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MIT14_30s09_lec24

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 24 Konrad Menzel May 14, 2009 Review Point Estimation Estimator function θ ˆ ( X 1 , . . . , X n ) of the sample bias of estimator is Bias( θ ˆ ) = E θ 0 [ θ ˆ ] θ 0 standard error of estimator given by σ ( θ ˆ ) = Var( θ ˆ ) Important criteria for assessing estimators are Unbiasedness Efficiency Consistency Methods for constructing Estimators: 1. Method of Moments: m th population moment is E θ [ X i m ] = µ m ( θ ) m th sample moment is X m = n 1 i n =1 X i m compute first k moments, equate X m = ! µ m ( θ ˆ ) for m = 1 , . . . , k , and solve for the estimate θ ˆ . 2. Maximum Likelihood write down likelihood function for the sample X 1 , . . . , X n , n L ( θ ) = f ( X 1 , . . . , X n θ ) = f ( X i θ ) | i =1 | find value of θ which maximizes L ( θ ) or log( L ( θ )). usually find maximum by setting first derivative to zero, but if support of random variable depends on θ may not be differentiable, so you should rather try to see what function looks like, and where the maximum should be. 1
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Confidence Intervals find functions of data A ( X 1 , . . . , X 2 ) and B ( X 1 , . . . , X n ) such that P θ 0 ( A ( X 1 , . . . , X n ) θ 0 B ( X 1 , . . . , X n )) = 1 α then [ A ( X 1 , . . . , X n ) , B ( X 1 , . . . , X n )] is a 1 α confidence interval for θ for given significance level 1 α many possible valid confidence intervals. In order to construct confidence intervals, usually proceed as follows: 1. find a ( θ 0 ) and b ( θ 0 ) such that P θ 0 ( a ( θ 0 ) T ( X 1 , . . . , X n ) b ( θ 0 )) = 1 α for some statistic T ( X 1 , . . . , X n ) (typically will use an estimator θ ˆ here). 2. rewrite the event inside the probability in form P ( A ( X 1 , . . . , X n ) θ 0 B ( X 1 , . . . , X n )) = 1 α 3. evaluate A ( ) and B ( ) at the sample values X 1 , . . . , X n to obtain confidence interval · · Some important cases: θ ˆ unbiased and normally distributed, Var( θ ˆ ) known: [ A ( X 1 , . . . , X n ) , B ( X 1 , . . . , X n )] = θ ˆ + Φ 1 α α Var( θ ˆ ) , θ ˆ + Φ 1 1 Var( θ ˆ ) 2 2 θ ˆ unbiased and normally distributed, Var( θ ˆ ) not known, but have estimator S ˆ : [ A ( X 1 , . . . , X n ) , B ( X 1 , . . . , X n )] = θ ˆ + t n 1 α α Var( θ ˆ ) , θ ˆ + t n 1 1 Var( θ ˆ ) 2 2 θ ˆ not normal, n > 30 or so: most estimators we have seen so far turn out to be asymptotically normally distributed, so
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