lecture_note_112916

# lecture_note_112916 - Econ 5280 Applied Econometrics...

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Chapter 5 Linear Models with Endogenous Errors 5.1 Introduction Note that the previous sections focus on the estimation of conditional expectation: E ( Y t X t ) , and the errors generated by the conditional expectation is orthogonal to the explanatory variable by its nature: E ( U t X t ) = 0 . This condition often does not hold for economic equations. In many cases, it is more appropriate eco- nomic equation that is not necessarily a conditional expectation. For example, demand function or supply function is not constructed by conditional expectation. It is an equation derived from economics that could be irrelevant to conditional expectation. For such a case, it is more appropriate to assume that errors are not orthogonal to the explanatory variables: E [ U t X t ] 0 . If this is the case, the OLS estimator cannot be a proper estimator, and the equation is often called the structural equation . The OLS estimator cannot consistently estimate the parameters in the structural equation. We may suppose that the following structural equation: Y t = X t β * + U t such that E [ U t X t ] 0 , and we may estimate the unknown parameter β * by the OLS estimation. Then, ̂ β n = β * + ( n - 1 n t = 1 X t X t ) - 1 ( n - 1 n t = 1 X t U t ) , where n - 1 n t = 1 X t X t a.s. E ( X t X t ) and n - 1 n t = 1 X t U t a.s. E ( X t U t ) 0 . 60
Thus, it follows that ̂ β n a.s. β * + E ( X t X t ) - 1 E ( U t X t ) β * . This implies that the OLS estimator is not consistent for β * . We often call this that the OLS estimator is asymptotically biased . Due to this asymptotic bias, the OLS estimator has to be replaced by another estimator, and we consider the replacement in this chapter. 5.2 Instrumental Variable Estimator The instrumental variable (IV) estimator can be a proper estimator for β * . Before examining the precise implication of the IV estimator, we first provide the following assumptions for the IV estimation and its inference. 1. Dateset is a sequence of IID observations: {( U t , X t , Z t ) R 1 + 2 k t = 1 , 2 , ,n } ; 2. For some unknown β * , Y t = X t β * + U t such that E ( U t Z t ) = 0 ; 3. E ( U 4 t ) < , E [ X 2 t,i ] < , and E [ Z 4 t,i ] < for i = 1 , 2 , ,k ; 4. E [ Z t X t ] and n t = 1 Z t X t are positive definite; and 5. Σ = E [ U 2 t Z t Z t ] is positive definite. 5.2.1 IV Estimator and Its Asymptotic Properties The IV estimator is defined by the orthogonality condition. Note that the orthogonality condition E ( U t Z t ) = 0 ) can be manipulated as follows: 0 = E ( Z t U t ) = E [ Z t ( Y t - X t β * )] , so that β * = E ( Z t X t ) - 1 E ( Z t Y t ) . Here, E ( Z t X t ) - 1 exists because E ( Z t X t ) is positive definite. Furthermore, E ( Z t Y t ) also exists mainly because E ( Z t X t ) exists and Y t = X t β * + U t . Thus, we may estimate β * by the sample analogs in the right 61

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side of this equation: ̃ β n = n - 1 n t = 1 Z t X t - 1 n - 1 n t = 1 Z t Y t = ( Z X ) - 1 Z Y , where Z = [ Z 1 , , Z n ] .
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• Fall '08
• JAKUBSON
• Econometrics, Estimation theory, zt, ut zt, Zt Z′t

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