(L12)DataTilt2(f)

(L12)DataTilt2(f) - Lecture#12 Data and Estimation Reading...

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Primbs/Investment Science 1 Lecture #12 Data and Estimation Reading: Luenberger Chapter 8, Sections 5 – 9
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Primbs/Investment Science 2 Data and Estimation Scaling Rules Effect of More Data Estimation of Parameters
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Primbs/Investment Science 3 Data and Statistics We have been talking a lot about means and variances and covariances. In practice, we have to estimate these values. Let’s see how well we can estimate means and variances...
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Primbs/Investment Science 4 Mean and Variance Estimators = = n i i r n r 1 1 ˆ Mean: n r σ σ = ˆ Error: = - - = n i i r r n s 1 2 2 ) ˆ ( 1 1 Variance: 1 2 2 2 - = n s σ σ Note that the error in the mean estimate depends on the standard deviation. But, the error for the variance estimate is a fraction of the variance that is being estimated.
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Primbs/Investment Science 5 Mean and Variance Estimators = = n i i r n r 1 1 ˆ Mean: = - - = n i i r r n s 1 2 2 ) ˆ ( 1 1 Variance: n r σ σ = ˆ Error: 1 2 2 2 - = n s σ σ Let’s consider estimating the monthly mean and variance for the market, and how many months we need for accurate estimates. # months 12 60 120 360 1200 mean error 1.25% 0.56% 0.40% 0.23% 0.13% variance error 8 x 10 -4 3.5 x 10 -4 2.4 x 10 -4 1.4 x 10 -4 7.6 x 10 -5
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Primbs/Investment Science 6 Mean and Variance Estimators = = n i i r n r 1 1 ˆ Mean: = - - = n i i r r n s 1 2 2 ) ˆ ( 1 1 Variance: n r σ σ = ˆ Error: 1 2 2 2 - = n s σ σ The mean is inherently more difficult to estimate than the variance. Using more data (daily for instance) will not improve your ability to estimate the mean. However, it will improve your ability to estimate the variance. Moral: (I) The mean return is difficult to estimate accurately. (II) The variance can be estimated will relative success.
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Primbs/Investment Science 7 Covariance Estimators = - - - = n i i i r r r r r r n s 1 ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) ˆ )( ˆ ( 1 1 2 1 To produce an unbiased estimate of the covariance use: Warning!: Excel divides by N instead of N-1 for its covariance estimator. This makes its covariance function inconsistent with its variance function!
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Primbs/Investment Science 8 Data and Estimation Scaling Rules Effect of More Data Estimation of Parameters
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