Bài giảng phân tích số liệu mảng chương 4 random effect model (rem)

6/6/2022 Chapter Random effect model (REM) Mr U_KHOA TOÁN KINH TẾ 57 Objectives (1) Introduce about Random Effect Model 58 (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 Notes There are too many parameters in the fixed effects model and the 59 loss of degrees of freedom can be avoided if the α*i can be assumed random 4.1 Introduce Random effect model yit = x itb + a *i + u it i = 1,N; t = 1,T (4.1) Here, α*i is assumed to be random If the individual effects α*i are supposed to have non zero mean, with E (α*i)= α0 Then we can define cross section units effects α*i= α0 + αi Pre Eq (4.1) Mr U_KHOA TOÁN KINH TẾ 6/6/2022 4.1.1 The assumptions on the components of errors About ( ) , ( ) ( ) ( ) ( ) ( ) ( ) 60 ( ) E a i = 0, ,V a i = E a 2i = s m2 , ,E a i x it = 0, ,E a ia j = About ( ) u it , E u it = 0, ,V u it = E u 2it = s 2u , ,E u it u js = for i ¹ j and t ¹ s The components of the error are not correlated E (αiuit) =0 Remark The αi are independent of the error term uit and the regressors xit, for all i and t 4.1.2 Mean and variance of errors The mean and variance of the component errors are ( ) ( ) ( ) 2 EMr eU_KHOA = 0, ,V e = V y = s + s TOÁN KINH TẾ it it it a u 6/6/2022 The covariance of the composite error, 61 Cov (εit, εjs ) = E(εitεjs)= E(αi+ uit) (αj+ ujs) = E (αiαj + uit αj + αi ujs + uitujs) Or Case Cov (εit, εjs ) = σ2α + σ2u Case Cov (εit, εjs ) = σ2α Case Cov (εit, εjs ) = Mr U_KHOA TOÁN KINH TẾ i = j, t= s i = j, t ≠ s i ≠ j, t ≠ s 6/6/2022 For cross section unit i, Eq (4.1) can be written as 62 The variance- covariance matrix of εi (for individual i) is ổổ ỗỗ ỗỗ ' E e ie i = E ỗ ỗ ỗỗ ỗ ỗố Mr U_KHOA TỐN KINH TẾ è ( ) e i1 ÷ e i2 ÷ ÷ ÷ e iT ÷ø (e i1 e i2 e iT ÷ ÷ ÷ ÷ ÷ ø ) 6/6/2022 æ e2 e i1e i2 e i1e iT i1 ỗ ỗ e i2 e i1 e 2i2 e i2 e iT = Eỗ ỗ ỗ e e e e e iT è iT i1 iT i2 ö ÷ ÷ ÷ ÷ ÷ ø æ s2 + s2 u ỗ a ỗ s a2 =ỗ ỗ ç s a è s s a2 s a s a2 + s 2u s a2 63 a s a2 + s 2u ÷ ÷ ÷ ÷ ÷ ø ỉ 'ư = U = s I + s ee = s ỗ Q + ee ữ + Ts a ee' T ø T è u T ( a ' u ) 2 = MrsU_KHOA Q + s + Ts P u u TẾ a TOÁN KINH (4.2) 6/6/2022 here P = ee' = e e'e T ( ) -1 (4.2) Þ U = Q + qP su ( -1 where q = (s Therefore, U Or, U -1/2 s 2u u ) ) = Q+ su a ổ ỗQ+ P = s u ỗố ( ( Mr U_KHOA TOÁN KINH TẾ ( s 2u + Ts a2 s 2u + Ts a2 2 And U = s 2(T-1) s + Ts u u a 64 (4.3) (4.4) + Ts -1/2 e' = I T - Q ) ) ) P (4.5) ÷ (4.6) ÷ø (4.7) 6/6/2022 By taking all cross section units in the sample, the variance - covariance 65 matrix of the error term (ε) will be of order NT x NT ổ ỗ E ee ' = E ỗ ỗ ỗố ( ) U 0 U 0 U ( ) ( ÷ ÷ ÷ ÷ø ) = U Ä I N = s 2u I T Ä I N + s a2 J T Ä I N = W (4.8) where J T = ee' Mr U_KHOA TOÁN KINH TẾ 6/6/2022 4.2 GLS estimation 66 Idea The generalised least squares (GLS) is used in estimating a random effects model when U is known Suppose that the variance – covariance matrix (U) is known Pre multiply Eq (4.1) by U-1/2 to get Homework Proving why reason Eq (4.10) equal with IT Mr U_KHOA TOÁN KINH TẾ 6/6/2022 With Eq (4.9) we can apply OLS to estimate parameter (γ) 67 The GLS estimators of γ are Eq (4.12) can be written in expanded form as ( Eq (4.12) = S W XX + qS B XX )( -1 B SW + qS Xy Xy ) (4.13) Homework Expanding detail (4.12) to finding why (4.12) can similar (4.13) Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Remark 68 - If θ = 1, then GLS estimator is equivalent to OLS pooled estimator - If θ = 0, then GLS estimator will be equal to LSDV - The parameter θ measures the weight given to between-group variation - If U is unknown, we can use a two-step GLS estimation known with name is called FGLS (Feasible Generalized Least Squares) 4.3 FGLS estimator Note When U is unknown as means as σ2α & σ2u are unidentifed We can use two-step GLS estimation known as FGLS Step We estimate the “within” estimation and “between” estimation model to find out Mr U_KHOA TOÁN KINH TẾ 6/6/2022 69 The σ2α & σ2u are obtained from the ”between ” effect estimation, the “within” effect estimation, etc Then, we have to caculate Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Step We have to estimate the following model: 70 4.4 Testing of Hypotheses Introduction In a panel regression model, either fixed or random effect is an issue of unobserved variables measuring heterogeneity across the entities which renders the bias in pooled regression estimation 4.4.1 Measuring of Goodness Fit Panel data can be utilities to calculate within-entity variation (R2W) , between-entity variation (R2B ) and overall variation (R2) Option Testing for Pooled Regression yi =α+ Xiβ+ εi (i = 1, , n) Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Pre Example (3.1) with model 71 ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε Option Testing for Fix Effects yi = eαi + Xiβ+ εi (i = 1, , n) Method Fix effects model is only valid when we could test the joint significance of the dummies by: H0: α1 = α2 =…= αN = H1: αi ≠ The F test is calculated by the following formular: Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Pre Example (3.1) With model 72 ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε Method F test can alse check by xtreg with option fe in Stata Option Testing of Random Effects H0: σ2α = H1: σ2α > To test this hypothesis, we can use Lagrange Multiplier (LM) test developed by Bresuch and Pagan (1980) Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Pre Example (3.1) With model 73 ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε 4.4.2 Fix or Random effect: Hausman Test yi = eαi + Xiβ+ εi (i = 1, , n) (FE) H0: E (εit|Xit) = H1: E (εit|Xit) ≠ Mr U_KHOA TOÁN KINH TẾ 6/6/2022 Using this fact, we have 74 Therefore, With Then The test statistic is Mr U_KHOA TOÁN KINH TẾ Hausman test (H) = q’ (Var(q))-1q 6/6/2022