This formula fits a linear model, provides a variety of options for robust standard errors, and conducts coefficient tests

lm_robust(formula, data, weights, subset, clusters, fixed_effects,
  se_type = NULL, ci = TRUE, alpha = 0.05, return_vcov = TRUE,
  try_cholesky = FALSE)

Arguments

Value

An object of class "lm_robust".

The post-estimation commands functions summary and tidy return results in a data.frame. To get useful data out of the return, you can use these data frames, you can use the resulting list directly, or you can use the generic accessor functions coef, vcov, confint, and predict. Marginal effects and uncertainty about them can be gotten by passing this object to margins from the margins, or to emmeans in the emmeans package.

Users who want to print the results in TeX of HTML can use the extract function and the texreg package.

If users specify a multivariate linear regression model (multiple outcomes), then some of the below components will be of higher dimension to accommodate the additional models.

An object of class "lm_robust" is a list containing at least the following components:

Details

This function performs linear regression and provides a variety of standard errors. It takes a formula and data much in the same was as lm does, and all auxiliary variables, such as clusters and weights, can be passed either as quoted names of columns, as bare column names, or as a self-contained vector. Examples of usage can be seen below and in the Getting Started vignette.

The mathematical notes in this vignette specify the exact estimators used by this function. The default variance estimators have been chosen largely in accordance with the procedures in this manual. The default for the case without clusters is the HC2 estimator and the default with clusters is the analogous CR2 estimator. Users can easily replicate Stata standard errors in the clustered or non-clustered case by setting `se_type` = "stata".

The function estimates the coefficients and standard errors in C++, using the RcppEigen package. By default, we estimate the coefficients using Column-Pivoting QR decomposition from the Eigen C++ library, although users could get faster solutions by setting `try_cholesky` = TRUE to use a Cholesky decomposition instead. This will likely result in quicker solutions, but the algorithm does not reliably detect when there are linear dependencies in the model and may fail silently if they exist.

If `fixed_effects` are specified, both the outcome and design matrix are centered using the method of alternating projections (Halperin 1962; Gaure 2013). Specifying fixed effects in this way will result in large speed gains with standard error estimators that do not need to invert the matrix of fixed effects. This means using "classical", "HC0", "HC1", "CR0", or "stata" standard errors will be faster than other standard error estimators. Be wary when specifying fixed effects that may result in perfect fits for some observations or if there are intersecting groups across multiple fixed effect variables (e.g. if you specify both "year" and "country" fixed effects with an unbalanced panel where one year you only have data for one country).

As with `lm()`, multivariate regression (multiple outcomes) will only admit observations into the estimation that have no missingness on any outcome.

References

Abadie, Alberto, Susan Athey, Guido W Imbens, and Jeffrey Wooldridge. 2017. "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments." arXiv Pre-Print. https://arxiv.org/abs/1710.02926v2.

Bell, Robert M, and Daniel F McCaffrey. 2002. "Bias Reduction in Standard Errors for Linear Regression with Multi-Stage Samples." Survey Methodology 28 (2): 169-82.

Gaure, Simon. 2013. "OLS with multiple high dimensional category variables." Computational Statistics \& Data Analysis 66: 8-1. https://doi.org/10.1016/j.csda.2013.03.024

Halperin, I. 1962. "The product of projection operators." Acta Scientiarum Mathematicarum (Szeged) 23(1-2): 96-99.

MacKinnon, James, and Halbert White. 1985. "Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties." Journal of Econometrics 29 (3): 305-25. https://doi.org/10.1016/0304-4076(85)90158-7.

Pustejovsky, James E, and Elizabeth Tipton. 2016. "Small Sample Methods for Cluster-Robust Variance Estimation and Hypothesis Testing in Fixed Effects Models." Journal of Business & Economic Statistics. Taylor & Francis. https://doi.org/10.1080/07350015.2016.1247004.

Samii, Cyrus, and Peter M Aronow. 2012. "On Equivalencies Between Design-Based and Regression-Based Variance Estimators for Randomized Experiments." Statistics and Probability Letters 82 (2). https://doi.org/10.1016/j.spl.2011.10.024.

Examples

library(fabricatr)
dat <- fabricate(
  N = 40,
  y = rpois(N, lambda = 4),
  x = rnorm(N),
  z = rbinom(N, 1, prob = 0.4)
)

# Default variance estimator is HC2 robust standard errors
lmro <- lm_robust(y ~ x + z, data = dat)

# Can tidy() the data in to a data.frame
tidy(lmro)
#>          term   estimate std.error  statistic      p.value  conf.low conf.high
#> 1 (Intercept)  3.9737898 0.4485498  8.8591939 1.118544e-10  3.064942 4.8826380
#> 2           x -0.5307197 0.4882497 -1.0869841 2.840745e-01 -1.520008 0.4585682
#> 3           z  0.2195192 0.9326624  0.2353683 8.152208e-01 -1.670234 2.1092726
#>   df outcome
#> 1 37       y
#> 2 37       y
#> 3 37       y
# Can use summary() to get more statistics
summary(lmro)
#> 
#> Call:
#> lm_robust(formula = y ~ x + z, data = dat)
#> 
#> Standard error type:  HC2 
#> 
#> Coefficients:
#>             Estimate Std. Error t value  Pr(>|t|) CI Lower CI Upper DF
#> (Intercept)   3.9738     0.4485  8.8592 1.119e-10    3.065   4.8826 37
#> x            -0.5307     0.4882 -1.0870 2.841e-01   -1.520   0.4586 37
#> z             0.2195     0.9327  0.2354 8.152e-01   -1.670   2.1093 37
#> 
#> Multiple R-squared:  0.04158 ,	Adjusted R-squared:  -0.01022 
#> F-statistic: 0.654 on 2 and 37 DF,  p-value: 0.5259
# Can also get coefficients three ways
lmro$coefficients
#> (Intercept)           x           z 
#>   3.9737898  -0.5307197   0.2195192 
coef(lmro)
#> (Intercept)           x           z 
#>   3.9737898  -0.5307197   0.2195192 
tidy(lmro)$estimate
#> [1]  3.9737898 -0.5307197  0.2195192
# Can also get confidence intervals from object or with new 1 - `alpha`
lmro$conf.low
#> (Intercept)           x           z 
#>    3.064942   -1.520008   -1.670234 
confint(lmro, level = 0.8)
#>                   10 %      90 %
#> (Intercept)  3.3884976 4.5590820
#> x           -1.1678144 0.1063751
#> z           -0.9974694 1.4365078

# Can recover classical standard errors
lmclassic <- lm_robust(y ~ x + z, data = dat, se_type = "classical")
tidy(lmclassic)
#>          term   estimate std.error  statistic      p.value  conf.low conf.high
#> 1 (Intercept)  3.9737898 0.4908831  8.0951850 1.039569e-09  2.979166 4.9684135
#> 2           x -0.5307197 0.4368420 -1.2149007 2.321003e-01 -1.415846 0.3544063
#> 3           z  0.2195192 0.8975742  0.2445694 8.081411e-01 -1.599139 2.0381773
#>   df outcome
#> 1 37       y
#> 2 37       y
#> 3 37       y

# Can easily match Stata's robust standard errors
lmstata <- lm_robust(y ~ x + z, data = dat, se_type = "stata")
tidy(lmstata)
#>          term   estimate std.error  statistic      p.value  conf.low conf.high
#> 1 (Intercept)  3.9737898 0.4517381  8.7966665 1.338896e-10  3.058481 4.8890982
#> 2           x -0.5307197 0.4798608 -1.1059867 2.758718e-01 -1.503010 0.4415707
#> 3           z  0.2195192 0.9172312  0.2393281 8.121720e-01 -1.638968 2.0780060
#>   df outcome
#> 1 37       y
#> 2 37       y
#> 3 37       y

# Easy to specify clusters for cluster-robust inference
dat$clusterID <- sample(1:10, size = 40, replace = TRUE)

lmclust <- lm_robust(y ~ x + z, data = dat, clusters = clusterID)
tidy(lmclust)
#>          term   estimate std.error  statistic      p.value  conf.low conf.high
#> 1 (Intercept)  3.9737898 0.6045112  6.5735587 0.0004326423  2.520725 5.4268549
#> 2           x -0.5307197 0.3732881 -1.4217430 0.2009822588 -1.426117 0.3646777
#> 3           z  0.2195192 0.7957097  0.2758785 0.7913574413 -1.699550 2.1385881
#>         df outcome
#> 1 6.479238       y
#> 2 6.541055       y
#> 3 6.382863       y

# Can also match Stata's clustered standard errors
lm_robust(
  y ~ x + z,
  data = dat,
  clusters = clusterID,
  se_type = "stata"
)
#>               Estimate Std. Error    t value    Pr(>|t|)  CI Lower  CI Upper DF
#> (Intercept)  3.9737898  0.6145043  6.4666589 0.000115865  2.583684 5.3638951  9
#> x           -0.5307197  0.3689754 -1.4383606 0.184182152 -1.365400 0.3039607  9
#> z            0.2195192  0.7855876  0.2794331 0.786227898 -1.557603 1.9966418  9

# Works just as LM does with functions in the formula
dat$blockID <- rep(c("A", "B", "C", "D"), each = 10)

lm_robust(y ~ x + z + factor(blockID), data = dat)
#>                    Estimate Std. Error    t value     Pr(>|t|)  CI Lower
#> (Intercept)       4.1595712  0.9428286  4.4117999 9.804957e-05  2.243513
#> x                -0.5860799  0.5340797 -1.0973641 2.801935e-01 -1.671460
#> z                 0.3870498  1.0611313  0.3647521 7.175555e-01 -1.769428
#> factor(blockID)B -0.3962175  1.2706935 -0.3118120 7.570875e-01 -2.978577
#> factor(blockID)C  0.2792504  1.3812823  0.2021675 8.409906e-01 -2.527853
#> factor(blockID)D -0.8210846  1.1856909 -0.6924946 4.933297e-01 -3.230698
#>                   CI Upper DF
#> (Intercept)      6.0756295 34
#> x                0.4993006 34
#> z                2.5435280 34
#> factor(blockID)B 2.1861423 34
#> factor(blockID)C 3.0863538 34
#> factor(blockID)D 1.5885293 34

# Weights are also easily specified
dat$w <- runif(40)

lm_robust(
  y ~ x + z,
  data = dat,
  weights = w,
  clusters = clusterID
)
#>               Estimate Std. Error    t value     Pr(>|t|)  CI Lower  CI Upper
#> (Intercept)  4.0120339  0.6209751  6.4608607 0.0004885974  2.517552 5.5065158
#> x           -0.4938056  0.4225456 -1.1686443 0.2893483806 -1.542660 0.5550485
#> z            0.3181113  1.0831727  0.2936847 0.7816242950 -2.528463 3.1646859
#>                   DF
#> (Intercept) 6.443394
#> x           5.666692
#> z           4.660184

# Subsetting works just as in `lm()`
lm_robust(y ~ x, data = dat, subset = z == 1)
#>              Estimate Std. Error   t value     Pr(>|t|)  CI Lower  CI Upper DF
#> (Intercept)  3.972133  0.7444527  5.335641 0.0003302924  2.313389 5.6308765 10
#> x           -2.601287  1.4320438 -1.816486 0.0993453585 -5.792079 0.5895056 10

# One can also choose to set the significance level for different CIs
lm_robust(y ~ x + z, data = dat, alpha = 0.1)
#>               Estimate Std. Error    t value     Pr(>|t|)  CI Lower  CI Upper
#> (Intercept)  3.9737898  0.4485498  8.8591939 1.118544e-10  3.217044 4.7305353
#> x           -0.5307197  0.4882497 -1.0869841 2.840745e-01 -1.354443 0.2930033
#> z            0.2195192  0.9326624  0.2353683 8.152208e-01 -1.353970 1.7930079
#>             DF
#> (Intercept) 37
#> x           37
#> z           37

# We can also specify fixed effects
# Speed gains with fixed effects are greatests with "stata" or "HC1" std.errors
tidy(lm_robust(y ~ x + z, data = dat, fixed_effects = ~ blockID, se_type = "HC1"))
#>   term   estimate std.error  statistic   p.value  conf.low conf.high df outcome
#> 1    x -0.5860799 0.5156647 -1.1365522 0.2636730 -1.634037 0.4618769 34       y
#> 2    z  0.3870498 1.0278841  0.3765501 0.7088478 -1.701862 2.4759618 34       y

# NOT RUN {
  # Can also use 'margins' or 'emmeans' package if you have them installed
  # to get marginal effects
  library(margins)
  lmrout <- lm_robust(y ~ x + z, data = dat)
  summary(margins(lmrout))

  # Can output results using 'texreg'
  library(texreg)
  texreg(lmrout)

  # Using emmeans to obtain covariate-adjusted means
  library(emmeans)
  fiber.rlm <- lm_robust(strength ~ diameter + machine, data = fiber)
  emmeans(fiber.rlm, "machine")
# }