Bài giảng Phân tích số liệu mảng - Chương 5: Dynamic panel model

Chia sẻ: Cao Ngữ Lam | Ngày: | Loại File: PDF | Số trang:32

Thêm vào BST

Báo xấu

8
lượt xem 5
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Bài giảng Phân tích số liệu mảng - Chương 5: Dynamic panel model. Chương này cung cấp cho sinh viên những nội dung gồm: giới thiệu về Dynamic panel model; ước tính hiệu ứng cố định và ngẫu nhiên; ước tính biến công cụ (phương pháp IV) (Anderson và Hsiao, 1982); 2SLS, phương pháp tiếp cận mô men tổng quát (GMM) (Arenalloand Bond, 1985);... Mời các bạn cùng tham khảo!

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Bài giảng Phân tích số liệu mảng - Chương 5: Dynamic panel model

6/6/2022 Chapter 5 Dynamic Panel Model Mr U_KHOA TOÁN KINH TẾ 1
Objectives 2 (1) Introduce about Dynamic Panel Model (2) Fixed and Random Effects Estimation (3) Instrumental Variable Estimation (IV approach) (Anderson and Hsiao, 1982) (4) 2SLS, Generalized Method of Moment (GMM) approach (Arenallo and Bond, 1985) Mr U_KHOA TOÁN KINH TẾ 6/6/2022
5.1 Introduction 3 Linear dynamic panel data models include lag dependent variables as covariates along with the unobserved effects, fixed or random, and exogenous regressor p p yit   0    j y t  j  x it   i  u it    j y t  j  x it  *  u it i (5.1) j1 j1 Notes: The presence of lagged dependent variable as a regressor incorporates the entire history of it, and any impact of xit on yit is conditioned on this history. We consider a dynamic panel model, in the sense that it contains (at least) one lagged variables. For simplicity, let us consider Mr U_KHOA TOÁN KINH TẾ yit = γ1yit-1+β’itxit +αi* + uit (5.2) 6/6/2022
yit = γ1yit-1+β’itxit +αi* + uit (5.2) 4 Eq. (5.2) requires that |γ |
By setting t = 1, 2,… and so on, the autoregressive process can be 5 expressed in the following way: yi ( t 1)   0   yi 0   i  ui 0 yi 2   0   yi1   i  ui 2   0   i   1   0   1 yi 0   i  ui 0   ui 2   0   0 1   i   i 1   12 yi 0   1ui1  ui 2 ............. t 1 yit   0 1   1  ...   t 1 1    1   i 1  ...   t 1 1  t 1 yi 0    1j ui ,t  j j 0 Or t 1 t 1 t 1 yit   0     i     yi 0    1j ui ,t  j 1 j 1 j t 1 j 0 j 0 j 0 Therefore t 2 t 2 t 2 yit 1   0     i     1 j 1 j t 1 1 y    1j ui ,t 1 j i0 j 0 Mr U_KHOA TOÁN KINH TẾ j 0 j 0 6/6/2022
For l arg e t , 1 1 6 E  yit /  i    0  i 1 1 1 1  2 V  yit /  i   1   12 5.2 Fixed and Random Effects Estimation yit = γ0 + γ1yit-1+ αi + uit (5.3) Remark: One possible cause for biasedness is the presence of the unknown individual effects αi, which creates a correlation between the explanatory variables and the residuals y it    y i   1 yit 1  y i ,1  uit  u i  Notes: yit 1  y i ,1  will be correlated  u it  ui  Mr U_KHOA TOÁN KINH TẾ 6/6/2022
    yit  y i   1  yit 1   y i ,1    uit   ui  7  depen on past value of uit  depen on past value of uit The within estimator or fix effects estimator is   y  y  N T it  yi it 1  y i ,1   1FE  i 1 t 1   y  N T 2 it 1  y i , 1 i 1 t 1   y   N T it 1  y i ,1 uit  u i  1  i 1 t 1   y  N T 2 it 1  y i ,1 i 1 t 1    i  yi   1FE y i ,1 Problem: Fixed effects the within transformation and LSDV produce biased estimates
  y   N T it 1  y i ,1 uit  u i / NT  8  1FE  1  i 1 t 1   y  N T 2 it 1  y i ,1 / NT i 1 t 1 Theorem. (Weak law of large numbers, Khinchine) If {Xi } for i=1,…, m is a sequence of i.i.d random variables with E(Xi ) = μ < ∞, then the sample mean coverges in probability to μ: 1 m 1 m  X i  E  X i     p m  X i  E  X i    m i 1 p m lim i 1
We have 9 p lim 1 N T   yit 1  y i ,1 uit  u i N  NT i 1 t 1   1 N T 1 N T  p lim  yit 1uit  p NT  yit 1 u i N  NT i 1 t 1 lim   i1 t 1 N   N1 N2 1 N T 1 N T  p lim  yi ,1uit  p NT  y i,1 u i N  NT i 1 t 1 lim   N i 1 t 1 N3 N4 N T 1 N1  p lim N  NT  y i 1 t 1 u  E  yit 1uit   0 it 1 it
N T N T 1 1 N 2  p lim N  NT  yit 1 u i  p NT i 1 t 1 N lim u  y i 1 i t 1 it 1 10 N 1 1 N  p lim N  NT  u iT yi,1  p N  u i yi,1 i 1 N lim i 1 N T N T N 1 1 1 N 3  p lim N  NT  y i ,1uit  p NT i 1 t 1 N lim  y i,1  uit  p N i 1 t 1 N lim y i 1 i ,1 ui N T N 1 1 1 N N 4  p lim N  NT  y i,1 u i  p NT T  y i ,1 u i  p N  y i ,1 u i i 1 t 1 N lim i 1 N lim i 1   y   N T 1 1 N p lim it 1  y i , 1 uit  u i   p lim  u i y i ,1 N  NT i 1 t 1 N  N i 1 1 N 1 N 1 N 0  p lim  u i y i ,1  p lim  u i y i ,1  p lim  u i y i ,1 N  N i 1 N  N i 1 N  N i 1
  y   N T it 1  y i ,1 uit  u i / NT  1 N 11  1FE   1  i 1 t 1   1  p lim  u i y i ,1   y  N T 2 N  N i 1 it 1  y i ,1 / NT i 1 t 1  If this plim is not null, then the  1,FE estimator is biased when N tends to infinity and T is fixed Fact. If T also tends to infinity, then the numerator converges to zero Fact. The problem is more prominent in the random effects model. The lagged dependent variable is correlated with the compound disturbance in the model. yit = γ1yit-1+αi* + uit= γ0 + γ1yit-1+ αi + uit (5.3)  t 2 j t 2 t 2  * E  yit 1   E   0   1   i   1   1 yi 0    1 ui ,t 1 j   i  0 * i j t 1 j Mr U_KHOA TOÁN KINH TẾ  j 0 j 0 j 0  6/6/2022
Pre Example (3.1) With model 12 ROAA = f(L.ROAA, HHI, L_A, SIZE, ASSET_GRO, GDP, INF) +ε 5.3 Instrumental Variable Estimation 5.3.1 Define the endogeneity bias and the smearing effect. Consider the (population) multiple linear regression model y = Xβ + ε - y is a Nx1 vector of observation for yj , j= 1,…,N - X is a NxK matrix of K explicative variables xjk for k=1,…,K and j=1,…,N - β = (β1 β2 … βK)’ is a Kx1 vector of parameters - ε is a Nx1 vector of error terms εi with V (ε/X) =σ2IN
Endogeneity we assume that the assumption A1 (exogeneity) is violated 13 E (ε/X) ≠0 With 1 p lim X '   E  x j j     0 K 1 N Theorem (Bias of the OLS estimator) If the regressors are endogenous, i.e. E (ε/X) 6= 0, the OLS estimator of β is biased    E  OLS / X   where β denotes the true value of the parameters. This bias is called the endogeneity bias. Theorem (Inconsistency of the OLS estimator) If the regressors are endogenous with plim N-1X’ε = γ the OLS estimator of β is inconsistent
 p lim  OLS    Q 1 1 14 where Q  p lim N X X ' Proof: Given the definition of the OLS estimator   OLS   X ' X  X ' y   X ' X  X '  X     1 1   X 'X   X '  1 We have 1  1  1  p lim OLS    p lim  X ' X   p lim  X '   N  N     Q 1   Notes. - The implication is that even though only one of the variables in X is  correlated with ε, all of the elements  OLS of are inconsistent, not just the estimator of the coefficient on the endogenous variable
Notes (cont.). 15 - This effects is called smearing effect: the inconsistency due to the endogeneity of the one variable is smeared across all of the least squares estimators 5.3.2 Instrustment variable Definition. Consider a set of H variables zh RN for h = 1, ..N. Denote Z the NxH matrix (z1 ... zH ). These variables are called instruments or instrumental variables if they satisfy two properties: (1) Exogeneity: They are uncorrelated with the disturbance. E(ε/Z)= 0Nx1 (2) Relevance: They are correlated with the independent variables, X E(xjkzjh) ≠ 0 for h {1, .., H} and k {1, ..,K}.
Assumptions: The instrumental variables satisfy the following 16 properties. Well behaved data: plimN-1Z’Z=QZZ a finite HxH positive definite matrix Relevance: plimN-1Z’X=QZX a finite HxK positive definite matrix Exogeneity: plimN-1Z’ε=0K1 Definition (Instrument properties) We assume that the H instruments are linearly independent E(Z’Z) is non singular Or equivalently rank (E(Z’Z)= H
(1) Exogeneity: They are uncorrelated with the disturbance. 17 E(εj/zj)= 0Nx1 E (εjzj)= 0 can expressed as an orthogonality condition or moment condition   E  zj  yj  xj   0 '   H ,1   H ,1  1,1  So, we have H equations and K unknown parameters Definition (Identification). The system is identified if there exists a unique vector β such that:   E  z j  yj  xj   0 '   H ,1   H ,1  1,1  where zj = (zj1..zjH )’ . For that, we have the following conditions:
(1) If H < K the model is not identifed. 18 (2) If H = K the model is just-identifed. (3) If H > K the model is over-identifed. Number of instrustments H H
5.3.3 Motivation of the IV estimator 19 By definition of the instruments: 1 1 p lim Z '   p lim Z '  y  X    0 K 1 N N so we have 1  1  p lim Z ' y   p lim Z ' X   N  N  or equivalently 1  1  1    p lim Z ' X  p lim Z ' y  N  N  If H = K, the Instrumental Variable (IV) estimator  IV of parameters β is defined as to be:    Z ' X  Z ' y 1 IV
Proof 20    Z ' X 1 Z ' y   Z ' X 1 Z '  X         Z ' X 1  Z '    IV 1        1 Z ' X   1 Z '  E  IV   N     N   so we have 1   1   1  p lim  IV     p lim Z ' X   p lim Z '    N   N  Under the assumption of exogeneity of the instruments 1 1 p lim Z '   p lim Z '  y  X    0 K 1 N N so we have  p lim  IV  