Bài giảng Phân tích số liệu mảng - Chương 3: Fix effect model (FEM)

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Bài giảng Phân tích số liệu mảng - Chương 3: Fix effect model (FEM). Chương này cung cấp cho sinh viên những nội dung gồm: giới thiệu về Fix Effect Model; ước tính các tham số độ dốc trong FEM bằng công cụ ước tính bên trong, công cụ ước tính giữa; ước tính FEM bằng phương pháp biến giả bình phương tối thiểu (LSDV);... Mời các bạn cùng tham khảo!

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Nội dung Text: Bài giảng Phân tích số liệu mảng - Chương 3: Fix effect model (FEM)

6/6/2022 Chapter 3 Fix Effect Model (FEM) Mr U_KHOA TOÁN KINH TẾ 37
Objectives 38 (1) Introduce about Fix Effect Model (2) Estimates the slope paramaters in FEM by Within Estimator, Between Estimator (3) Estimates FEM by Least Square Dummy Variables (LSDV) method Mr U_KHOA TOÁN KINH TẾ 6/6/2022
3.1 Introduce about FEM 39 Notations Let us denote æ yi1 ö æ x x 2i1 ... x Ki1 ö æ b1 ö 1,i,1 ç ÷ ç ÷ ç ÷ ç yi2 ÷ ç x1i2 x 2i2 ... x Ki2 ÷ ç b2 ÷ yi = ç ÷ ;X i = ç ÷ ;b = ç ÷ ç ... ÷ ç ... ... ... ... ÷ ç ... ÷ ç è yiT ÷ ø T´1 ç x1iT è x 2iT ... x KiT ÷ ø T´K ç è bK ÷ø K´1 Let us denote e a unit vector and εi the vector of errors æ ö æ e i1 ö 1 ç ÷ ç ÷ e i2 ÷ e=ç 1 ÷ ;e i = ç ç ÷ ç ... ÷ ç ... ÷ ç è ÷ ø T´1 ç Mr U_KHOA TOÁN KINH TẾ 1 è e iT ÷ ø T´1 6/6/2022
We consider the fix effect model: 40 yi = α0 + eαi + Xiβ+ εi. (i = 1, .., n) (3.1) where αi is assumed to be a constant term or have correlation with the explanatory variables Assumption 3.1 The error term εit are i.i.d ( it) with: • E (εit) = 0 • E (εitεis) = σ2ε when t =s and = 0 if t ≠ s or E(εiε’i) = σ2εIT here IT denotes the identity matrix (T,T) • E (εitεjs) = 0 , i ≠ j, (ts), or E(εiε’j)= 0T here 0T denotes the identity matrix (T,T) Theorem 3.1 Under assumption (3.1), OLS estimator of parameters (β) is the U_KHOA linear unbiased estimator (BLUE) Mr best TOÁN KINH TẾ 6/6/2022
3.2 Estimates the slope paramaters 41 Case 1. Single regression Method 1. Within Estimator yi = eαi + Xiβ+ εi. (i = 1, .., n) (3.1) yit = αi + xit β+ εit ( it) (3.1) Taking mean of this equation (3.1) over time for each cross section unit i, we have yi = a i + x ib + e i (3.2) Again by taking average Eq. (3.2) across individuals, we have y.. = a i + x ..b + e.. (3.3) Subtracting Eq. (3.2) from Eq. (3.1) for each t to get (y it ) ( ) ( - yi = b x it - x i + e it - e i Mr U_KHOA TOÁN KINH TẾ ) (3.4) 6/6/2022
Remark. (3.4) can be estimated by applying OLS, also calling the name 42 “Within Estimator” Method 2. Between estimator (BE) Subtracting (3.2) from (3.3) for each t to get ( y - y ) = b( x - x ) + (e - e ) i .. i .. i .. (3.6) Between estimation (3.6) by OLS Example 3.1 Using file “Data_Ch1.xlsx” to run the following model it = 0 + 2 _ it + 𝑖𝑡 Mr U_KHOA TOÁN KINH TẾ 6/6/2022
Remark. OLS estimates by Pooled OLS can be looked as the 45 weight sum of within estimates and between estimates Case 2. Multiple regression Method 1. Within estimation yit = αi + xit β + εit (3.8) here - β’= (β1, β2, …, βk) ; - x’it = (x1it, x2it, …, xkit); - αi is a scalar intercepts representing the unobserved effects which are same over time; Mr U_KHOA TOÁN KINH TẾ 6/6/2022
- The error term, εit , represents the effects of omitted variables 46 that will change across the individual units and time periods. Assumption. εit is not uncorrelate with xit and εit ~ N(0, σ2ε) In vector form, (3.8) can expressed for unit i as æ yi1 ö æ a i ö æ x1i1 x 2i1 ... x ki1 ö æ b1 ö æ e i1 ö ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç yi2 ÷ ç a i ÷ ç x1i2 x 2i2 ... x ki2 ÷ ç b2 ÷ ç e i2 ÷ ç ÷ =ç ÷ +ç ÷ç ÷ +ç ÷ ç ... ÷ ç ... ÷ ç ... ... ... ... ÷ ç ... ÷ ç ... ÷ ç è yiT ÷ ç a i ÷ ç x1iT ø è ø è x 2iT ... x kiT ÷ ç øè bk ÷ ç ø è e iT ÷ ø Or, yi = ea i + X ib + e i (3.9) e is a vector of oder T, e’ = (1, 1, …,1) Mr U_KHOA TOÁN KINH TẾ 6/6/2022
1 Set Q = I T - ee' T 47 Pre multiplying Eq. (3.9) by Q, we have Qyi = Qeαi + QXiβ+ Qεi Now, æ 1 1 1 ö ç 1- - ... - ÷æ æ y -y ö ç T T T ÷ yi1 ö ç i1 i. ÷ ç 1 1 1 ÷ç ÷ æ 1 ö - 1- ... - ÷ç yi2 ÷ ç yi2 - yi. ÷ Qyi = ç I T - ee'÷ yi = ç T T T ÷ =ç ÷ è T ø ç ÷ç ... ÷ ç ... ÷ ç ... ... ... ... ÷ç ç ÷çè yiT ÷ ç y - y ø è iT ÷ 1 1 1 i. ø ç - - ... 1- ÷ è T T T ø Eq. (3.9) can show that Qe =0 → Qyi = QXiβ+ Qεi (3.10) WeU_KHOA TOÁN KINH TẾOLS to find β parameter of Eq. (3.10) Mr can apply 6/6/2022
48 For all cross section units N and over time T, Eq. (3.10) can be expressed in the following matrix form: QY = QDα + QXβ+ Qε = QXβ+ Qε (3.12) Here, æ y1 ö æ ö æ X1 ö æ e1 ö ç ÷ e 0 ... 0 ç ÷ ç ÷ y2 ÷ ç ÷ X2 ÷ e2 ÷ ç 0 e ... 0 ÷ ,X = ç ç Y=ç ÷ ,D = ç ç ... ... ... ... ÷ ç ÷ ,e = ç ÷ ç ... ÷ ç ... ÷ ç ... ÷ ç ç è ÷ ø è yN ÷ø 0 0 ... e ç è XN ÷ø ç è eN ÷ø The OLS obtained from Eq. (3.12) is Mr U_KHOA TOÁN KINH TẾ 6/6/2022
49 Substituting (3.13) into (3.12) Therefore, Mr U_KHOA TOÁN KINH TẾ 6/6/2022
Pre Example 3.1 With model 50 ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε Requirement: - The within estimates by OLS - The between estimates by OLS Remark. The within estiamates can also obtained by panel regression by using xtreg command in Stata with option fixed efects denoting by fe Mr U_KHOA TOÁN KINH TẾ 6/6/2022
3.3 Least Squares Dummy Variable (LSDV) Regression 51 yit = αi + xit β+ εit æ yi1 ö æ a i ö æ x1i1 x 2i1 ... x ki1 ö æ b1 ö æ e i1 ö ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç yi2 ÷ ç a i ÷ ç x1i2 x 2i2 ... x ki2 ÷ ç b2 ÷ ç e i2 ÷ ç ÷ =ç ÷ +ç ÷ç ÷ +ç ÷ ç ... ÷ ç ... ÷ ç ... ... ... ... ÷ ç ... ÷ ç ... ÷ ç è yiT ÷ ç a i ÷ ç x1iT ø è ø è x 2iT ... x kiT ÷ ç øè bk ÷ ç ø è e iT ÷ ø Or, yi = ea i + X ib + e i (3.9) In vector form, for all cross section units , Eq. (3.9) can be expressed as Mr U_KHOA TOÁN KINH TẾ 6/6/2022
æ y1 ö æ e ö æ ö æ ö æ X1 ö æ b1 ö æ e1 ö ç ÷ ç 0 0 ç ÷ç ÷ ç 52 ÷ ç y2 ÷ ÷ ç ÷ ç ÷ X2 ÷ ç b2 ÷ ç e2 ÷ ç =ç 0 ÷ ç ... ÷ a1 + ç e ÷ a 2 + ...+ ç 0 ÷ aN + ç ç ÷ç ÷ +ç ÷ ç ... ÷ ÷ ç ... ÷ ç ... ÷ ç ... ÷ ç ... ÷ ç ... ÷ ç ÷ ç ÷ ç ÷ ç è yN ÷ è 0 ø ø è 0 ø è e ø ç è XN ÷ ç øè bk ÷ ç ø è eN ÷ø Here, æ yi1 ö æ ö æ x x 2i1 ... x ki1 ö æ e i1 ö ç ÷ 1 ç 1i1 ÷ ç ÷ ç yi2 ÷ ç ÷ ç x1i2 x 2i2 ... x ki2 ÷ ç e i2 ÷ 1 yi = ç ÷ ;e = ç ç ... ÷ ;X i = ç ÷ ÷ ;e = ç ÷ ç ... ÷ ç ... ... ... ... ÷ ç ... ÷ ç ç è ÷ ø T´1 è yiT ÷ ø T´1 1 ç x1iT è x 2iT ... x kiT ÷ ø T´K ç è e iT ÷ ø T´1 Eq. reduces to Y = α’D + X β + ε Here, D is the NT x N matrix for dummy regressor and can be expressed as D = IN Ä eT Mr U_KHOA TOÁN KINH TẾ 6/6/2022
Kronecker product AÄ B of two matrices A = (aij)nm and B = 53 (bkl)n1m1 is defined by æ a B a B ... a 1m B ö 11 12 ç ÷ ç a 21B a 22 B ... a 2m B ÷ AÄB= ç ÷ ç ... ... ... ... ÷ ç a n1B a n2 B è ... a nm B ÷ ø nn ´mm 1 1 ( A Ä B) ' = A 'Ä B' ( A Ä B) = A Ä B -1 -1 -1 ( A Ä B)(C Ä D ) = AC Ä BD Mr U_KHOA TOÁN KINH TẾ 6/6/2022
Assumption 3.1 The error term εit are i.i.d ( it) with: 54 • E (εit) = 0 • E (εitεis) = σ2ε when t =s and = 0 if t ≠ s or E(εiε’i) = σ2εIT here IT denotes the identity matrix (T,T) • E (εitεjs) = 0 , i ≠ j, (ts), or E(εiε’j)= 0T here 0T denotes the identity matrix (T,T) Theorem 3.1 Under assumption (3.1), OLS estimator of parameters (β) is the best linear unbiased estimator (BLUE) Mr U_KHOA TOÁN KINH TẾ 6/6/2022
The OLS estimator of αi and β are obtained by minmising55 N N Su = å e e = å ( yi - eai - Xib ) ( yi - ea i - X ib ) ' ' i i (3.15) i =1 i =1 Taking partial derivates Eq. (3.15) with respect to αi and β setting them to zero, we have Pre Example (3.1) With model ROAA TOÁN KINH TẾ L_A, SIZE, ASSET_GRO, GDP, INF) + ε Mr U_KHOA = f(HHI, 6/6/2022
Remark. There are too many parameters in the fixed effects model and 56 the loss of degrees of freedom can be avoided if the αi can be assumed random Homework. With three research include: 1. Grunfeld Investment Equation (p.21) 2. Gasoline Demand (p.23) 3. Public Capital Productivity (p.25) a) Estimate parameters ahead of the explanatory variables in those study by Within estimator, Between estimator, LSDV and FEM by command xtreg. b) Comparing results receiving from those models c) Explanatory about αi parameters in method LSDV 6/6/2022 Mr U_KHOA TOÁN KINH TẾ