Advanced Econometrics - Part II - Chapter 2: Hypothesis Testing

Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 1 University of New England Chapter 2 HYPOTHESIS TESTING I. MAXIMUM LIKELIHOOD ESTIMATORS: 1 ( ) ( , ) n i i f Zθ θ = =∏ ˆ arg max ( )MLE θ θ θ→ =  i 1 ( ) ln ( ) ln ( , ) n iL f Zθ θ θ = = = ∑ • Asymptotic normality: Solve: MLEθˆ for 0 L θ ∂ = ∂ 12 ˆ ~ N , -E 'MLE L θ θ θ θ −  ∂   ∂ ∂   2 ( ) E ' L L LI Eθ θ θ θ θ ′   ∂ ∂ ∂   = = −    ∂ ∂ ∂ ∂       θ vector )1( ×k 1 2 k L L L L θ θ θ θ ∂   ∂   ∂ ∂  ∂=  ∂     ∂   ∂               = kθ θ θ θ  2 1 2 2 2 2 1 2 11 2 2 2 2 2 2 1 22 2 2 2 2 21 k k k k k L L L L L L L L L L θ θ θ θθ θ θ θ θθ θ θ θ θ θ θ θ  ∂ ∂ ∂  ∂ ∂ ∂ ∂∂   ∂ ∂ ∂  ∂ = ∂ ∂ ∂ ∂∂ ′∂ ∂      ∂ ∂ ∂  ∂ ∂ ∂ ∂ ∂         • For the linear model: )1()1)(()1( ×××× += nkknn XY εβ Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 2 University of New England ),0(~ ˆ 2IN eXY σε β +=→ )()( 2 1ln 2 2ln 2 ),( 2 22 ββ σ σπσβ XYXYnnL −′−−−−= 2 2 2 2 1 ( ) 1 ( ) ( ) 2 2 L X Y X X L n Y X Y X β β σ β β σ σ σ ∂ ′ ′= − − +∂  ∂ ′= − + − −∂ )0( )0( = =     =−′−= = → − n eeXYXY n YXXX ')ˆ()ˆ(1ˆ ')'(ˆ 2 1 ββσ β             = ne e e e  2 1 2 1 12 4 ( ' ) 0 20 X X LE n σ σθ θ − −    ∂  − =   ′∂ ∂     • We consider maximum likelihood estimator θ & the hypothesis: qc =)(θ II. WALD TEST • Let θˆ be the vector of parameter estimator obtained without restrictions. • We test the hypothesis: qcH =)(:0 θ θˆ is restriction MLE of θ • If the restriction is valid, then qc −)ˆ(θ should be close to zero. We reject the hypothesis of this value significantly different from zero. • The Wald statistic is: ( ) ])ˆ([])ˆ([])ˆ([ 1 qcqcVarqcW −−′−= − θθθ Under: qcH =)(:0 θ • W has chi-squared distribution with degree of freedom equal to the number of restrictions (i.e number of equations in 0)ˆ( =− qc θ ) [ ] 2 JX~W Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 3 University of New England III. LIKELIHOOD RATIO TEST: • qcH =)(:0 θ  Let Uθˆ be the maximum likelihood estimator of θ obtained without restriction.  Let Rθˆ be the MLE of θ with restrictions.  If RU LL ˆ&ˆ are the likelihood functions evaluated at these two estimate.  The likelihood ratio: U R L L ˆ ˆ =λ ( )10 ≤≤ λ  If the restriction qc =)(θ is valid then RLˆ should be close to ULˆ . • Under [ ] 2 J0 X~ln2)(: λθ −→= qcH is chi-squared, with degree of freedom equal to the number of restrictions imposed. [ ] 2 JX~ln2 λ−=LR IV. LAGRANGE MULTIPLIER TEST (OR SCORE TEST): qcH =)(:0 θ Let λ be a vector of Lagrange Multipliers, define the Lagrange function: ( ) ( ) [ ]qcLL −′+= )(* θλθθ The FOC is: ( ) ( ) ( ) ( ) ( ) * * 0 0 L L c L c q θ θ θ λ θ θ θ θ θ λ ′∂ ∂ ∂  = + =  ′∂ ∂ ∂  ∂ = − = ∂ Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 4 University of New England 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 ˆ( ) ˆ R R L θ θ ∂ ∂ is close to 0 or λ is close to 0. We can test this hypothesis: →= qcH )(:0 θ leads to LM test. 1 2ˆ ˆ ˆ( ) ( ) ( ) ˆ ˆ ˆ ˆ R R R R R R R L L LLM Eθ θ θ θ θ θ θ −′       ∂ ∂ ∂ = −           ′∂ ∂ ∂ ∂        Under the null hypothesis 0:0 =λH 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: ( )1)1()1)(()1( ˆ ××××× +=+= nnkknn eXXY βεβ ( ) ( )1 0 : ×× = jkj qqRH β ( )kj R × 1. Wald test: βˆ is an MEE of β (unrestriction) ( ) ( )[ ] ( ) [ ]2J112 X~ˆˆˆ qRRXXRqRW −′′′−= −− βσβ βˆ is an unrestriction estimator of β : n ee′ =2σˆ It can be shown that: [ ] 2 JX~ )( ee eeeenW RR ′ ′−′ = (1) With RR XYe βˆ−= Rβˆ is an estimator subject to the restriction βR . Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 5 University of New England 2. LR test: qRH =β:0 ),ˆ( ),ˆ( XL XL R β β λ = [ ] [ ]2JX~)ˆ(ln)ˆ(ln2ln2 RLLLR ββλ −=−= It can be shown: )ln(ln eeeenLR RR ′−′= RR XYe βˆ−= 3. LM test: 0 :H R qβ = It can be shown: RR RR RR RRRR ee eeeen ee eXXXXeneXXXXneLM ′′ ′−′ = ′ ′′′ = ′′ = −− )()( ˆ )( 1 2 1 σ (3) It can be shown: 22 2 )( 2 )(       ′ ′−′ + ′ ′−′ =      ′ ′−′ − ′ ′−′ = RR RR RR RRRRRR ee eeeen ee eeeen ee eeeen ee eeeenLR (2) From (1), (2), (3) we have: For the linear models: LMLRW ≥≥ 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. Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 6 University of New England 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: OLSβˆ & SLS2βˆ estimators. Where SLS2βˆ uses instruments to control for possible endogeneity of the regressor: SLSOLSH 20 ˆˆ: ββ ≈ Hausman’s statistic: [ ] [ ]2r2122 ~)ˆˆ()ˆˆ()ˆˆ( χββββββ OLSSLSOLSSLSOLSSLS VarCov −−′− − r: the number of endogenous regressors. Model general: consider two estimators θˆ and θ~ We consider the test situation where: H0 : ˆplim( ) 0θ θ− = HA : ˆplim( ) 0θ θ− ≠ Assume under H0 : )) ~ˆ(,0()~ˆ( θθθθ −→− VarNn The Hausman test statistic: [ ] 2 q 1 ~)~ˆ()~ˆ(1)~ˆ( χθθθθθθ −      −−= − Var n H q is rank of )~-ˆ( θθVar For the linear model: )ˆ()ˆ()ˆˆ( 22 OLSSLSOLSSLS VarCovVarCovVarCov ββββ −=− Advanced Econometrics - Part II Chapter 2: Hypothesis Testing Nam T. Hoang UNE Business School 7 University of New England 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, 05.0=α 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.