This function is intended for use with large datasets with multiple group
effects of large cardinality. If dummy-encoding the group effects results
in a manageable number of coefficients, you are probably better off by using
lm().
The formula specification is a response variable followed by a four part
formula. The first part consists of ordinary covariates, the second part
consists of factors to be projected out. The third part is an
IV-specification. The fourth part is a cluster specification for the
standard errors. I.e. something like y ~ x1 + x2 | f1 + f2 | (Q|W ~ x3+x4) | clu1 + clu2 where y is the response, x1,x2 are
ordinary covariates, f1,f2 are factors to be projected out, Q
and W are covariates which are instrumented by x3 and
x4, and clu1,clu2 are factors to be used for computing cluster
robust standard errors. Parts that are not used should be specified as
0, except if it's at the end of the formula, where they can be
omitted. The parentheses are needed in the third part since | has
higher precedence than ~. Multiple left hand sides like y|w|x ~ x1 + x2 |f1+f2|... are allowed.
Interactions between a covariate x and a factor f can be
projected out with the syntax x:f. The terms in the second and
fourth parts are not treated as ordinary formulas, in particular it is not
possible with things like y ~ x1 | x*f, rather one would specify
y ~ x1 + x | x:f + f. Note that f:x also works, since R's
parser does not keep the order. This means that in interactions, the factor
must be a factor, whereas a non-interacted factor will be coerced to
a factor. I.e. in y ~ x1 | x:f1 + f2, the f1 must be a factor,
whereas it will work as expected if f2 is an integer vector.
In older versions of lfe the syntax was `felm(y ~ x1 + x2 + G(f1)
G(f2), iv=list(Q ~ x3+x4, W ~ x3+x4), clustervar=c('clu1','clu2'))`. This
syntax still works, but yields a warning. Users are strongly
encouraged to change to the new multipart formula syntax. The old syntax
will be removed at a later time.
The standard errors are adjusted for the reduced degrees of freedom coming
from the dummies which are implicitly present. (An exception occurs in the
case of clustered standard errors and, specifically, where clusters are
nested within fixed effects; see
here.)
In the case of two factors,
the exact number of implicit dummies is easy to compute. If there are more
factors, the number of dummies is estimated by assuming there's one
reference-level for each factor, this may be a slight over-estimation,
leading to slightly too large standard errors. Setting exactDOF='rM'
computes the exact degrees of freedom with rankMatrix() in package
Matrix.
For the iv-part of the formula, it is only necessary to include the
instruments on the right hand side. The other explanatory covariates, from
the first and second part of formula, are added automatically in the
first stage regression. See the examples.
The contrasts argument is similar to the one in lm(), it is
used for factors in the first part of the formula. The factors in the second
part are analyzed as part of a possible subsequent getfe() call.
The cmethod argument may affect the clustered covariance matrix (and
thus regressor standard errors), either directly or via adjustments to a
degrees of freedom scaling factor. In particular, Cameron, Gelbach and Miller
(CGM2011, sec. 2.3) describe two possible small cluster corrections that are
relevant in the case of multiway clustering.
The first approach adjusts each component of the cluster-robust
variance estimator (CRVE) by its own \(c_i\) adjustment factor. For
example, the first component (with \(G\) clusters) is adjusted by
\(c_1=\frac{G}{G-1}\frac{N-1}{N-K}\),
the second component (with \(H\) clusters) is adjusted
by \(c_2=\frac{H}{H-1}\frac{N-1}{N-K}\), etc.
The second approach applies the same adjustment to all CRVE components:
\(c=\frac{J}{J-1}\frac{N-1}{N-K}\), where
\(J=\min(G,H)\) in the case of two-way clustering, for example.
Any differences resulting from these two approaches are likely to be minor,
and they will obviously yield exactly the same results when there is only one
cluster dimension. Still, CGM2011 adopt the former approach in their own
paper and simulations. This is also the default method that felm uses
(i.e. cmethod = 'cgm'). However, the latter approach has since been
adopted by several other packages that allow for robust inference with
multiway clustering. This includes the popular Stata package
reghdfe, as well as the
FixedEffectModels.jl
implementation in Julia. To match results from these packages exactly, use
cmethod = 'cgm2' (or its alias, cmethod = 'reghdfe'). It is
possible that some residual differences may still remain; see discussion
here.
The old syntax with a single part formula with the G() syntax for the
factors to transform away is still supported, as well as the
clustervar and iv arguments, but users are encouraged to move
to the new multi part formulas as described here. The clustervar and
iv arguments have been moved to the ... argument list. They
will be removed in some future update.