Panel Data Analysis Part III Modern Moment Estimation James J. - PowerPoint PPT Presentation

Panel Data Analysis Part III – Modern Moment Estimation James J. Heckman University of Chicago Econ 312, Spring 2019 Heckman Part III

Review Moments and Identification: • Y = X β + U • E ∗ ( U | X ) = 0 ⇒ Cov ( Y − X β, X ) = 0 • ⇒ ˆ β = ( X ′ X ) − 1 X ′ Y Heckman Part III

Review Moments and Identification: • ( T × 1) = Y X β + U T × 1 ( T × K )( K × 1) • E ∗ ( U | X ) � = 0 • E ∗ ( U | Z ) = 0 • Z = M × K ( M ≥ K ) • E ∗ ( X | Z ) non-degenerate • ∴ Cov ( Z ′ X ) rank = K • Z ′ ( Y − X β ) = 0: These are the moments in GMM. • Z ′ Y = ( Z ′ X ) β if M = K • ˆ β = ( Z ′ X ) − 1 Z ′ Y otherwise GMM Heckman Part III

• Suppose y it = β X it + η i + v it i = 1 , .., I • U it = η i + v it t = 1 , ..., T • X it is strictly exogenous if E ∗ ( U it | X T i ) = 0 ∀ t X T = ( X i 1 , ..., X iT ) i • .. OLS identifies β and E ∗ ( η i | X T . i ) = 0. • E ∗ is linear projection. Heckman Part III

Panel Data Setting: • X it is strictly exogenous given η i if E ∗ ( v it | X T i , η i ) = 0 t = 1 , ..., T for all X T but not necessarily for U it i Heckman Part III

Consequence: • First difference eliminates fixed effects: E ∗ ( v it − v i , t − 1 | X T i ) = 0. • Multivariate regression with cross equation restrictions. • Assume that this is essentially all the information. • Partial Adjustment Model With Strictly Exogenous Variable y it = α y i , ( t − 1) + β 0 X i , t + β 1 X i , t − 1 + η i + v it . • Assume E ∗ ( v it | X T i ) = 0 , t = 2 , ..., T . • Does not restrict serial correlation in v it . Heckman Part III

Restrictions: • E ∗ (∆ v it | X T i ) = 0 . • Model identified for T ≥ 3 . • T = 3 case; acquire orthogonality restrictions E ( X is (∆ y i 3 − α ∆ y i 2 − β 0 ∆ X i 3 − β 1 ∆ X i 2 )) = 0 ⇔ E ( X is (∆ v i 3 )) = 0 , s = 1 , 2 , 3 • Use these in GMM to identify model. • 3 equations in 3 unknowns and we acquire exact identification. • Note: Strict exogeneity enables us to identify dynamic effect of X on y with arbitrary serial correlation in the errors; • Price: Assumes X not influenced by past values of y and v . Heckman Part III

• Definition: X is predetermined if • E ∗ ( v it | X t i , y t − 1 ) = 0 , t = 2 , ..., T (A) i i ) , y t − 1 i , ..., y t − 1 • X t i = ( X 1 i , ..., X t = ( y 1 ). i i • Current shocks are uncorrelated with past values of y and current and past values of X . • Feedback from lagged dependent variables to future X not ruled out. • E.g., Euler equations. (Information set of agents uncorrelated with current and future idiosyncratic shocks but not past shocks). Heckman Part III

Example: Euler Equation: U t � c ( c t ) � ( c t − 1 ) β R t | I t − 1 = 1 E U t − 1 c • β = subjective discount rate • R t = 1 + r t = period plus interest rate Heckman Part III

Special Case Power Utility: U t = ( c t ) 1 − γ − 1 ; U c , t = ( c t ) − γ 1 − γ �� c t � � − γ E β R t | I t − 1 = 1 c t − 1 Heckman Part III

• Z t − 1 is in the information set. • Crucial that instruments don’t include variables that cause the innovation. � � �� � − γ � • E t − 1 C t Z t − 1 β R t − 1 = 0 C t − 1 • β C − γ t − 1 R t − 1 = ε t t C − γ � � � � C − γ C − γ • ε t = β − t − 1 R t +1 − 1 ( C t − 1 ) − γ R t − 1 t E t − 1 t C − γ • ε t is forecast error. • E ( ε t Z t − 1 ) = 0. • Z t has to be relevant in forecasting future returns or consumption growth. • Need at least 2 instruments for ( β, γ ) parameters. Heckman Part III

Implication of Predeterminedness: • E ∗ ( v i , t − v i , t − 1 | X t − 1 , y t − 2 ) = 0, t = 3 , ..., T i i • For T = 3 , we acquire   y i 1 0 = E X i 1 (∆ y i 3 − α ∆ y i 2 − β 0 ∆ X i 3 − β 1 ∆ X i 2 )   X i 2 • This condition is not the same as that in strictly exogenous models: • We acquire 3 moments only 2 in common with last (across strictly exogenous and these models). • Standard errors are consistent with arbitrary serial correlation. Heckman Part III

• If we had ruled out arbitrary serial correlation in the first model of strictly exogenous regressors, by it | X T 1 , y t − 1 E ∗ ( v ∗ ) = 0 t = 2 , ..., T i we acquire superset of all conditions. (A) and previous ones. • (A) = ⇒ E (∆ v it , ∆ v i , t − j ) = 0 j > 1 because we have that the covariances are zero • Cov ( v i , t y t − 1 ) = 0 i • . .. Cov ( v it , v i , t − 1 ) = 0 generically. Heckman Part III

• Observe that in the predetermined case we can have special cases of serial correlation. • e.g. for T = 4 E (∆ v i , t ∆ v i , t − j ) = 0 j > 2 • Valid for first order MA . • Valid orthogonality conditions derived from: • ∆ y i , 3 − α ∆ y i , 2 − β 0 ∆ X i , 3 − β 1 ∆ X i , 2 = v i , 3 − v i , 2 • ∆ y i , 4 − α ∆ y i , 3 − β 0 ∆ X i , 4 − β 1 ∆ X i , 3 = v i , 4 − v i , 3 Heckman Part III

• Orthogonality conditions: E ( y i 1 ∆ v i , 4 ) = 0 E ( x i 1 ∆ v i , 4 ) = 0 E ( x i 2 ∆ v i , 4 ) = 0 . Heckman Part III

• Other orthogonality conditions from: y i , 4 = α y i , 3 + β 0 X i , 4 + β 1 X i , 3 + η i + v i , 4 y i , 3 = α y i , 2 + β 0 X i , 2 + β 1 X i , 1 + η i + v i , 3 ∆ y i , 4 = α ( y i , 3 − y i , 2 ) + β 0 ( X i , 4 − X i , 3 ) + β 1 ( X i 32 − X i , 2 ) + ( v i , 4 − v i , 3 ) Heckman Part III

Suppose Uncorrelated Fixed Effects • Some X it uncorrelated with η i E [ X T i ( y i 2 − α y i 1 − β 0 X i 2 − β 1 X i 1 )] = 0 • T orthogonality conditions for each regressor. • Predetermined variables could be uncorrelated with fixed effects X it = ρ X i , ( t − 1) + γ v i , ( t − 1) + ϕη i + ε i , t if φ = 0 , X would be uncorrelated with η. Heckman Part III

• Adds more orthogonality restrictions: E ( X i 1 ( y i 2 − α y i 1 − β 0 X i 2 − β 1 X i 1 )) = 0 E ( X it ( y it − α y i , t − 1 − β 0 X it − β 1 X i , t − 1 )) = 0 , t = 2 , ..., T . • Only identified when T ≥ 3. Heckman Part III

Statistical Definitions: • Strict Exogeneity: E ∗ ( y it | X T i , η i ) = E ∗ ( y it | X t i , η i ) E ∗ ( X i , t +1 | X t i , y t ⇐ ⇒ i , η i ) = E ∗ ( X i , ( t +1) | X t i , η i ) • ( y does not Granger cause X ). Heckman Part III

• Let X ( t +1) T = ( X i , t +1 , ..., X i , T ) if i t X ( t +1) T • E ∗ ( y it | X T i , η i ) = β ′ t X t i + δ ′ + γ t η i and i • E ∗ ( X i ( t +1) | X t i , y t i , η i ) = ψ ′ X t i + φ ′ t y t i + ε t η i δ t = 0 ⇐ ⇒ φ t = 0. Heckman Part III

AR-1 Models Balestra - Nerlove Problem • y it = α y i , t − 1 + η i + v i , t • i = 1 , ..., I ; t = 2 , ..., T • (A-1) E ∗ ( v it | y t − 1 ) = 0 t = 2 , ..., T i E ( η i ) = γ , E ( v 2 it ) = σ 2 t Var ( η i ) = σ 2 η • η i and v it freely correlated • E ( v 2 it | y t − 1 ) need not coincide with σ 2 t . i Heckman Part III

• We get ( T − 1)( T − 2) / 2 moment restrictions: E ( y t − 2 (∆ y it − α ∆ y i , t − 1 )) = 0 i • Using minimum discrepancy (CMD) methods take y it = α y i , t − 1 + η i + v it . Heckman Part III

• For s < t , we obtain: y i , t y i , s = α y i , t − 1 , y i , s + η i y i , s + y i , s v i , t E ( y it y is ) = α E ( y i , t − 1 y i , s ) + E ( η i y i , s ) + E ( y i , s v i , t ) =0 E ( y it y is ) = ω ts [ E ( y i , t − 1 y is ) = ω t − 1 , s ] E ( y is η i , t ) = c s Heckman Part III

� T + 1 � • We take T × distinct elements of 2 Ω = E ( y i y ′ i ). • For T = 3, we obtain ω 31 = αω 21 + c 1 • ω 21 = αω 11 + c 1 α = ω 31 − ω 21 = α ( ω 21 − ω 11 ) ω 21 − ω 11 ( ω 21 − ω 11 ) c 1 = ω 31 − αω 21 c 2 = ω 32 − αω 22 • . .. model just identified. • Fit discrepancies between the population moments and fitted moments. Heckman Part III

y t = α y t − 1 + β X t + η i + U it U it = ρ U it − 1 + ε it ⊥ ε i , t ′ ∀ t , t ′ ε i , t ⊥ ⊥ X t ′ ∀ t , t ′ ε it ⊥ η i ⊥ ⊥ X t ′ ε it ? maybe y t = α y t − 1 + β X t + η i + ρ U i , t − 1 + ε i , t y t = α y t − 1 + β X t + η i + ρ ( y t − 1 − α t − 2 − β X t − 1 − η i ) + ε it = ( α + ρ ) y t − 1 + β X t − ρβ X t − 1 − ρα y t − 2 + (1 − ρ ) η i + ε i , t • What parameters are identified? • Suppose we work with ∆ y it : eliminates fixed effect. Heckman Part III

I. Suppose ρ = 1: (errors are random walks) y t = ( α + 1) y t − 1 − α y t − 2 | + β ( X t − X t − 1 ) + ε it II. Suppose α = 1 y t = (1 + ρ ) y t − 1 − ρ y t − 2 | + β X t − ρβ X t − 1 + (1 − ρ ) + ε it • In I., η i vanishes • In II., it does not Heckman Part III

Other Restrictions Heckman Part III

• Lack of correlation between effects and errors: E ∗ ( v it | y t − 1 , η i ) = 0 , t = 2 , ..., T i 0 = E [( y it − α y i , t − 1 ) [∆ y i , t − 1 − α ∆ y i , t − 2 ]] quadratic (in α ) restrictions: because E ( η i ∆ v i , t − 1 ) = 0. Heckman Part III