vignettes/standard_errors.Rmd
standard_errors.Rmd
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.
For example, when performing the exact same estimation across various software, it is not uncommon to obtain different standard-errors. If your first thought is: there must be a bug… well, put that thought aside because there ain’t no bug. It often boils down to the choices the developer made regarding small sample correction which, maybe surprisingly, has many degrees of freedom when it comes to implementation.
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 of this vignette describes how standard-errors are
computed in fixest
’s estimations. In particular, it details
all the possible choices surrounding small sample correction. 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 0.10.0 or
higher.
fixest
There are two components defining the standard-errors in
fixest
. The main type of standard-error is given by the
argument vcov
, the small sample correction is defined by
the argument ssc
.
Here’s an example, the explanations follow in the next two sections:
library(fixest)
data(trade)
# OLS estimation
gravity = feols(log(Euros) ~ log(dist_km) | Destination + Origin + Product + Year, trade)
# Two-way clustered SEs
summary(gravity, vcov = "twoway")
#> OLS estimation, Dep. Var.: log(Euros)
#> Observations: 38,325
#> Fixed-effects: Destination: 15, Origin: 15, Product: 20, Year: 10
#> Standard-errors: Clustered (Destination & Origin)
#> Estimate Std. Error t value Pr(>|t|)
#> log(dist_km) -2.16988 0.171367 -12.6621 4.6802e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1.74337 Adj. R2: 0.705139
#> Within R2: 0.219322
# Two-way clustered SEs, without small sample correction
summary(gravity, vcov = "twoway", ssc = ssc(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: Clustered (Destination & Origin)
#> Estimate Std. Error t value Pr(>|t|)
#> log(dist_km) -2.16988 0.165494 -13.1115 2.9764e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1.74337 Adj. R2: 0.705139
#> Within R2: 0.219322
vcov
The argument vcov
can be equal to either:
"iid"
, "hetero"
, "cluster"
,
"twoway"
, "NW"
, "DK"
, or
"conley"
.
If vcov = "iid"
, then the standard-errors are based on
the assumption that the errors are non correlated and homoskedastic. If
vcov = "hetero"
, this corresponds to the classic
hereoskedasticity-robust standard-errors (White correction), where it is
assumed that the errors are non correlated but the variance of their
generative law may vary.
If vcov = "cluster"
, then arbitrary correlation of the
errors within clusters is accounted for. Same for
vcov = "twoway"
: arbitrary correlation within each of the
two clusters is accounted for.
In the context of panel data or time series, vcov = "NW"
(Newey-West, 1987) or vcov = "DK"
(Driscoll-Kraay, 1998)
account for temporal correlation between the errors; the two differing
on how to account for heterogeneity between units. Their implementation
is based on Millo (2017).
Finally, vcov = "conley"
accounts for spatial
correlation of the errors.
ssc
The type of small sample correction applied is defined by the
argument ssc
which accepts only objects produced by the
function ssc
. 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 ssc
is cluster.adj
.
This argument is only relevant when the standard-errors are clustered or
when they are corrected for serial correlation (Newey-West or
Driscoll-Kraay). 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 adjustment 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 \(G_{id,time}=10\).
When standard-errors are corrected for serial correlation, the corresponding adjustment applied is \(G_{time} / (G_{time} - 1)\).
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
adjustment 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 and confidence
intervals are 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) minus one. 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.
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
.
Using the Grunfeld data set from the plm
package, here
are some comparisons when the estimation doesn’t contain
fixed-effects.
library(sandwich)
library(plm)
data(Grunfeld)
# Estimations
res_lm = lm(inv ~ capital, Grunfeld)
res_feols = feols(inv ~ capital, Grunfeld)
# Same standard-errors
rbind(se(res_lm), se(res_feols))
#> (Intercept) capital
#> [1,] 15.63927 0.0383394
#> [2,] 15.63927 0.0383394
# Heteroskedasticity-robust covariance
se_lm_hc = sqrt(diag(vcovHC(res_lm, type = "HC1")))
se_feols_hc = se(res_feols, vcov = "hetero")
rbind(se_lm_hc, se_feols_hc)
#> (Intercept) capital
#> se_lm_hc 17.05558 0.06633144
#> se_feols_hc 17.05558 0.06633144
Note that Stata’s reg inv capital, 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 “iid” standard-errors between
lm
and plm
.
# Estimations
est_lm = lm(inv ~ capital + as.factor(firm) + as.factor(year), Grunfeld)
est_plm = plm(inv ~ capital + as.factor(year), Grunfeld, index = c("firm", "year"), model = "within")
# we use panel.id so that panel VCOVs can be applied directly
est_feols = feols(inv ~ capital | firm + year, Grunfeld, panel.id = ~firm + year)
#
# "iid" standard-errors
#
# By default fixest clusters the SEs when FEs are present,
# so we need to ask for iid SEs explicitly.
rbind(se(est_lm)["capital"], se(est_plm)["capital"], se(est_feols, vcov = "iid"))
#> capital
#> [1,] 0.02597821
#> [2,] 0.02597821
#> [3,] 0.02597821
# p-values:
rbind(pvalue(est_lm)["capital"], pvalue(est_plm)["capital"], pvalue(est_feols, vcov = "iid"))
#> capital
#> [1,] 1.519204e-35
#> [2,] 1.519204e-35
#> [3,] 1.519204e-35
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 firm
se_lm_firm = se(vcovCL(est_lm, cluster = ~firm, type = "HC1"))["capital"]
se_plm_firm = se(vcovHC(est_plm, cluster = "group"))["capital"]
se_stata_firm = 0.06328129 # vce(cluster firm)
se_feols_firm = se(est_feols) # By default: clustered according to firm
rbind(se_lm_firm, se_plm_firm, se_stata_firm, se_feols_firm)
#> capital
#> se_lm_firm 0.06493478
#> se_plm_firm 0.05693726
#> se_stata_firm 0.06328129
#> se_feols_firm 0.06328129
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_firm_lm = se(est_feols, ssc = ssc(fixef.K = "full"))
rbind(se_lm_firm, se_feols_firm_lm)
#> capital
#> se_lm_firm 0.06493478
#> se_feols_firm_lm 0.06493478
# How to get the plm version
se_feols_firm_plm = se(est_feols, ssc = ssc(fixef.K = "none", cluster.adj = FALSE))
rbind(se_plm_firm, se_feols_firm_plm)
#> capital
#> se_plm_firm 0.05693726
#> se_feols_firm_plm 0.05693726
And finally let’s look at Newey-West and Driscoll-Kray standard-errors:
#
# Newey-west
#
se_plm_NW = se(vcovNW(est_plm))["capital"]
se_feols_NW = se(est_feols, vcov = "NW")
rbind(se_plm_NW, se_feols_NW)
#> capital
#> se_plm_NW 0.08390222
#> se_feols_NW 0.08629896
# we can replicate plm's by changing the type of SSC:
rbind(se_plm_NW,
se(est_feols, vcov = NW ~ ssc(adj = FALSE, cluster.adj = FALSE)))
#> capital
#> se_plm_NW 0.08390222
#> 0.08390222
#
# Driscoll-Kraay
#
se_plm_DK = se(vcovSCC(est_plm))["capital"]
se_feols_DK = se(est_feols, vcov = "DK")
rbind(se_plm_DK, se_feols_DK)
#> capital
#> se_plm_DK 0.08359734
#> se_feols_DK 0.08800884
# Replicating plm's
rbind(se_plm_DK,
se(est_feols, vcov = DK ~ ssc(adj = FALSE, cluster.adj = FALSE)))
#> capital
#> se_plm_DK 0.08359734
#> 0.08359734
As we can see, the type of small sample correction we choose can have a non-negligible impact on the standard-error.
Now a specific comparison with lfe
(version 2.8-7) 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
:
library(lfe)
# lfe: clustered by firm
est_lfe = felm(inv ~ capital | firm + year | 0 | firm, Grunfeld)
se_lfe_firm = se(est_lfe)
# The two are different, and it cannot be directly replicated by feols
rbind(se_lfe_firm, se_feols_firm)
#> capital
#> se_lfe_firm 0.06016851
#> se_feols_firm 0.06328129
# You have to provide a custom VCOV to replicate lfe's VCOV
my_vcov = vcov(est_feols, ssc = ssc(adj = FALSE))
se(est_feols, vcov = my_vcov * 199/198) # Note that there are 200 observations
#> capital
#> 0.06016851
# Differently from feols, the SEs in lfe are different if year is not a FE:
# => now SEs are identical.
rbind(se(felm(inv ~ capital + factor(year) | firm | 0 | firm, Grunfeld))["capital"],
se(feols(inv ~ capital + factor(year) | firm, Grunfeld))["capital"])
#> capital
#> [1,] 0.06328129
#> [2,] 0.06328129
# Now with two-way clustered standard-errors
est_lfe_2way = felm(inv ~ capital | firm + year | 0 | firm + year, Grunfeld)
se_lfe_2way = se(est_lfe_2way)
se_feols_2way = se(est_feols, vcov = "twoway")
rbind(se_lfe_2way, se_feols_2way)
#> capital
#> se_lfe_2way 0.06213837
#> se_feols_2way 0.06041290
# To obtain the same SEs, use cluster.df = "conventional"
sum_feols_2way_conv = summary(est_feols, vcov = twoway ~ ssc(cluster.df = "conv"))
rbind(se_lfe_2way, se(sum_feols_2way_conv))
#> capital
#> se_lfe_2way 0.06213837
#> 0.06213837
#> capital
#> [1,] 9.273982e-05
#> [2,] 9.273982e-05
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.
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_ssc()
and setFixest_vcov()
.
For example, if you want to remove the small sample adjustment, just use:
setFixest_ssc(ssc(adj = FALSE))
By default, the standard-errors are clustered in the presence of fixed-effects and in the presence of a panel. You can change this behavior with, e.g.:
setFixest_vcov(no_FE = "iid", one_FE = "iid",
two_FE = "hetero", panel = "driscoll_kraay")
which changes the way the default standard-errors are computed when the estimation contains no fixed-effects, one fixed-effect, two or more fixed-effects, or is a panel.
Version 0.10.0 brings about many important changes:
The arguments se
and cluster
have been
replaced by the argument vcov
. Retro compatibility is
ensured.
The argument dof
has been renamed to
ssc
for clarity (since it was dealing with small sample
correction). This is not retro compatible.
Three new types of standard-errors are added: Newey-West and Driscoll-Kraay for panel data; Conley to account for spatial correlation.
The argument ssc
can now be directly summoned in the
vcov
formula.
The functions setFixest_dof
and
setFixest_se
have been renamed into
setFixest_ssc
and setFixest_vcov
. Retro
compatibility is not ensured.
The default standard-error name has changed from
"standard"
to "iid"
(thanks to Grant McDermott
for the suggestion!).
Since lfe
has returned to CRAN (good news!), the
code chunks involving it are now re-evaluated.
The illustration is now based on the Grunfeld data set from the
plm
package (to avoid problems with RNG).
Version 0.8.0. Evaluation of the chunks related to
lfe
have been removed since its archival on the CRAN. Hard
values from the last CRAN version are maintained.
Version 0.7.0 introduces the following important modifications:
To increase clarity, se = "white"
becomes
se = "hetero"
. Retro-compatibility is ensured.
The default values for computing clustered standard-errors become
similar to reghdfe
to avoid cross-software confusion. That
is, now by default cluster.df = "min"
and
t.df = "min"
(whereas in the previous version it was
cluster.df = "conventional"
and
t.df = "conventional"
).
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.
Cameron AC, Gelbach JB, Miller DL (2011). “Robust Inference with Multiway Clustering”, Journal of Business & Ecomomic Statistics, 29(2), 238–249.
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.
MacKinnon JG, White H (1985). “Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties” Journal of Econometrics, 29(3), 305–325.
Millo G (2017). “Robust Standard Error Estimators for Panel Models: A Unifying Approach” Journal of Statistical Software, 82(3).
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.