Kolm 6 the pairwise bootstrap steps 1 randomly draw n

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Unformatted text preview: d cases where we may not have an analytical form of the standard error, the bootstrap is an extremely powerful tool • Bootstrapping can be useful in understanding small sample biases [For you: Why?] RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 6 The Pairwise Bootstrap Steps: 1. Randomly draw N pairs of “data” from (y1, x 1,1, x 1,2, …, x 1,k )′ , ..., (yN , x N ,1, x N ,2, …, x N ,k )′ , assuming a probability of 1/N for each (yi , x i′) -pair, with replacement (thus it is possible to pick the same (yi , x i′) -pair more than once). ∗ * * * In this way we obtain a sequence of N “samples” (y1 , x 1,1, x 1,2, …, x 1,k )′ , ..., ∗ * * * (yN , x N ,1, x N ,2, …, x N ,k )′ ∗ ∗ ∗ ∗ 2. We apply OLS to the bootstrapped data set {(y1 , x 1 ′ ),...,(yN , x N ′ )} , calling the estimate β ∗ . This is our bootstrap parameter estimate 3. We repeat steps 1 and 2 for M times. Every time we perform step 2 we use ˆ ˆ ˆ new random draws, obtaining M different β ∗ ’s {β ∗(1),..., β ∗(M ) } ˆ 4. Calculate the bootstrap standard error estimate of β j (i.e., the j-th element of ˆ ˆ β ) using the formula Std∗ (β j ) = 1 M ⎛ *(k ) ⎜β − 1 ∑ ⎜ ˆj M ⎝ k =1 ⎜ M 2 ⎞ ˆ*(k ) ⎟ ⎟ ∑ βj ⎠ ⎟ ⎟ l =1 M RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 7 Remark: • Pairwise bootstrap produces standard errors that are robust to heteroscedasticity, so it is a bootstrap alternative to the robust standard errors RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 8 End Notes 1 M should be large. Depending on the application, we would choose M equal to 1000 or larger. RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 9...
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