# fwildclusterboot

library(fwildclusterboot)

The fwildclusterboot package implements the “fast” wild cluster bootstrap algorithm developed in Roodman et al (2019) for regression objects in R. The “fast” algorithm makes it feasible to calculate test statistics based on a large number of bootstrap draws even for large samples - as long as the number of bootstrapping clusters is not too large.

For linear regression models, fwildclusterboot supports almost all features of Stata’s boottest package. This means that a set of different bootstrap distributions, regression weights, fixed effects, and both restricted (WCR) and unrestricted (WCU) bootstrap inference are supported. The main difference is that it currently only supports univariate hypothesis tests of regression paramters of the form $$H_{0}: R\beta = r$$ vs $$H_{1}: R\beta \neq r$$, where r is scalar.

Further, fwildclusterboot serves as an R port to WildBootTests.jl, which implements the fast wild cluster bootstrap in Julia at extreme speed. Beyond being significantly faster than fwildclusterboot's native R implementation of the wild cluster bootstrap for OLS (in particular for more demanding problems), WildBootTests.jl offers support for the WRE bootstrap for IV models ((Davidson & MacKinnon, 2010)) and functionality for tests of multiple hypothesis.

A description of the “fast” algorithm is beyond the scope of this vignette. It is very clearly presented in Roodman et al. (2019). For technical details of the implementation in fwildclusterboot, have a look at the technical vignette (tba).

# The boottest() function

The fwildclusterboot package consists of one key function, boottest(). It implements the fast wild bootstrap and works with regression objects of type lm, felm and fixest from base R and the lfe and fixest packages.

To start, we create a random data set with two cluster variables (group_id1 & group_id2), two fixed effects and a set of covariates. The icc_ arguments control the cluster variable’s intra-cluster correlation.

# set seed via dqset.seed for boot_algo = "R" & Rademacher, Webb & Normal weights
dqrng::dqset.seed(2352342)
# set 'familiar' seed for all other algorithms and weight types
set.seed(23325)

# load data set voters included in fwildclusterboot
data(voters)

# estimate the regression model via lm
lm_fit <- lm(proposition_vote ~ treatment + ideology1 + log_income + Q1_immigration , data = voters)

# model with interaction
lm_fit_interact <- lm(proposition_vote ~ treatment + ideology1 + log_income:Q1_immigration , data = voters)

The boottest() function has 4 required and several optional arguments. The required objects are

• object: a regression object of type lm, fixest or felm
• clustid: a character vector that defines the clustering variables
• param: a character vector of length one - the model parameter to be tested
• B: the number of bootstrap iterations
# boottest on an object of type lm
boot_lm <- boottest(lm_fit, clustid = "group_id1", param = "treatment", B = 9999)

To tests for an interaction, it is important to use the coefficient names that are internally created by the modeling function.

#names(coef(lm_fit_interact))
boot_lm_interact <- boottest(lm_fit_interact, clustid = "group_id1", param = "log_income:Q1_immigration1", B = 9999)

boottest() further allows for multivariable tests. Suppose we’re interested in testing the null hypothesis $$0.6*treatment + 0.2*ideology1 = 0.02$$. To test such a hypothesis, one would have to specify the hypothesis via the param, R and r arguments:

boot_multi <- boottest(lm_fit, clustid = "group_id1", param = c("treatment", "ideology1"), R = c(0.6, 0.2), r = 0.02, B = 9999)

To access the estimation results, boottest() comes with summary(), tidy() and glance() methods. The tidy() method returns the estimation results in a data.frame. summary() returns additional information on top of the test statistics reported by tidy(). Theglance() method enables the use of output formatting tools from the modelsummary package.

# fwildclusterboot's internal summary() method
summary(boot_lm)
#> boottest.lm(object = lm_fit, param = "treatment", B = 9999, clustid = "group_id1")
#>
#>  Hypothesis: 1*treatment = 0
#>  Observations: 300
#>  Clustering: 1-way
#>  Confidence Sets: 95%
#>  Number of Clusters: 40
#>
#>              term estimate statistic p.value conf.low conf.high
#> 1 1*treatment = 0    0.073     3.709   0.001    0.033     0.113
summary(boot_multi)
#> boottest.lm(object = lm_fit, param = c("treatment", "ideology1"),
#>     B = 9999, clustid = "group_id1", R = c(0.6, 0.2), r = 0.02)
#>
#>  Hypothesis: 0.6*treatment+0.2*ideology1 = 0.02
#>  Observations: 300
#>  Clustering: 1-way
#>  Confidence Sets: 95%
#>  Number of Clusters: 40
#>
#>                                 term estimate statistic p.value conf.low
#> 1 0.6*treatment+0.2*ideology1 = 0.02    0.048     2.346   0.025    0.024
#>   conf.high
#> 1     0.072

if(requireNamespace("modelsummary")){
# summary via the modelsummary package
library(modelsummary)
msummary(list(boot_lm, boot_lm_interact),
estimate = "{estimate} ({p.value})",
statistic = "[{conf.low}, {conf.high}]")
}
#> Loading required namespace: modelsummary
Model 1 Model 2
1*treatment = 0 0.073 (0.001)
[0.033, 0.113]
1*log_income × Q1_immigration1 = 0 -0.038 (0.000)
[-0.057, -0.019]
Num.Obs. 300 300
R2 0.316 0.339
AIC -82.1 -92.2
BIC -30.2 -40.4
Log.Lik. 55.025 60.102

A plot() method allows the user to inspect the bootstrap t-statistics:

plot(boot_lm)

## Multiway Clustering

The boottest() function supports clustering of any dimension. E.g. for two-way clustering, one simply needs to specify the names of the cluster variables in a character vector.

boot_lm <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999)
summary(boot_lm)
#> boottest.lm(object = lm_fit, param = "treatment", B = 9999, clustid = c("group_id1",
#>     "group_id2"))
#>
#>  Hypothesis: 1*treatment = 0
#>  Observations: 300
#>  Clustering: 2-way
#>  Confidence Sets: 95%
#>  Number of Clusters: 40 20 251
#>
#>              term estimate statistic p.value conf.low conf.high
#> 1 1*treatment = 0    0.073     3.845   0.005     0.03     0.115

## The Heteroskedastic Bootstrap

If you drop the clustid argument, boottest() will run a heteroskedasticity robust wild bootstrap via the ‘R-lean’ algorithm. At the moment, the null hypothesis is always imposed, only Rademacher and Webb weights are supported, and no confidence intervals are computed. Further, no regression weights are supported. As all algorithms in fwildclusterboot, p-values are calculated based on pivotal t-statistics.

boot_lm <- boottest(lm_fit, param = "treatment", B = 9999)
summary(boot_lm)
#> boottest.lm(object = lm_fit, param = "treatment", B = 9999)
#>
#>  Hypothesis: 1*treatment = 0
#>  Observations: 300
#>  Clustering: 0-way
#>  Confidence Sets: 95%
#>
#>              term estimate statistic p.value conf.low conf.high
#> 1 1*treatment = 0    0.073     3.096   0.002       NA        NA
boot_lm$boot_algo #> [1] "R-lean" ## Choice of Bootstrap Weights Furthermore, the user can choose among four different weighting distribution via the type argument: Rademacher, Mammen, Normal and Webb. By default, boottest() uses the Rademacher distribution. boot_lm_rade <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 999, type = "rademacher") boot_lm_webb <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 999, type = "webb") if(requireNamespace("modelsummary")){ library(modelsummary) msummary(list(boot_lm_rade, boot_lm_webb), estimate = "{estimate} ({p.value})", statistic = "[{conf.low}, {conf.high}]") } Model 1 Model 2 1*treatment = 0 0.073 (0.003) 0.073 (0.004) [0.028, 0.117] [0.031, 0.114] Num.Obs. 300 300 R2 0.316 0.316 R2 Adj. 0.288 0.288 AIC -82.1 -82.1 BIC -30.2 -30.2 Log.Lik. 55.025 55.025 ## Other function arguments Via the function argument sign_level, the user can control the significance level of the test. The default value is sign_level = 0.05, which corresponds to a 95% confindence interval. boot_lm_5 <- boottest(lm_fit, clustid = c("group_id1"), param = "treatment", B = 9999, sign_level = 0.05) boot_lm_10 <- boottest(lm_fit, clustid = c("group_id1"), param = "treatment", B = 9999, sign_level = 0.10) if(requireNamespace("modelsummary")){ library(modelsummary) msummary(list(boot_lm_5, boot_lm_10), estimate = "{estimate} ({p.value})", statistic = "[{conf.low}, {conf.high}]") } Model 1 Model 2 1*treatment = 0 0.073 (0.000) 0.073 (0.001) [0.033, 0.113] [0.040, 0.106] Num.Obs. 300 300 R2 0.316 0.316 R2 Adj. 0.288 0.288 AIC -82.1 -82.1 BIC -30.2 -30.2 Log.Lik. 55.025 55.025 In the case of multiway clustering, the user might want to specify the bootstrap clustering level. By default, boottest chooses the clustering level with the highest number of clusters as bootcluster = "max". Other choices are the minimum cluster, or independent clustering variables. boot_lm1 <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999, bootcluster = "min") boot_lm2 <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999, bootcluster = "group_id1") if(requireNamespace("modelsummary")){ library(modelsummary) msummary(list(boot_lm1, boot_lm2), estimate = "{estimate} ({p.value})", statistic = "[{conf.low}, {conf.high}]") } Model 1 Model 2 1*treatment = 0 0.073 (0.004) 0.073 (0.008) [0.031, 0.110] [0.027, 0.116] Num.Obs. 300 300 R2 0.316 0.316 R2 Adj. 0.288 0.288 AIC -82.1 -82.1 BIC -30.2 -30.2 Log.Lik. 55.025 55.025 ## Fixed Effects Last, boottest() supports out-projection of fixed effects in the estimation stage via lfe::felm() and fixest::feols(). Within the bootstrap, the user can choose to project out only one fixed effect, which can be set via the fe function argument. All other fixed effects specified in either felm() or feols() are treated as sets of binary regressors.  if(requireNamespace("fixest")){ # estimate the regression model via feols library(fixest) feols_fit <- feols(proposition_vote ~ treatment + ideology1 + log_income | Q1_immigration , data = voters) boot_feols <- boottest(feols_fit, clustid = "group_id1", param = "treatment", B = 9999, fe = "Q1_immigration") } #> Loading required namespace: fixest ## The Subcluster Bootstrap In the case of few treated clusters, MacKinnon and Webb (2018) suggest to use subclusters to form the bootstrap distribution. boottest() allows the user to specify subclusters via the bootcluster argument. boot_min <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999, bootcluster = "min") boot_var <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999, bootcluster = "group_id1") boot_2var <- boottest(lm_fit, clustid = c("group_id1", "group_id2"), param = "treatment", B = 9999, bootcluster = c("group_id1", "Q1_immigration")) if(requireNamespace("modelsummary")){ library(modelsummary) msummary(model = list(boot_min, boot_2var), estimate = "{estimate} ({p.value})", statistic = "[{conf.low}, {conf.high}]") } Model 1 Model 2 1*treatment = 0 0.073 (0.003) 0.073 (0.008) [0.032, 0.110] [0.029, 0.118] Num.Obs. 300 300 R2 0.316 0.316 R2 Adj. 0.288 0.288 AIC -82.1 -82.1 BIC -30.2 -30.2 Log.Lik. 55.025 55.025 ## Regression Weights / Weighted Least Squares (WLS) If regression weights are specified in the estimation stage via lm(), feols() or felm(), boottest() incorporates the weights into the bootstrap inference: # regression with weights / WLS lm_w_fit <- lm(proposition_vote ~ treatment + ideology1 + log_income, weights = voters$weights, data = voters)

boot_w_lm <- boottest(lm_w_fit,
clustid = "group_id1",
param = "treatment",
B = 9999)

## Parallel Execution

A major bottleneck for the performance of boottest() is a large matrix multiplication, which includes the bootstrap weights matrix on the right. In order to speed up the computation, this multiplication calls the c++ Eigen library, which allows for parallelization of dense matrix products. By default, boottest() uses one thread. Note that there is a cost of parallelization due to communication overhead. As a rule of thumb, if boottest() takes more than 10 seconds per execution, using a second thread might speed up the bootstrap.

The number of threads can be specified via the nthreads argument of boottest():

boot_lm <- boottest(lm_fit,
clustid = "group_id1",
param = "treatment",
B = 9999,
nthreads = 2)

# Running the wild cluster bootstrap via WildBootTests.jl

fwildclusterboot serves as an R port to WildBootTests.jl package.

For guidance on how to install Julia and WildBootTests.jl and how to connect R and Julia, please take a look at the running WildBootTests.jl through fwildclusterboot vignette.

You can tell boottest() to run WildBootTests.jl by using the boot_algo function argument:

boot_lm <- boottest(lm_fit,
clustid = "group_id1",
param = "treatment",
B = 9999,
boot_algo = "WildBootTests.jl")
generics::tidy(boot_lm)
#             term   estimate statistic    p.value   conf.low conf.high
#1 1*treatment = 0 0.07290769  3.709435 0.00060006 0.03326969 0.1134117

The seed used within Julia is linked to R’s global seed, which you can set through the familiar set.seed() function.

If you decide to run all your bootstraps through WildBootTests.jl, you can set a global variable via

setBoottest_boot_algo("WildBootTests.jl")

Calling boottest() without specifying boot_algo = "WildBootTests.jl" will now automatically run the bootstrap through WildBootTests.jl.

## The WRE bootstrap for IV models

Through WildBootTests.jl, fwildclusterboot supports the WRE bootstrap by Davidson & MacKinnon, 2010 for IV (instrumental variables) models for objects of type ivreg via the boottest() function:

library(ivreg)

data("SchoolingReturns", package = "ivreg")

# drop all NA values from SchoolingReturns
SchoolingReturns <- SchoolingReturns[rowMeans(sapply(SchoolingReturns, is.na)) == 0,]
ivreg_fit <- ivreg(log(wage) ~ education + age + ethnicity + smsa + south + parents14 |
nearcollege + age  + ethnicity + smsa + south + parents14, data = SchoolingReturns)

boot_ivreg <- boottest(object = ivreg_fit,
B = 999,
param = "education",
clustid = "kww",
type = "mammen",
impose_null = TRUE)
generics::tidy(boot_ivreg)
#              term  estimate statistic   p.value    conf.low conf.high
# 1 1*education = 0 0.0638822  1.043969 0.2482482 -0.03152655 0.2128746

## Tests of multiple joint hypotheses (q > 1)

Through WildBootTests.jl, you can also test multiple joint hypotheses via the mboottest() function. With minor differences, mboottest()'s syntax largely mirrors boottest().

To jointly test the null hypothesis $$H_0: treatment = 0 \text{ and } ideology1 = 0$$ vs $$H_0: treatment \neq 0 \text{ and } ideology1 \neq 0$$ via a wild cluster bootstrap, you can run

library(clubSandwich)
R <- clubSandwich::constrain_zero(2:3, coef(lm_fit))
wboottest <-
mboottest(object = lm_fit,
clustid = "group_id1",
B = 999,
R = R)
generics::tidy(wboottest)
#   teststat p_val
# 1 8.469086     0

# Miscellanea

## A sanity check if fwildclusterboot::boottest() works as intended

A sanity check to see if fwildclusterboot::boottest() works as intended is to look at its t_stat return value. For both the WCR and WCU bootstrap, boottest() re-calculates the “original” - hence non-bootstrapped - t-statistic from its input regression model. The t-stat computed in boottest() and the t-stats reported by either lm(), feols() and lfe() under the same error clustering structure and small-sample adjustments should be identical. If you find that they differ, please report a bug on github. Note that fwildclusterboot explicitly tests for t-stat equality against fixest::feols() here.

data <-
fwildclusterboot:::create_data(N = 1000,
N_G1 = 20,
icc1 = 0.81,
N_G2 = 10,
icc2 = 0.01,
numb_fe1 = 10,
numb_fe2 = 10,
seed = 8769)

# oneway clustering
feols_fit <- fixest::feols(proposition_vote ~ treatment + ideology1 + log_income,
data = data,
cluster = ~group_id1,
cluster.df = 'conventional')
)

feols_tstats <- fixest::coeftable(feols_fit)[c("treatment", "log_income", "ideology1"), 3]

boot_tstats <-
lapply(c("treatment", "log_income", "ideology1"), function(x){
boot1 <- fwildclusterboot::boottest(feols_fit,
clustid = c("group_id1"),
B = 999,
param = x,
cluster.df = 'conventional'),
impose_null = TRUE)\$t_stat
})

df <- cbind(feols_tstats, unlist(boot_tstats))
colnames(df) <- c("feols tstat", "boottest tstat")
df
#>            feols tstat boottest tstat
#> treatment    0.9958071      0.9958071
#> log_income  -2.9129869     -2.9129869
#> ideology1    1.4169933      1.4169933

## Small Sample Corrections

boottest() offers several options for small sample adjustments via the ssc function argument which need to be specified via the boot_ssc() function. boot_ssc() has 4 arguments and is intentionally designed to mimic fixest's ssc() function. For more information on the default choices and alternative options, see ?fwildclusterboot::boot_ssc().

## Memory & the ‘lean’ implementation of the wild cluster bootstrap

Because the R-implementation of the fast algorithm is memory-intensive, fwildclusterboot further supports a Rcpp-based ‘lean’ implementation of the wild cluster bootstrap in case that memory demands get prohibitively large. In general, the ‘lean’ algorithm is much slower: its main feature is that it requires much less memory. The algorithm is equivalent to the ‘wild2’ algorithm in the “Fast & Wild” paper by Roodman et al. Note that the implementation in WildBootTests.jl is, in general, very memory-efficient.

library(bench)

dt <- fwildclusterboot:::create_data(
N = 10000,
N_G1 = 250,
icc1 = 0.01,
N_G2 = 10,
icc2 = 0.01,
numb_fe1 = 10,
numb_fe2 = 10,
seed = 7645
)

lm_fit <- lm(proposition_vote ~ treatment + ideology1 + log_income + Q1_immigration , data = dt)

res <-
bench::mark(
"R" = boottest(lm_fit,
clustid = "group_id1",
param = "treatment",
B = 9999,
boot_algo = "R",
"R-lean" = boottest(lm_fit,
clustid = "group_id1",
param = "treatment",
B = 9999,
boot_algo = "R-lean",
"WildBootTests.jl" =
boottest(lm_fit,
clustid = "group_id1",
param = "treatment",
B = 9999,
boot_algo = "WildBootTests.jl"),
iterations = 1,
check = FALSE
)

res

## Seeds

For Rademacher, Normal and Webb weights and boot_algo = "R", you need to set a global seed via dqrng::dqset.seed(). For all other options - Mammen weights with boot_algo = "R", boot_algo = "R-lean" or boot_algo = "WildBootTests.jl", you can set a global seed via the familiarset.seed().

## Treatment of Invalid Test Statistics for multiway clustering

In case of multi-way clustering, it is not guaranteed that the covariance matrix is positive definite, in which case the resulting bootstrap test statistics are invalid. boottest() follows the implementation in STATA and deletes invalid tests statistics, and informs the user with a note.

## On the handling of missing values

boottest() retrieves both the design matrix $$X$$, the dependent variable $$y$$ and the cluster variables from the input object of type lm, fixest or felm. Because boottest() allows to add or delete clustering variables that are not employed in lm(), feols() and felm(), it may occur that a cluster variable is added in boottest() that is not included in the regression model, either as a cluster variable or covariate.

In this case, boottest by default deletes the respective rows in the dependent variable, design matrix and in the cluster variables. In consequence, estimation (in the modeling step) and inference (via boottest()) are done on a different sample. boottest() returns a warning.

This in turn has a consequence for the use of boottest() and modelsummary. boottest() simply calls the glance() methods for objects of types fixest, felm and lm from the broom package, and therefore, the number of observations reported via msummary() is the number of observations used in the modeling stage.

The default behavior of boottest() - to delete missings with a warning - can be set off via the na_omit function argument. If na_omit is set to FALSE, boottest() will not allow for missing values in the added cluster variables and throw an error.

## A note of caution

The feols() function from fixest introduces several useful formula shortcuts. E.g. one can fit several regressions at once. All these advanced formula tools are not supported in boottest(). boottest() tries to catch any use of advanced formulas, but might fail to return errors in some cases.