCAPM turns a single factor model with state-dependent weights into multi-factor model ft with constant weights: Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1 = a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 ) f1,t+1 f2,t+1 f3,t+1 ′ Multiple risk factors ft ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ). Scaled consumption models have multiple, constant betas for each factor, rather than a single time-varying beta for ∆ ln Ct+1 . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Deriving the “beta”-representation Let F = (1 f′ )′ , M = b′ F, ignore time indices 1 = E[MRi ] = E[Ri F′ ]b ⇒ unconditional moments = E[Ri ]E[F′ ]b + Cov(Ri , F′ )b ⇒ 1 − Cov(Ri , F′ )b E[Ri ] = E [F′ ]b 1 − Cov(Ri , f′ )b = E [F′ ]b 1 − Cov(Ri , f′ )Cov(f, f′ )−1 Cov(f, f′ )b = E [F′ ]b = R0 − R0 β′ Cov(f, f′ )b = R0 − β′ λ ⇒ multiple, constant betas Estimate cross-sectional model using Fama-MacBeth (see Brandt lecture). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Fama-MacBeth Methodology: Preview–See Brandt Step 1: Estimate β’s in time-series regression for each portfolio i: βi ≡ Cov(ft+1 , ft+1 )−1 Cov(ft+1 , Ri,t+1 ) ′ Step 2: Cross-sectional regressions (T of them): Ri,t+1 − R0,t = αi,t + βi′ λt T T λ = 1/T ∑ λt ; σ2 (λ) = 1/T ∑ (λt − λ)2 t= 1 t= 1 Note: report Shanken t-statistics (corrected for estimation error of betas in first stage) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled models: conditioning done in SDF: Mt+1 = at + bt ∆ ln Ct+1 , not in Euler equation: E(MR) = 1N . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled models: conditioning done in SDF: Mt+1 = at + bt ∆ ln Ct+1 , not in Euler equation: E(MR) = 1N . Gives rise to a restricted conditional consumption beta model: Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 t Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled models: conditioning done in SDF: Mt+1 = at + bt ∆ ln Ct+1 , not in Euler equation: E(MR) = 1N . Gives rise to a restricted conditional consumption beta model: Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 t Rewrite as Ri = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1 t time-varying beta Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Scaled models: conditioning done in SDF: Mt+1 = at + bt ∆ ln Ct+1 , not in Euler equation: E(MR) = 1N . Gives rise to a restricted conditional consumption beta model: Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 t Rewrite as Ri = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1 t time-varying beta Unlikely the same time-varying beta as obtained from modeling conditional mean Et (Mt+1 Rt+1 ) = 1. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Conditioning in SDF: theory provides guidance: typically a few variables that capture risk-premia. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Conditioning in SDF: theory provides guidance: typically a few variables that capture risk-premia. Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Conditioning in SDF: theory provides guidance: typically a few variables that capture risk-premia. Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ). Latter may require variables beyond a few that capture risk-premia. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Conditioning in SDF: theory provides guidance: typically a few variables that capture risk-premia. Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ). Latter may require variables beyond a few that capture risk-premia. Approximating condition mean well requires large number of instruments (misspecified information sets) Results sensitive to chosen conditioning variables, may fail to span information sets of market participants. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Distinction is important. Conditioning in SDF: theory provides guidance: typically a few variables that capture risk-premia. Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ). Latter may require variables beyond a few that capture risk-premia. Approximating condition mean well requires large number of instruments (misspecified information sets) Results sensitive to chosen conditioning variables, may fail to span information sets of market participants. Partial solution: summarize information in large number of time-series with few estimated dynamic factors (e.g., Ludvigson and Ng ’07, ’09). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Bottom lines: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Bottom lines: 1 Conditional moments of Mt+1 Rt+1 difficult to model ⇒ reason to focus on unconditional moments E[Mt+1 Rt+1 ] = 1. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Bottom lines: 1 Conditional moments of Mt+1 Rt+1 difficult to model ⇒ reason to focus on unconditional moments E[Mt+1 Rt+1 ] = 1. 2 Models are misspecified: interesting question is whether state-dependence of Mt+1 on consumption growth ⇒ less misspecification than standard, fixed-weight CCAPM. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Scaled Consumption-Based Models Distinguishing two types of conditioning, or state dependence Bottom lines: 1 Conditional moments of Mt+1 Rt+1 difficult to model ⇒ reason to focus on unconditional moments E[Mt+1 Rt+1 ] = 1. 2 Models are misspecified: interesting question is whether state-dependence of Mt+1 on consumption growth ⇒ less misspecification than standard, fixed-weight CCAPM. 3 As before, can compare models on basis of HJ distances, using White ”reality check” method to compare statistically. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Growing interest in asset pricing models with recursive preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Growing interest in asset pricing models with recursive preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW). Two reasons recursive utility is of interest: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Growing interest in asset pricing models with recursive preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW). Two reasons recursive utility is of interest: More flexibility as regards attitudes toward risk and intertemporal substitution. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Growing interest in asset pricing models with recursive preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW). Two reasons recursive utility is of interest: More flexibility as regards attitudes toward risk and intertemporal substitution. Preferences deliver an added risk factor for explaining asset returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Growing interest in asset pricing models with recursive preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW). Two reasons recursive utility is of interest: More flexibility as regards attitudes toward risk and intertemporal substitution. Preferences deliver an added risk factor for explaining asset returns. But, only a small amount of econometric work on recursive preferences ⇒ gap in the literature. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Here discuss two examples of estimating EZW models: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Here discuss two examples of estimating EZW models: 1 For general stationary, consumption growth and cash flow dynamics: Chen, Favilukis, Ludvigson ’07. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Here discuss two examples of estimating EZW models: 1 For general stationary, consumption growth and cash flow dynamics: Chen, Favilukis, Ludvigson ’07. 2 When restricting cash flow dynamics (e.g., “long-run risk”): Bansal, Gallant, Tauchen ’07. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Here discuss two examples of estimating EZW models: 1 For general stationary, consumption growth and cash flow dynamics: Chen, Favilukis, Ludvigson ’07. 2 When restricting cash flow dynamics (e.g., “long-run risk”): Bansal, Gallant, Tauchen ’07. In (1) DGP is left unrestricted, as is joint distribution of consumption and returns (distribution-free estimation procedure). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • Asset Pricing Models With Recursive Preferences Here discuss two examples of estimating EZW models: 1 For general stationary, consumption growth and cash flow dynamics: Chen, Favilukis, Ludvigson ’07. 2 When restricting cash flow dynamics (e.g., “long-run risk”): Bansal, Gallant, Tauchen ’07. In (1) DGP is left unrestricted, as is joint distribution of consumption and returns (distribution-free estimation procedure). In (2) DGP and distribution of shocks explicitly modeled. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstein-Zin-Weil basics Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)): 1 1− ρ + β R t ( Vt + 1 ) 1 − ρ 1− ρ Vt = ( 1 − β ) Ct 1 1− 1− θ Rt (Vt+1 ) = E Vt+1θ |Ft Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstein-Zin-Weil basics Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)): 1 1− ρ + β R t ( Vt + 1 ) 1 − ρ 1− ρ Vt = ( 1 − β ) Ct 1 1− 1− θ Rt (Vt+1 ) = E Vt+1θ |Ft Vt+1 is continuation value, θ is RRA, 1/ρ is EIS. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstein-Zin-Weil basics Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)): 1 1− ρ + β R t ( Vt + 1 ) 1 − ρ 1− ρ Vt = ( 1 − β ) Ct 1 1− 1− θ Rt (Vt+1 ) = E Vt+1θ |Ft Vt+1 is continuation value, θ is RRA, 1/ρ is EIS. Rescale utility function (Hansen, Heaton, Li ’05): 1 1− ρ 1− ρ Vt Vt + 1 Ct + 1 = ( 1 − β ) + β Rt Ct Ct + 1 Ct Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstein-Zin-Weil basics Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)): 1 1− ρ + β R t ( Vt + 1 ) 1 − ρ 1− ρ Vt = ( 1 − β ) Ct 1 1− 1− θ Rt (Vt+1 ) = E Vt+1θ |Ft Vt+1 is continuation value, θ is RRA, 1/ρ is EIS. Rescale utility function (Hansen, Heaton, Li ’05): 1 1− ρ 1− ρ Vt Vt + 1 Ct + 1 = ( 1 − β ) + β Rt Ct Ct + 1 Ct C1−θ Special case: ρ = θ ⇒ CRRA separable utility Vt = β 1−θ . t Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics The MRS is pricing kernel (SDF) with added risk factor:   ρ−θ Vt+1 Ct+1 Ct + 1 − ρ  Ct+1 Ct  Mt + 1 = β Ct R Vt+1 Ct+1 t Ct+1 Ct Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics The MRS is pricing kernel (SDF) with added risk factor:   ρ−θ Vt+1 Ct+1 Ct + 1 − ρ  Ct+1 Ct  Mt + 1 = β Ct R Vt+1 Ct+1 t Ct+1 Ct Difficulty: MRS a function of V/C, unobservable, embeds Rt (·). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics The MRS is pricing kernel (SDF) with added risk factor:   ρ−θ Vt+1 Ct+1 Ct + 1 − ρ  Ct+1 Ct  Mt + 1 = β Ct R Vt+1 Ct+1 t Ct+1 Ct Difficulty: MRS a function of V/C, unobservable, embeds Rt (·). Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth return Rw,t : 1− θ θ −ρ −ρ 1− ρ Ct + 1 1 1− ρ Mt + 1 = β Ct Rw,t+1 where Rw,t proxied by stock market return. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics The MRS is pricing kernel (SDF) with added risk factor:   ρ−θ Vt+1 Ct+1 Ct + 1 − ρ  Ct+1 Ct  Mt + 1 = β Ct R Vt+1 Ct+1 t Ct+1 Ct Difficulty: MRS a function of V/C, unobservable, embeds Rt (·). Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth return Rw,t : 1− θ θ −ρ −ρ 1− ρ Ct + 1 1 1− ρ Mt + 1 = β Ct Rw,t+1 where Rw,t proxied by stock market return. Problem: Rw,t+1 represents a claim to future Ct , itself unobservable. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. 2 If returns, Ct are jointly lognormal and homoscedastic, risk premia are approx. log-linear functions of COV between returns, and news about current and future Ct growth. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. 2 If returns, Ct are jointly lognormal and homoscedastic, risk premia are approx. log-linear functions of COV between returns, and news about current and future Ct growth. But.... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. 2 If returns, Ct are jointly lognormal and homoscedastic, risk premia are approx. log-linear functions of COV between returns, and news about current and future Ct growth. But.... EIS=1 ⇒ consumption-wealth ratio is constant, contradicting statistical evidence. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. 2 If returns, Ct are jointly lognormal and homoscedastic, risk premia are approx. log-linear functions of COV between returns, and news about current and future Ct growth. But.... EIS=1 ⇒ consumption-wealth ratio is constant, contradicting statistical evidence. Joint lognormality strongly rejected in quarterly data. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
    • EZW Recursive Preferences Epstien-Zin-Weil basics 1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series process, log(V/C) has an analytical solution. 2 If returns, Ct are jointly lognormal and homoscedastic, risk premia are approx. log-linear functions of COV between returns, and news about current and future Ct growth. But.... EIS=1 ⇒ consumption-wealth ratio is constant, contradicting statistical evidence. Joint lognormality strongly rejected in quarterly data. Points to need for estimation method feasible under less restrictive assumptions. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models