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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 | {z } (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 | {z } (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 | {z } (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 | {z } (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 ) , | {z } 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 ) , | {z } t= 1 (r × 1 ) yT ≡ wT′ , wT′ −1 , ...w1′ Sydney C. Ludvigson ′ is a T · h × 1 vector of observations. Methods Lecture: GMM and Consumption-Based Models GMM Review (Hansen, 1982) Sample moments: T g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) , | {z } t= 1 (r × 1 ) yT ≡ wT′ , wT′ −1 , ...w1′ ′ is a T · h × 1 vector of observations. b minimizes the scalar The GMM estimator θ ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], ( 1× r ) (r×r) (1) ( 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 ) , | {z } t= 1 (r × 1 ) yT ≡ wT′ , wT′ −1 , ...w1′ ′ is a T · h × 1 vector of observations. b minimizes the scalar The GMM estimator θ ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], ( 1× r ) (r×r) (1) ( 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 ) , | {z } t= 1 (r × 1 ) yT ≡ wT′ , wT′ −1 , ...w1′ ′ is a T · h × 1 vector of observations. b minimizes the scalar The GMM estimator θ ′ Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], ( 1× r ) (r×r) (1) ( 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): b θ consistent, asym. normal. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models GMM Review (Hansen, 1982) Asym. properties (Hansen 1982): b θ consistent, asym. normal. Optimal weighting WT = S−1 ∞ S = r×r ∑ E j=− ∞ Sydney C. Ludvigson ′ . [h (θo , wt )] h θo , wt−j {z } | {z } | r×1 1× r Methods Lecture: GMM and Consumption-Based Models GMM Review (Hansen, 1982) Asym. properties (Hansen 1982): b θ consistent, asym. normal. Optimal weighting WT = S−1 ∞ S = r×r ∑ E j=− ∞ ′ . [h (θo , wt )] h θo , wt−j {z } | {z } | 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) b T . Employ an b T depends on b θT which depends on S S iterative procedure: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models GMM Review (Hansen, 1982) b T . Employ an b T depends on b θT which depends on S S iterative procedure: 1 2 3 4 (1) Obtain an initial estimate of θ = b θT , by minimizing Q (θ; yT ) subject to arbitrary weighting matrix, e.g., W = I. (1) (1) b . Use b θT to obtain initial estimate of S = S T (1 ) b ; obtain Re-minimize Q (θ; yT ) using initial estimate S T (2) b new estimate θT . 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 Sydney C. Ludvigson " ∞ ∑ β i u (C t + i ) i= 0 # 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 u (C t ) = 1− γ Ct 1− γ γ>0 (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 u (C t ) = 1− γ Ct 1− γ γ>0 (2) u (Ct ) = ln(Ct ) γ = 1 N assets => N first-order conditions n o −γ −γ Ct = βEt (1 + ℜi,t+1 ) Ct+1 Sydney C. Ludvigson i = 1, ..., N. (3) Methods Lecture: GMM and Consumption-Based Models Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions ( " 0 = Et 1 − β (1 + ℜi,t+1 ) Sydney C. Ludvigson −γ Ct + 1 −γ Ct #) . (4) Methods Lecture: GMM and Consumption-Based Models Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions ( " 0 = Et 1 − β (1 + ℜi,t+1 ) −γ Ct + 1 −γ Ct #) . (4) 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 ( " 0 = Et 1 − β (1 + ℜi,t+1 ) −γ Ct + 1 −γ Ct #) . (4) 2 params to estimate: β and γ, so θ = ( β, γ) ′ . xt∗ denotes info set of investors 0=E nh n oi o −γ −γ 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt∗ Sydney C. Ludvigson i = 1, ...N (5) Methods Lecture: GMM and Consumption-Based Models Asset Pricing Example: Hansen and Singleton (1982) Re-write moment conditions ( " 0 = Et 1 − β (1 + ℜi,t+1 ) −γ Ct + 1 −γ Ct #) . (4) 2 params to estimate: β and γ, so θ = ( β, γ) ′ . xt∗ denotes info set of investors 0=E nh n oi o −γ −γ 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt∗ i = 1, ...N (5) ∗ xt ⊂ xt . Conditional model (5) => unconditional model: 0=E (" 1− ( β (1 + ℜi,t+1 ) −γ Ct + 1 −γ Ct )# ) xt i = 1, ...N (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 Ct+1 xt 1 − β (1 + ℜ1,t+1 ) Ct−γ − γ 1 − β (1 + ℜ2,t+1 ) Ct−+γ1 xt Ct · h (θ, wt ) = r×1 · · −γ Ct+1 1 − β (1 + ℜN,t+1 ) −γ xt Ct Sydney C. Ludvigson (7) 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 Ct+1 xt 1 − β (1 + ℜ1,t+1 ) Ct−γ − γ 1 − β (1 + ℜ2,t+1 ) Ct−+γ1 xt Ct · h (θ, wt ) = r×1 · · −γ Ct+1 1 − β (1 + ℜN,t+1 ) −γ xt Ct (7) 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 Ct+1 xt 1 − β (1 + ℜ1,t+1 ) Ct−γ − γ 1 − β (1 + ℜ2,t+1 ) Ct−+γ1 xt Ct · h (θ, wt ) = r×1 · · −γ Ct+1 1 − β (1 + ℜN,t+1 ) −γ xt Ct (7) 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 | {z } r×r | {z } 1× r Sydney C. Ludvigson r×1 Methods Lecture: GMM and Consumption-Based Models Asset Pricing Example: Hansen and Singleton (1982) Test of Over-identifying (OID) restrictions: a TQ b 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 b 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 b 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 b 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 b 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 b 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 gj,t (γ, β) = gj,t (γ) = Sydney C. Ludvigson 1 T 1 T j ∑Tt=1 eR,t j ∑Tt=1 eX,t 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 )] RMSE = q 1 N j 2 ∑N j=1 [eX ] , RMSR = q j = 1, ..., N 1 N j f 2 ∑N j=1 [E(Rt+1 − Rt+1 )] 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 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected (2) 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 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected (2) 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 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected (2) 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 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected (2) 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: DistT (θ) = r mingT (θ)′ GT−1 gT (θ), θ 1 gT (θ) ≡ T Sydney C. Ludvigson T GT ≡ ∑ [Mt (θ)Rt − 1N ] 1 T T ′ t Rt ∑ R|{z} t= 1 N ×N t= 1 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: DistT (θ) = r mingT (θ)′ GT−1 gT (θ), θ 1 gT (θ) ≡ T T GT ≡ ∑ [Mt (θ)Rt − 1N ] 1 T T ′ t Rt ∑ R|{z} t= 1 N ×N 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: DistT (θ) = r mingT (θ)′ GT−1 gT (θ), θ 1 gT (θ) ≡ T GT ≡ T ∑ [Mt (θ)Rt − 1N ] 1 T T ′ t Rt ∑ R|{z} t= 1 N ×N 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: DistT (θ) = r mingT (θ)′ GT−1 gT (θ), θ 1 gT (θ) ≡ T GT ≡ T ∑ [Mt (θ)Rt − 1N ] 1 T T ′ t Rt ∑ R|{z} t= 1 N ×N 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: DistT (θ) = r mingT (θ)′ GT−1 gT (θ), θ 1 gT (θ) ≡ T GT ≡ T ∑ [Mt (θ)Rt − 1N ] 1 T T ′ t Rt ∑ R|{z} t= 1 N ×N 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 2 3 4 5 Take benchmark model, e.g., model with smallest squared distance d21,T ≡ min{d2j,T }K j =1 . Null: d21,T − d22,T ≤ 0, where d22,T is competing model with the next smallest squared distance. √ Test statistic T W = T (d21,T − d22,T ). If null is true, test statistic should not be unusually large, given sampling error. Given distribution for T W , reject null if historical value Tb 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 2 3 4 5 Take benchmark model, e.g., model with smallest squared distance d21,T ≡ min{d2j,T }K j =1 . Null: d21,T − d22,T ≤ 0, where d22,T is competing model with the next smallest squared distance. √ Test statistic T W = T (d21,T − d22,T ). If null is true, test statistic should not be unusually large, given sampling error. Given distribution for T W , reject null if historical value Tb 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 ). [E (MR) − 1N ]′ E RR′ −1 [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 ). [E (MR) − 1N ]′ E RR′ −1 [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 , Sydney C. Ludvigson ct+1 ≡ ln(Ct+1 ) 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 u ( Ct ) = 1− γ Ct ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ |{z} |{z} at = a0 Sydney C. Ludvigson bt = b0 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 u ( Ct ) = 1− γ Ct ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ |{z} |{z} at = a0 bt = b0 Model with habit and time-varying risk aversion: Campbell and Cochrane ’99, Menzly et. al ’04 u ( Ct , S t ) = ( Ct S t ) 1 − γ , 1−γ St + 1 ≡ C t − Xt Ct ⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1 | {z } | {z } at Sydney C. Ludvigson bt 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 u ( Ct ) = 1− γ Ct ⇒ Mt+1 ≈ β − βγ ∆ct+1 1−γ |{z} |{z} at = a0 bt = b0 Model with habit and time-varying risk aversion: Campbell and Cochrane ’99, Menzly et. al ’04 u ( Ct , S t ) = ( Ct S t ) 1 − γ , 1−γ St + 1 ≡ C t − Xt Ct ⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1 | {z } | {z } 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 U(Ct , Ht ) = e 1− σ C t 1− 1 σ ε −1 ε ε −1 ε −1 ε ε e Ct = χCt + (1 − χ) Ht , ⇒ ln Mt+1 ≈ at + bt ∆ ln Ct+1 + dt ∆ ln St+1 , St + 1 ≡ pC t Ct pC t Ct + pH t Ht 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 2 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 : Mt+1 Sydney C. Ludvigson = bt′ ft+1 with bt = b0 + b1 zt = b′ ft+1 ⊗ (1 zt )′ 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 2 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 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 2 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1 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-basedmodels 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 ) |{z} | {z } | {z } f1,t+1 Sydney C. Ludvigson f2,t+1 f3,t+1 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 ) |{z} | {z } | {z } 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 ) |{z} | {z } | {z } 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 = E[MRi ] 1 = E[Ri F′ ]b ⇒ unconditional moments = E[Ri ]E[F′ ]b + Cov(Ri , F′ )b ⇒ E[Ri ] = 1 − Cov(Ri , F′ )b 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 λ = 1/T ∑ λt ; t= 1 T σ2 (λ) = 1/T ∑ (λt − λ)2 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: Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 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: Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 Rewrite as Rit = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1 | {z } 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: Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1 Rewrite as Rit = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1 | {z } 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)): h i 1−1 ρ 1− ρ Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ h i 1−1 θ Rt (Vt+1 ) = E Vt1+−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)): h i 1−1 ρ 1− ρ Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ h i 1−1 θ Rt (Vt+1 ) = E Vt1+−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)): h i 1−1 ρ 1− ρ Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ h i 1−1 θ Rt (Vt+1 ) = E Vt1+−1θ |Ft Vt+1 is continuation value, θ is RRA, 1/ρ is EIS. Rescale utility function (Hansen, Heaton, Li ’05): " # 1−1 ρ Vt Vt + 1 Ct + 1 1 − ρ = ( 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)): h i 1−1 ρ 1− ρ Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ h i 1−1 θ Rt (Vt+1 ) = E Vt1+−1θ |Ft Vt+1 is continuation value, θ is RRA, 1/ρ is EIS. Rescale utility function (Hansen, Heaton, Li ’05): " # 1−1 ρ Vt Vt + 1 Ct + 1 1 − ρ = ( 1 − β ) + β Rt Ct Ct + 1 Ct C1−θ t Special case: ρ = θ ⇒ CRRA separable utility Vt = β 1− θ. 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 C C t+1 t Mt + 1 = β Vt+1 Ct+1 Ct R t Sydney C. Ludvigson Ct+1 Ct 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 C C t+1 t Mt + 1 = β Vt+1 Ct+1 Ct R 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 C C t+1 t Mt + 1 = β Vt+1 Ct+1 Ct R 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 : ( ) 11−−ρθ 1θ −−ρρ 1 Ct + 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 C C t+1 t Mt + 1 = β Vt+1 Ct+1 Ct R 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 : ( ) 11−−ρθ 1θ −−ρρ 1 Ct + 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 EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Loglinearizing the model. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Loglinearizing the model. Parametric restrictions on law of motion or joint dist. of Ct and Ri,t , or on value of key preference parameters. Obtain estimates of β, RRA θ, EIS ρ−1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Loglinearizing the model. Parametric restrictions on law of motion or joint dist. of Ct and Ri,t , or on value of key preference parameters. Obtain estimates of β, RRA θ, EIS ρ−1 Evaluate EZW model’s ability to fit asset return data relative to competing model specifications. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Loglinearizing the model. Parametric restrictions on law of motion or joint dist. of Ct and Ri,t , or on value of key preference parameters. Obtain estimates of β, RRA θ, EIS ρ−1 Evaluate EZW model’s ability to fit asset return data relative to competing model specifications. Investigate implications for Rw,t+1 and return to human wealth. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 CFL: semiparametric approach to estimate EZW model without: Need to proxy Rw,t+1 with observable returns. Loglinearizing the model. Parametric restrictions on law of motion or joint dist. of Ct and Ri,t , or on value of key preference parameters. Obtain estimates of β, RRA θ, EIS ρ−1 Evaluate EZW model’s ability to fit asset return data relative to competing model specifications. Investigate implications for Rw,t+1 and return to human wealth. Semiparametric approach is sieve minimum distance (SMD) procedure. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First order conditions for optimal consumption choice: ρ−θ −ρ Vt + 1 C t + 1 Ct + 1 Ct + 1 Ct Et β Ri,t+1 − 1 = 0 Vt + 1 C t + 1 Ct R t Ct + 1 Ct Sydney C. Ludvigson (8) Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First order conditions for optimal consumption choice: ρ−θ −ρ Vt + 1 C t + 1 Ct + 1 Ct + 1 Ct Et β Ri,t+1 − 1 = 0 Vt + 1 C t + 1 Ct R t Ct + 1 Ct CFL: plug Et β Ct+1 Ct Vt Ct (8) 1−ρ 1−1 ρ t + 1 Ct + 1 = ( 1 − β ) + β Rt V into (8): Ct + 1 Ct − ρ n h 1 β ρ−θ Vt+1 Ct+1 Ct+1 Ct Vt 1 − ρ Ct Sydney C. Ludvigson − (1 − β ) io 1−1 ρ Ri,t+1 − 1 =0 i = 1, ..., N. (9) Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First order conditions for optimal consumption choice: ρ−θ −ρ Vt + 1 C t + 1 Ct + 1 Ct + 1 Ct Et β Ri,t+1 − 1 = 0 Vt + 1 C t + 1 Ct R t Ct + 1 Ct CFL: plug Et β Ct+1 Ct Vt Ct (8) 1−ρ 1−1 ρ t + 1 Ct + 1 = ( 1 − β ) + β Rt V into (8): Ct + 1 Ct − ρ n h 1 β ρ−θ Vt+1 Ct+1 Ct+1 Ct Vt 1 − ρ Ct − (1 − β ) io 1−1 ρ Ri,t+1 − 1 =0 i = 1, ..., N. (9) N test asset returns, {Ri,t+1 }N i=1 . (9) is a x-sect asset pricing model. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First order conditions for optimal consumption choice: ρ−θ −ρ Vt + 1 C t + 1 Ct + 1 Ct + 1 Ct Et β Ri,t+1 − 1 = 0 Vt + 1 C t + 1 Ct R t Ct + 1 Ct CFL: plug Et β Ct+1 Ct Vt Ct (8) 1−ρ 1−1 ρ t + 1 Ct + 1 = ( 1 − β ) + β Rt V into (8): Ct + 1 Ct − ρ n h 1 β ρ−θ Vt+1 Ct+1 Ct+1 Ct Vt 1 − ρ Ct − (1 − β ) io 1−1 ρ Ri,t+1 − 1 =0 i = 1, ..., N. (9) N test asset returns, {Ri,t+1 }N i=1 . (9) is a x-sect asset pricing model. Moment restrictions (9) form the basis of empirical investigation. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First order conditions for optimal consumption choice: ρ−θ −ρ Vt + 1 C t + 1 Ct + 1 Ct + 1 Ct Et β Ri,t+1 − 1 = 0 Vt + 1 C t + 1 Ct R t Ct + 1 Ct CFL: plug Et β Ct+1 Ct Vt Ct (8) 1−ρ 1−1 ρ t + 1 Ct + 1 = ( 1 − β ) + β Rt V into (8): Ct + 1 Ct − ρ n h 1 β ρ−θ Vt+1 Ct+1 Ct+1 Ct Vt 1 − ρ Ct − (1 − β ) io 1−1 ρ Ri,t+1 − 1 =0 i = 1, ..., N. (9) N test asset returns, {Ri,t+1 }N i=1 . (9) is a x-sect asset pricing model. Moment restrictions (9) form the basis of empirical investigation. Empirical model is semiparametric: δ ≡ ( β, θ, ρ)′ denote finite dimensional parameter vector; Vt /Ct unknown function. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic; and F(·) is such that the process{Vt /Ct : t = 1, ...} is asymptotically stationary ergodic. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic; and F(·) is such that the process{Vt /Ct : t = 1, ...} is asymptotically stationary ergodic. Justified if, for example, Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic; and F(·) is such that the process{Vt /Ct : t = 1, ...} is asymptotically stationary ergodic. Justified if, for example, ∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden first-order Markov process xt . Under general assumptions, information in xt is summarized by Vt−1 /Ct−1 and Ct /Ct−1 . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic; and F(·) is such that the process{Vt /Ct : t = 1, ...} is asymptotically stationary ergodic. Justified if, for example, ∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden first-order Markov process xt . Under general assumptions, information in xt is summarized by Vt−1 /Ct−1 and Ct /Ct−1 . With a nonlinear Markov process for xt , F(·) can display nonmonotonicities in both arguments. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Assume Vt Ct an unknown function F: R2 → R of form Vt Vt − 1 Ct =F , , Ct Ct − 1 Ct − 1 Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic; and F(·) is such that the process{Vt /Ct : t = 1, ...} is asymptotically stationary ergodic. Justified if, for example, ∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden first-order Markov process xt . Under general assumptions, information in xt is summarized by Vt−1 /Ct−1 and Ct /Ct−1 . Note: Markov assumption only a motivation for arguments of F(·). Econometric methodology itself leaves LOM for ∆ ln Ct unspecified. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Ft denotes agents information set at time t. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Ft denotes agents information set at time t. zt+1 contains all observations at t + 1 and γi (zt+1 , δ, F) ≡ β Ct+1 − ρ t Ct+ 1 F V Ct , Ct+1 Ct Ct+1 1 n o1−ρ Ct 1− ρ Vt − 1 Ct 1 F , − 1 − β ) ( β C C t−1 Sydney C. Ludvigson t−1 ρ−θ Ri,t+1 − 1 Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Ft denotes agents information set at time t. zt+1 contains all observations at t + 1 and γi (zt+1 , δ, F) ≡ β Ct+1 − ρ t Ct+ 1 F V Ct , Ct+1 Ct Ct+1 1 n o1−ρ Ct 1− ρ Vt − 1 Ct 1 F , − 1 − β ) ( β C C t−1 t−1 ρ−θ Ri,t+1 − 1 δo ≡ ( βo , θo , ρo )′ , Fo ≡ Fo (zt , δo ) denote true parameters that uniquely solve the conditional moment restrictions (Euler equations): E {γi (zt+1 , δo , Fo (·, δo ))|Ft } = 0 Sydney C. Ludvigson i = 1, ..., N, (10) Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Ft denotes agents information set at time t. zt+1 contains all observations at t + 1 and γi (zt+1 , δ, F) ≡ β Ct+1 − ρ t Ct+ 1 F V Ct , Ct+1 Ct Ct+1 1 n o1−ρ Ct 1− ρ Vt − 1 Ct 1 F , − 1 − β ) ( β C C t−1 t−1 ρ−θ Ri,t+1 − 1 δo ≡ ( βo , θo , ρo )′ , Fo ≡ Fo (zt , δo ) denote true parameters that uniquely solve the conditional moment restrictions (Euler equations): E {γi (zt+1 , δo , Fo (·, δo ))|Ft } = 0 i = 1, ..., N, (10) Let wt ⊆ Ft . Equation (10) ⇒ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. Sydney C. Ludvigson i = 1, ..., N. Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. Sydney C. Ludvigson i = 1, ..., N. (11) Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N. (11) Since mt = 0, mt must have zero variance, mean. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N. (11) Since mt = 0, mt must have zero variance, mean. Thus can find params by minimizing variance or quadratic b t. norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N. (11) Since mt = 0, mt must have zero variance, mean. Thus can find params by minimizing variance or quadratic b t. norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m Since (11) is cond. mean, must hold for each observation, t. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N. (11) Since mt = 0, mt must have zero variance, mean. Thus can find params by minimizing variance or quadratic b t. norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m Since (11) is cond. mean, must hold for each observation, t. Obs > params, need way to weight each obs; using sample mean is one way: min ET [(mt )2 ]. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Intuition behind minimum distance procedure: Theory ⇒ mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N. (11) Since mt = 0, mt must have zero variance, mean. Thus can find params by minimizing variance or quadratic b t. norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m Since (11) is cond. mean, must hold for each observation, t. Obs > params, need way to weight each obs; using sample mean is one way: min ET [(mt )2 ]. Minimum distance procedure useful for distribution-free estimation involving conditional moments. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Minimum distance procedure useful for distribution-free estimation involving conditional moments: min ET [(mt )2 ]. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Minimum distance procedure useful for distribution-free estimation involving conditional moments: min ET [(mt )2 ]. Contrast with GMM, used for unconditional moments: E[f (xt , α)] = 0. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Minimum distance procedure useful for distribution-free estimation involving conditional moments: min ET [(mt )2 ]. Contrast with GMM, used for unconditional moments: E[f (xt , α)] = 0. With GMM take sample counterpart to population mean: gT = ∑Tt=1 f (xt , α) = 0. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Minimum distance procedure useful for distribution-free estimation involving conditional moments: min ET [(mt )2 ]. Contrast with GMM, used for unconditional moments: E[f (xt , α)] = 0. With GMM take sample counterpart to population mean: gT = ∑Tt=1 f (xt , α) = 0. Then choose parameters α to min gT′ WgT . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Minimum distance procedure useful for distribution-free estimation involving conditional moments: min ET [(mt )2 ]. Contrast with GMM, used for unconditional moments: E[f (xt , α)] = 0. With GMM take sample counterpart to population mean: gT = ∑Tt=1 f (xt , α) = 0. Then choose parameters α to min gT′ WgT . With GMM we average and then square. With SMD, we square and then average. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 True parameters δo and Fo (·, δo ) solve: min inf E m(wt , δ, F)′ m(wt , δ, F) , δ∈D F∈V where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt } γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F)) Sydney C. Ludvigson ′ Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 True parameters δo and Fo (·, δo ) solve: min inf E m(wt , δ, F)′ m(wt , δ, F) , δ∈D F∈V where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt } γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F)) ′ For any candidate δ ≡ ( β, θ, ρ) ′ ∈ D , define V ∗ ≡ F∗ (zt , δ) ≡ F∗ (·, δ) as: F∗ (·, δ) = arg infE m(wt , δ, F)′ m(wt , δ, F) F∈V Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 True parameters δo and Fo (·, δo ) solve: min inf E m(wt , δ, F)′ m(wt , δ, F) , δ∈D F∈V where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt } γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F)) ′ For any candidate δ ≡ ( β, θ, ρ) ′ ∈ D , define V ∗ ≡ F∗ (zt , δ) ≡ F∗ (·, δ) as: F∗ (·, δ) = arg infE m(wt , δ, F)′ m(wt , δ, F) F∈V It is clear that Fo (zt , δo ) = F∗ (zt , δo ) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two-Step Procedure Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First step: For any candidate δ ∈ D , an initial estimate of F∗ (·, δ) obtained using SMD that consists of two parts: (Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two-Step Procedure Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First step: For any candidate δ ∈ D , an initial estimate of F∗ (·, δ) obtained using SMD that consists of two parts: (Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07). 1 Replace the conditional expectation with a consistent, nonparametric estimator (specified later). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two-Step Procedure Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First step: For any candidate δ ∈ D , an initial estimate of F∗ (·, δ) obtained using SMD that consists of two parts: (Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07). 1 Replace the conditional expectation with a consistent, nonparametric estimator (specified later). 2 Approximate the unknown function F by a sequence of finite dimensional unknown parameters (sieves) FKT . Approximation error decreases as KT increases with T. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two-Step Procedure Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 First step: For any candidate δ ∈ D , an initial estimate of F∗ (·, δ) obtained using SMD that consists of two parts: (Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07). 1 Replace the conditional expectation with a consistent, nonparametric estimator (specified later). 2 Approximate the unknown function F by a sequence of finite dimensional unknown parameters (sieves) FKT . Approximation error decreases as KT increases with T. Second step: estimates of δo is obtained by solving a sample minimum distance problem such as GMM. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Approximate F Vt Ct Vt − 1 Ct , ;δ Ct − 1 Ct − 1 =F Vt−1 Ct Ct−1 , Ct−1 ; δ with a bivariate sieve: KT ≈ FKT (·, δ) = a0 (δ) + ∑ aj (δ)Bj j= 1 Vt − 1 Ct , Ct − 1 Ct − 1 Sieve coefficients {a0 , a1 , ..., aKT } depend on δ Basis functions {Bj (·, ·) : j = 1, ..., KT } have known functional forms independent of δ V0 t Initial value for V Ct at time t = 0, denoted C0 , taken as a unknown scalar parameter to be estimated. KT KT 0 Given V , aj j=1 , Bj j=1 and data on consumption C0 oT n n oT Vi Ct , use F to generate a sequence . KT Ct−1 Ci t= 1 Sydney C. Ludvigson i= 1 Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Recall m(wt , δo , F∗ (·, δo )) ≡ E {γ(zt+1 , δo , F∗ (·, δo ))|wt } = 0. First-step SMD estimate b F (·) for F∗ (·) based on 1 b F (·, δ) = arg min T FK T T ∑ mb (wt , δ, FK T t= 1 b (wt , δ, FKT (·, δ)), (·, δ))′ m b (wt , δ, FKT (·, δ)) any nonpara. estimator of m. m Do this for a three dimensional grid of values of δ = ( β, θ, ρ)′ . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Example of nonparametric estimator of m: Let p0j (wt ), j = 1, 2, ..., JT , Rdw → R be instruments. pJT (·) ≡ (p01 (·) , ..., p0JT (·))′ ′ Define T × JT matrix P ≡ pJT (w1 ) , ..., pJT (wT ) . Then: ! T b (w, δ, F) = m ∑ γ(zt+1, δ, F)pJ T (wt )′ (P′ P)−1 pJT (w) t= 1 b (·) a sieve LS estimator of m(w, δ, F). m Procedure equivalent to regressing each γi on instruments and taking fitted values as estimate of conditional mean. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 b (·) a sieve LS estimator of m(w, δ, F). m Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 b (·) a sieve LS estimator of m(w, δ, F). m Attractive feature of this estimator of F∗ : implemented as GMM h i′ i −1 h b FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P } gT (δ,FT ; yT ) , | {z } FT ∈VT W where yT = zT′ +1, ...z2′ , wT′ , ...w1′ gT (δ,FT ; yT ) = Sydney C. Ludvigson 1 T T ′ (12) denotes vector of all obs and ∑ γ(zt+1, δ,FT )⊗pJT (wt ) (13) t=1 Methods Lecture: GMM and Consumption-Based Models EZW Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 b (·) a sieve LS estimator of m(w, δ, F). m Attractive feature of this estimator of F∗ : implemented as GMM h i′ i −1 h b FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P } gT (δ,FT ; yT ) , | {z } FT ∈VT W where yT = zT′ +1, ...z2′ , wT′ , ...w1′ gT (δ,FT ; yT ) = 1 T T ′ (12) denotes vector of all obs and ∑ γ(zt+1, δ,FT )⊗pJT (wt ) (13) t=1 Weighting gives greater weight to moments more highly correlated with instruments pJT (·). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: First Step SMD Est of F∗ Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 b (·) a sieve LS estimator of m(w, δ, F). m Attractive feature of this estimator of F∗ : implemented as GMM h i′ i −1 h b FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P } gT (δ,FT ; yT ) , | {z } FT ∈VT W where yT = zT′ +1, ...z2′ , wT′ , ...w1′ gT (δ,FT ; yT ) = 1 T T ′ (12) denotes vector of all obs and ∑ γ(zt+1, δ,FT )⊗pJT (wt ) (13) t=1 Weighting gives greater weight to moments more highly correlated with instruments pJT (·). Weighting can be understood intuitively by noting that variation in conditional mean m(wt , δ, F) is what identifies F∗ (·, δ). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sydney C. Ludvigson i = 1, ..., N. Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Regardless the model is correctly or incorrectly specified, estimate δ by minimizing GMM objective: i′ h i h F (·, δ) ; yT ) W gT (δ, b F (·, δ) ; yT ) δb = arg min gT (δ, b δ ∈D Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Regardless the model is correctly or incorrectly specified, estimate δ by minimizing GMM objective: i′ h i h F (·, δ) ; yT ) W gT (δ, b F (·, δ) ; yT ) δb = arg min gT (δ, b δ ∈D Examples: W = I, W = GT−1 . Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Regardless the model is correctly or incorrectly specified, estimate δ by minimizing GMM objective: i′ h i h F (·, δ) ; yT ) W gT (δ, b F (·, δ) ; yT ) δb = arg min gT (δ, b δ ∈D Examples: W = I, W = GT−1 . b F (·, δ) not held fixed in this step: depends on δ! Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Regardless the model is correctly or incorrectly specified, estimate δ by minimizing GMM objective: i′ h i h F (·, δ) ; yT ) W gT (δ, b F (·, δ) ; yT ) δb = arg min gT (δ, b δ ∈D Examples: W = I, W = GT−1 . b F (·, δ) not held fixed in this step: depends on δ! b (·, δ) obtained using min. dist over a grid of values δ. Estimator F Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences: Second Step GMM Est of δo Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Under correct specification, δo satisfies : E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, Sample moments: b (·, δ); yT ) ≡ gT (δ, F 1 T i = 1, ..., N. b (·, δ)) ⊗ xt . ∑Tt=1 γ(zt+1 , δ, F Regardless the model is correctly or incorrectly specified, estimate δ by minimizing GMM objective: i′ h i h F (·, δ) ; yT ) W gT (δ, b F (·, δ) ; yT ) δb = arg min gT (δ, b δ ∈D Examples: W = I, W = GT−1 . b F (·, δ) not held fixed in this step: depends on δ! b (·, δ) obtained using min. dist over a grid of values δ. Estimator F Choose the δ and corresponding b F (·, δ) that minimizes GMM criterion. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Why estimate in two steps? All params could be estimated in one step by minimizing the SMD criterion. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Why estimate in two steps? All params could be estimated in one step by minimizing the SMD criterion. Less desirable for asset pricing: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Why estimate in two steps? All params could be estimated in one step by minimizing the SMD criterion. Less desirable for asset pricing: 1 Want estimates of RRA and EIS to reflect values required to match unconditional risk premia. Not possible using SMD which emphasizes conditional moments. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Why estimate in two steps? All params could be estimated in one step by minimizing the SMD criterion. Less desirable for asset pricing: 1 Want estimates of RRA and EIS to reflect values required to match unconditional risk premia. Not possible using SMD which emphasizes conditional moments. 2 SMD procedure effectively changes set of test assets–linear combinations of original portfolio returns. But we may be interested in explaining original returns! Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Why estimate in two steps? All params could be estimated in one step by minimizing the SMD criterion. Less desirable for asset pricing: 1 Want estimates of RRA and EIS to reflect values required to match unconditional risk premia. Not possible using SMD which emphasizes conditional moments. 2 SMD procedure effectively changes set of test assets–linear combinations of original portfolio returns. But we may be interested in explaining original returns! 3 Linear combinations may imply implausible long and short positions, do not necessarily deliver a large spread in unconditional mean returns. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Euler equation need not hold with equality. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Euler equation need not hold with equality. As before, compare models by relative magnitude of misspecification, rather than... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Euler equation need not hold with equality. As before, compare models by relative magnitude of misspecification, rather than... ...asking whether each model individually fits data perfectly (given sampling error). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Euler equation need not hold with equality. As before, compare models by relative magnitude of misspecification, rather than... ...asking whether each model individually fits data perfectly (given sampling error). Use W = G−1 in second step, compute HJ distance. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Recursive Preferences: Two Step Estimation Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07 Procedure allows for model misspecification: Euler equation need not hold with equality. As before, compare models by relative magnitude of misspecification, rather than... ...asking whether each model individually fits data perfectly (given sampling error). Use W = G−1 in second step, compute HJ distance. Test whether HJ distances of competing models are statistically different (White reality check–Chen and Ludvigson ’09). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Bansal, Gallant, Tauchen ’07: SMM estimation of LRR model: Bansal & Yaron ’04. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Bansal, Gallant, Tauchen ’07: SMM estimation of LRR model: Bansal & Yaron ’04. Structural estimation of EZW utility, restricting to specific law of motion for cash flows (“long-run risk”LRR). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Bansal, Gallant, Tauchen ’07: SMM estimation of LRR model: Bansal & Yaron ’04. Structural estimation of EZW utility, restricting to specific law of motion for cash flows (“long-run risk”LRR). Cash flow dynamics in BGT version of LRR model: ∆ct+1 = µc + xc,t + σt ε c,t+1 ∆dt+1 = µd + φx xc,t + φs st + σε d σt ε d,t+1 |{z} LR risk xc,t = φxc,t−1 + σε x σε xc,t σt2 = σ2 + ν(σt2−1 − σ2 ) + σw wt st = (µd − µc ) + dt − ct ε c,t+1 , ε d,t+1 , ε xc,t , wt ∼ N.i.i.d (0, 1) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps 1 Solve the model over grid of values of deep parameters: ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps 1 Solve the model over grid of values of deep parameters: ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′ 2 For each value of ρd on the grid, combine solutions with long simulation of length N of model. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps 1 Solve the model over grid of values of deep parameters: ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′ 2 For each value of ρd on the grid, combine solutions with long simulation of length N of model. 3 Simulation: Monte Carlo draws from the Normal distribution for primitive shocks. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps 1 Solve the model over grid of values of deep parameters: ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′ 2 For each value of ρd on the grid, combine solutions with long simulation of length N of model. 3 Simulation: Monte Carlo draws from the Normal distribution for primitive shocks. 4 Form obs eqn for simulated and historical data, e.g., yt = (dt − ct , ct − ct−1 , pt − dt , rd,t )′ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh, Tauchen ’97, Tauchen ’97. Outline of SMM steps 1 Solve the model over grid of values of deep parameters: ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′ 2 For each value of ρd on the grid, combine solutions with long simulation of length N of model. 3 Simulation: Monte Carlo draws from the Normal distribution for primitive shocks. 4 Form obs eqn for simulated and historical data, e.g., yt = (dt − ct , ct − ct−1 , pt − dt , rd,t )′ 5 Choose value ρd that most closely “matches” moments between dist of simulated and historical data ( “match” made precise below.) Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Let {b yt }N t=1 denote simulated data (in obs eqn). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Let {b yt }N t=1 denote simulated data (in obs eqn). Let {ỹt }Tt=1 denote historical data on same variables. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Let {b yt }N t=1 denote simulated data (in obs eqn). Let {ỹt }Tt=1 denote historical data on same variables. Auxiliary model of hist. data: e.g., VAR, with density f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Let {b yt }N t=1 denote simulated data (in obs eqn). Let {ỹt }Tt=1 denote historical data on same variables. Auxiliary model of hist. data: e.g., VAR, with density f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model. Score function of f -model: sf (yt |yt−L , ...yt−1 , α) = Sydney C. Ludvigson ∂ ln[f (yt |yt−L , ..., yt−1 , α)] ∂α Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Let {b yt }N t=1 denote simulated data (in obs eqn). Let {ỹt }Tt=1 denote historical data on same variables. Auxiliary model of hist. data: e.g., VAR, with density f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model. Score function of f -model: sf (yt |yt−L , ...yt−1 , α) = ∂ ln[f (yt |yt−L , ..., yt−1 , α)] ∂α QMLE estimator of auxiliary model on historical data α̃ = arg maxLT (α, {ỹt }Tt=1 ) α LT (α, {ỹt }Tt=1 ) = Sydney C. Ludvigson 1 T T ∑ t= L+ 1 ln f (ỹt |ỹt−L , ..., ỹt−1 , α) Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 First-order-condition: ∂ LT (α̃, {ỹt }Tt=1 ) = 0 ∂α Sydney C. Ludvigson or, 1 T T ∑ t= L+ 1 sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0. Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 First-order-condition: ∂ LT (α̃, {ỹt }Tt=1 ) = 0 ∂α or, 1 T T ∑ t= L+ 1 sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0. Idea: since above, good estimator for ρd is one that sets 1 N sf (b yt (ρd )|b yt−L (ρd ), ..., b yt−1 (ρd ), α̃) ≈ 0. N t=∑ L+1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 First-order-condition: ∂ LT (α̃, {ỹt }Tt=1 ) = 0 ∂α or, 1 T T ∑ t= L+ 1 sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0. Idea: since above, good estimator for ρd is one that sets 1 N sf (b yt (ρd )|b yt−L (ρd ), ..., b yt−1 (ρd ), α̃) ≈ 0. N t=∑ L+1 If dim(α) >dim(ρd ), use GMM: 1 N b T ( ρd , α ) = m sf (b yt (ρd )|b yt−L (ρd ), ..., b yt−1 (ρd ), α̃) | {z } N t=∑ L+1 dim( α)×1 Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 First-order-condition: ∂ LT (α̃, {ỹt }Tt=1 ) = 0 ∂α or, 1 T T ∑ t= L+ 1 sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0. Idea: since above, good estimator for ρd is one that sets 1 N sf (b yt (ρd )|b yt−L (ρd ), ..., b yt−1 (ρd ), α̃) ≈ 0. N t=∑ L+1 If dim(α) >dim(ρd ), use GMM: 1 N b T ( ρd , α ) = m sf (b yt (ρd )|b yt−L (ρd ), ..., b yt−1 (ρd ), α̃) | {z } N t=∑ L+1 dim( α)×1 The GMM estimator is b T (ρd , α̃ )′ Ĩ −1 m b T (ρd , α̃) ρbd = arg min{m ρd Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 b T (ρd , α̃)′ Ĩ −1 m b T (ρd , α̃)}. GMM: ρbd = arg min{m ρd Ĩ −1 is inv. of var. of score, data determined from f -model T Ĩ = ∑ t= 1 ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )] ∂α̃ Sydney C. Ludvigson ′ ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)] ∂α̃ Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 b T (ρd , α̃)′ Ĩ −1 m b T (ρd , α̃)}. GMM: ρbd = arg min{m ρd Ĩ −1 is inv. of var. of score, data determined from f -model T Ĩ = ∑ t= 1 ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )] ∂α̃ ′ ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)] ∂α̃ y }N Sims {b t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note: no closed-form for p(·|ρd ). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 b T (ρd , α̃)′ Ĩ −1 m b T (ρd , α̃)}. GMM: ρbd = arg min{m ρd Ĩ −1 is inv. of var. of score, data determined from f -model T Ĩ = ∑ t= 1 ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )] ∂α̃ ′ ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)] ∂α̃ y }N Sims {b t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note: no closed-form for p(·|ρd ). as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 b T (ρd , α̃)′ Ĩ −1 m b T (ρd , α̃)}. GMM: ρbd = arg min{m ρd Ĩ −1 is inv. of var. of score, data determined from f -model T Ĩ = ∑ t= 1 ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )] ∂α̃ ′ ∂ ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)] ∂α̃ y }N Sims {b t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note: no closed-form for p(·|ρd ). as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt ⇒ use Monte Carlo compute expect. of s(·) under p(·|ρd ). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt If f = p above is mean of scores of likelihood. Should be zero, given f.o.c for MLE estimator. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt If f = p above is mean of scores of likelihood. Should be zero, given f.o.c for MLE estimator. Thus, if data do follow the structural model p(·|ρd ), then m(ρod , αo ) = 0, forms basis of a specification test. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 as b T (ρd , α) → m(ρd , α) as N → ∞, where Intuition: m m ( ρd , α ) = Z ··· Z s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt If f = p above is mean of scores of likelihood. Should be zero, given f.o.c for MLE estimator. Thus, if data do follow the structural model p(·|ρd ), then m(ρod , αo ) = 0, forms basis of a specification test. Summary: solve model for many values of ρd , store long simulations of model each time, do one-time estimation of auxiliary f -model. Choose ρd to minimize GMM criterion above. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: 1 Computational: one-time estimation of structural model; useful if f -model is nonlinear. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: 1 Computational: one-time estimation of structural model; useful if f -model is nonlinear. 2 If f -model good description of data, under null, MLE efficiency is obtained. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: 1 Computational: one-time estimation of structural model; useful if f -model is nonlinear. 2 If f -model good description of data, under null, MLE efficiency is obtained. If dim(α) >dim(ρd ), score-based SMM is consistent, asymptotically normal, assuming: Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: 1 Computational: one-time estimation of structural model; useful if f -model is nonlinear. 2 If f -model good description of data, under null, MLE efficiency is obtained. If dim(α) >dim(ρd ), score-based SMM is consistent, asymptotically normal, assuming: That the auxiliary model is rich enough to identify non-linear structural model. Sufficient conditions for identification unknown. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models EZW Preferences With Restricted Dynamics: Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07 Advantages of using score functions as moments: 1 Computational: one-time estimation of structural model; useful if f -model is nonlinear. 2 If f -model good description of data, under null, MLE efficiency is obtained. If dim(α) >dim(ρd ), score-based SMM is consistent, asymptotically normal, assuming: That the auxiliary model is rich enough to identify non-linear structural model. Sufficient conditions for identification unknown. Big issue: are these the economically interesting moments? Regards both choice of moments, and weighting function. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Little work linking financial markets to macroeconomic risks, given by primitives in the IMRS over consumption. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Little work linking financial markets to macroeconomic risks, given by primitives in the IMRS over consumption. No model that relates returns to other returns can explain asset prices in terms of primitive economic shocks. Such models of SDF only describe asset prices. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Little work linking financial markets to macroeconomic risks, given by primitives in the IMRS over consumption. No model that relates returns to other returns can explain asset prices in terms of primitive economic shocks. Such models of SDF only describe asset prices. So far many consumption-based models have been evaluated using calibration exercises ⇒ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Little work linking financial markets to macroeconomic risks, given by primitives in the IMRS over consumption. No model that relates returns to other returns can explain asset prices in terms of primitive economic shocks. Such models of SDF only describe asset prices. So far many consumption-based models have been evaluated using calibration exercises ⇒ A crucial next step in evaluating consumption-based models is structural econometric estimation. But... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Little work linking financial markets to macroeconomic risks, given by primitives in the IMRS over consumption. No model that relates returns to other returns can explain asset prices in terms of primitive economic shocks. Such models of SDF only describe asset prices. So far many consumption-based models have been evaluated using calibration exercises ⇒ A crucial next step in evaluating consumption-based models is structural econometric estimation. But... ...models are imperfect and will never fit data infallibly. Argue here for need to move away from testing if models are true, towards comparison of models based on magnitude of misspecification. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Example: scaled consumption models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Example: scaled consumption models. Rather than ask whether scaled models are true... Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Example: scaled consumption models. Rather than ask whether scaled models are true... ...ask whether allowing for state-dependence of SDF on consumption growth reduces misspecification over the analogous non-state-dependent model. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. ⇒ Can’t expect such models to perform as well as financial factor models of SDF. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. ⇒ Can’t expect such models to perform as well as financial factor models of SDF. True systematic risk factors are macroeconomic in nature; asset prices derived endogenously from these. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. ⇒ Can’t expect such models to perform as well as financial factor models of SDF. True systematic risk factors are macroeconomic in nature; asset prices derived endogenously from these. Financial factors could represent projection of true Mt on portfolios (i.e., mimicking portfolios). Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. ⇒ Can’t expect such models to perform as well as financial factor models of SDF. True systematic risk factors are macroeconomic in nature; asset prices derived endogenously from these. Financial factors could represent projection of true Mt on portfolios (i.e., mimicking portfolios). In which case, they will always perform at least as well, or better than, mismeasured macro factors from true Mt ⇒ Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Macroeconomic data, unlike financial measured with error. ⇒ Can’t expect such models to perform as well as financial factor models of SDF. True systematic risk factors are macroeconomic in nature; asset prices derived endogenously from these. Financial factors could represent projection of true Mt on portfolios (i.e., mimicking portfolios). In which case, they will always perform at least as well, or better than, mismeasured macro factors from true Mt ⇒ Not sensible to run horse races between financial factor models and macro models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models Consumption-Based Asset Pricing: Final Thoughts Goal: not to find better factors, but rather to explain financial factors from deeper economic models. Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models