Advanced Econometrics - Part I - Chapter 5: Inference & Prediction

Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 5 INFERENCE & PREDICTION I. WALD TESTS: • Nested models: If we can obtain one model from another by imposing restrictions (on the parameters), we say that the Z models are nested. • Non-nested model: If neither model is obtained as a restriction on the parameters of the other model. There two models are non-nested. Example: A Wald test is for choosing between nested models.    ++= +++= iii iiii XY XXY εββ εβββ 221 33221 A Wald test is for choosing between non-nested models.    +++= +++= iiii iiii MHY ZXY εααα εβββ 321 321 We'll be concerned with (several) possible restrictions on β in the usual model. εβ += XY ),0(~ 2 IN σε X: non-stochastic ran(X) = k. The general form of r restrictions: 11 ××× = rkkr qR β R & q re known & non-random, assume rank(R) = r (<k).           −1100 1110 0001             4 3 2 1 β β β β =           0 1 0 Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam →      = =++ = 43 432 1 1 0 ββ βββ β r: number of restrictions. II. LEAST SQUARES DISCREPANCY: • Suppose just estimate the model by OLS & obtain: YXXXu ')'( ˆ 1−=β (u: unrestriction). Let qRm u −= βˆ • Let's consider the sampling distribution of m: qRm ur −=× βˆ1 (a linear function of uβˆ ) =−= )ˆ()( qREmE uβ qRE u −)ˆ(β qR −= β → 0)( =mE if qR =β =−= )ˆ()( qRVarCovmVarCov uβ uVarCovRβˆ 'ˆ RRV uβ= ')'( 12 RXXR −= εσ ')'( 12 RXXR −= εσ So )')'(,0(~ 12 1 RXXRNm rrr × − × ε σ if qR =β • From the theorem for construction of Hausman's test we have: ),0(~1 Σ× Nmr → ~)}({' 1−mVarCovm 2 ][rχ So: ] ˆ[]')'([]'ˆ[ 121 qRRXXRqRW uu −−= −− βσβ ε ~ 2 ][rχ Under H0: qR =β if 2εσ is known. • Usually we don't know 2 εσ , we can replace 2 εσ by any consistent estimator of 2 εσ , say 2 εσ with 2ˆlim εσp = 2 εσ then will get the same asymptotic test distribution 2 ][rχ • Reject H0 if W > critical value (this test is asymptotic test). Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam F-statistics: We can modify the Wald test statistics slightly and get the exact test (not asymptotic) in finite sample: Note that: 222 ' εεε σ εε σσ MeeESS uuu = ′ = =       ′       −×− εε σ ε σ ε )()( knkn M ~ 2 ][ kn−χ We have: )( . ]ˆ[]')'([]'ˆ[ 2 2 121 kn ESS r qRRXXRqR F u uu r kn − −− = −− − ε ε ε σ σ βσβ = )( 2 ][ 2 ][ kn r kn r − −χ χ ~ F(r, n-k) Note: rank(M) = trace(M) = (n-k). • Calculate F. Reject H0 in favour of qR ≠β if F > Fcritical. • We also can prove that: ]ˆ[]')'([]'ˆ[ 121 qRRXXRqRW uu −−= −− βσβ ε UURRUR eeeeESSESS ′−′=−= III. THE RESTRICTED LEAST SQUARES ESTIMATOR: If we test the validity of certain linear restrictions on βˆ and we can't reject them, how might we incorporate them into the estimator? Problem: Minimize e'e st: qR u =βˆ (R: restriction). ]'ˆ[2]ˆ[]'ˆ[ qRXYXY RRR −+−−=℘ βλββ λββββ ]' ˆ[2ˆ'ˆˆ2' qRXXYY RRRR −+′+−= 0'2ˆ'2'2ˆ =++−=∂ ∂℘ λβ β RXXYX R R (i) Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam = ∂ ∂℘ λ ]ˆ[2 qR R −β (ii) → → − )(*)'( 1 iXXR → 0')'(ˆ')'(')'( 11 ˆ 1 =++− −−− λβ β RXXRXXXXRYXXXR R Iu  → λββ ')'( ˆˆ 1 RXXRRR Ru −=− → ] ˆ[]')'([ 11 qRRXXR u −= −− βλλ Put into (i): 0]ˆ[]')'(['ˆ'' 11 =−++−→ −− qRRXXRRXXYX uR βλβ → 0]ˆ[]')'([')'(ˆ')'(')'( 1111 ˆ 1 =−++ −−−−− qRRXXRRXXXXXXYXXX uR Iu βλβ β  → ] ˆ[]')'([')'(ˆˆ 111 qRRXXRRXX uuR −−= −−− βββ Theorem: - The RLS estimator Rβˆ is unbiased if qR =β otherwise it is biased. - The covariance matrix of Rβˆ is: ])'([]')'(['[)'()ˆ( 11112 −−−− −= XXRRXXRRIXXVarCov R εσβ { }])'(]')'([')'()ˆ( 11112 −−−− −= XXRRXXRRIXXVarCov R εσβ Proof: (Exercise) • If the restrictions are valid ( qR =β ) then the RLS estimator Rβˆ is more efficient than OLS estimator (has smaller variance). • If the restriction are false the Rβˆ is not only unbiased, it is also inconsistent → it's good to know how to construct the test (uniform most power) of H0: qR =β . Back to Wald test: (A): )( ]ˆ[]')'([]'ˆ[ 121 kn ee r qRRXXRqR F uu uu r kn − ′ −− = −− − βσβ ε Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam (B): ]ˆ[]')'([')'(ˆˆ 111 qRRXXRRXX uuR −−=− −−− βββ Ru e uRR XXXYXYe u ββββ ˆˆˆˆ −+−=−=  )ˆˆ( Ruu Xe ββ −+= → RRee′ 0)ˆˆ(')'ˆˆ( +−−+′= uRuRuu XXee ββββ (since )0=′ Xeu → RRee′ uuee′− = ]ˆ[]')'([')')('()'(]')'([]'ˆ[ 111111 qRRXXRRXXXXXXRRXXRqR uu −− −−−−−− ββ Using (B): → RRee′ uuee′− = ]ˆ[]')'([]'ˆ[ 11 qRRXXRqR uu −− −− ββ Put into (A): → ( ) ),(~ )( knrF kn ee r eeee F uu uuRR r kn − − ′ ′−′ =− (same as original F-statistics) → ( ) ~2 εσ r eeee F uuRR r kn ′−′ =− r r 2 ][χ → ~r knrF − 2 ][rχ same as original 2 ][rχ statistics. IV. NON-LINEAR RESTRICTIONS: All of the discussion and results so far relate to linear restrictions qR =β . What about non-linear restrictions: 1 )( × = r qC β C is a function. Recall Taylor series expansions: ...)(')(!2 1)(')()()( 0 2 0000 +−+−+= xfxxxfxxxfxf Advanced Econometrics Chapter 5: Inference & Prediction Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam in our vector case: ( ) ...ˆ)()()ˆ( 1 +− ′       ∂ ∂ += × ββ β β ββ CCC k where βˆ is some estimator of β (consistent) [ ] ( ) =      ∂ ∂ − ′       ∂ ∂ = β β ββ β β β )(ˆ)()ˆ( CVCCV ( )       ∂ ∂ ′       ∂ ∂ = β β β β β )(ˆ)( CVC so we can form a Wald test statistics: )ˆ()}ˆcov({)'ˆ( 1 qccasyestqcW −−= − ( ) )ˆ()(ˆ)()'ˆ( 1 ˆˆ qcCVCqcW −               ∂ ∂′      ∂ ∂ −= − ββ β ββ β β where ) ˆ(ˆ βcc = and → dW 2 ][Jχ if any consistent estimator of β and if )ˆ(βV is used. Warning: The value of W is not in variant to the way the non-linear restrictions are written. Ex: β1/β2 = β3 or β1 = β2β3 so, it is possible to get conflicting result → Wald test has this weakness when we have non- linear restrictions. Of course, in the non-linear case, there is no exact, finite-sample test → The F-test does not apply.