Three-stage least squares

Three-stage least squares Econometric Methods Lecture 7, PhD Warsaw School of Economics

Outline 1 Introduction 2 3

3-Stage Least Squares: basic idea 1-stage (OLS): inconsistent for systems with simultaneous equation models 2-stage (2SLS): let's clean endogenous regressors step 1: regress endogenous regressors against all predetermined variables of the system, take theoretical value step 2: do OLS with these theoretical (instead of empirical) values in case of endogenous regressors but still: a limited-information method! focus on single equation simultaneous correlations between various equations' error terms: ignored! result: ineciency

solution do 2SLS (step 1 and 2) add step 3 to account for these correlations the dierence between 2SLS and 3SLS analogous to dierence between OLS and GLS in the single-equation context whereby the latter takes into account inter-temporal (and not simultaneous) correlations between error terms

Notation (1) Consider the entire m-equation model: y 1 = Z1 F1 + ε 1 y 2 = Z2 F2 + ε 2 y m = Zm Fm + ε m where Zj = [ ] Ỹ j Xj vector of explanatory variables (endoand exogeneous respectively) in j-th equation Let ^F 2SLS j vector of j-th equation's parameter estimates via 2SLS

Notation (2) Denote the model as matrix equation: y 1 Z 1 0 0 F 1 ε 1 y 2 0 Z 2 0 F 2 ε 2 = + y m 0 0 Z m F m ε m }{{}}{{}}{{}}{{} y Z F ε where ε 1, ε 2, are vertical vectors sized T 1

Covariances of error terms Knowing 2SLS parameter estimated (and hence 2SLS residuals), we can estimate the variance-covariance matrix of the random disturbances Var (ε t ) = E ( ) ε t ε T t element-by-element in a standard way: ˆΣ = [ˆσ ij ] ( ) ( ) T ˆσ ij = /T y i Z i^f2mnk i y j Z j^f2mnk j where T number of observations Alternatively, instead of T, one can divide the product sum in the nominator by the geometrically averaged degrees of freedom for equations i and j: ( T M i K i ) (T M j K j ) M i ( M j ) number of endogenous regressors in i-th (j-th) equation K i ( K j ) number of exogeneous regressors in i-th (j-th) equation

Kronecker product [ ] [ ] a11 a 12 b11 b 12 = a 21 a 22 b 21 b [ ] 22 [ b11 b a 11 12 b11 b a b 21 b 12 12 [ 22 ] [ b 21 b 22 b11 b a 21 12 b11 b a b 21 b 22 12 22 b 21 b 22 It can be demonstrated easily that: (A B) 1 = A 1 B 1 (A B) (C D) = AC BD ] ]

3SLS step 1 Step 1 Reduced-form estimation and theoretical values for j-th equation: ( ^~ Z j = XΠ j = X X T 1 X) X T ~ Zj In the matrix notation, for the entire system: ^Z = ( X X T 1 X) X T Z1 0 0 ( 0 X X T 1 X) X T Z2 0 ( 0 0 X X T 1 X) X T Zm { I m [ X ( X T X) 1 X T ]} Z =

3SLS step 2 Step 2 Estimation of parameters for indivual equations in the structural form (2SLS): empirical endogenous explanatory variables replaced with theoretical values from step 1: ^F2SLS = ( ^Z T ^Z) 1 ^ZT y

3SLS step 3 (1) Step 3 Acknowledging the simultaneous correlations of error terms in the model If j-th equation's error term is spherical, its variance-covariance σ 2 jj 0 0 0 σjj 2 0 matrix is 0 0 σjj 2 What's the variance-covariance matrix of the entire vector ε?

3SLS step 3 (2) σ 2 11 σ 2 12 σ 2 1m σ 2 11 σ 2 12 σ 2 1m σ 2 11 σ 2 12 σ 2 1m σ 2 12 σ 2 22 σ 2 2m σ 2 12 σ 2 22 σ 2 2m σ 2 12 σ 2 22 σ 2 2m σ 2 1m σ 2 2m σ 2 mm σ 2 1m σ 2 2m σ 2 mm σ 2 1m σ 2 2m σ 2 mm

Kronecker product in 3SLS σ 2 11 σ 2 12 σ 2 1m σ 2 11 σ 2 12 σ 2 1m σ 2 11 σ 2 12 σ 2 1m σ 2 12 σ 2 22 σ 2 2m σ 2 12 σ 2 22 σ 2 2m σ 2 12 σ 2 22 σ 2 2m σ 2 1m σ 2 2m σ 2 mm σ 2 1m σ 2 2m σ 2 mm = = σ 2 11 σ 2 12 σ 2 1m σ 2 12 σ 2 22 σ 2 2m σ1m 2 σ2m 2 σmm 2 σ1m 2 σ2m 2 σmm 2 1 0 0 0 1 0 0 0 1 T T = Σ I T

Final formula for 3SLS estimator (1) Knowing the entrie variance-covariance matrix ˆΣ, we can apply GLS to the model y = ZF + ε, ε (0, Ω) ^FGLS = ( Z T Ω 1 1 Z) Z T Ω 1 y This is done jointly with using 2SLS theoretical values for endogenous regressors (obtained in step 1): ( 1^Z) 1 ^Z T Ω ^ZT Ω 1 y ^F3SLS =

Final formula for 3SLS estimator (2) We already know that ˆΩ = ˆΣ IT and ^Z = ^F3SLS consequence: = { [ [Z T Im [ Z {Im T X X ( ) X T 1 ]} T 1 { T X X (ˆΣ IT) ( ) X T 1 ]} T ) 1 T X X (ˆΣ IT y { [ ( ) Im X X T 1 ]} T X X Z In [ Im From the properties of Kronecker product: ^F3SLS [ ( ) = [Z {ˆΣ T 1 X X T 1 ]} 1 [ T X X Z] Z {ˆΣ T 1 X ( ) X T 1 ]} 1 T X X Z] X ( ) X T 1 ]} T X X y

Matrix of predetermined variables: 1 0 6 1 X = [ ] 1 1 5 2 x 0 x 1 x 2 x 3 = 1 1 3 2 1 2 3 1 1 2 2 0 Matrix of endogenous variables: 1 3 4 Y = [ ] 2 3 6 y 1 y 2 y 3 = 1 2 7 2 1 5 3 1 3 Estimate with 3SLS the structural parameters of the following model: y 1t = α 0 + α 1x 1t + α 2y 2t + ε 1t y 2t = β 0 + β 1x 2t + β 2y 3t + ε 2t y 3t = γ 0 + γ 1x 3t + γ 2y 1t + ε 3t