Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Chapter HYPOTHESIS TESTING I MAXIMUM LIKELIHOOD ESTIMATORS: n (θ ) = ∏ f ( Z i , θ ) → θˆMLE = arg max (θ ) θ i =1 n ∑ ln f (Z ,θ ) = L(θ ) ln= (θ ) • i i =1 Asymptotic normality: ∂L Solve: θˆMLE for =0 ∂θ −1 ∂2 L ∂θ∂θ ' θˆMLE ~ N θ , -E ∂L ∂L ′ ∂2 L I (θ ) = E = −E ∂θ ∂θ ∂θ∂θ ' ∂L ∂θ 1 ∂L ∂L = ∂θ ∂θ ∂L ∂θ k θ1 θ θ = 2 θ k ∂2 L ∂θ1 ∂2 L ∂2 L = ∂θ ∂θ1 ∂θ∂θ ′ ∂ L ∂θ k vector (k ì1) L ∂θ1∂θ ∂2 L ∂θ 22 ∂ L ∂θ k ∂θ ∂2 L ∂θ1∂θ k ∂2 L ∂θ ∂θ k ∂2 L ∂θ k2 For the linear model: Y = Xβ + ε ( n×1) Nam T Hoang UNE Business School ( n×k )( k ×1) ( n×1) University of New England Advanced Econometrics - Part II Chapter 2: Hypothesis Testing → Y = Xβˆ + e ε ~ N (0,σ I ) n n L( β ,σ ) = − ln 2π − ln σ − (Y − Xβ )′(Y − Xβ ) 2 2σ ∂L − (− X ′Y + X ′X β ) ∂β = σ n ∂L = − + (Y − X β )′(Y − X β ) 2σ 2σ ∂σ βˆ = ( X ' X ) −1 X 'Y → ˆ ˆ e' e σˆ = (Y − Xβ )′(Y − Xβ ) = n n σ ( X ' X ) −1 −1 ∂2 L −E = ′ ∂ ∂ θ θ (= 0) (= 0) e1 e e = 2 en 2σ 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 : c(θ ) = q • If the restriction is valid, then c(θˆ) − q should be close to zero We reject the θˆ is restriction MLE of θ 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 : 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 = ) W ~ X [2J ] Nam T Hoang UNE Business School University of New England Advanced Econometrics - Part II III • Chapter 2: Hypothesis Testing LIKELIHOOD RATIO TEST: H : 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: λ= Lˆ R LˆU (0 ≤ λ ≤ 1) If the restriction c(θ ) = q is valid then Lˆ R should be close to LˆU • Under H : 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 : 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 ) ∂c (θ ) ∂L (θ ) = + λ= ∂θ ∂θ ∂θ ′ ∂L* (θ ) = c (θ ) − = q ∂λ Nam T Hoang UNE Business School 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 in the maximized value of the likelihood function This means ∂L(θˆR ) ∂θˆ is close to or λ R is close to We can test this hypothesis: H : c(θ ) = q → leads to LM test −1 ′ ∂L(θˆR ) ∂ L(θˆR ) ∂L(θˆR ) = −E LM ∂θˆ ∂θˆ ∂θˆ′ ∂θˆ R R R R Under the null hypothesis H : λ = 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) H : Rβ = q q ( j ×k ) ( n×1) ( n×1) ( n×k )( k ×1) ( j×1) R ( j ×k ) Wald test: βˆ is an MEE of β (unrestriction) ( )[ ]( ) ′ −1 −1 W = Rβˆ − q Rσˆ ( X ′X ) R′ Rβˆ − q ~ X [2J ] βˆ is an unrestriction estimator of β : σˆ = e′e n It can be shown that: W = n(e′R eR − e′e) ~ 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 University of New England Advanced Econometrics - Part II Chapter 2: Hypothesis Testing LR test: H : Rβ = q λ= L( βˆ R , X ) L( βˆ , X ) ] [ LR = −2 ln λ = ln L( βˆ ) − ln L( βˆR ) ~ X [2J ] It can be shown: LR = n(ln e′R eR − ln e′e) eR = Y − XβˆR LM test: H : 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 = = = e′R eR e′R e′R σˆ (3) It can be shown: 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′R eR e′e e′e e′R eR (2) 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 University of New England Advanced Econometrics - Part II VI Chapter 2: Hypothesis Testing 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 : 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 ) = HA : plim(θˆ − θ ) ≠ 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 University of New England Advanced Econometrics - Part II VII Chapter 2: Hypothesis Testing 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] = - Pr[accept H0/H0 wrong] = - Pr[Type II error] Monte-Carlo: set H0 wrong, → see the probability of reject H0 → power size Nam T Hoang UNE Business School University of New England ...Advanced Econometrics - Part II Chapter 2: Hypothesis Testing → Y = Xβˆ + e ε ~ N (0,σ I ) n n L( β ,σ ) = − ln 2π − ln σ − (Y − Xβ )′(Y −... UNE Business School University of New England Advanced Econometrics - Part II III • Chapter 2: Hypothesis Testing LIKELIHOOD RATIO TEST: H : c(θ ) = q Let θˆU be the maximum likelihood estimator... Hoang UNE Business School 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