# Ex can no longer be replaced by x also we will

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: in terms of how Yi ’s enter the estimator expressions Unbiased; expectation of the estimator equals the true value of the parameter Estimator. Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Today’s Lecture Important Note We have altered the model to accommodate stochastic regressors. This change also alters checking the properties of an estimator. In particular checking whether estimator is biased or not needs extra care. (E(X ) can no longer be replaced by X ) Also we will introduce a new property called “consistency”. Introduction Measurement Error Simultaneous Equations Today’s Lecture Today’s Pathology Of Least Squares Some estimation issues in the context of model B: Measurement Error (Chapter 8) Simultaneous Equations (Chapter 9) Past Exam Practice Question Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Today’s Lecture Today’s Pathology Of Least Squares Some estimation issues in the context of model B: Measurement Error (Chapter 8) Simultaneous Equations (Chapter 9) What assumption of Model B is violated? B.7 as the disturbance term will not be independent of regressors. Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Today’s Lecture Today’s Pathology Of Least Squares Some estimation issues in the context of model B: Measurement Error (Chapter 8) Simultaneous Equations (Chapter 9) What assumption of Model B is violated? B.7 as the disturbance term will not be independent of regressors. What links these topics? Stochastic regressors context. For consistency of basic OLS need generally cov (X , u ) = 0. Measurement Error and Simultaneous Equations Bias are two reasons for why this may not hold. Detection and remedy tools similar. Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Today’s Lecture Consistency Of OLS & cov (X , u ) = 0 Model: Yi = β1 + β2 Xi + ui b2 = = = n ¯ ¯ i =1 (Xi − X )(Yi − Y ) n ¯2 i =1 (Xi − X ) n ¯ ¯ i =1 (Xi − X )(β2 (Xi − X ) + n ¯2 i =1 (Xi − X ) n ¯ ¯ (Xi − X )(ui − u ) β2 + i =1 n ¯ )2 i =1 (Xi − X (1) ¯ (ui − u )) (2) (3) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Today’s Lecture Consistency Of OLS & cov (X , u ) = 0 Key for consistency: Is the regressor correlated with error term? plim(b2 ) = β2 + = β2 + n ¯ ¯ i =1 (Xi − X )(ui − u )) n 1 ¯ )2 ) plim( n i =1 (Xi − X (4) cov (X , u ) Var (X ) (5) 1 plim( n Need =0 for consistency Make sure you are clear on: 1 Relation between cov (X , u ) = 0 assumption and B7. (Latter stronger) 2 Is cov (X , u ) = 0 enough to show OLS unbiased as well? (No) 3 Implicit regularity assumptions on X. (Var (X ) = 0) 4 Intuitive extension to multiple regression analysis. Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Deﬁnition Disease 4: Measurement Error True model: Yi = β1 + β2 Zi + vi Observe regressor with measurement error: Xi = Zi + wi Yi = β1 + β2 (Xi − wi ) + vi (6) = β1 + β2 Xi + vi − β2 wi (7) = β1 + β2 Xi + ui (8) Consider covariance between X and u: Cov (X , u ) = Cov (Z + w , v − β2 w ) = Cov...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online