Why is lm_robust() HC3 standard error smaller than coeftest() HC0 standard error?

It appears that the vcovHC() method for plm automatically estimates cluster-robust standard errors, while for lm_robust(), it does not. Therefore, the HC1 estimation of the standard error for plm will appear inflated compared to lm_robust (of lm for that matter).

Using some toy data:

library(sandwich)
library(tidyverse)
library(plm)
library(estimatr)
library(lmtest)

set.seed(1981)
x <- sin(1:1000)
y <- 1 + x + rnorm(1000)
f <- as.character(sort(rep(sample(1:100), 10)))
t <- as.character(rep(sort(sample(1:10)), 100))

dat <- tibble(y = y, x = x, f = f, t = t)

lm_fit <- lm(y ~ x + f + t, data = dat)
plm_fit <- plm(y ~ x, index = c("f", "t"), model = "within", effect = "twoways", data = dat)
rb_fit <- lm_robust(y ~ x, fixed_effects = ~ f + t, data = dat, se_type = "HC1", return_vcov = TRUE)

sqrt(vcovHC(lm_fit, type = "HC1")[2, 2])
#> [1] 0.04752337
sqrt(vcovHC(plm_fit, type = "HC1"))
#>            x
#> x 0.05036414
#> attr(,"cluster")
#> [1] "group"
sqrt(rb_fit$vcov)
#>            x
#> x 0.04752337

rb_fit <- lm_robust(y ~ x, fixed_effects = ~ f + t, data = dat, se_type = "HC3", return_vcov = TRUE)
sqrt(vcovHC(lm_fit, type = "HC3")[2, 2])
#> [1] 0.05041177
sqrt(vcovHC(plm_fit, type = "HC3"))
#>            x
#> x 0.05042142
#> attr(,"cluster")
#> [1] "group"
sqrt(rb_fit$vcov)
#>            x
#> x 0.05041177

There does not appear to be equivalent cluster-robust standard error types in the two packages. However, the SEs get closer when specifying cluster-robust SEs in lm_robust():

rb_fit <- lm_robust(y ~ x, fixed_effects = ~ f + t, clusters = f, data = dat, se_type = "CR0")
summary(rb_fit)
#> 
#> Call:
#> lm_robust(formula = y ~ x, data = dat, clusters = f, fixed_effects = ~f + 
#>     t, se_type = "CR0")
#> 
#> Standard error type:  CR0 
#> 
#> Coefficients:
#>   Estimate Std. Error t value  Pr(>|t|) CI Lower CI Upper DF
#> x    0.925    0.05034   18.38 1.133e-33   0.8251    1.025 99
#> 
#> Multiple R-squared:  0.3664 ,    Adjusted R-squared:  0.2888
#> Multiple R-squared (proj. model):  0.3101 ,  Adjusted R-squared (proj. model):  0.2256 
#> F-statistic (proj. model): 337.7 on 1 and 99 DF,  p-value: < 2.2e-16
coeftest(plm_fit, vcov. = vcovHC(plm_fit, type = "HC1"))
#> 
#> t test of coefficients:
#> 
#>   Estimate Std. Error t value  Pr(>|t|)    
#> x 0.925009   0.050364  18.366 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

^{Created on 2020-04-16 by the reprex package (v0.3.0)}