Can I wild cluster bootstrap this? =)

[s3alfisc]

Hi Grant,

Thanks for sharing this package - it looks really cool! Naturally, I am wondering if I can wild (cluster) bootstrap this. =)

After taking a very brief (really very, very brief) glance at both code and the Wooldridge paper, my very topline understanding of the estimation procedure is that etwfe implements a linear regression / glm with a set of interacted covariates and a few estimated properties (e.g. equation 5.7 in the Wooldridge paper attached below, where $\bar{x}_{1}$ is an estimated mean and therefore has a sampling variance for which one has to correct inference for).

After estimating the 'full' model, one then computes marginal effects, e.g. for the effect on a specific cohort, you compute the marginal effect for a given cohort. If we assume that there are $C$ cohorts, for linear models, inference for all $C$ 'cohort effects' should then boil down to C linear multi-parameter tests of the form $R_{c}'\beta_{c} = r$ vs $R_{c}'\beta_{c} \neq r$, where $R_{c}$ is a vector of dimension $k_{c}$, with $k_{c}$ denoting all regression parameters that depend on cohort $c$?

Here's a quick formulaic example to (hopefully) clarify:

We estimate a linear model

$y = \beta_0 + \beta_{1} X_{1} + \beta_{2} X_{1} X_{2} + \epsilon_{i}$

via OLS.

We then compute the average marginal effect at the mean for $X_{1}$ as

$E[\frac{\partial E(y|X)}{\partial X_{1}}] = \beta_{1} + \beta_{2} \bar{X}_{2}.$

Inference for the marginal effect then follows a standard $R\beta = r$ schema, where $R = [1, \bar{X}_{2}]$, $\beta = [\beta_1, \beta_2]$ and $r = 0$.

Do I get this right? If inference can be squeezed into the $R\beta = r$ schema, it will be fairly straightforward to make fwildclusterboot work with etwfe - but only under the assumption that $\bar{x}_{1}$ is a noiseless estimate of $x_1$.

Best, Alex

[grantmcdermott]

Hi Alex. Sorry for the super slow response. I've been on vacation and then straight into work travel...

I think you're intuition is correct and everything should pass through correctly. However, I must confess, that I haven't thought about this passed reading your comment and the time it's taken to type up this response. I'll let you test (simulate) and decide when is best to close/update ;-)

[s3alfisc]

Hi Grant, thanks for your feedback! No worries about the delay, I hope you had a relaxing vacation =) I will close this for now and will update you once I manage to implement support for the wcb and etwfe + have some simulations at hand that look promising!

[jtorcasso]

Hi all! Checking into this issue to see if there was any progress. I tried doing this myself but got an error.

[s3alfisc]

Hi @jtorcasso - I have never looked into this in more detail. But it should definitely be possible, and if I understand everything correctly, implementing support for the wild cluster bootstrap / etwfe should be around one Saturday of work for me (I am alwazys optimistic with these things, though). But I am committed to too many projects at the moment, so I cannot promise to take a closer look in the next weeks.

In principle, there are two options how to add support for wild cluster bootstrapping to etwfe - one of them is sketched out above and requires to loop over boottest() multiple times, similar as done for the boot_aggregate() implementation of the wcb for fixest::sunab() here. The other option is to simply pass a bootstrapped covariance matrix to the feols call / marginaleffects(). You could do this via sandwich::vcovBS (potentially slow, and note that this way you would not compute a percentile-t bootstrap and would not impose the null on the bootstrap dgp). Alternatively, I have a rough implementation of the classical pairs bootstrap in the dev branch of fwildclusterboot, which computes a bootstrapped vcov matrix (but I have not really tested it). Beyond, I unfortunately don't think it is currently possible to get a full bootstrap vcov matrix out of fwildclusterboot::boottest(). If you post your code as an issue / PR to fwildclusterboot, I can take a look tomorrow evening, in case you think it might help. You can also send me an email if you'd prefer that (you could find it e.g. here). =)

[s3alfisc]

Also, just to point this out again - if the Sun-Abrams method is also appropriate for your use case, boot_aggregate does support the wcb for fixest::sunab(). =)

[s3alfisc]

Ok, I took a quick look at this, there are (at least) two issues:

• when trying to fetch the cluster variable, boottest() does not find the data internally created by etwfe(). This should not be too hard to fix, but I will have to read up on environments in advanced R.
• boottest() does not handle varying slopes fixed effects syntax. In fact, it can project out at most one fixed effect in each bootstrap operation. For all other fe's, it needs to construct dummy variables. fixest currently does not provide a custom method for this task. The solution here is to wait for the inclusion of this PR into fixest, which allows to transform all sorts of fixed effects into dummies.
• With those two things solved, it should be possible to just plug in an etwfe object into boot_aggregate().

[s3alfisc]

Actually, here is a hacky preliminary solution, in five steps, which works without the PR to fixest mentioned above:

• run etwfe. get the model formula.
• transform all fixed effects into dummies. change the fixed effects varying slopes syntax from var1[[var2]] to var1 / var2
• from etwfe(), export data, i.e. by assigning it as a global variable etwfe_data (very hacky, sorry!)
• fit feols(new_fml, etwfe_data)
• run the wcb via fwildclusterboot::boot_aggregate(). Done.

library(etwfe)
library(fixest)
library(fwildclusterboot)

data("mpdta", package="did")
head(mpdta)

mod = etwfe(
  fml = lemp ~ lpop,
  tvar  = year,
  gvar = first.treat,
  data = mpdta,
  se = "hetero"
)

# get it from assigning a global var data_etwfe <<- data in `etwfe()`
data_etwfe

# get formulas, write one part formula, varying slopes as var1 / var2
mod$fml_all
# new formula
fml <- lemp ~ .Dtreat:i(first.treat, i.year, ref = 0, ref2 = 2003)/lpop_dm + first.treat + first.treat/lpop + year + year/lpop
# fit model
fit <- feols(fml, data_etwfe, cluster ~ countyreal, ssc = ssc(adj = FALSE, fixef.K = "none"))

#aggregate
agg <- aggregate(
  fit, 
  c("ATT"="^.Dtreat:first.treat::[0-9]{4}:year::[0-9]{4}$")
)
agg
#       Estimate Std. Error t value    Pr(>|t|)
# ATT 0.09483085 0.02676169 3.54353 0.000431911

boot_aggregate(
  fit, 
  c("ATT"="^.Dtreat:first.treat::[0-9]{4}:year::[0-9]{4}$"), 
  B = 999,
  boot_ssc = ssc(adj = FALSE),
  clustid = ~countyreal 
)
# Estimate  t value Pr(>|t|)    [0.025%   0.975%]
# [1,] 0.09483085 3.547078        0 0.04099321 0.1487666

t-stats, pvalue and confidence interval of boot_aggregate() are computed via boottest(). Note that the reported non-bootstrapped t-stat of aggregate() and boot_aggregate() should match exactly, differences are likely / potentially due to incorrect handling of small sample adjustments in boot_aggregate (partially fixed in the dev branch).

Keep in mind that none of this is properly tested, that fixed effect varying slopes syntax does not work (wrong results), and that there is a bug for boot_aggregate() and the HC bootstrap (fixed in dev). Anyways, I hope this helps!