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If I have a regression and I am bootstrapping the coefficients, if I end up with many calculated coefficients how do I calculate standard errors?

Is it the std of the samples? Or the std divided by the square root of the boot sample size? Or do I compute N samples of size B and compute the std of each sample and take the mean of those stds? (Where std = standard deviation)?

I have found references suggesting we can simply take the std of the bootstrapped samples but this is for normally distributed bootstrap sample data, if the bootstrap samples are not normal then what can we do to get estimates of standard errors?

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    $\begingroup$ "I have found references suggesting we can simply take the std of the bootstrapped samples but this is for normally distributed data." Why do you think this is only for normally distributed data? $\endgroup$ Commented Jul 9, 2020 at 17:25
  • $\begingroup$ I would imagine that the distribution of your bootstrapped statistics - regression coefficients, in your case - will always converge to normal, the more bootstrap iterations (samples) you have, somehow a manifestation of the central limit theorem. Am I confused? $\endgroup$
    – ouranos
    Commented Jul 9, 2020 at 17:26
  • $\begingroup$ Oh sorry Mathew Drury, I should update the question, I mean only for normally distributed bootstrapped samples, not only for normally distributed sample data. $\endgroup$
    – MHall
    Commented Jul 9, 2020 at 19:07
  • $\begingroup$ Ouranos, the bootstrap I'm doing gives really large outliers, like -5000 or +3000 when the coefficients estimated at all under 2 in absolute value. It is an experimental model, causal/noncausal VAR and I'm trying to get the bootstrap to make sense. With 1000 boot samples the dist is not near to normal. I may have to try larger sample sizes. $\endgroup$
    – MHall
    Commented Jul 9, 2020 at 19:25

2 Answers 2

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You get the regression standard errors for the coefficients from the standard deviation of the bootstrapped coefficients from each rep:

. sysuse auto
(1978 Automobile Data)

. bs, reps(101) saving("bs_reg.dta", replace): reg price foreign mpg weight
(running regress on estimation sample)

Bootstrap replications (101)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 
..................................................    50
..................................................   100
.

Linear regression                               Number of obs     =         74
                                                Replications      =        101
                                                Wald chi2(3)      =      58.55
                                                Prob > chi2       =     0.0000
                                                R-squared         =     0.4996
                                                Adj R-squared     =     0.4781
                                                Root MSE          =  2130.7695

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     foreign |    3673.06   622.7476     5.90   0.000     2452.498    4893.623
         mpg |    21.8536   81.81615     0.27   0.789    -138.5031    182.2103
      weight |   3.464706   .7507974     4.61   0.000      1.99317    4.936242
       _cons |  -5853.696   3816.206    -1.53   0.125    -13333.32    1625.931
------------------------------------------------------------------------------

. use "bs_reg.dta", clear
(bootstrap: regress)

. summarize

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
  _b_foreign |        101    3599.271    622.7476   2129.934   4902.719
      _b_mpg |        101    36.65845    81.81615  -175.5054   288.2288
   _b_weight |        101    3.512753    .7507974   1.817149   5.068236
     _b_cons |        101   -6261.379    3816.206  -16303.59   2266.739

Of course, this relies on independence of observations or clusters of observations to work. In your time series context, unless you adapt the procedure, the basic bootstrap will not work, even asymptotically. Here's a nice answer that gets into the various ways to adapt the BS to TS setting.

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In many contexts the distribution of coefficient estimates is normal. In that case taking the standard deviation among the coefficient estimates makes sense.

It seems, however, that your particular application is not leading to such a distribution of coefficient estimates. You can calculate an empirical standard deviation among the coefficient estimates but it won't necessarily have the usual interpretation in terms of coverage. You can't necessarily think of +/- 1 SD among the estimates from the bootstrapped samples as representing a 68.3% confidence interval about the mean. This happens if the coefficient value being calculated from the bootstrap samples isn't pivotal.

This answer is one of many on this site that discusses this problem, and ways to deal with it. This search provides links to more. Briefly, there are ways that can deal with bias and skew in values calculated from bootstrap samples although sometimes there can be intractable problems.

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