Bài giảng phân tích số liệu mảng chương 3 fix effect model (fem)

6/6/2022 Chapter Fix Effect Model (FEM) Mr U_KHOA TOÁN KINH TẾ 37 Objectives (1) Introduce about Fix Effect Model 38 (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 ổ ỗ ç yi = ç ç çè ỉ x yi1 1,i,1 ỗ ữ yi2 ữ ỗ x1i2 ữ ;X i = ỗ ữ ỗ ỗố x1iT yiT ữứ T1 ổ x 2i1 x Ki1 ữ ỗ x 2i2 x Ki2 ữ ỗ ữ ;b = ç ÷ ç çè x 2iT x KiT ÷ø T´K b1 ÷ b2 ÷ ÷ ÷ b K ÷ø K´1 Let us denote e a unit vector and εi the vector of errors Mr U_KHOA TON KINH T ổ ỗ e=ỗ ỗ ỗố 1 ổ ỗ ữ ữ ;e i = ỗỗ ữ ỗ ữứ ỗố T1 e i1 ÷ e i2 ÷ ÷ ÷ e iT ÷ø T´1 6/6/2022 We consider the fix effect model: yi = α0 + eαi + Xiβ+ εi (i = 1, , n) (3.1) 40 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) = • E (εitεis) = σ2ε when t =s and = if t ≠ s or E(εiε’i) = σ2εIT here IT denotes the identity matrix (T,T) • E (εitεjs) = , 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 Mr U_KHOA TOÁN KINH TẾ the best linear unbiased estimator (BLUE) 6/6/2022 3.2 Estimates the slope paramaters 41 Case Single regression Method 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 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 Mr U_KHOA TOÁN KINH TẾ = + _ it + 𝑖𝑡 6/6/2022 Remark OLS estimates by Pooled OLS can be looked as the 45 weight sum of within estimates and between estimates Case Multiple regression Method 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 ổ ỗ ỗ ç ç çè yi1 ỉ a i ỉ x1i1 ữ ỗ ữ ỗ yi2 ữ ỗ a i ữ ỗ x1i2 ữ =ỗ ữ +ỗ ữ ỗ ữ ỗ yiT ữứ ỗố a i ữứ çè x1iT x 2i1 x 2i2 x 2iT x ki1 ổ ữỗ x ki2 ữ ỗ ữỗ ữ ỗ x kiT ữứ ỗố b1 ổ ữ ỗ b2 ữ ỗ ữ +ỗ ữ ỗ b k ữứ ỗố e i1 ÷ e i2 ÷ ÷ ÷ 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 Set Q = I T - ee' T 47 Pre multiplying Eq (3.9) by Q, we have Qyi = Qeαi + QXiβ+ Qεi Now, æ 1 ç 1T T T ç 1 ç æ Qyi = ỗ I T - ee'ữ yi = ỗ T T T ỗ T ứ ố ỗ ỗ 1 1ỗ T T T ố ữổ ữ ữỗ ữỗ ữỗ ữỗ ữ ỗố ữ ứ ổ y -y yi1 i1 i ỗ ữ yi2 ữ ỗ yi2 - yi ữ =ỗ ữ ỗ yiT ữứ ç y - y i è iT Eq (3.9) can show that Qe =0 → Qyi = QXiβ+ Qεi (3.10) We can apply OLS to find β parameter of Eq (3.10) Mr U_KHOA TỐN KINH TẾ 6/6/2022 ÷ ÷ ÷ ÷ ÷ ø 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+ Q = QX+ Q (3.12) Here, ổ ỗ ỗ Y=ỗ ç çè y1 ỉ ÷ ç y2 ÷ ÷ ,D = ỗỗ ữ ỗố y N ữứ e 0 e 0 e ổ ỗ ữ ữ ,X = ỗỗ ữ ỗ ữứ ỗố ổ X1 ữ ỗ X2 ữ ỗ ữ ,e = ỗ ữ ỗ ỗố X N ữứ e1 ữ e2 ÷ ÷ ÷ e N ÷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 = i + xit + it ổ ỗ ỗ ỗ ỗ çè yi1 ỉ a i ỉ x1i1 ÷ ç ÷ ç yi2 ÷ ç a i ÷ ç x1i2 ữ =ỗ ữ +ỗ ữ ỗ ữ ç yiT ÷ø çè a i ÷ø çè x1iT x 2i1 x 2i2 x 2iT x ki1 ổ ữỗ x ki2 ữ ỗ ữỗ ữ ỗ x kiT ữứ ỗố b1 ổ ữ ỗ b2 ữ ỗ ữ +ỗ ữ ç b k ÷ø çè e i1 ÷ e i2 ÷ ÷ ÷ 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 ổ ỗ ỗ ỗ ỗ ỗố y1 ổ e ữ ỗ y2 ữ ỗ = ữ ỗ ữ ỗ y N ữứ ố ổ ữ ỗ ữ a1 + ỗ ữ ỗ ữứ çè e ỉ ÷ ç ÷ a + + ỗ ữ ỗ ữứ ỗố 0 e ổ ỗ ữ ữ a N + ỗỗ ữ ỗ ữứ ỗố X1 ổ ữỗ X2 ữ ỗ ữỗ ữ ỗ X N ữứ çè b1 ỉ ÷ ç b2 ÷ ç ÷ +ỗ ữ ỗ b k ữứ ỗố e1 52 ÷ e2 ÷ ÷ ÷ e N ÷ø Here, ổ ỗ ỗ yi = ỗ ỗ ỗố yi1 ổ ữ ỗ yi2 ữ ữ ;e = ỗỗ ữ ỗố yiT ữứ T1 1 ổ x 1i1 ỗ ữ x1i2 ỗ ữ ;X i = ỗ ữ ỗ ữứ ỗố x1iT T1 x 2i1 x 2i2 x 2iT æ x ki1 ữ ỗ x ki2 ữ ỗ ;e = ữ ỗ ữ ỗ ỗố x kiT ÷ø T´K e i1 ÷ e i2 ÷ ÷ ÷ 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 TỐN KINH TẾ 6/6/2022 Kronecker product Ậ B of two matrices A = (aij)nm and B = 53 (bkl)n1m1 is defined by æ a B a B 11 12 ç ç a 21B a 22 B AÄB= ç ç çè a n1B a n2 B a 1m B ÷ a 2m B ÷ ÷ ÷ a nm B ÷ø nn ´mm ( A Ä B) ' = A 'Ä B' ( A Ä B) = A Ä B ( A Ä B)(C Ä D ) = AC Ä BD -1 Mr U_KHOA TOÁN KINH TẾ -1 1 -1 6/6/2022 Assumption 3.1 The error term εit are i.i.d ( it) with: 54 • E (εit) = • E (εitεis) = σ2ε when t =s and = if t ≠ s or E(εiε’i) = σ2εIT here IT denotes the identity matrix (T,T) • E (εitεjs) = , 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 =1 ' i i ' (3.15) 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 Mr U_KHOA TOÁN KINH TẾ ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε 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: Grunfeld Investment Equation (p.21) Gasoline Demand (p.23) 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 Mr U_KHOA TOÁN KINH TẾ 6/6/2022