Quantitative Methods in Economics Panel data and generalized least squares

Transcription

1 Quantitative Methods in Economics Panel data and generalized least squares Maximilian Kasy Harvard University, fall / 22

2 Roadmap, Part I 1. Linear predictors and least squares regression 2. Conditional expectations 3. Some functional forms for linear regression 4. Regression with controls and residual regression 5. Panel data and generalized least squares 2 / 22

3 Takeaways for these slides Panel data: more than one outcome Estimator: GLS generalizes LS to multidimensional outcome Example 1: Siblings earnings and education Example 2: Agricultural output over time 3 / 22

4 Panel data Data: Y 1 X 1 X X 1K Y =.., X =.. =... Y M X M1... X MK X M Example: Population of families with Y t = log(earnings) and Z t =(education, age) for family member t, K 1 X t is constructed (polynomials, etc.) from Z t (t = 1,...,M). 4 / 22

5 Generalized linear predictor Linear predictor with weighting: β 1 β =.. = arg min E[(Y Xc) Φ(Y Xc)], c R β K K Φ is a M M symmetric, positive-definite matrix. Notation: EΦ (Y X) = Xβ. 5 / 22

6 Inner Product: If U and V are M 1 random vectors, and Φ is a (nonrandom) M M positive-definite matrix, U,V Φ = E(U ΦV). Norm: V Φ = V,V 1/2 Φ. β = argmin c Y Xc 2 Φ. Questions for you Find the first order conditions for this minimization problem. 6 / 22

7 Solution: Let Xβ = ( X (1)... X (K )) β = X (1) β X (K ) β K, Orthogonal projection: Y Xβ,X (k) Φ = 0 (k = 1,...,K ) E[(Y Xβ) ΦX] = 0 Solving for β : and thus E(Y ΦX) β E(X ΦX) = 0 E(X ΦX)β = E(X ΦY ), β = [E(X ΦX)] 1 E(X ΦY ). 7 / 22

8 Sample version - generalized least squares (GLS) Data: ( ) x i = x (1) i... x (K ) i y i1 y i =.., y = y im y 1.. y n, x i11... x i1k =.., x = x im1... x imk y is nm 1 and x is nm K. Let A be a nm nm symmetric, positive-definite matrix. GLS estimator: b = argmin(y xc) A(y xc). c x 1.. x n. 8 / 22

9 Inner Product: u,v R nm, Norm: Orthogonal projection: u,v A = u Av. v A = v,v 1/2 A. b = argmin c y xc A. y xb,x (k) A = 0 (k = 1,...,K ) (y xb) Ax = 0 Solving for b: y Ax b (x Ax) = 0 (x Ax)b = x Ay, and thus b = (x Ax) 1 x Ay. 9 / 22

10 Rewriting the GLS estimator Factorization of A : A = C C, where C is nm nm and nonsingular. Define ỹ = Cy, x = Cx. Note that with ṽ = Cv, v 2 A = v Av = v C Cv = ṽ ṽ = ṽ 2 I, 10 / 22

11 thus b = argmin y xc A = argmin ỹ xc I = ( x x) 1 x ỹ. c c We can obtain a GLS fit using a least-squares program. In Matlab, b = x\ỹ. 11 / 22

12 Panel data: Application to siblings Consider siblings in families with two children Y it = log(earnings) of sibling t in family i Z it = education of sibling t in family i (t = 1,2) Data: W i = (Y i1,y i2,z i1,z i2 ) i.i.d. (i = 1,...,n). 12 / 22

13 Latent variable model Assume that earnings are determined by E (Y i1 Z i1,z i2,a i ) = γ 1 + γ 2 Z i1 + γ 3 A i, E (Y i2 Z i1,z i2,a i ) = γ 1 + γ 2 Z i2 + γ 3 A i. Here A i is a family background variable that is not observed. Note that, by assumption, sibling education does not show up given A i! Now consider a regression of earnings on own- and sibling education: E (Y i1 1,Z i1,z i2 ) = γ 1 + γ 2 Z i1 + γ 3 E (A i 1,Z i1,z i2 ), E (Y i2 1,Z i1,z i2 ) = γ 1 + γ 2 Z i2 + γ 3 E (A i 1,Z i1,z i2 ). 13 / 22

14 Next, consider a hypothetical regression of A i on both siblings education: E (A i 1,Z i1,z i2 ) = λ 0 + λ 1 Z i1 + λ 2 Z i2. Questions for you Assume λ 1 = λ 2 and combine these equations to derive an expression for E (Y it 1,Z it,z i1 + Z i2 ). 14 / 22

15 Solution: E (Y i1 1,Z i1,z i1 + Z i2 ) = (γ 1 + γ 3 λ 0 ) + γ 2 Z i1 + γ 3 λ 1 (Z i1 + Z i2 ), E (Y i2 1,Z i2,z i1 + Z i2 ) = (γ 1 + γ 3 λ 0 ) + γ 2 Z i2 + γ 3 λ 1 (Z i1 + Z i2 ). If the latent variable model is true, then the coefficient on own-education equals γ 2. Generalized Linear Predictor: ( ) Yi1 Y i =, X i = Y i2 E Φ (Y i X i ) = X i β = ( ) 1 Zi1 (Z i1 + Z i2 ), 1 Z i2 (Z i1 + Z i2 ) ( ) β1 + β 2 Z i1 + β 3 (Z i1 + Z i2 ). β 1 + β 2 Z i2 + β 3 (Z i1 + Z i2 ) 15 / 22

16 Panel data: Application to production functions Suppose output is determined by Q it = L γ it F iv it (i = 1,...,n; t = 1,...,T ), Q it = output for farm i in year t, L it = labor input, 0 < γ < 1 F i = soil quality, V it = rainfall. Profit maximization at time t, where V it is uncertain: max L Questions for you E[P t Q it W t L Info it ] = maxp t [L γ F i E(V it Info it )] W t L L Derive the first order condition for the firms profit maximization problem. Solve for the implied demand for labor. 16 / 22

17 Solution: First order condition: Derived demand for labor: γp t L γ 1 F i E(V it Info it ) = W t. logl it = 1 1 γ [logγ log W t P t + logf i + loge(v it Info it )]. 17 / 22

18 We can write the production function as logq it = γ logl it + logf i + logv it. Equivalently Y it = γz it + A i + logv it, with Y it = logq it, Z it = logl it, and A i = logf i. Hypothetical regression of output at time t on labor input at all times and soil quality (unobserved): E(Y it Z i1,...,z it,a i ) = γz it + A i + E(logV it Z i1,...,z it,a i ) 18 / 22

19 One more assumption: strict exogeneity of Z conditional on A: E(logV it Z i1,...,z it,a i ) = constant, then Questions for you Take first differences : Calculate Then calculate E(Y it Z i1,...,z it,a i ) = γz it + A i + constant E(Y it Y i,t 1 Z i1,...,z it,a i ). E(Y it Y i,t 1 Z i1,...,z it ). 19 / 22

20 Solution: Strict exogeneity conditional on A E(Y it Y i,t 1 Z i1,...,z it,a i ) = γz it + A i (γz i,t 1 + A i ) = γ(z it Z i,t 1 ), and thus E(Y it Y i,t 1 Z i1,...,z it ) = γ(z it Z i,t 1 ). 20 / 22

21 Generalized Linear Predictor: Y i2 Y i1 Z i2 Z i1 Ydif i =., X i =., Y it Y i,t 1 Z it Z i,t 1 EΦ (Ydif i X i ) = X i β = β(z i2 Z i1 ). β(z it Z i,t 1 ). If the latent variable model with the strict exogeneity assumption is true, then β = γ. 21 / 22

22 Within group estimator Strict exogeneity conditional on A where E[ 1 T T t=1y it Z i1,...,z it,a i ] = 1 T Z i = 1 T T t=1 (γz it + A i + constant) = γ Zi + A i + constant, T t=1z it, Ȳ i = 1 T Consider de-meaned regression: T t=1 Y it. E(Y it Ȳ i Z i1,...,z it,a i ) = γz it + A i (γ Zi + A i ) = γ(z it Zi ), and thus E(Y it Ȳ i Z i1,...,z it ) = γ(z it Zi ). 22 / 22

23 Generalized Linear Predictor Define demeaned variables, Y i1 Ȳ i Z i1 Zi Ydev i =., X i =.. Y it Ȳ i Z it Zi Generalized linear predictor: β(z i1 Zi ) EΦ (Ydev i X i ) = X i β =.. β(z it Zi ) If the latent variable model with the strict exogeneity assumption is true, then β = γ. 23 / 22

24 Will return to panel data when talking about differences in differences! 24 / 22