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,fixestorfelm - 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
#> Bootstr. Type: rademacher
#> 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
#> Bootstr. Type: rademacher
#> 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 |
| R2 Adj. | 0.288 | 0.311 |
| 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
#> Bootstr. Type: rademacher
#> 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 0.115The 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
#> Bootstr. Type: rademacher
#> 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: fixestThe 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)