Advanced Econometrics - Chapter 12: Simultaneous equations models

Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 12 SIMULTANEOUS EQUATIONS MODELS I. MODEL: Many economic problems involve the interaction of multiple endogenous variables within a system of equations. Estimating the parameters of such as system is typically not as simple as doing OLS equation-by-equation. Issues such as identification (wether the parameters are even estimable) and endogeneity bias are then primary topic in this chapter. 1. Example: Demand and supply for peanuts at t , 1 , 2 d t t t s t t t t q p q p R α ε β γ ε = +  = + + qt = quantity, pt = price, Rt = input price. Equilibrium: qd,t = qs,t = qt ⇔ 1 2t t t t tp p Rα ε β γ ε+ = + + ⇔ 2 1( ) t t t tp Rα β γ ε ε− = + − ⇔ 2 1 ( ) ( ) t t t tp R γ ε ε α β α β − = + − − The model is the joint determination of price and quantity qt and pt are endogenous variables, Rt is assume determined outside the model, it is exogenous variable. All three equations are needed to determine the equilibrium price and quantity. Note: The completeness of the system requires that the number of the equations equals the number of endogenous variables. 2. General model: The simultaneous system can be written as: Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam ( ) ( )(1 ) (1 ) (1 ) 0t t t M M K MM K M Y X ε × ×× × × Γ + Β + = Where: [ ]1 2t M tY Y Y Y=  [ ]1 2t M tX X X X=  [ ]1 2t M tε ε ε ε=  11 12 1 21 22 2 ( ) 1 2 M M M M M M MM γ γ γ γ γ γ γ γ γ ×      Γ =              11 12 1 21 22 2 ( ) 1 2 M M M M M M KM β β β β β β β β β ×      Β =              * 1 11 12 13 1 2 21 2 t t t t t t t t q p I p q p α β β β ε α β ε  = + + + +  = + + It: Income; *tp : price of an alternative; qt, pt: Endogenous (determined within the model) It, Rt, *tp : Exogenous (determined outside the model) Matrix form: [ ] [ ] [ ] 1 2 * 12 1 2 11 21 13 1 1 1 0 0 0 0t tt t t t t t t Y X q p I p ε α α β ε ε β β β Γ Β + +  − −    + + + =     + +  +        3. Reduced form: Each endogenous variable is expressed in terms of all exogenous variables: t t tY X V= Π + Where: 1 ( ) ( )K M K M − × × Π = − Β Γ Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam 1 ( )(1 ) (1 ) t t M MM M V ε − ×× × = Γ If the system is completeness → Γ squared matrix is non-singular and Γ−1 exists. [ ] [ ] 11 12 * 21 22 1 2 31 32 1 t t t t t t t t Y X q p I p V V π π π π π π     = +         • Reduced form can always be estimated, so reduced form equations are identified. • If πij are given we can find structural parameters: 11 12 1 2 21 22 12 11 21 31 32 13 1 1 0 0 π π α α π π β β β π π β Γ Β + +    − −    ΠΓ = −Β ⇔ = − +    + +  +        For equation (1): demand 11 12 11 1 21 22 11 12 31 32 11 13 π π β α π π β β π π β β − + = − − + = − − + = − This is “Indirect least squares” estimation (ILS) Over-identification → ILS cannot be applied? ⇒ 3 equations, 4 unknown: 11β , 1α , 12β , 13β . So parameters of 1 st structure equation (demand) are not identified. This equation cannot be estimated. For equation (2): supply 11 12 21 2 21 22 21 31 32 21 0 0 π π β α π π β π π β − + = − − + = − + = ⇒ ( ) 21 21 22 11 12 21 22 / / β π π α π π π π = = − II. RANK AND ORDER CONDITIONS FOR IDENTIFICATION: 1. Classification of variables: + Current endogenous (unlagged) = jointly dependent + Predetermined: a. Exogenous (unlagged & lagged) Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam b. Lagged endogenous Ex: variables 1tq − or 1tp − on RHS equation Notation: for a given equation M* = number of endogenous variables in this equation M** = number of endogenous variables in system (excluded) K* = number of included predetermined variables in this equation KK* = number of excluded M = M∗ + M∗∗; K = K∗ + K∗∗ 2. Order conditions: 0t t tY X εΓ + Β + = 1 1 0t t tY X ε − −= − ΒΓ − Γ = ( )(1 ) (1 ) (1 ) t t t K MM K M Y X V ×× × × + Π + 1 ( ) ( )k M K M − × × Π = − Β Γ Π can be known by OLS because reduced form always identified. Question: Can ijβ 's and ijγ 's (Β and Γ) be fund in terms of ijΠ 's (Π) We have: 1−Π = −ΒΓ ⇒ ΠΓ = −Β ⇒ 0ΠΓ + Β = ⇒  ( )( ) ( ) ( ) ( ) 0K K MK M K M AK M K M K M I ×× × × + + Γ   Π =   Β     Consider the first equation: Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam ⇒ [ ]  1 ( 1) 1( ) ( )1 0K K K M K A M K I γ β × × + +   Π =       We have K equations, K+M unknown variables → It is impossible to be identification without additional information. ∗ Theorem: A necessary condition for a given equation to be identified is that the number of predetermined variables excluded (K**) from this equation be greater than or equal to the number of jointly dependent variables included in this equation mins 1. ** * 1K M≥ − ** ** * ** 1K M M M+ ≥ + − ** ** 1K M M+ ≥ − Excluded variables # Dependent variables in system -1 Ex. Demand: ** 0K =  * , 2 1 1 t tp q M = − = ⇒ not identified Supply:  * ** , 2 t tI P K =  * , 2 1 1 t tp q M = − = ⇒ order condition is satisfied, not identified yet. If over identified: ** * 1K M> − If exactly identified: ** * 1K M= − 3. Rank conditions: [ ]  1 ( 1) 1( ) ( ) 1 0K K K M K M K a I γ β × × + + ×   Π =        K equations, (M + K) unknown parameters Added information in the form of general linear restrictions on a :  1 1 1 1 ( 1) ( ) 0 r r M K R a × × + = [ ]  [ ]  1 1 1 ( ) ( ) 1r M K M K R a × + + × r1: number of restrictions on exogenous & endogenous variables. Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam 1 0 0 1 0 0 0 1 1 1 1 R  =    3rd coefficient = 0, sum of coefficients on variables 2 → 5 = 0.  1 11 1 1 ( ) 1 1 ( ) ( ) ( ) 1 0 M K K Kr r KR r K M K M K a I γ β + × + × + + ×    Π =           ∗ Theorem: A necessary and sufficient condition for a solution a to this system of homogenous equations is that the rank of 1 KK I R Π         be equals to (M + K – 1) this solution is nontrivial and unique up to a factor of proportionality. ∗ An equivalent condition is that the rank of: (R1A) = M - 1 Because 1 1 ( ) KIRank R A K Rank R  Π + =     • Rank condition: necessary and sufficient condition for identification: 1( ) 1Rank R A M= − • Order condition: necessary: 1 1r M≥ − Number of rows of R1 is a number of endogenous variables and exogenous variables outside the equation (1) K** + M** → order condition now: 1 1r M≥ − : 1 11 2 12 3 11 12 2 1 2 22 3 21 22 2 23 3 2 3 31 2 31 32 2 33 3 3 t t t t t t t t t t t t t t t Y Y Y X Y Y X X Y Y X X γ γ β β ε γ β β β ε γ β β β ε = + + + + = + + + + + = + + + + 32 33( )β β= Yjt: endogenous variables, Xjt: Exogenous (predetermined) Matrix form: Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam [ ] [ ] [ ] [ ] 11 21 31 12 1 2 3 11 31 1 2 3 22 32 12 21 23 33 1 2 3 1 0 0 0 0 1 1 0 1 0 0 0 0 t t t t t t t t t Y Y Y X X X β β β β γ γ β β γ γ β β ε ε ε   −       − +    −       + = • For equation (1): write restrictions as: 1 1 0R a =  Where 1a  is the first column of A Γ  = Β  . Then: [ ] 1 2 ( ) 0 0 1 0 0 0 0 1K M R × +   =       1 2 1 3 1 2r M= ≥ − = − = ⇒ Order condition: yes. Rank condition: yes 22 321 23 33 0 ( ) 0 R A β β β β   =     1( ) 2 1Rank R A M= = − → eq(1) is identified. • For equation (2): write restrictions as: 2 2 0R a =  2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 R  =     2 2 1 3 1 2r M= ≥ − = − = ⇒ Order condition: yes. Rank condition: yes 2 12 1 0 0 ( ) 0 0 R A β −  =     2( ) 1 1Rank R A M= ≠ − → eq(2) is not identified. • For equation (3): 32 33( )β β= 3 3 0R a =  . Then: 3 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 R    =   −   Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 8 University of Economics - HCMC - Vietnam 3 3 1 2r M= ≥ − = ⇒ Order condition: yes. Rank condition: yes 3 12 22 23 32 33 1 0 0 ( ) 0 0 ( ) ( )(0 0) R A β β β β β −   =   − − −  3( ) 2 1Rank R A M= = − → eq(3) is identified. Example 2: 0 1 1 2 1 0 1 2 3 ( )t t t t t t t t t t t t I Y Y i C Y Y C I α α α ε β β ε ε −= + − + + = + + = + + Matrix form: [ ] [ ] [ ] [ ] 0 0 1 2 1 2 3 1 1 3 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 t t t t t t t tI C Y i Y α β α ε ε ε α β α − −       − + + =    −       • For equation (1): 1 2 1 3 1 2r M= = − = − = ⇒ Order condition: yes. 1 0 1 0 0 0 0 0 0 1 0 0 1 R  =     Rank condition: yes 1 1 3 1 1 0 1 1 0 1 1 ( ) ( ) ( 0) ( 1 0) 0 1 R A α α β β − −    = =   + + − + −   1( ) 2 1Rank R A M= = − → eq(1) is identified. • For equation (2): write restrictions as: 2 2 0R a =  2 3 1r M= ≥ − ⇒ Order condition: yes. Rank condition: 22 3 1 0 1 ( ) 0 0 0 0 R A Aα α −   =      Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 9 University of Economics - HCMC - Vietnam 1( ) 2 1Rank R A M= = − → eq(1) is identified. • For equation (3): 3 3 1 2r M= ≥ − = ⇒ Order condition: yes. 3( ) 2 1Rank R A M= = − → eq(3) is identified. III. ESTIMATION OF A SIMULTANEOUS EQUATION SYSTEM: 1. Under-identified: Order or rank condition fails. 2. Identified: • Exactly identified: r = M – 1 and rank condition is met. • Over-identified: r > M – 1 and rank condition is met. Problem: How do we estimate Β, Γ consistently? a) If a system is under-identified (there are some equations which are under-identified) → there is no way to estimate it consistently. b) If a system is exact-identified: One possible way is to use indirect least squares estimation to solve. 1 0KI R γ β   Π =         c) If a system is over-identified: ILS does not work because there will be more than one possible estimator and no obvious means of choosing among them. OLS → inconsistent estimators because of edogeneity, also call: “Simultaneous equation bias” of least squares. We will discuss various methods of consistent and efficient estimation (“All of methods in general use can be placed under the umbrella of instrument variable (I.V) estimator” Greene). Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 10 University of Economics - HCMC - Vietnam ∗ Note: generalised I.V estimator of β: 1/ /ˆ IV Z ZX M X X M Yβ −    =     where / 1 /( )ZM Z Z Z Z −= ( )n L Z × are instrumental variable for ( )n k X × with L > k. If L = k → Z/X is a squared matrix and: ( ) 11/ / / /ˆIV Z ZX M X X M Y Z X Z Yβ −−    = =    where / 1 /( )ZM Z Z Z Z −= (simple IV). And 1/ /ˆ IV Z ZX M X X M Yβ −    =     is also the 2SLS (two-stage least square) estimator: • First stage: get Xˆ Z= Π with / 1( )Z Z ZX−Π = → / 1ˆ ( )X Z Z Z ZX−= • Second stage: 1/ / 2 ˆ ˆ ˆ ˆ SLS X X X Yβ −    =     = 1/ / Z ZX M X X M Y −        = ˆ IVβ when use Xˆ as an IV. We can prove that ˆIVβ generalized is a consistent estimator of β. 1/ /ˆ IV Z ZX M X X M Yβ −    =     11 1 / / / / / /1 1 1 1 1 1ˆ IV X Z Z Z Z X X Z Z Z Zn n n n n n β β ε −− −          = +                        So ( ) 11 1ˆlim .0IV XZ ZZ ZX XZ ZZp Q Q Q Q Qβ β β−− − = + =  General IV estimator is also 2SLS estimator. ∗ Requirement: number of instrumental variables has to be larger than number of endogenous variables in the regression (≥) for 2SLS can be applied. General structural model: (1) ( ) ( ) (1 )(1 ) (1 ) (1 ) 0 1,2 ,t t t M M K M MM K M Y X t Tε × × ×× × × Γ + Β + = =  (1’) ( ) ( ) ( ) ( ) ( ) ( ) 0 T M M M T K K M T M T M Y X E × × × × × × Γ + Β + = (2) ( ) (1 )(1 ) 0t MM E ε ×× = Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 11 University of Economics - HCMC - Vietnam (3) ( )/ ( )( ) t t M MM M E ε ε ×× = Σ (4) ( )/ ( ) ( ) 0t s M M M M E ε ε × × = t≠s (no auto) (5) / ( )( 1) (1 ) 0t s K Mk M E X ε ×× ×   =    Xt predetermined (6) ( )/t t XXE X X = Σ finite, non-singular Well-behave data: X →sample statistics converge to corresponding population. (7) / ( ) ( ) 1lim t t XX k kt k k p X X T × × = Σ∑ (8) / ( )( 1) (1 ) 1lim 0t t k Mk Mt p X T ε ×× × =∑ (9) / ( )( 1) (1 ) 1lim t t M MM Mt p T ε ε ×× × = Σ∑ (10) Γ is non-singular We can use the set X as instrumental variables for Y • First equation (1st column) * *1 1 1 ( ) ( 1) ( ) ( 1) ( 1) ( 1) 0 T M M T K K T T Y Xγ β ε × × × × × × + + = ↔ 1 * * 11 ** ** 1 ( 1) 0 ( ) 0 0 M K Y X M M β εγ −           −    + + =                     * ** * ** M M M K K K  + =  + = ↔ * * * * 1 1 1 1 1 1 ( 1) ( 1)[ ( 1)] [( 1) 1] ( ) ( 1)T TT M M T K K y Y Xγ β ε × ×× − − × × × = + + Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 12 University of Economics - HCMC - Vietnam ↔ 1 * * 1 * * 1 1 1 1 1 ( 1) ( 1)1 [ ( 1)] ( 1) [ ] T T Z T K M K M y Y X δ γ ε β× × × + − + −   = +    ↔ 1 1 1 1 ( 1) ( 1)T T y Z δ ε × × = + a. OLS estimation: / 1 / / 1 / 1 1 1 1 1 1 1 1 1 1 ˆ ( ) ( ) OLS Z Z Z y Z Z Zδ δ ε− −= = + / 1 / 1 1 1 1 1 1 1 1ˆlim( ) lim( ) lim( ) OLS P p Z Z p Z T T δ δ ε−= + / 1 1 1 1 1 1 / 1 1 1lim( ) 0 1 0lim( ) ZZ p Y T p X T ε δ δ ε − −     →≠ + Σ ≠  →=     b. Two-stage least squares: Stage 1 (step1): Reduced form equation for Y1: * 1 2 3 * ( 1) ( 1)( 1)[ ( 1)] M T TTT M Y y y y × ××× −   =      2 ( 1) 2 2 2( )( 1) ( 1) ( 1) ˆ ˆˆ T T KT K T y y X V × ×× × × = Π +  3 ( 1) 3 3 3( )( 1) ( 1) ( 1) ˆ ˆˆ T T KT K T y y X V × ×× × × = Π +  * * * ˆˆ M M My X V= Π + 1 2 3 * 2 3 * ˆ ˆ ˆˆ ˆ ˆ M MY X X X V V V  = Π Π Π +     1 1 1 2 3 * 2 3 * ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ M M V Y X V V V Π   = Π Π Π +       Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 13 University of Economics - HCMC - Vietnam 1 1 1 1 ˆ ˆ ˆˆY X V Y V= Π + = + (All M* – 1 reduced form equations for Y1) All estimated by OLS / /1 1ˆ ˆ ˆ0, 0X V Y V→ = = Put expression for Y1 into 1st structural equation: 1 1 1 1 1 1 1 1 1 1 1 1 ˆ ˆ[ ]y Y X Y V Xγ β ε γ β ε= + + = + + + 1 1 1 1 1 1 1 1 ˆ ˆy Y X Vγ β γ ε= + + +  1 1 1 1 1 1 1 1 1 1 ˆ ˆ ˆ( ) Z y Y X V δ γ γ ε β   = + +     1 1 1 1 1 1 ˆ ˆ( )y Z Vδ γ ε= + + Step 2 (Stage 2): Estimate by OLS: 1 2 ˆ SLS δ / 1 /1 1 1 1ˆ ˆ ˆ( ) (( )Z Z Z y −= = / 1 /1 1 1 1 1 1 1 0 ˆ ˆ ˆ ˆ( ) ( )Z Z Z Vδ γ ε− → + +  1Zˆ are instrument instrumental variables of [Y1 X1] 1 2 ˆ SLS δ is consistent. 1Zˆ = W1 → we get the same 1 2 ˆ SLS δ 3. Instrumental variables estimators: Let W1 be a * *( 1)T M K× + − matrix such that: (1) /1 1 1lim WZp W ZT = Σ finite, non-singular . (1) /1 1 1lim 0p W T ε = finite, non-singular . 1ˆ IV δ / 1 /1 1 1 1( ) (( )W Z W y −= 1ˆlim IV p δ 1δ= 1 / 1 1 1limp W Z T −       / 1 1 1limp W T ε     1δ= Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 14 University of Economics - HCMC - Vietnam → 1ˆ IV δ is consistent. Note: we can use W1 = 1 1 1ˆ ˆZ Y X =   as an instrumental variables set → get 2-SLS (2-SLS as IV with W1 = Z�1). 4. Indirect least squares: Requires: exactly identified → ** * 1K M= − or ** ** 1K M M+ = − 1ˆ ILS δ / 1 /1 1 1 1( ) ( )X Z X y −= Note: * * 1 [ ( 1)]T K M Z × + − * * 1 1 [ ( 1)]T K M X W × + − = → same as estimator of δ1 found by algebraically solving for γ�1 and �̂�1from Π�1 Note: 2-SLS as IV with W1 = 1 1 1ˆ ˆZ Y X =   1 _ 2 ˆ IV SLS δ / 1 /1 1 1 1ˆ ˆ( ) (( )Z Z Z y −= / 1 /1 1 1 1ˆ ˆ ˆ( ) (( )Z Z Z y −= because of orthogonolity condition: / 1 1Zˆ Z / 1 1 ˆ ˆZ Z= Note: 2-SLS computational formula 1 1 1 ˆ ˆZ Y X =   / 1 / 1 1 1 ˆ ˆ ( )Y X X X X X Y−= Π = → 1 1 1ˆ ˆZ Y X =   = / 1 / 1 1( )X X X X Y X −   = [ ] / 1 / 1 1( )X X X X Y X − Because: ( ) ( )/ 1 /1 2 1 2( ) I X X X X X X X X−=   / 1 / / 1 / 1 2( ) ( )X X X X X X X X X X − −=  So: [ ]/ 1 / / 1 /1 1 1 1ˆ ( ) ( )Z X X X X Y X X X X X Z− −= = Then: 1 2 ˆ SLS δ / 1 /1 1 1 1ˆ ˆ ˆ( ) (( )Z Z Z y −= Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 15 University of Economics - HCMC - Vietnam / / 1 / / 1 / 1 / 1 / 11 1 1 1( ( ) ( ) ) ( ( ) )Z X X X X X X X X Z X X X X Z y − − − − −= / / 1 / 1 / / 1 /1 1 1 1( ( ) ) ( ( ) )Z X X X X Z Z X X X X y − − −= 1 2 ˆ SLS δ / 1 /1 1 1 1( ) ( )X XZ M Z Z M y −= 5. Three-stage Least Squares Estimation: 1 1 1 1y Z δ ε= + 2 2 2 2y Z δ ε= + [ ]1 1 1Z Y X= M M M My Z δ ε= + [ ]M M MZ Y X= Using SUR framework: 1 2 M y y y              * * 1 [ ( 1)] 2 0 0 0 0 0 0 T K M M Z Z Z × + −       =                1 1 2 2 M M δ ε δ ε δ ε            +               y Zδ ε= + ( 1)( 1) TMTM y Zδ ε ×× = + ( ) ( ) MT MT E εε × ′ 1 ( 1) 2 1 2 (1 ) (1 ) ( 1) T N T T N T ε ε ε ε ε ε × × × ×             =                   11 12 1 21 22 2 ( ) 1 2 M M TM M M M MM I I I I I I I I I I σ σ σ σ σ σ σ σ σ ×      = = Σ ⊗              { }ij MMσΣ = Define: Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 16 University of Economics - HCMC - Vietnam X = ( ) 0 0 0 0 0 0 M T K X X I X X ×       = ⊗              Pre-multiply system by / ( )TM KM X × / / / (*)X y X Z Xδ ε= + ( )/ / / / /( ) [ ]( )[ ] ( )E X X X I X I X I I X X Xεε = Σ ⊗ = ⊗ Σ ⊗ ⊗ = Σ ⊗ Apply GLS to (*): 1 3 ˆ SLS δ 1/ / 1( ( )Z X X X XZ −− = Σ ⊗  / / 1 /( ( )Z X X X X y− = Σ ⊗  Σ is estimated by ˆ ˆij M Mσ × Σ =   where / ˆ i jij e e T σ = for all i, j pairs and 2 ˆ i i i i SLS e y z δ − = − (i = 1, 2,, M) Different way:  1st stage: Perform 2-SLS for each equation of the system. Save ( 1) i T e × s (i = 1,2,, M); ( 1) ( 1) 2 ˆ i i i i T T SLS e y z δ × × − = −  2nd stage: Estimate Σ by ˆ ˆij M Mσ × Σ =   where / ˆ i jij e e T σ = for all i, j pairs  3rd stage: (1) → ( 1)( 1) TMTM y Zδ ε ×× = + Estimate δ by using instrumental variable. Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 17 University of Economics - HCMC - Vietnam * * 1 [ ( 1)] 2 ˆ 0 0 ˆˆ 0 0 ˆ0 0 T K M M Z ZZ Z × + −       =                 Where 1 1 1ˆ ˆZ Y X =   ˆ ˆ M M MZ Y X =   and using GLS method: 1 3 ˆ SLS δ 1/ 1 / 1ˆ ˆˆ ˆ( ( ) ( ( )Z I Z Z I y − − −   = Σ ⊗ Σ ⊗    Note: / 1 /1 1ˆ ( )Z X X X X Z −= / 1 /ˆ ( )M MZ X X X X Z −= → / / / 1 /ˆ ( )Z Z X X X X−= ⊗ 1 3 ˆ SLS δ 1/ 1 / 1 / / 1 / 1 /ˆ ˆ( ( ( ) ) ( ( ( ) )Z X X X X Z Z X X X X y −− − − −   = Σ ⊗ Σ ⊗    If we use: OLS IˆVδ = 1/ /ˆ ˆZ Z Z y −    =     is simply equation-by-equation 2SLS. The improvement of 3SLS is using GLS to gain more efficiency, we need to have one more stage → that is calculate ˆ ˆij M Mσ × Σ =   Note: In the 2-SLS procedure: The matrix: 1 1 1ˆ ˆZ Y X =   has (M* + K* - 1) columns, all columns are linear function of the K column of X (Because / 1 /1 1ˆ ( )Z X X X X Z −= ). There exist, at most, K linear independent combination of the columns of X. if the equation is not identified then ** * 1K M< − → * ** * * 1K K K M+ < + − → * * 1K K M< + − Advanced Econometrics Chapter 12: Simultaneous equations models Nam T. Hoang University of New England - Australia 18 University of Economics - HCMC - Vietnam but 1 1 1ˆ ˆZ Y X =   only has maximum K independent columns. → 1Zˆ will not have full rank, 1/ 1 1 ˆ ˆZ Z −     does not exist → 2-SLS estimators cannot compute. If, however, the order condition but not the rank condition is met, then although the 2- SLS estimator can be computed, it is not a consistent estimator. Note: For a system of exactly identified equation: * * 1K K M< + − → / ( ) ( ) i K T T K X Z × × = squared matrix (K×K) 2 iˆ SLS δ − 1/ / 1 / / / 1 /( ( ) ) ( ( ) )i i i iZ X X X X Z Z X X X X y −− −   =     / 1 / / 1 / / 1 /( ) ( )( ) ( )( )i i i iX Z X X Z X Z X X X X y − − −= / 1 /( )i iX Z X y −= → 2 iˆ SLS δ − ˆ ILS δ= when * * 1K K M= + − for every equation in the system.