Two-stage least squares estimation

In statistics and econometrics which is two-stage least squares estimation or two-stage least squares estimation ( ZSKQ estimate ), and two-stage least squares ( English Two Stage Least Squares , short TSLS or 2SLS ), is established by the econometricians Estimation method developed by Henri Theil with limited information . In this two-step process, the endogenous (i.e., the variables correlated with the disturbance variable) are first regressed on all exogenous variables of the equation and all instruments . Second, the estimated values ​​obtained in this way for the endogenous regressors , which as a linear combination of exogenous variables are not correlated with the disturbance term , are then inserted into the original model and the resulting new model is estimated. The two-stage least squares estimator can be interpreted as an instrument variable estimator . The ZSKQ estimate is second to the common least squares method in estimating linear equations in applied econometrics.

The procedure

Consider a typical multiple linear regression model ( real model ), with the vector of unknown regression parameters , which - experimental design matrix , the vector of the dependent variable and the vector of disturbances . The generalized least squares (VKQ) estimator can be expressed in different ways. Each of these expressions has its own interpretation. A well-known specification  is the so - called two - stage least-squares estimation , which was developed by Henri Theil. For the derivation of the two-stage least squares estimator , the generalized least squares estimator can be expressed as follows: ${\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle (p\times 1)}$ ${\displaystyle (n\times p)}$ ${\displaystyle \mathbf {X} }$ ${\displaystyle (n\times 1)}$ ${\displaystyle \mathbf {y} }$ ${\displaystyle (n\times 1)}$ ${\displaystyle {\boldsymbol {\varepsilon }}}$ ${\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}}$

${\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}=(\mathbf {Z} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Z} _{i})^{-1}\mathbf {Z} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}={\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}$

The reduced form is . The -th equation of the reduced form can be partitioned as follows: ${\displaystyle \mathbf {Y} =\mathbf {X} \mathbf {\Pi } +\mathbf {V} }$ ${\displaystyle i}$

${\displaystyle [\mathbf {y} _{i}\;\;\mathbf {Y} _{i}\;\;\mathbf {Y} _{i}^{*}]=\mathbf {X} [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]+[\mathbf {v} _{i}\;\;\mathbf {V} _{i}\;\;\mathbf {V} _{i}^{*}]}$,

where the -vector is the -th commonly dependent variable that includes other commonly dependent variables in the -th equation, the -Matrix of the commonly dependent variables not appearing in the -th equation, and the partitioned matrix of coefficients of the reduced ones Shape is. The least squares estimator of is and therefore holds with the aid of the prediction matrix , where is the matrix of the predicted values ​​of . By the fact that , one can also write: ${\displaystyle \mathbf {y} _{i}}$ ${\displaystyle (T\times 1)}$ ${\displaystyle i}$ ${\displaystyle \mathbf {Y} _{i}^{*}}$ ${\displaystyle i}$ ${\displaystyle \mathbf {Y} _{i}^{*}}$ ${\displaystyle (T\times m_{i}^{*})}$ ${\displaystyle i}$ ${\displaystyle [\pi _{i}\;\;\mathbf {\Pi } \;\;\mathbf {\Pi } _{i}^{*}]}$ ${\displaystyle \mathbf {\Pi } _{i}}$ ${\displaystyle {\hat {\mathbf {\Pi } }}_{i}=(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}}$ ${\displaystyle \mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}=\mathbf {X} {\hat {\mathbf {\Pi } }}_{i}={\hat {\mathbf {Y} }}_{i}=\mathbf {Y} _{i}-{\hat {\mathbf {V} }}_{i}}$ ${\displaystyle {\hat {\mathbf {Y} }}_{i}}$ ${\displaystyle (T\times (m_{i}-1))}$ ${\displaystyle \mathbf {Y} _{i}}$ ${\displaystyle (\mathbf {X} ^{\top }\mathbf {X} )^{-1}(\mathbf {X} ^{\top }\mathbf {X} )=\mathbf {I} }$

${\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}={\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }(\mathbf {X} ^{\top }\mathbf {X} )^{-1}(\mathbf {X} ^{\top }\mathbf {X} )\mathbf {Y} _{i}&\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {Y} _{i}&\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}\mathbf {Y} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}$

or.

${\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}={\begin{pmatrix}{\hat {\mathbf {Y} }}_{i}^{\top }{\hat {\mathbf {Y} }}_{i}&{\hat {\mathbf {Y} }}_{i}^{\top }\mathbf {X} _{i}\\\\\mathbf {X} _{i}^{\top }{\hat {\mathbf {Y} }}_{i}&\mathbf {X} _{i}^{\top }\mathbf {X} _{i}\\\\\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}{\hat {\mathbf {Y} }}_{i}^{\top }\mathbf {y} _{i}\\\\\mathbf {X} _{i}^{\top }\mathbf {y} _{i}\\\\\end{pmatrix}}}$

Once defined, the two-stage least squares estimator can be specified as follows ${\displaystyle {\hat {\mathbf {Z} }}_{i}=[{\hat {\mathbf {Y} }}_{i}\;\;\mathbf {X} _{i}]}$

${\displaystyle {\tilde {\boldsymbol {\delta }}}_{i}=({\hat {\mathbf {Z} }}_{i}^{\top }{\hat {\mathbf {Z} }}_{i})^{-1}{\hat {\mathbf {Z} }}_{i}^{\top }\mathbf {y} _{i}}$.

literature

• George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , TC Lee. Introduction to the Theory and Practice of Econometrics. 2nd Edition. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4 .

Individual evidence

1. ↑ George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , TC Lee. Introduction to the Theory and Practice of Econometrics. 2nd Edition. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4 , p. 645.
2. ↑ George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl, TC Lee. Introduction to the Theory and Practice of Econometrics. 2nd Edition. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4 , p. 645.

Web links

• Springer Gabler Verlag, Gabler Wirtschaftslexikon, keyword: smallest square method, two-stage