MIT14_30s09_lec18

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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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14.30 Introduction to Statistical Methods in Economics Lecture Notes 18 Konrad Menzel April 23, 2009 1 Properties of Estimators (continued) 1.1 Standard Error Often we also want to make statements about the precision of the estimator - we can always state the value of the estimate, but how confident are we that it is actually close to the true parameter? Definition 1 The standard error σ ( θ ˆ ) of an estimate is the standard deviation (or estimated standard deviation) of the estimator, SE ( θ ˆ ) = Var( θ ˆ ( X 1 , . . . , X n )) Should recall that an estimator is a function of the random variables, and therefore a random variable for which we can calculate expectation, variance and other moments. Example 1 The mean X ¯ n of an i.i.d. sample X 1 , . . . , X n where Var( X i ) = σ 2 has variance σ n 2 . There- fore, the standard error is σ ¯ SE ( X n ) = n If we don’t know σ 2 , we calculate the estimated standard error SE ˆ ( X ¯ n ) = σ ˆ n The standard error is a way of comparing the precision of estimators, and we’d obviously favor the estimator which has the smaller variance/standard error. Definition 2 If θ ˆ A and θ ˆ B are unbiased estimators for θ , i.e. E θ 0 [ θ ˆ A ] = E θ 0 [ θ ˆ B ] = θ 0 , then we say that θ ˆ A is efficient relative to θ ˆ B if Var( θ ˆ B ) Var( θ ˆ A ) Sometimes we look at an entire class of estimators Θ = { θ ˆ 1 , θ ˆ 2 , . . . } , and we say that θ ˆ A is efficient in that class if it has the lowest variance of all members of Θ. Example 2 Suppose that X and Y are scores from two different Math tests. You are interested in some underlying ”math ability”, and the two scores are noisy (and possibly correlated) measurements with E [ X ] = E [ Y ] = µ , Var( X ) = X , Var( Y ) = σ Y 2 , and Cov( X, Y ) = . Instead of using only one of the σ 2 σ XY measurements, you decide to combine them into a weighted average pX + (1 p ) Y instead. What is the 1
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expectation of this weighted average? Which value of p minimizes the variance of the weighted average?
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