MIT14_30s09_lec24

MIT14_30s09_lec24 - MIT OpenCourseWare http://ocw.mit.edu...

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Unformatted text preview: 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 . • 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 [ 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 Confidence Intervals • find functions of data A ( X 1 , . . . , X 2 ) and B ( X 1 , . . . , X n ) such that P θ ( 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 ) and b ( θ ) such that P θ ( a ( θ ) ≤ T ( X 1 , . . . , X n ) ≤ b ( θ )) = 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 )...
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MIT14_30s09_lec24 - MIT OpenCourseWare http://ocw.mit.edu...

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