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:
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{\ displaystyle \ mathbf {y} = \ mathbf {X} {\ boldsymbol {\ beta}} + {\ boldsymbol {\ varepsilon}}}
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{\ 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} ^ {\ to p} \ mathbf {y} _ {i} \\\\\ end {pmatrix}}}
The reduced form is . The -th equation of the reduced form can be partitioned as follows:
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{\ displaystyle \ mathbf {Y} = \ mathbf {X} \ mathbf {\ Pi} + \ mathbf {V}}
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{\ 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} ^ {*}]}
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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:
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{\ displaystyle [\ pi _ {i} \; \; \ mathbf {\ Pi} \; \; \ mathbf {\ Pi} _ {i} ^ {*}]}
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{\ displaystyle {\ hat {\ mathbf {\ Pi}}} _ {i} = (\ mathbf {X} ^ {\ top} \ mathbf {X}) ^ {- 1} \ mathbf {X} ^ {\ top} \ mathbf {Y} _ {i}}
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{\ 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} - {\ has {\ mathbf {V}}} _ {i}}
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{\ displaystyle (\ mathbf {X} ^ {\ top} \ mathbf {X}) ^ {- 1} (\ mathbf {X} ^ {\ top} \ mathbf {X}) = \ mathbf {I}}
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{\ 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.
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{\ displaystyle {\ tilde {\ varvec {\ 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
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{\ displaystyle {\ hat {\ mathbf {Z}}} _ {i} = [{\ hat {\ mathbf {Y}}} _ {i} \; \; \ mathbf {X} _ {i}]}
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{\ displaystyle {\ tilde {\ boldsymbol {\ delta}}} _ {i} = ({\ hat {\ mathbf {Z}}} _ {i} ^ {\ top} {\ hat {\ mathbf {Z} }} _ {i}) ^ {- 1} {\ hat {\ mathbf {Z}}} _ {i} ^ {\ top} \ mathbf {y} _ {i}}
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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
↑ 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.
↑ 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
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