Bootstrap

# 1 every time we perform step 2 using new random draws

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Unformatted text preview: + … + βk *x k * + ε* RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 4 4. We repeat steps 2 and 3 for M times.1 Every time we perform step 2 using new ˆ ˆ ˆ random draws, obtaining M different β ∗ ’s {β ∗(1),..., β ∗(M ) } ˆ 5. Calculate the bootstrap standard error estimate of β j (i.e., the j-th element of ⎛ ˆ ˆ ) using the formula Std∗ (β ) = 1 ∑ ⎜ β *(k ) − 1 β ⎜ ˆj j M k =1 ⎜ M ⎝ M 2 ⎞ ˆ*(k ) ⎟ ⎟ ∑ βj ⎠ ⎟ ⎟ l =1 M RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 5 Remark: ˆ • In the case of OLS (as described above), Std∗ (β j ) converges to the “standard ˆˆ ε ′ε as error estimate without heteroscedasticity adjustment” (i.e., N (X ′X )−1 jj M → ∞ ) [For you: Why?] • In our simple example there is no point in bootstrapping as we can simply calculate the standard error directly using the standard formula. Note however, that in more complicate...
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