# bootstrap standard error for regression model

I'm trying to approximate by bootstrap the standard error of a regression-type estimator for which the asymptotic distribution is difficult to calculate. I have a 0-1 coded dummy variable in the model that equals one in only a few cases. In some iterations, none of the observations sampled include the dummy variable valued at one, so the corresponding column of the data matrix contains all zeroes, and the covariance matrix for the sampled data is singular (because the model also includes an intercept). As a result, I cannot estimate the value of the parameter.

I'm trying to figure out a way around this problem. Won't removing the regressor from the estimated model on these iterations potentially introduce a bias in the remaining coefficients and overstate their standard errors?

Any suggestions are appreciated!

## 1 Answer

In your simulations, you can justifiably code the effect for those runs as zero. (There is perfect prediction in the instances when none where observed, and this has no dependence on the variable of interest.) That said, I would still tally how often that happens in your runs, and report that when presenting the SE estimate from the simulation.

• So what do you recommend I do when that effect is zero? Say I want to calculate the sample variance of this parameter. Do I leave out the draws when this parameter is not included (in other words, valued at zero) in the model? Or what if I want to calculate the sample covariance between this parameter and another? Do I leave the zero draws? It seems to me including it at zero would overstate the probability of the estimated parameter falling in a range near zero. On the other hand, ignoring them would not be true to the bootstrap methodology, too. – Alex Mar 31 '18 at 13:33
• In the bootstrap simulation, you generate the parameter(s). Then, with your data set of parameters, you can calculate the standard deviation of those parameters, which would be interpreted as a reasonable estimate of the standard error of that parameter for the given sample size. Having zeros in that data set may "deflate" the SD you calculate, but no more so than if you had other repeated values show up in the simulations. Likewise, the covariance between two parameters would be calculated by taking the covariance of the estimated parameters. Having zeros doesn't prohibit this. – Gregg H Mar 31 '18 at 13:52