10232012 p kolm 3 the residual bootstrap steps 1

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Unformatted text preview: ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 3 The Residual Bootstrap Steps: ˆˆ ˆ 1. We start by estimating the regression coefficients β0, β1, …, βk , for the linear regression equation y = β0 + β1x 1 + … + βk x k + ε ˆ ˆ using OLS. Then calculate the regression residuals εi ≡ y; − yi for i = 1,..., N ˆ ˆ 2. Randomly draw N “samples” from {ε1,..., εN } , assuming a probability of 1/N ˆ ˆ for each εi , with replacement (thus it is possible to pick the same εi more than once). In this way we obtain a sequence of N “samples.” Call the bootstrapped ˆ∗ ˆ∗ residuals ε1 ,..., εN ˆˆ 3. Generate the bootstrapped dependent variable yi∗ ≡ yi + εi∗ for i = 1,..., N . Now we N have pairs of “bootstrapped samples” ∗ * * * (y1 , x 1,1, x 1,2, …, x 1,k )′ , ..., ∗ * * * (yN , x N ,1, x N ,2, …, x N ,k )′ We can use OLS to calculate our bootstrap parameter ˆˆ ˆ estimates β *, β *, …, β * for the regression equation 0 1 k y * = β0* + β1*x 1*...
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