Advanced Econometrics - Part I - Chapter 4: Estimation By Instrumental Variables

Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 4 ESTIMATION BY INSTRUMENTAL VARIABLES (Instrumental Variable Estimators) I. ENDOGENEITY: Now suppose ε, X are not independently generated: Cov(ε,X)≠ 0 and E(ε|X) ≠ 0. There are 4 sources of this problem: 1. Errors in measurement of independent variables: Suppose that the true regression equation is given by: yi = β0 + β1xi + εi where E(εi) = E(εixi) = 0 Note: ),(),(),()]([),( 0 iiiiiiiii xExExExxExCov εεεεε =−=−=  So if 0),(0),( =↔= iiii xExCov εε Suppose ii exx i += * Assume: E(ei) = E(eixi) = 0 → estimate: yi = β0 + β1xi* + ui where: ui = εi - β1ei correlated with ii exx i += * through terms ei → 0),( * ≠ii xuCov Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam 2. Variables on both sides of regression equation are jointly determined (endogenous) → RHS variables are endogenous.    ++= ++= iii iii he ueh εαα ββ 10 10 → iii ue εβαβα α βα βαα 1111 1 11 010 1 1 11 − + − + − + = → 0),( ≠ii euCov 3. Omitted variables: iiii asw εβββ +++= 210 Estimate: iii usw ++= 10 ββ Where: iii au εβ += 2 , if ai and si are correlated → 0),( ≠ii suCov 4. Lagged dependent variables (Yt-1) as a regressor and auto correlated errors. 0),( 1 1 1 ≠→    += +++= − − − tt ttt tttt YCov u YXY ε ρεε ελβα because Yt-1 and εt both contain εt-1. Model: (1) εβ += ×kn XY (2) X and ε are not generated independently (3) E(ε|X) ≠ 0 (4) E(εε'|X) = σ2I (5) X consists of stationary random variables with: =      ′ ×× k i kn i XXE 1 =′ )1lim( XX n p XXΣ Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam Now 0)1lim( ≠=′ γεX n p and βγβεββ ≠Σ+=′Σ+= − ≠ − 1 0 1 )1lim(ˆlim XXXX Xn pp  → βˆ is an inconsistent estimator. βˆ is also no longer unbiased βεββ ≠′′+= ≠ −  0 1 )()()ˆ( XEXXXXE II. ESTIMATION BY INSTRUMENTAL VARIABLES: Suppose we can find a set of k variables kn W × that have two properties: 1. Exogeneity (validity): They are uncorrelated with the disturbance ε. 2. Relevance: They are correlated with the independent variable X. Such that:            Σ= Σ=′=      = =→= WW WWii XW n p WWEWW n E W n p wEWE '1lim )('1 0'1lim 0)'(0)( ε εε (W & X are stationary random variables). Then W is a set of instrumental variables and we define: YWXWIV ')'( ˆ 1−=β Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam IVβˆ : IV estimator. Consistency: IV estimator IVβˆ is consistent: YWXWIV ')'(ˆ 1−=β )(')'( 1 εβ += − XWXW IVβˆ            += − n W n XW ε β '' 1 (Slutsky theorem). IVp βˆlim  0 1 'lim'lim            += − n Wp n XWp εβ ββ =Σ+= − 0.1WX IV estimator is unbiased. ( )IVWE βˆ ( ) ( ) ( ) βεβ =+= −Σ −  0 1 '' 1 WEWEXWE WX III. TWO-STAGE LEAST SQUARES ESTIMATION:           Σ= Σ= == ≠ × singular non'1lim '1lim '1lim0)( 0)( WX WW kn XW n p WW n p W n pWE XE W εε ε Now we have a set of instruments qn Z × , that are unrelated to ε. X consists two parts:       = ×−×× rnrknkn XXX 2 )( 1 Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam X1: exogenous variables X2: endogenous variables Note: q must be ≥ r (if q < r → (W'W)-1 doesn't exist. Z includes X1, We can define reduced form equations for X2: rnrqqnrn VZX ×××× +Π=2    = ×× r nkn XXXX 2 2 2 1 1 22        ΠΠΠ=Π ×× r qrq 2 1 1 So:    = ×× r nrn VVVV 2 1 1        +Π= +Π= +Π= ×××× 111 2 22 2 2 11 1 2 n r q rqnn r VZX VZX VZX  Estimate this system by OLS, rq× Π are estimators:     × r n XXX 2 2 2 1 1 2  = rnrqqn VZ ××× +Π ˆˆ =     ΠΠΠ × r q Z ˆˆˆ 2 1 1  [ ]rVVV ˆˆˆ 21 + Then we get: Π= ˆˆ 2 ZX → 2Xˆ is a good instrument. ( 2Xˆ is correlated with X2 but not correlated with ε because Π= ˆˆ 2 ZX , )0),( =εZCov • After the first stage we get the set W = [ ]21 XˆX . Apply OLS on [ ]WY we have: Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam YWWWSLS ')'( ˆ 1 2 −=β → two-stage least squares estimator. • We can also use W = [ ]21 XˆX as an instrument variable and get: YWXWIV ')'( ˆ 1−=β 2 1 ')'( XZZZ −=Π We can show that: SLSIV 2ˆˆ ββ = IV. ASYMPTOTIC DISTRIBUTION OF IVβˆ nW n XW n n IV            =− − εββ '1'1)ˆ( 1 nW nWX      Σ= − ε'11 ∑ = = n i iiWW 1 ' εε Wi =                 ik i i w w w  3 2 1 0)()()( == wEwEwE iiii εε WWiiiiiiii wwEwwEwVar Σ=′=′= 22 )()()( εε σσεεε So by the central limit theorem: nW n       ε'1 ~ ),0( 2 XXN Σεσ )ˆ( ββ −IVn → ),0( 21 WWNWX ΣΣ − εσ Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam → d ( ) ),0( 211 εσ−− Σ′ΣΣ WXWWWXN IVβˆ → asy ( )             Σ′ΣΣ −−    )ˆ( 2 11, IVAsyVarCov WXWWWX n N β εσβ Note: ββ =)ˆ( IVE → IVβˆ is also an unbiased estimator. OLSβˆ is asymptotically efficient to IVβˆ . V. HAUSMAN SPECIFICATION TEST AND AN APPLICATION TO IV ESTIMATION: 1. Theorem: Let )1( ×r Z ~ ),0( 1 rr XXr N ×× Σ then: 2 ][ 1 ~' rZZ χ −Σ Proof: Recall: for λj: eigenvalue             =Λ nλ λ λ     00 00 00 2 1 1×r jC : eigenvector [ ]rCCCC 21= Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 8 University of Economics - HCMC - Vietnam we have: 2/12/1' ΛΛ=Λ=Σ × CC rr               =Λ nλ λ λ     00 00 00 2 1 2/1 → )()'(' 2/12/1 ΛΛ=ΣCC → =ΛΣΛ  DD CC )(')'( 2/1 ' 2/1 I=ΛΛΛΛ −− ))(()'()'( 2/12/12/12/1 → IDD =Σ' with 2/1−Λ= CD → 1111 )'(')'( −−−− =Σ DDDDDD → 11)'( −−=Σ DD → Σ='DD Note: C' = C-1, CC' = I Let 11 ' ××× = rrrr ZDW → W ~ N(0,DΣD') = N(0,I) 1×r W ~ N(0,I) → 1 ' ×r WW 2 ][~ rχ → )'()''( ZDZD 2 ][~ rχ → ZDDZ 1 '' −Σ 2 ][~ rχ Finally: ZZ 1' −Σ 2 ][~ rχ 2. Hausman Test: εββεβ ++=+= ×rn XXXY 2211 Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 9 University of Economics - HCMC - Vietnam H0: 12 0)( ×= rXE ε H0: 0)( 2 ≠XE ε Two alternative estimators: OLSβˆ : consistent under H0 but not under HA IVβˆ : consistent under both H0, HA (but inefficient compare to OLSβˆ )     += += − − εββ εββ ')'(ˆ ')'(ˆ 1 1 WXW XXX IV OLS Under H0: OLSIV ββ ˆˆ = Construct the Hausman's test statistic: ( ) ( )[ ] ( )OLSIVOLSIVOLSIV VarCov ββββββ ˆˆˆˆˆˆ 1 −−′− − 2 ][~ rχ (Note: Z ~ ),0( ΣN → ZZ 1' −Σ 2 ][~ rχ ) ( ) ( ) ( ) ( )OLSIVOLSIVOLSIV CovVarCovVarCovVarCov ββββββ ˆ,ˆ2ˆˆˆˆ −+=− ( )OLSIVCov ββ ˆ,ˆ ( )( )[ ]{ }XWE OLSIV ,'ˆˆ ββββ −−= ( )[ ]{ }XWXXXWXWE ,)'('')' 11 −−= εε 11 )'()'(')'( 2 −−= XXXEWXW I  εσ εε 211 )'(')'( εσ −−= XXXWXW 21)'( εσ −= XX ( )OLSVarCov βˆ= So ( ) ( ) ( )OLSIVOLSIV VarCovVarCovVarCov ββββ ˆˆˆˆ −=− Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 10 University of Economics - HCMC - Vietnam Then, the Hausman's test statistic is: ( ) ( ) ( )[ ] ( )OLSIVOLSIVOLSIV VarCovVarCov ββββββ ˆˆˆˆˆˆ 1 −−′− − 2 ][~ rχ Under H0: H 2 ][~ rχ 3. Wu's approach: εββ ++= 2211 XXY Do we have: 0)( 2 =XE ε In the first stage of IV estimation: rn X rqqnrn VZX ×××× +Π=  2 ˆ 2 ˆ r ≤ q → we get 2Xˆ * 22211 ˆ εγββ +++= XXXY → * 22211 )ˆ( εγββ +−++= VXXXY → * 2211 ˆ)( εγγββ +−++= VXXY Test: H0: 11 0 ×× = rr γ If reject H0 → 0)( 2 ≠XE ε VI. CHOOSING THE INSTRUMENTS: 1. If we are working with time-series data, lagged values of regressors will generally provide appropriate instruments. EX: εβββ +++= 33221 xxy Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 11 University of Economics - HCMC - Vietnam           = 21 1312 1 1 nn xx xx X                = −− 3,12,1 1312 0302 1 1 1 nn xx xx xx W  2. Choice of Z affects asymptotic efficiency of IVβˆ . Generally want to choose instruments to be highly corrected with the regressors (but uncorrelated with the errors). 3. With the cross-section data, not always easy. One option is to use the ranks of the data to form Z. Example: εββ ++= ii xy 21                       = 101 31 81 51 11 21 141 X                       = 61 31 51 41 11 21 71 Z Appendix: Measurement Error in Linear Regression εβ += ×× knn XY 1 (1) We don't observe X, but observe X* knknkn VXX ××× +=* (2) Where: 1 0)( × = n XE ε nn IXE × = 2)'( σεε Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 12 University of Economics - HCMC - Vietnam Put (2) into (1) yields: εβ +−= ×× 1 * 1 )( kn VXY )(*  u VXY βεβ −+= The error term u = ε - Vβ is correlated with the regressor X* through the measurement error V. Formally, we have: uX n p ′*1lim )(1lim * βε VX n p −′= )()(1lim βε VVX n p −′+= )'1lim'1lim'1lim'1lim 000 VV n pVX n pV n pX n p ββεε −−+=  0≠Σ−= VVβ An OLS regression of Y on X will lead to an inconsistent estimate of β. **1lim XX n p ′ )()(1lim VXVX n p +′+= VV n pXV n pVX n pXX n p '1lim'1lim'1lim'1lim +++= vvXX Σ+Σ= OLSβˆ =     + ′       ′=      ′       ′ )( ******* uXXXXYXXX β = +β uXXX ′      ′ *** Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 13 University of Economics - HCMC - Vietnam βˆlimp = +β       ′       ′ − uX n XX n p * 1 ** 11lim = −β ( ) βVVVVXX ΣΣ+Σ −1 βˆlimp = −β ( ) βVVVVXX ΣΣ+Σ −1             = 4 5 3 2 β Clearly, OLS is inconsistent as long as there are measurement errors and ΣVV ≠ 0 1) 1 ˆ ×k β is inconsistent as long as ΣVV ≠ 0. 2) If there are some variables which are correctly measured. → Their coefficient estimators are also inconsistent. A badly measured variable contaminates all the least squares estimates. → The effect of measurement errors is also called: "contamination bias". 3) For example if only one regressor is measured with errors             =Σ 200 000 000 v VV σ    → the bias and inconsistent of all correctly measured variables depend on the form of ΣXX → unknown. 4) In practice, it seems that the coefficients of the correctly measured variables are consistent but this depends on the special form of ΣXX Research questions: In practice → What kind of ΣXX we will count on the coefficients of correctly measured variables? Advanced Econometrics Chapter 4: Estimation By Instrumental Variables Nam T. Hoang University of New England - Australia 14 University of Economics - HCMC - Vietnam → If we cannot find a good instrumental variable: omit wrongly measured variables or don't omit? Which form of ΣXX. Computer programs could answer these questions (I guess). The form of ΣXX can be tested by simulations. 5) There are other cases that endogeneity is a problem → what is the role of ΣXX in affecting the inconsistency of the coefficients in those cases. 6) Endogeneity by measurement errors is a serious problem.