# Bootstrap Non Parametric Regression standard error comparisons

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 ?

• 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. – jsk Jun 14 at 15:43