zunia.vn

Tuyển sinh 2024 dành cho Gen-Z

zunia.vn

» Kinh Tế - Quản Lý

» Kinh tế học

Lecture Advanced Econometrics (Part II) - Chapter 2: Hypothesis testing

Chia sẻ: Tran Nghe | Ngày: | Loại File: PDF | Số trang:7

Báo xấu

56
lượt xem 3
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Lecture "Advanced Econometrics (Part II) - Chapter 2: Hypothesis testing" presentation of content: Maximum likelihood estimators, wald test, likelihood ratio test, lagrange multiplier test, application of tests procedures to linear models, hausman specification test, power and size of tests.

Chủ đề:

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

Lưu

Nội dung Text: Lecture Advanced Econometrics (Part II) - Chapter 2: Hypothesis testing

Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Chapter 2 HYPOTHESIS TESTING I. MAXIMUM LIKELIHOOD ESTIMATORS: n (θ ) = ∏ f ( Z i , θ ) → θˆMLE = arg max (θ ) i =1 θ n L(θ ) ln= = (θ ) ∑ ln f (Z ,θ ) i =1 i • Asymptotic normality: ∂L Solve: θˆMLE for =0 ∂θ   ∂2 L   −1 θˆMLE ~ N  θ , -E      ∂θ∂θ '     ∂L ∂L ′   ∂2 L     I (θ ) = E    = −E   θ vector (k ×1)  ∂θ  ∂θ    ∂θ∂θ '     ∂L   ∂θ   1 θ1   ∂L  θ  ∂L   =  ∂θ 2  θ =  2 ∂θ          ∂L  θ k   ∂θ k   ∂2 L ∂2 L ∂2 L      ∂θ1 2 ∂θ1∂θ 2 ∂θ1∂θ k   ∂2 L ∂2 L ∂2 L  ∂2 L    =  ∂θ 2 ∂θ1 ∂θ 22 ∂θ 2 ∂θ k  ∂θ∂θ ′          ∂ L 2 ∂ L2 ∂2 L      ∂θ k ∂θ 1 ∂θ k ∂θ 2 ∂θ k2  • For the linear model: Y = Xβ + ε ( n×1) ( n×k )( k ×1) ( n×1) Nam T. Hoang UNE Business School 1 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing → Y = Xβˆ + e ε ~ N (0,σ 2 I ) n n 1 L( β ,σ 2 ) = − ln 2π − ln σ 2 − (Y − Xβ )′(Y − Xβ ) 2 2 2σ 2  ∂L 1 (= 0)  ∂β = − 2 (− X ′Y + X ′X β ) σ   ∂L = n 1 − 2 + 2 (Y − X β )′(Y − X β )  ∂σ 2 2σ 2σ (= 0)  e1  βˆ = ( X ' X ) −1 X 'Y e   → 2 1 e =  2 ˆ ˆ e' e  σˆ = (Y − Xβ )′(Y − Xβ ) =  n n   en  −1 σ 2 ( X ' X ) −1 0   ∂2 L    −E   = 2σ 4  ∂θ∂θ ′  0    n  • We consider maximum likelihood estimator θ & the hypothesis: c(θ ) = q II. WALD TEST • Let θˆ be the vector of parameter estimator obtained without restrictions. • We test the hypothesis: H 0 : c(θ ) = q θˆ is restriction MLE of θ • If the restriction is valid, then c(θˆ) − q should be close to zero. We reject the hypothesis of this value significantly different from zero. • The Wald statistic is: ( −1 ) W = [c(θˆ) − q ]′ Var[c(θˆ) − q ] [c(θˆ) − q ] Under: H 0 : c(θ ) = q • W has chi-squared distribution with degree of freedom equal to the number of restrictions (i.e number of equations in c(θˆ) − q = 0 ) W ~ X [2J ] Nam T. Hoang UNE Business School 2 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing III. LIKELIHOOD RATIO TEST: • H 0 : c(θ ) = q  Let θˆU be the maximum likelihood estimator of θ obtained without restriction.  Let θˆR be the MLE of θ with restrictions.  If LˆU & Lˆ R are the likelihood functions evaluated at these two estimate.  The likelihood ratio: Lˆ R λ= LˆU (0 ≤ λ ≤ 1)  If the restriction c(θ ) = q is valid then Lˆ R should be close to LˆU . • Under H 0 : c(θ ) = q → −2 ln λ ~ X [2J ] is chi-squared, with degree of freedom equal to the number of restrictions imposed. LR = −2 ln λ ~ X [2J ] IV. LAGRANGE MULTIPLIER TEST (OR SCORE TEST): H 0 : c(θ ) = q Let λ be a vector of Lagrange Multipliers, define the Lagrange function: L* (θ ) = L(θ ) + λ ′[c(θ ) − q ] The FOC is:  * ′  ∂L (θ ) =∂L (θ )  ∂c (θ )   ∂θ +  λ=0 ∂θ  ∂θ ′    ∂L* (θ )  = c (θ ) − = q 0  ∂λ Nam T. Hoang UNE Business School 3 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing If the restrictions are valid, then imposing them will not lead to a significant difference ∂L(θˆR ) in the maximized value of the likelihood function. This means is close to 0 or λ ∂θˆR is close to 0. We can test this hypothesis: H 0 : c(θ ) = q → leads to LM test. ′ −1  ∂L(θˆR )    ∂ 2 L(θˆR )    ∂L(θˆR )  =LM  −E  ∂θˆ    ∂θˆ ∂θˆ′    ∂θˆ   R    R R    R  Under the null hypothesis H 0 : λ = 0 LM has a limiting chi-squared distribution with degrees of freedom equal to the number of restrictions. Graph V. APPLICATION OF TESTS PROCEDURES TO LINEAR MODELS Model: Y = Xβ + ε = Xβˆ + e ( n×1) ( n×k )( k ×1) ( n×1) ( n×1) H 0 : Rβ = q q R ( j ×k ) ( j×1) ( j ×k ) 1. Wald test: βˆ is an MEE of β (unrestriction) (′ )[ W = Rβˆ − q Rσˆ 2 ( X ′X ) R′ Rβˆ − q ~ X [2J ] −1 −1 ]( ) e′e βˆ is an unrestriction estimator of β : σˆ 2 = n It can be shown that: n(e′R eR − e′e) W = ~ X [2J ] e′e (1) With eR = Y − Xβˆ R βˆR is an estimator subject to the restriction Rβ . Nam T. Hoang UNE Business School 4 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing 2. LR test: H 0 : Rβ = q L( βˆ R , X ) λ= L( βˆ , X ) [ ] LR = −2 ln λ = 2 ln L( βˆ ) − ln L( βˆR ) ~ X [2J ] It can be shown: LR = n(ln e′R eR − ln e′e) eR = Y − XβˆR 3. LM test: H 0 : Rβ = q It can be shown: neR X ( X ′X ) −1 X ′eR ne′R X ( X ′X ) −1 X ′eR n(e′R eR − e′e) LM = = = (3) σˆ 2 e′R eR e′R e′R It can be shown: 2 2 n(e′R eR − e′e) n  e′R eR − e′e  n(e′R eR − e′e) n  e′R eR − e′e  LR = −   = +   (2) e′e 2 e′e  e′R eR 2  e′R eR  From (1), (2), (3) we have: For the linear models: W ≥ LR ≥ LM The tests are asymptotically equivalent but in general will give different numerical results in finite samples. Which test should be used? The choice among would, LR & LM is typically made on the Basic of ease of computation. LR require both restrict & unrestrict. Wald require only unrestrict & LM requires only restrict estimators. Nam T. Hoang UNE Business School 5 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing VI. HAUSMAN SPECIFICATION TEST: - Consider a test for endogeneity of a regressor in linear model. - Test based on comparisons between two different estimators are called Hausman Test. - Two alternative estimators are: βˆOLS & βˆ2 SLS estimators. Where βˆ2 SLS uses instruments to control for possible endogeneity of the regressor: H 0 : βˆOLS ≈ βˆ2 SLS Hausman’s statistic: [ ] −1 ( βˆ2 SLS − βˆOLS )′ VarCov( βˆ2 SLS − βˆOLS ) ( βˆ2 SLS − βˆOLS ) ~ χ [2r ] r: the number of endogenous regressors. Model general: ~ consider two estimators θˆ and θ We consider the test situation where: H0 : plim(θˆ − θ ) = 0 HA : plim(θˆ − θ ) ≠ 0 ~ ~ Assume under H0 : n (θˆ − θ ) → N (0,Var (θˆ − θ )) The Hausman test statistic: −1 ~ 1 ~  ~ H = (θˆ − θ ) Var (θˆ − θ )  (θˆ − θ ) ~ χ [2q ] n  ~ q is rank of Var (θˆ - θ ) For the linear model: VarCov( βˆ2 SLS − βˆOLS ) = VarCov( βˆ2 SLS ) − VarCov( βˆOLS ) Nam T. Hoang UNE Business School 6 University of New England
Advanced Econometrics - Part II Chapter 2: Hypothesis Testing VII. POWER AND SIZE OF TESTS: Size of a test: Size = Pr[type I error] = Pr[reject H0 | H0 true] Common choices: 0.01, 0.05 or 0.1, α = 0.05 Monte-Carlo: set H0 true, → see the probability of reject H0 → size Power of a test: Power = Pr [reject H0/H0 wrong] = 1 - Pr[accept H0/H0 wrong] = 1 - Pr[Type II error] Monte-Carlo: set H0 wrong, → see the probability of reject H0 → power size. Nam T. Hoang UNE Business School 7 University of New England

CÓ THỂ BẠN MUỐN DOWNLOAD

CEO.18: Bộ Tài Liệu Tiêu Chuẩn ISO Trong Ngành Công Nghệ Thông Tin

26 tài liệu

1748 lượt tải

THÔNG TIN

TRỢ GIÚP

HỖ TRỢ KHÁCH HÀNG

Theo dõi chúng tôi

Chịu trách nhiệm nội dung:

Nguyễn Công Hà - Giám đốc Công ty TNHH TÀI LIỆU TRỰC TUYẾN VI NA

LIÊN HỆ

Địa chỉ: P402, 54A Nơ Trang Long, Phường 14, Q.Bình Thạnh, TP.HCM

Hotline: 093 303 0098

Email: support@tailieu.vn

Giấy phép Mạng Xã Hội số: 670/GP-BTTTT cấp ngày 30/11/2015 Copyright © 2022-2032 TaiLieu.VN. All rights reserved.