Nonparametric Estimation in Panel Data Models with Heterogeneity and - PowerPoint PPT Presentation

Nonparametric Estimation in Panel Data Models with Heterogeneity and Time–Varyingness Jiti Gao † , Fei Liu † , Yanrong Yang ‡ Monash University † Australian National University ‡ Dec 12, 2019

An Econometric Problem Panel Data Analysis 1. Data Structure: Dependent Variable y it and Independent Variable x it = ( X 1, it , X 2, it , . . . , X p , it ) with i = 1, 2, . . . , N and t = 1, 2, . . . , T . 2. Aim: Accurately model and estimate the relation between y it and x it for all cross-sections i = 1, 2, . . . , N and time-periods t = 1, 2, . . . , T . 3. Major Benefit: Homogeneity (Blessing of Dimensionality). 4. Challenge: Heterogeneity (Curse of Dimensionality). 1 / 43

Literature Review Bai (2009, Econometrica) Common factor models are widely used to capture cross-sectional dependence in panel data sets: y it = x ⊤ e it = λ ⊤ it β + e it , i F t + ε it (1) for i = 1, . . . , N and t = 1, . . . , T , where ◮ β is a p -dimensional unknown parameter; ◮ { F t } are unknown r -dimensional common factors; ◮ { λ i } are corresponding factor loadings. Advantages of factor models: ◮ heterogenous effects of common shocks; ◮ Appropriate flexibility. 2 / 43

Literature Review Bai (2009, Econometrica) ◮ Bai (2009) proposes an iterative numerical method to approximate the minimizer of the least squares objective function: � � 2 N T it β − λ ⊤ y it − x ⊤ SSR = ∑ ∑ i F t (2) i = 1 t = 1 ◮ Estimate β by least squares method; ◮ Estimate λ i and F t by PCA method; ◮ Repeat until convergence. ◮ Extensions: ◮ Ando and Bai (2014). ◮ Challenges: ◮ Poor performance with endogenous factors (see Jiang et al., 2017). 3 / 43

Literature Review Pesaran (2006, Econometrica) ◮ Pesaran (2006) proposes valid proxies for F t in the following model:        λ ⊤ i + β ⊤  ε it + β ⊤ i γ ⊤  y it i η it  = i  F t +  , (3) γ ⊤ x it η it i where { γ i } are unknown factor loadings. ◮ Extensions: Chudik and Pesaran (2015). ◮ Challenges: ◮ Rank condition r ≤ p + 1, ◮ No estimators for F t , λ i . 4 / 43

Literature Review Time-varying panel data models ◮ Limitations of time-constant slope coefficients: ◮ The risk of model misspecification; ◮ The time-variation in parameters has been well recognized in many fields: ◮ Silvapulle et al. (2017). ◮ Existing time-varying panel data models: ◮ Li et al. (2011): y it = x ⊤ it β t + f t + α i + ε it ; (4) where β t = β ( τ t ) and f t = f ( τ t ) with τ t = t T . 5 / 43

Literature Review Heterogeneous panel data models ◮ Existing heterogeneous panel data models: ◮ Pesaran (2006)’s random coefficient assumption: β i = β + u i . (5) ◮ Su et al. (2016)’s unknown group pattern: K β ( k ) 1 { i ∈ G k } , ∑ β i = (6) k = 1 where K is known and fixed but G k is unknown. ◮ Gao et al. (2019)’s complete heterogeneity: y it = x ⊤ it β i + f it + α i + ε it , (7) where f it = f i ( τ t ) . 6 / 43

Proposed Model Our model ◮ We consider the following model: it β it + λ ⊤ y it = x ⊤ i F t + ε it , (8) where ◮ x it and y it are observable; ◮ β it = β i ( τ t ) is an unknown deterministic function; ◮ x it can be correlated with { λ i , F t } . 7 / 43

Outline of Contribution 1. Generality of Model: Heterogeneous and Time-varying coefficients. 2. Unified Estimation Approach: observed, unobserved or partially observed factors. 3. Asymptotic Theory: reconcile computational elements (iteration steps) with statistical properties. 4. Empirical Application: relation between health care expenditure and income elasticity. 8 / 43

Proposed Estimation Approach Recall the heterogeneous model: y it = x ⊤ it β i ( τ t ) + λ ⊤ i F t + ε it . The idea of iteration: ◮ With given F t , we can estimate β i ( τ ) and λ i by a profile method. ◮ With β i ( τ ) and λ i , F t can be estimated by OLS method. 9 / 43

Estimation Procedure F ( 0 ) = ( � F ( 0 ) F ( 0 ) (1) Find an initial estimator � 1 , . . . , � T ) ⊤ . F ( n ) (2) With � and by regarding λ i as known, β i ( τ ) can be estimated by local linear t method. For τ ∈ ( 0, 1 ) � � � t − τ T � �� 2 � t − τ T � T F ( n ) y it − λ ⊤ i � − x ⊤ ∑ min a i ( τ ) + b i ( τ ) K , (9) t it Th Th a i ( τ ) , b i ( τ ) t = 1 we have � � − 1 � � ( n + 1 ) � M i ( τ ) ⊤ W ( τ ) M i ( τ ) M i ( τ ) ⊤ W ( τ ) F ( n ) λ i y i − � β ( τ , λ i ) = [ I p , 0 p ] . (10) i (3) With � β i ( τ , λ i ) , we can estimate λ i by the least squares method: � � 2 T ( n + 1 ) F ( n ) y it − x ⊤ it � ( τ , λ i ) − λ ⊤ ∑ i � min β . (11) i t λ i t = 1 See notation 10 / 43

Estimation Procedure We have � F ( n ) � − 1 � ( n + 1 ) F ( n ) ⊤ ( I − S i ) ⊤ ( I − S i ) � F ( n ) ⊤ ( I − S i ) ⊤ ( I − S i ) y i , � � = λ (12) i where S i = ( s i ( 1/ T ) ⊤ x i 1 , . . . , s i ( T / T ) ⊤ x iT ) ⊤ , with s i ( τ ) = [ I p , 0 p ][ M i ( τ ) ⊤ W ( τ ) M i ( τ )] − 1 M i ( τ ) ⊤ W ( τ ) . After plugging � λ i back into � β i ( τ , λ i ) , we have � � � � − 1 ( n + 1 ) ( n + 1 ) � M i ( τ ) ⊤ W ( τ ) M i ( τ ) M i ( τ ) ⊤ W ( τ ) y i − � F ( n ) � ( τ ) = [ I p , 0 p ] β λ (13) i i for i = 1, . . . , N . 11 / 43

Estimation Procedure ( n + 1 ) ( n + 1 ) (4) With � ( τ ) and � β λ , we can estimate F t by OLS method: i i � ( n + 1 ) � − 1 � ( n + 1 ) ⊤ � ( n + 1 ) ⊤ R ( n + 1 ) F ( n + 1 ) � � = Λ Λ Λ t 1, t � � ⊤ ( n + 1 ) ( n + 1 ) where R ( n + 1 ) y 1 t − x ⊤ 1 t � ( τ t ) , . . . , y Nt − x ⊤ Nt � = β β ( τ t ) . 1, t 1 N (5) Repeat Steps 2-4 until convergence. 12 / 43

Asymptotic Properties Assumption 1 (i-v) Regularity assumptions on weak serial and cross-sectional dependence and kernel estimation. F ( n ) − F 0 . For the initial estimator � (vi) Let R ( n ) = � F ( 0 ) , suppose that F T − 1/2 � R ( 0 ) ( Th ) − 1/2 � W ( τ ) ⊤ R ( 0 ) F � = O P ( δ F ,0 ) F � = O P ( δ F ,0 ) , and where δ F ,0 satisfies that NTh 4 δ 2 F ,0 → 0, δ 2 F ,0 / h → 0 and max { N , T } δ 4 F ,0 / h → 0, as N , T → ∞ . Assumption 2 (i-iv) Regularity assumptions on positive definiteness of asymptotic covariance matrices. See Assumptions 13 / 43

Asymptotic Properties Theorem 2.1 (Consistency) Under Assumption 1, as N , T → ∞ simultaneously, (1) N − 1/2 � � ( n ) − Λ � � �� � = O p ( max { δ F ,0 , δ NT } ) ; Λ (2) T − 1/2 � � � � F ( n ) − F �� � = O p ( max { δ F ,0 , δ NT } ) , √ √ T } − 1 . where δ NT = min { N , 14 / 43

Asymptotic Properties Assume that x it = g i ( τ t ) + v it . (14) Notations: � � � � � � � � v i 1 v ⊤ F 0 1 F 0 ⊤ v it F 0 ⊤ v it λ 0 ⊤ Σ v , i = E , Σ F = E , Σ v , F , i = E , Σ v , λ , i = E , i 1 1 t i � 1 Σ X , i ( τ ) = g i ( τ ) g ⊤ Ω F , i = Σ F − Σ ⊤ 0 Σ − 1 i ( τ ) + Σ v , i , X , i ( τ ) d τ Σ v , F , i , v , F , i N z it = F 0 t − Σ ⊤ v , F , i Σ − 1 N → ∞ N − 1 λ 0 i λ 0 ⊤ ∑ σ ij , ts = E [ ε it ε js ] , X , i ( τ t ) x it , Σ λ = lim , i i = 1 � � Σ v , λ , i ( τ ) + g i ( τ ) λ 0 ⊤ ∆ F , i = Σ v , F , i Ω − 1 F , i Σ ⊤ λ † i ( τ ) = Σ − 1 X , i ( τ ) v , F , i , , i � � � � N N Ω 1 ( t , s ) = N − 1 i λ 0 ⊤ x ⊤ it Σ − 1 Ω 2 ( t , s ) = N − 1 i λ 0 ⊤ z ⊤ it Ω − 1 ∑ λ 0 ∑ λ 0 X , i ( τ t ) x is E , E F , i z is , i i i = 1 i = 1 Ω 3 ( t , s ) = Σ − 1 λ ( h − 1 K s ,0 ( τ t ) Ω 1 ( t , s ) + Ω 2 ( t , s )) , 15 / 43

Asymptotic Properties Theorem 2.2 (CLT, n ≥ 2 ) Let Assumptions 1 and 2 hold. Then, as N , T → ∞ simultaneously, (1) if N / T → c 1 < ∞ , for any given t , we have � � √ → N ( √ c 1 d F , t , Σ F , t ) , F ( n ) t − b † ( n ) D � − F 0 − − N t F , t where Σ F , t = Σ − 1 F , t Σ − 1 λ Σ 0 λ , n − 1 T b † ( n ) Ω 3 ( s j , s j + 1 )) R ( 0 ) = T − n ∑ ∏ Ω 3 ( t , s 1 ) F , s n , F , t s 1 , s 2 ,..., s n = 1 j = 1 √ N T T ) Σ − 1 Ω − 1 F , i Σ ⊤ v , F , i Σ − 1 d F , t = N , T → ∞ 1/ ( N ∑ ∑ X , i ( τ s ) g i ( τ s ) σ ii , ts . lim λ s = 1 i = 1 See Assumptions 16 / 43

Asymptotic Properties Theorem 2.2 (CLT, n ≥ 2 ) Let Assumptions 1 and 2 hold. Then, and as N , T → ∞ simultaneously, (2) if T / N → c 2 < ∞ , for any given i , we have � � √ → N ( √ c 2 d λ , i , Σ λ , i ) , ( n ) i − b † ( n ) D � − λ 0 T λ − − i λ , i where Σ λ , i = Ω − 1 λ , i Ω − 1 F , i Σ 0 F , i , T b † ( n ) i ( τ t ) b † ( n − 1 ) = T − 1 Ω − 1 F , i Σ ⊤ ∑ λ † , v , F , i λ , i F , t t = 1 √ N d ∗ N Ω − 1 F , i Σ − 1 ∑ λ , i = 1/ σ ij ,11 . λ µ λ j = 1 See Assumptions 17 / 43