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Panel Data Models with Heterogeneous Marginal Effects (based on the article presented by Kirill Evdokimov at the NES 20th Anniversary Conference).

Author: Kirill Evdokimov, Princeton University

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- 1. Panel Data Models with Heterogeneous Marginal E¤ects Kirill Evdokimov Princeton University The NES 20th Anniversary Conference December 14, 2012Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 1 / 16
- 2. Based on two papers of mine: “Identi…cation and Estimation of a Nonparametric Panel Data Model with Unobserved Heterogeneity” “Nonparametric Quasi-Di¤erencing with Applications” Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 2 / 16
- 3. Model Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . , T i - individual, t - time period Observed: Yit and Xit Unobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks function m (x, α) is not known ∂m (Xit , αi ) /∂x depends on αi ) heterogeneous marginal e¤ects will ID m (x, α) and Fαi (αjXit ) distributions of the outcomes, counterfactuals "what percentage of the treated would be better/worse o¤?"Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16
- 4. Model Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . , T i - individual, t - time period Observed: Yit and Xit Unobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks function m (x, α) is not known ∂m (Xit , αi ) /∂x depends on αi ) heterogeneous marginal e¤ects will ID m (x, α) and Fαi (αjXit ) distributions of the outcomes, counterfactuals "what percentage of the treated would be better/worse o¤?"Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16
- 5. Model Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . , T i - individual, t - time period Observed: Yit and Xit Unobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks function m (x, α) is not known ∂m (Xit , αi ) /∂x depends on αi ) heterogeneous marginal e¤ects will ID m (x, α) and Fαi (αjXit ) distributions of the outcomes, counterfactuals "what percentage of the treated would be better/worse o¤?"Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16
- 6. Model Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . , T i - individual, t - time period Observed: Yit and Xit Unobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks function m (x, α) is not knownSome assumptions: E [Uit jXit ] = 0 and Uit ? αi jXit no restrictions on Fαi (αjXit = x ) ) …xed e¤ects no parametric assumptions; T = 2 is su¢ cient Yit must be cont.; also, no lagged dep. variablesKirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 4 / 16
- 7. MotivationUnion Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck m(1,αi ) m(0,αi ) is union wage premium When m (Xit , αi ) = Xit β + αi , it becomes m (1, αi ) m (0, αi ) = β = const!Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 5 / 16
- 8. MotivationUnion Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck m(1,αi ) m(0,αi ) is union wage premium may be NOT monotone in αi When m (Xit , αi ) = Xit β + αi , it becomes m (1, αi ) m (0, αi ) = β = const!Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 5 / 16
- 9. Motivation (Yit = m (Xit , αi ) + Uit )Life Cycle Models of Consumption and Labor Supply Heckman and MaCurdy (1980), MaCurdy (1981). Under the assumptions of these papers and ρ = r : Cit = C (Wit , λi ) + Uit | {z } |{z} unknown fn. meas .err . Cit and Wit - (log) consumption and hourly wage λi - Lagrange multiplier Parametric speci…cation of utility ) additively separable λi Instead note that: C (w , λ) is unknown C (w , λ) is strictly increasing in λ (unobserved) λi is likely to be correlated with Wit can handle this nonparametrically Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16
- 10. Motivation (Yit = m (Xit , αi ) + Uit )Life Cycle Models of Consumption and Labor Supply Heckman and MaCurdy (1980), MaCurdy (1981). Under the assumptions of these papers and ρ = r : Cit = C (Wit , λi ) + Uit | {z } |{z} unknown fn. meas .err . Cit and Wit - (log) consumption and hourly wage λi - Lagrange multiplier Parametric speci…cation of utility ) additively separable λi Instead note that: C (w , λ) is unknown C (w , λ) is strictly increasing in λ (unobserved) λi is likely to be correlated with Wit can handle this nonparametrically Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16
- 11. Motivation (Yit = m (Xit , αi ) + Uit )Life Cycle Models of Consumption and Labor Supply Heckman and MaCurdy (1980), MaCurdy (1981). Under the assumptions of these papers and ρ = r : Cit = C (Wit , λi ) + Uit | {z } |{z} unknown fn. meas .err . Cit and Wit - (log) consumption and hourly wage λi - Lagrange multiplier Parametric speci…cation of utility ) additively separable λi Instead note that: C (w , λ) is unknown C (w , λ) is strictly increasing in λ (unobserved) λi is likely to be correlated with Wit can handle this nonparametrically Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16
- 12. Motivation (Yit = m (Xit , αi ) + Uit )Life Cycle Models of Consumption and Labor Supply Heckman and MaCurdy (1980), MaCurdy (1981). Under the assumptions of these papers and ρ = r : Cit = C (Wit , λi ) + Uit | {z } |{z} unknown fn. meas .err . Cit and Wit - (log) consumption and hourly wage λi - Lagrange multiplier Parametric speci…cation of utility ) additively separable λi Instead note that: C (w , λ) is unknown C (w , λ) is strictly increasing in λ (unobserved) λi is likely to be correlated with Wit can handle this nonparametrically Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16
- 13. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 14. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 15. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 16. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 17. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 18. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 19. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 20. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 21. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 22. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 23. Identi…cation, Yit = m (Xit , αi ) + Uit 1 Subpopulation Xi 1 = Xi 2 = x: then m (Xi 1 , αi ) = m (Xi 2 , αi ) = m (x, αi ) Yi 1 = m(x, αi ) + Ui 1 Yi 2 = m(x, αi ) + Ui 2 Yi 1 Yi 2 = Ui 1 Ui 2 =) Obtain distr L (Uit jXit = x ) 2 Deconvolve: Yit = m (Xit , αi )+Uit , αi ?Uit given Xit = x, thus L (Yit jXit = x ) = L (m (x, αi ) jXit = x ) L (Uit jXit = x ), | {z } | {z } from data from step 1 where " " means "convolution" and "L" means "probability Law". Then obtain L (m (x, αi ) jXit = x ) by deconvolution 3 Use L (m (x, αi ) jXit = x ), eg. quantiles Qm (x ,αi )jX it (q jx ) [for nonparametric between- or within- variation] more Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16
- 24. More things in the paper Time E¤ects: Yit = m (Xit , αi ) + η t (Xit ) + Uit , η 1 (x ) 0 Dynamic models, e.g., Yit = m (Xit , αi ) + Uit , Uit = ρUit 1 + εit Measurement error in covariates Multivariate unobserved heterogeneity Yit = m Xit , βi0 Wit + Uit Estimation - conditional deconvolution Simple. No optimization. Just run many kernel regressions and compute a one-dimensional integral Rates of convergence Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 8 / 16
- 25. Empirical Illustration - Union Wage Premium Data: Current Population Survey (CPS), T = 2 matching migration measurement error limited set of variables 1987/1988, n = 17, 756 (replicating Card, 1996) Most related papers: Card (1996) and Lemieux (1998)Union Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck. ∆(α) = m (1, α) m (0, α) is the union wage premium. Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16
- 26. Empirical Illustration - Union Wage Premium Data: Current Population Survey (CPS), T = 2 matching migration measurement error limited set of variables 1987/1988, n = 17, 756 (replicating Card, 1996) Most related papers: Card (1996) and Lemieux (1998)Union Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck. ∆(α) = m (1, α) m (0, α) is the union wage premium. Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16
- 27. Empirical Illustration - Union Wage Premium Data: Current Population Survey (CPS), T = 2 matching migration measurement error limited set of variables 1987/1988, n = 17, 756 (replicating Card, 1996) Most related papers: Card (1996) and Lemieux (1998)Union Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck. ∆(α) = m (1, α) m (0, α) is the union wage premium. Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16
- 28. Empirical Illustration - Union Wage Premium Data: Current Population Survey (CPS), T = 2 matching migration measurement error limited set of variables 1987/1988, n = 17, 756 (replicating Card, 1996) Most related papers: Card (1996) and Lemieux (1998)Union Wage Premium Yit = m (Xit , αi ) + Uit Yit is (log)-wage, Xit is 1 if member of a union, 0 o/w, αi is skill and Uit is luck. ∆(α) = m (1, α) m (0, α) is the union wage premium. Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16
- 29. Empirical Illustration - Union Wage Premium∆ (Qαi (0.25)) = 7.0%, ∆ (Qαi (0.5)) = 3.2%, ∆ (Qαi (0.75)) = 0.7% Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 10 / 16
- 30. Empirical Illustration - Union Wage Premium∆ (Qαi (0.25)) = 7.0%, ∆ (Qαi (0.5)) = 3.2%, ∆ (Qαi (0.75)) = 0.7% Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 11 / 16
- 31. Empirical Illustration - Propensity Score Also interested in P (Xit = 1jαi = a). Examples: Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 12 / 16
- 32. Empirical Illustration - Propensity Score Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 13 / 16
- 33. Alternative Model, Quasi-Di¤erencing We had Yit = m (Xit , αi ) + Uit (1) Identi…cation is based on the NP …rst-di¤erencing Now consider Yit = gt (Xit , αi + Uit ) (2) Identi…cation is based on the NP quasi-di¤erencing Applications: Nonparametric panel (transformation) models Duration model with multiple spells/states A class of static games of incomplete information (e.g., Common Value Auctions, Cournout with common demand parameter) Also: Yit = gt (Xit , βi Wit + Uit ) (20 ) Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 14 / 16
- 34. Related Literature (incomplete list) Linear and Semiparametric models:Chamberlain(1984,1992), Horowitz & Markatou (1996),Wooldridge(1997), Blundell, Gri¢ th, and Windmeijer (2002), Graham andPowell(2008), Graham, Hahn, and Powell (2009), Bonhomme(2010),Honore and Tamer (2006) Nonparametric Panel Models:Altonji and Matzkin (2005), Arellano and Bonhomme (2011a,gb), Besterand Hansen (2007, 2009), Chernozhukov, Fernandez-Val, Hahn, andNewey (2009), Cunha, Heckman, and Schennach (2010), Hoderlein andWhite (2009), Hu and Shum (2009) Transformation and Multiple Spell Duration Models:Abbring and Ridder (2003, 2010), Abrevaya (1999), van den Berg (2001),Flinn and Heckman (1982,1983), Honoré (1993), Horowitz and Lee(2004), Lee (2008), Khan and Tamer (2009), Ridder (1990), and a list ofempirical applications Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 15 / 16
- 35. Summary Generalization of …rst-di¤erencing and quasi-di¤erencing approaches to nonlinear panel data models. Nonparametric identi…cation Allow identi…cation of distributional, policy, and counterfactual e¤ects Are useful in a range of applied models Estimation: First-Di¤erencing: a simple estimation procedure Quasi-Di¤erencing: sieve estimation More details in the papersKirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 16 / 16

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