It is an euphemism to say that standard-errors are a critical element of your estimations: literally your paper’s results depend on them. It is therefore unfortunate that no conventional “best” way exists to compute them. Multiple definitions can create confusion and the purpose of this document is to lay bare the fiddly details of standard-error computation in this package.

The first part describes how standard-errors are computed in fixest’s estimations. Please note that here I don’t discuss the why, but only the how. For a thorough introduction to the topic, see the excellent paper by Zeileis, Koll and Graham (2020). The second part illustrates how to replicate some standard-errors obtained from other estimation methods with fixest.

This document applies to fixest version 0.7.0 or higher.

How standard-errors are computed in fixest

There are two components defining the standard-errors in fixest. The main type of standard-error is given by the argument se, the small sample correction is defined by the argument dof (degree of freedom).

Here’s an example, the explanations follow in the next two sections:

# OLS estimation
gravity = feols(log(Euros) ~ log(dist_km) | Destination + Origin + Product + Year, trade)
# Two-way clustered SEs
summary(gravity, se = "twoway")
#> OLS estimation, Dep. Var.: log(Euros)
#> Observations: 38,325 
#> Fixed-effects: Destination: 15,  Origin: 15,  Product: 20,  Year: 10
#> Standard-errors: Two-way (Destination & Origin) 
#>              Estimate Std. Error t value Pr(>|t|)    
#> log(dist_km)  -2.1699   0.171367 -12.662 4.68e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> Log-likelihood: -75,682.54   Adj. R2: 0.70514 
#>                            R2-Within: 0.21932
# Two-way clustered SEs, without DOF correction
summary(gravity, se = "twoway", dof = dof(adj = FALSE, cluster.adj = FALSE))
#> OLS estimation, Dep. Var.: log(Euros)
#> Observations: 38,325 
#> Fixed-effects: Destination: 15,  Origin: 15,  Product: 20,  Year: 10
#> Standard-errors: Two-way (Destination & Origin) 
#>              Estimate Std. Error t value Pr(>|t|)    
#> log(dist_km)  -2.1699   0.165494 -13.112 2.98e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> Log-likelihood: -75,682.54   Adj. R2: 0.70514 
#>                            R2-Within: 0.21932

The argument se

The argument se can be equal to either: "standard", "hetero", "cluster", "twoway", or "threeway".

If se = "standard", then the standard-errors are based on the assumption that the errors are all independent and generated with the same law (in particular, same variance). If se = "hetero", this corresponds to the classic hereoskedasticity-robust standard-errors (White correction), where it is assumed that the errors are independent but the variance of their generative law may vary. If se = "cluster", then arbitrary correlation of the errors within clusters is accounted for. Same for se = "twoway" (resp. "threeway"): arbitrary correlation within each of the two (resp. three) clusters is accounted for.

The argument dof

The type of small sample correction applied is defined by the argument dof which accepts only objects produced by the function dof. The main arguments of this function are adj, fixef.K and cluster.adj. I detail each of them below.

Say you have \(\tilde{V}\) the variance-covariance matrix (henceforth VCOV) before any small sample adjustment. Argument adj can be equal to TRUE or FALSE, leading to the following adjustment:

When the estimation contains fixed-effects, the value of \(K\) in the previous adjustment can be determined in different ways, governed by the argument fixef.K. To illustrate how \(K\) is computed, let’s use an example with individual (variable id) and time fixed-effect and with clustered standard-errors. The structure of the 10 observations data is:

The standard-errors are clustered with respect to the cluster variable, further we can see that the variable id is nested within the cluster variable (i.e. each value of id “belongs” to only one value of cluster; e.g. id could represent US counties and cluster US states).

The argument fixef.K can be equal to either "none", "nested" or "full". Then \(K\) will be computed as follows:

Where \(K_{vars}\) is the number of estimated coefficients associated to the variables. fixef.K="none" discards all fixed-effects coefficients. fixef.K="nested" discards all coefficients that are nested (here the 5 coefficients from id). Finally fixef.K="full" accounts for all fixed-effects coefficients (here 6: equal to 5 from id, plus 2 from time, minus one used as a reference [otherwise collinearity arise]). Note that if fixef.K="nested" and the standard-errors are not clustered, this is equivalent to using fixef.K="full".

The last argument of dof is cluster.adj. This argument is only relevant when the standard-errors are clustered. Let \(M\) be the sandwich estimator of the VCOV without adjustment. Then for one-way clustered standard errors:

With \(G\) the number of unique elements of the cluster variable (in the previous example \(G=2\) for cluster).

The effect of the adjusment for two-way clustered standard-errors is as follows:

Using the data from the previous example, here the standard-errors are clustered by id and time, leading to \(G_{id}=5\), \(G_{time}=2\), and \(G_{id,time}=10\).

Yet more details

You’re already fed up about about these details? I’m sorry but there’s more, so far you’ve only seen the main arguments! I now come to detail three more elements: fixef.force_exact, cluster.df and t.df.

Argument fixef.force_exact is only relevant when there are two or more fixed-effects. By default all the fixed-effects coefficients are accounted for when computing the degrees of freedom. In general this is fine, but in some situations it may overestimate the number of estimated coefficients. Why? Because some of the fixed-effects may be collinear, the effective number of coefficients being lower. Let’s illustrate that with an example. Consider the following set of fixed-effects:

There are 6 different values of id and 4 different values of time. By default, 9 coefficients are used to compute the degrees of freedom (6 plus 4 minus one reference). But we can see here that the “effective” number of coefficients is equal to 8: two coefficients should be removed to avoid collinearity issues (any one from each color set). If you use fixef.force_exact=TRUE, then the function fixef is first run to determine the number of free coefficients in the fixed-effects, this number is then used to compute the degree of freedom.

Argument cluster.df is only relevant when you apply two-way clustering (or higher). It can have two values: either "conventional", or "min" (the default). This affects the adjustments for each clustered matrix. The "conventional" way to make the adjustment has already been described in the previous equation. If cluster.df="min" (again, the default), and for two-way clustered standard errors, the adjusment becomes:

Now instead of having a specific adjustment for each matrix, there is only one adjustment of \(G_{min}/(G_{min}-1)\) where \(G_{min}\) is the minimum cluster size (here \(G_{min}=\min(G_{id},G_{time})\)).

Argument t.df is only relevant when standard-errors are clustered. It affects the way the p-value is computed. It can be equal to: either "conventional", or "min" (the default). By default, when standard-errors are clustered, the degrees of freedom used in the Student t distribution is equal to the minimum cluster size (among all clusters used to cluster the VCOV). If t.df="conventional", the degrees of freedom used to find the p-value from the Student t distribution is equal to the number of observations minus the number of estimated coefficients.

Replicating standard-errors from other methods

This section illustrates how the results from fixest compares with the ones from other methods. It also shows how to replicate the latter from fixest.

First, let’s generate some data:

# Data generation
N = 20 ; n_id = N/5; n_time = N/n_id
base = data.frame(y = rnorm(N), x = rnorm(N), id = rep(1:n_id, n_time), 
                  time = rep(1:n_time, each = n_id))

Here are some comparisons when the estimation doesn’t contain fixed-effects.


# Estimations
res_lm    = lm(y ~ x, base)
res_feols = feols(y ~ x, base)

# Same standard-errors
rbind(se(res_lm), se(res_feols))
#>      (Intercept)         x
#> [1,]     0.23693 0.3347682
#> [2,]     0.23693 0.3347682
# Heteroskedasticity-robust covariance
se_lm_hc    = sqrt(diag(vcovHC(res_lm, type = "HC1")))
se_feols_hc = se(res_feols, se = "hetero")
rbind(se_lm_hc, se_feols_hc)
#>             (Intercept)        x
#> se_lm_hc      0.2341859 0.215208
#> se_feols_hc   0.2341859 0.215208

Note that Stata’s reg y x, robust also leads to similar results (same SEs, same p-values).

The most important differences arise in the presence of fixed-effects. Let’s first compare “standard” standard-errors between lm and plm.


# Estimations
est_lm    = lm(y ~ x + as.factor(id) + as.factor(time), base)
est_plm   = plm(y ~ x + as.factor(time), base, index = c("id", "time"), model = "within")
est_feols = feols(y ~ x | id + time, base)

# "Standard" standard-errors

# By default fixest clusters the SEs when FEs are present,
#  so we need to ask for standard SEs explicitly.
rbind(se(est_lm)["x"], se(est_plm)["x"], se(est_feols, se = "standard"))
#>              x
#> [1,] 0.3956029
#> [2,] 0.3956029
#> [3,] 0.3956029
# p-values:
rbind(pvalue(est_lm)["x"], pvalue(est_plm)["x"], pvalue(est_feols, se = "standard"))
#>              x
#> [1,] 0.3638934
#> [2,] 0.3638934
#> [3,] 0.3638934

The standard-errors and p-values are identical, note that this is also the case for Stata’s xtreg.

Now for clustered SEs:

# Clustered by id
se_lm_id    = sqrt(vcovCL(est_lm, cluster = base$id, type = "HC1")["x", "x"])
se_plm_id   = sqrt(vcovHC(est_plm, cluster = "group")["x", "x"])
se_stata_id = 0.165385      # vce(cluster id)
se_feols_id = se(est_feols) # By default: clustered according to id

rbind(se_lm_id, se_plm_id, se_stata_id, se_feols_id)
#>                     x
#> se_lm_id    0.1865794
#> se_plm_id   0.1229459
#> se_stata_id 0.1653850
#> se_feols_id 0.1653850

As we can see, there are three different versions of the standard-errors, feols being identical to Stata’s xtreg clustered SEs. By default, the p-value is also identical to the one from Stata (from fixest version 0.7.0 onwards).

Now let’s see how to replicate the standard-errors from lm and plm:

# How to get the lm version
se_feols_id_lm = se(est_feols, dof = dof(fixef.K = "full"))
rbind(se_lm_id, se_feols_id_lm)
#>                        x
#> se_lm_id       0.1865794
#> se_feols_id_lm 0.1865794
# How to get the plm version
se_feols_id_plm = se(est_feols, dof = dof(fixef.K = "none", cluster.adj = FALSE))
rbind(se_plm_id, se_feols_id_plm)
#>                         x
#> se_plm_id       0.1229459
#> se_feols_id_plm 0.1229459

As we can see, the type of small sample correction we choose can have a non-negligible impact on the standard-error.

Other multiple fixed-effects methods

Now a specific comparison with lfe (version 2.8-5.1) and Stata’s reghdfe which are popular tools to estimate econometric models with multiple fixed-effects.

From fixest version 0.7.0 onwards, the standard-errors and p-values are computed similarly to reghdfe, for both clustered and multiway clustered standard errors. So the comparison here focuses on lfe.

Here are the differences and similarities with lfe:


# lfe: clustered by id
est_lfe = felm(y ~ x | id + time | 0 | id, base)
se_lfe_id = se(est_lfe)

# The two are different, and it cannot be directly replicated by feols
rbind(se_lfe_id, se_feols_id)
#>                     x
#> se_lfe_id   0.1458559
#> se_feols_id 0.1653850
# You have to provide a custom VCOV to replicate lfe's VCOV
my_vcov = vcov(est_feols, dof = dof(adj = FALSE))
se(est_feols, .vcov = my_vcov * 19/18) # Note that there are 20 observations
#>         x 
#> 0.1458559
# Differently from feols, the SEs in lfe are different if time is not a FE:
# => now SEs are identical. (The warning is from lfe.)
rbind(se(felm(y ~ x + factor(time) | id | 0 | id, base))["x"],
      se(feols(y ~ x + factor(time) | id, base))["x"])
#> Warning in chol.default(mat, pivot = TRUE, tol = tol): the matrix is either
#> rank-deficient or indefinite
#>             x
#> [1,] 0.165385
#> [2,] 0.165385
# Now with two-way clustered standard-errors
est_lfe_2way = felm(y ~ x | id + time | 0 | id + time, base)
se_lfe_2way  = se(est_lfe_2way)
se_feols_2way = se(est_feols, se = "twoway")
rbind(se_lfe_2way, se_feols_2way)
#>                       x
#> se_lfe_2way   0.3268584
#> se_feols_2way 0.3080378
# To obtain the same SEs, use cluster.df = "conventional"
sum_feols_2way_conv = summary(est_feols, se = "twoway", dof = dof(cluster.df = "conv"))
rbind(se_lfe_2way, se(sum_feols_2way_conv))
#>                     x
#> se_lfe_2way 0.3268584
#>             0.3268584
# We also obtain the same p-values
rbind(pvalue(est_lfe_2way), pvalue(sum_feols_2way_conv))
#>              x
#> [1,] 0.3347851
#> [2,] 0.3347851

As we can see, there is only slight differences with lfe when computing clustered standard-errors. For multiway clustered standard-errors, it is easy to replicate the way lfe computes them.

Defining how to compute the standard-errors once and for all

Once you’ve found the preferred way to compute the standard-errors for your current project, you can set it permanently using the functions setFixest_dof() and setFixest_se().

For example, if you want to remove the small sample adjustment, just use:

setFixest_dof(dof(adj = FALSE))

By default, the standard-errors are clustered in the presence of fixed-effects. You can change this behavior with, e.g.:

setFixest_se(no_FE = "standard", one_FE = "standard", two_FE = "standard")

which changes the way the default standard-errors are computed when the estimation contains no fixed-effects, one fixed-effect, or two or more fixed-effects.


References & acknowledgments

I wish to thank Karl Dunkle Werner, Grant McDermott and Ivo Welch for raising the issue and for helpful discussions. Any error is of course my own.

Zeileis A, Koll S, Graham N (2020). “Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R.” Journal of Statistical Software, 95(1), 1–36.

MacKinnon, JG, White, H (1985). “Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties” Journal of Econometrics, 29(3), 305–325.

Kauermann G, Carroll RJ (2001). “A Note on the Efficiency of Sandwich Covariance Matrix Estimation”, Journal of the American Statistical Association, 96(456), 1387–1396.

Cameron AC, Gelbach JB, Miller DL (2011). “Robust Inference with Multiway Clustering”, Journal of Business & Ecomomic Statistics, 29(2), 238–249.