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Suppose that we use a Non parametric bootstrap regression.If I'm not wrong ,that means that we have to sample with replacment from the residuals, for an amount of times. Lets assume that we have $i=1,..,50$ , $\epsilon_{i}^{j}$ residuals (which are computed as $\epsilon_{i}=y_{i}-\hat{a}-\hat{b}x_{i}$,without assuming that they follow a normal distribution) to sample from and we want to apply bootstrap $j=1,...,1000$ times.That means that we have to compute the following

$y_{i}^{j}=\hat{a}+\hat{\beta}x_{i}+\epsilon_{i}^{j}$

where $\hat{a}$ and $\hat{\beta}$ are the coefficients estimated with the use of OLS.

Next, when we have computed all the $y_{i}^{j}$ for $i=1,...,50$ and $j=1,...,1000$ then we will run a regression model with the use of

the first (column) $(y_{1}^{1},...,y_{i}^{1},...,y_{50}^{1})$ and $(x_{i})_{i=1}^{50}$ and compute the coefficients $\hat{a^{1}}$ and $\hat{\beta^{1}}$.We will repeat this whole procedure for $1000$ times and then estimate

$\alpha^{*}=\frac{1}{1000}\sum_{j=1}^{1000}\alpha^{j}$

$\beta^{*}=\frac{1}{1000}\sum_{j=1}^{1000}\beta^{j}$

To summarize my question:

The standard deviation of the coefficients, estimated with the use of a parametric linear regression, have to be smaller that the ones that are estimated with the use of bootstrap ?

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    $\begingroup$ Generally the point is to get a bootstrap estimate of the standard error of $\hat{\beta}$, which is $sd(\beta^j)$, not to produce new estimates of $\beta$, though perhaps that's not what you meant by $\beta^*$. I don't know if you're guaranteed to get a larger standard error from the bootstrap than the parametric linear regression. $\endgroup$ – jsk Jun 14 at 15:43

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