Estimates GLM models with any number of fixed-effects.

## Usage

```
feglm(
fml,
data,
family = "gaussian",
vcov,
offset,
weights,
subset,
split,
fsplit,
split.keep,
split.drop,
cluster,
se,
ssc,
panel.id,
start = NULL,
etastart = NULL,
mustart = NULL,
fixef,
fixef.rm = "perfect",
fixef.tol = 1e-06,
fixef.iter = 10000,
fixef.algo = NULL,
collin.tol = 1e-10,
glm.iter = 25,
glm.tol = 1e-08,
nthreads = getFixest_nthreads(),
lean = FALSE,
warn = TRUE,
notes = getFixest_notes(),
verbose = 0,
only.coef = FALSE,
data.save = FALSE,
combine.quick,
mem.clean = FALSE,
only.env = FALSE,
env,
...
)
feglm.fit(
y,
X,
fixef_df,
family = "gaussian",
vcov,
offset,
split,
fsplit,
split.keep,
split.drop,
cluster,
se,
ssc,
weights,
subset,
start = NULL,
etastart = NULL,
mustart = NULL,
fixef.rm = "perfect",
fixef.tol = 1e-06,
fixef.iter = 10000,
fixef.algo = NULL,
collin.tol = 1e-10,
glm.iter = 25,
glm.tol = 1e-08,
nthreads = getFixest_nthreads(),
lean = FALSE,
warn = TRUE,
notes = getFixest_notes(),
mem.clean = FALSE,
verbose = 0,
only.env = FALSE,
only.coef = FALSE,
env,
...
)
fepois(
fml,
data,
vcov,
offset,
weights,
subset,
split,
fsplit,
split.keep,
split.drop,
cluster,
se,
ssc,
panel.id,
start = NULL,
etastart = NULL,
mustart = NULL,
fixef,
fixef.rm = "perfect",
fixef.tol = 1e-06,
fixef.iter = 10000,
fixef.algo = NULL,
collin.tol = 1e-10,
glm.iter = 25,
glm.tol = 1e-08,
nthreads = getFixest_nthreads(),
lean = FALSE,
warn = TRUE,
notes = getFixest_notes(),
verbose = 0,
combine.quick,
mem.clean = FALSE,
only.env = FALSE,
only.coef = FALSE,
data.save = FALSE,
env,
...
)
```

## Arguments

- fml
A formula representing the relation to be estimated. For example:

`fml = z~x+y`

. To include fixed-effects, insert them in this formula using a pipe: e.g.`fml = z~x+y|fixef_1+fixef_2`

. Multiple estimations can be performed at once: for multiple dep. vars, wrap them in`c()`

: ex`c(y1, y2)`

. For multiple indep. vars, use the stepwise functions: ex`x1 + csw(x2, x3)`

. The formula`fml = c(y1, y2) ~ x1 + cw0(x2, x3)`

leads to 6 estimation, see details. Square brackets starting with a dot can be used to call global variables:`y.[i] ~ x.[1:2]`

will lead to`y3 ~ x1 + x2`

if`i`

is equal to 3 in the current environment (see details in`xpd`

).- data
A data.frame containing the necessary variables to run the model. The variables of the non-linear right hand side of the formula are identified with this

`data.frame`

names. Can also be a matrix.- family
Family to be used for the estimation. Defaults to

`gaussian()`

. See`family`

for details of family functions.- vcov
Versatile argument to specify the VCOV. In general, it is either a character scalar equal to a VCOV type, either a formula of the form:

`vcov_type ~ variables`

. The VCOV types implemented are: "iid", "hetero" (or "HC1"), "cluster", "twoway", "NW" (or "newey_west"), "DK" (or "driscoll_kraay"), and "conley". It also accepts object from`vcov_cluster`

,`vcov_NW`

,`NW`

,`vcov_DK`

,`DK`

,`vcov_conley`

and`conley`

. It also accepts covariance matrices computed externally. Finally it accepts functions to compute the covariances. See the`vcov`

documentation in the vignette.- offset
A formula or a numeric vector. An offset can be added to the estimation. If equal to a formula, it should be of the form (for example)

`~0.5*x**2`

. This offset is linearly added to the elements of the main formula 'fml'.- weights
A formula or a numeric vector. Each observation can be weighted, the weights must be greater than 0. If equal to a formula, it should be one-sided: for example

`~ var_weight`

.- subset
A vector (logical or numeric) or a one-sided formula. If provided, then the estimation will be performed only on the observations defined by this argument.

- split
A one sided formula representing a variable (eg

`split = ~var`

) or a vector. If provided, the sample is split according to the variable and one estimation is performed for each value of that variable. If you also want to include the estimation for the full sample, use the argument`fsplit`

instead. You can use the special operators`%keep%`

and`%drop%`

to select only a subset of values for which to split the sample. E.g.`split = ~var %keep% c("v1", "v2")`

will split the sample only according to the values`v1`

and`v2`

of the variable`var`

; it is equivalent to supplying the argument`split.keep = c("v1", "v2")`

. By default there is partial matching on each value, you can trigger a regular expression evaluation by adding a`'@'`

first, as in:`~var %drop% "@^v[12]"`

which will drop values starting with`"v1"`

or`"v2"`

(of course you need to know regexes!).- fsplit
A one sided formula representing a variable (eg

`fsplit = ~var`

) or a vector. If provided, the sample is split according to the variable and one estimation is performed for each value of that variable. This argument is the same as`split`

but also includes the full sample as the first estimation. You can use the special operators`%keep%`

and`%drop%`

to select only a subset of values for which to split the sample. E.g.`fsplit = ~var %keep% c("v1", "v2")`

will split the sample only according to the values`v1`

and`v2`

of the variable`var`

; it is equivalent to supplying the argument`split.keep = c("v1", "v2")`

. By default there is partial matching on each value, you can trigger a regular expression evaluation by adding an`'@'`

first, as in:`~var %drop% "@^v[12]"`

which will drop values starting with`"v1"`

or`"v2"`

(of course you need to know regexes!).- split.keep
A character vector. Only used when

`split`

, or`fsplit`

, is supplied. If provided, then the sample will be split only on the values of`split.keep`

. The values in`split.keep`

will be partially matched to the values of`split`

. To enable regular expressions, you need to add an`'@'`

first. For example`split.keep = c("v1", "@other|var")`

will keep only the value in`split`

partially matched by`"v1"`

or the values containing`"other"`

or`"var"`

.- split.drop
A character vector. Only used when

`split`

, or`fsplit`

, is supplied. If provided, then the sample will be split only on the values that are not in`split.drop`

. The values in`split.drop`

will be partially matched to the values of`split`

. To enable regular expressions, you need to add an`'@'`

first. For example`split.drop = c("v1", "@other|var")`

will drop only the value in`split`

partially matched by`"v1"`

or the values containing`"other"`

or`"var"`

.- cluster
Tells how to cluster the standard-errors (if clustering is requested). Can be either a list of vectors, a character vector of variable names, a formula or an integer vector. Assume we want to perform 2-way clustering over

`var1`

and`var2`

contained in the data.frame`base`

used for the estimation. All the following`cluster`

arguments are valid and do the same thing:`cluster = base[, c("var1", "var2")]`

,`cluster = c("var1", "var2")`

,`cluster = ~var1+var2`

. If the two variables were used as fixed-effects in the estimation, you can leave it blank with`vcov = "twoway"`

(assuming`var1`

[resp.`var2`

] was the 1st [resp. 2nd] fixed-effect). You can interact two variables using`^`

with the following syntax:`cluster = ~var1^var2`

or`cluster = "var1^var2"`

.- se
Character scalar. Which kind of standard error should be computed: “standard”, “hetero”, “cluster”, “twoway”, “threeway” or “fourway”? By default if there are clusters in the estimation:

`se = "cluster"`

, otherwise`se = "iid"`

. Note that this argument is deprecated, you should use`vcov`

instead.- ssc
An object of class

`ssc.type`

obtained with the function`ssc`

. Represents how the degree of freedom correction should be done.You must use the function`ssc`

for this argument. The arguments and defaults of the function`ssc`

are:`adj = TRUE`

,`fixef.K="nested"`

,`cluster.adj = TRUE`

,`cluster.df = "min"`

,`t.df = "min"`

,`fixef.force_exact=FALSE)`

. See the help of the function`ssc`

for details.- panel.id
The panel identifiers. Can either be: i) a one sided formula (e.g.

`panel.id = ~id+time`

), ii) a character vector of length 2 (e.g.`panel.id=c('id', 'time')`

, or iii) a character scalar of two variables separated by a comma (e.g.`panel.id='id,time'`

). Note that you can combine variables with`^`

only inside formulas (see the dedicated section in`feols`

).- start
Starting values for the coefficients. Can be: i) a numeric of length 1 (e.g.

`start = 0`

), ii) a numeric vector of the exact same length as the number of variables, or iii) a named vector of any length (the names will be used to initialize the appropriate coefficients). Default is missing.- etastart
Numeric vector of the same length as the data. Starting values for the linear predictor. Default is missing.

- mustart
Numeric vector of the same length as the data. Starting values for the vector of means. Default is missing.

- fixef
Character vector. The names of variables to be used as fixed-effects. These variables should contain the identifier of each observation (e.g., think of it as a panel identifier). Note that the recommended way to include fixed-effects is to insert them directly in the formula.

- fixef.rm
Can be equal to "perfect" (default), "singleton", "both" or "none". Controls which observations are to be removed. If "perfect", then observations having a fixed-effect with perfect fit (e.g. only 0 outcomes in Poisson estimations) will be removed. If "singleton", all observations for which a fixed-effect appears only once will be removed. Note, importantly, that singletons are removed in just one pass, there is no recursivity implemented. The meaning of "both" and "none" is direct.

- fixef.tol
Precision used to obtain the fixed-effects. Defaults to

`1e-6`

. It corresponds to the maximum absolute difference allowed between two coefficients of successive iterations.- fixef.iter
Maximum number of iterations in fixed-effects algorithm (only in use for 2+ fixed-effects). Default is 10000.

- fixef.algo
`NULL`

(default) or an object of class`demeaning_algo`

obtained with the function`demeaning_algo`

. If`NULL`

, it falls to the defaults of`demeaning_algo`

. This arguments controls the settings of the demeaning algorithm. Only play with it if the convergence is slow, i.e. look at the slot`$iterations`

, and if any is over 50, it may be worth playing around with it. Please read the documentation of the function`demeaning_algo`

. Be aware that there is no clear guidance on how to change the settings, it's more a matter of try-and-see.- collin.tol
Numeric scalar, default is

`1e-10`

. Threshold deciding when variables should be considered collinear and subsequently removed from the estimation. Higher values means more variables will be removed (if there is presence of collinearity). One signal of presence of collinearity is t-stats that are extremely low (for instance when t-stats < 1e-3).- glm.iter
Number of iterations of the glm algorithm. Default is 25.

- glm.tol
Tolerance level for the glm algorithm. Default is

`1e-8`

.- nthreads
The number of threads. Can be: a) an integer lower than, or equal to, the maximum number of threads; b) 0: meaning all available threads will be used; c) a number strictly between 0 and 1 which represents the fraction of all threads to use. The default is to use 50% of all threads. You can set permanently the number of threads used within this package using the function

`setFixest_nthreads`

.- lean
Logical, default is

`FALSE`

. If`TRUE`

then all large objects are removed from the returned result: this will save memory but will block the possibility to use many methods. It is recommended to use the arguments`se`

or`cluster`

to obtain the appropriate standard-errors at estimation time, since obtaining different SEs won't be possible afterwards.- warn
Logical, default is

`TRUE`

. Whether warnings should be displayed (concerns warnings relating to convergence state).- notes
Logical. By default, three notes are displayed: when NAs are removed, when some fixed-effects are removed because of only 0 (or 0/1) outcomes, or when a variable is dropped because of collinearity. To avoid displaying these messages, you can set

`notes = FALSE`

. You can remove these messages permanently by using`setFixest_notes(FALSE)`

.- verbose
Integer. Higher values give more information. In particular, it can detail the number of iterations in the demeaning algoritmh (the first number is the left-hand-side, the other numbers are the right-hand-side variables). It can also detail the step-halving algorithm.

- only.coef
Logical, default is

`FALSE`

. If`TRUE`

, then only the estimated coefficients are returned. Note that the length of the vector returned is always the length of the number of coefficients to be estimated: this means that the variables found to be collinear are returned with an NA value.- data.save
Logical scalar, default is

`FALSE`

. If`TRUE`

, the data used for the estimation is saved within the returned object. Hence later calls to predict(), vcov(), etc..., will be consistent even if the original data has been modified in the meantime. This is especially useful for estimations within loops, where the data changes at each iteration, such that postprocessing can be done outside the loop without issue.- combine.quick
Logical. When you combine different variables to transform them into a single fixed-effects you can do e.g.

`y ~ x | paste(var1, var2)`

. The algorithm provides a shorthand to do the same operation:`y ~ x | var1^var2`

. Because pasting variables is a costly operation, the internal algorithm may use a numerical trick to hasten the process. The cost of doing so is that you lose the labels. If you are interested in getting the value of the fixed-effects coefficients after the estimation, you should use`combine.quick = FALSE`

. By default it is equal to`FALSE`

if the number of observations is lower than 50,000, and to`TRUE`

otherwise.- mem.clean
Logical, default is

`FALSE`

. Only to be used if the data set is large compared to the available RAM. If`TRUE`

then intermediary objects are removed as much as possible and`gc`

is run before each substantial C++ section in the internal code to avoid memory issues.- only.env
(Advanced users.) Logical, default is

`FALSE`

. If`TRUE`

, then only the environment used to make the estimation is returned.- env
(Advanced users.) A

`fixest`

environment created by a`fixest`

estimation with`only.env = TRUE`

. Default is missing. If provided, the data from this environment will be used to perform the estimation.- ...
Not currently used.

- y
Numeric vector/matrix/data.frame of the dependent variable(s). Multiple dependent variables will return a

`fixest_multi`

object.- X
Numeric matrix of the regressors.

- fixef_df
Matrix/data.frame of the fixed-effects.

## Value

A `fixest`

object. Note that `fixest`

objects contain many elements and most of them
are for internal use, they are presented here only for information. To access them,
it is safer to use the user-level methods (e.g. `vcov.fixest`

, `resid.fixest`

,
etc) or functions (like for instance `fitstat`

to access any fit statistic).

- nobs
The number of observations.

- fml
The linear formula of the call.

- call
The call of the function.

- method
The method used to estimate the model.

- family
The family used to estimate the model.

- data
The original data set used when calling the function. Only available when the estimation was called with

`data.save = TRUE`

- fml_all
A list containing different parts of the formula. Always contain the linear formula. Then, if relevant:

`fixef`

: the fixed-effects.- nparams
The number of parameters of the model.

- fixef_vars
The names of each fixed-effect dimension.

- fixef_id
The list (of length the number of fixed-effects) of the fixed-effects identifiers for each observation.

- fixef_sizes
The size of each fixed-effect (i.e. the number of unique identifier for each fixed-effect dimension).

- y
(When relevant.) The dependent variable (used to compute the within-R2 when fixed-effects are present).

- convStatus
Logical, convergence status of the IRWLS algorithm.

- irls_weights
The weights of the last iteration of the IRWLS algorithm.

- obs_selection
(When relevant.) List containing vectors of integers. It represents the sequential selection of observation vis a vis the original data set.

- fixef_removed
(When relevant.) In the case there were fixed-effects and some observations were removed because of only 0/1 outcome within a fixed-effect, it gives the list (for each fixed-effect dimension) of the fixed-effect identifiers that were removed.

- coefficients
The named vector of estimated coefficients.

- coeftable
The table of the coefficients with their standard errors, z-values and p-values.

- loglik
The loglikelihood.

- deviance
Deviance of the fitted model.

- iterations
Number of iterations of the algorithm.

- ll_null
Log-likelihood of the null model (i.e. with the intercept only).

- ssr_null
Sum of the squared residuals of the null model (containing only with the intercept).

- pseudo_r2
The adjusted pseudo R2.

- fitted.values
The fitted values are the expected value of the dependent variable for the fitted model: that is \(E(Y|X)\).

- linear.predictors
The linear predictors.

- residuals
The residuals (y minus the fitted values).

- sq.cor
Squared correlation between the dependent variable and the expected predictor (i.e. fitted.values) obtained by the estimation.

- hessian
The Hessian of the parameters.

- cov.iid
The variance-covariance matrix of the parameters.

- se
The standard-error of the parameters.

- scores
The matrix of the scores (first derivative for each observation).

- residuals
The difference between the dependent variable and the expected predictor.

- sumFE
The sum of the fixed-effects coefficients for each observation.

- offset
(When relevant.) The offset formula.

- weights
(When relevant.) The weights formula.

- collin.var
(When relevant.) Vector containing the variables removed because of collinearity.

- collin.coef
(When relevant.) Vector of coefficients, where the values of the variables removed because of collinearity are NA.

## Details

The core of the GLM are the weighted OLS estimations. These estimations are performed
with `feols`

. The method used to demean each variable along the fixed-effects
is based on Berge (2018), since this is the same problem to solve as for the Gaussian
case in a ML setup.

## Combining the fixed-effects

You can combine two variables to make it a new fixed-effect using `^`

.
The syntax is as follows: `fe_1^fe_2`

. Here you created a new variable which is the combination
of the two variables fe_1 and fe_2. This is identical to doing `paste0(fe_1, "_", fe_2)`

but more convenient.

Note that pasting is a costly operation, especially for large data sets.
Thus, the internal algorithm uses a numerical trick which is fast, but the drawback is
that the identity of each observation is lost (i.e. they are now equal to a meaningless
number instead of being equal to `paste0(fe_1, "_", fe_2)`

). These “identities”
are useful only if you're interested in the value of the fixed-effects (that you can
extract with `fixef.fixest`

). If you're only interested in coefficients of the variables,
it doesn't matter. Anyway, you can use `combine.quick = FALSE`

to tell the internal
algorithm to use `paste`

instead of the numerical trick. By default, the numerical
trick is performed only for large data sets.

## Varying slopes

You can add variables with varying slopes in the fixed-effect part of the formula.
The syntax is as follows: `fixef_var[var1, var2]`

. Here the variables var1 and var2 will
be with varying slopes (one slope per value in fixef_var) and the fixed-effect
fixef_var will also be added.

To add only the variables with varying slopes and not the fixed-effect,
use double square brackets: `fixef_var[[var1, var2]]`

.

In other words:

`fixef_var[var1, var2]`

is equivalent to`fixef_var + fixef_var[[var1]] + fixef_var[[var2]]`

`fixef_var[[var1, var2]]`

is equivalent to`fixef_var[[var1]] + fixef_var[[var2]]`

In general, for convergence reasons, it is recommended to always add the fixed-effect and avoid using only the variable with varying slope (i.e. use single square brackets).

## Lagging variables

To use leads/lags of variables in the estimation, you can: i) either provide the argument
`panel.id`

, ii) either set your data set as a panel with the function
`panel`

, `f`

and `d`

.

You can provide several leads/lags/differences at once: e.g. if your formula is equal to
`f(y) ~ l(x, -1:1)`

, it means that the dependent variable is equal to the lead of `y`

,
and you will have as explanatory variables the lead of `x1`

, `x1`

and the lag of `x1`

.
See the examples in function `l`

for more details.

## Interactions

You can interact a numeric variable with a "factor-like" variable by using
`i(factor_var, continuous_var, ref)`

, where `continuous_var`

will be interacted with
each value of `factor_var`

and the argument `ref`

is a value of `factor_var`

taken as a reference (optional).

Using this specific way to create interactions leads to a different display of the
interacted values in `etable`

. See examples.

It is important to note that *if you do not care about the standard-errors of
the interactions*, then you can add interactions in the fixed-effects part of the formula,
it will be incomparably faster (using the syntax `factor_var[continuous_var]`

, as explained
in the section “Varying slopes”).

The function `i`

has in fact more arguments, please see details in its associated help page.

## On standard-errors

Standard-errors can be computed in different ways, you can use the arguments `se`

and `ssc`

in `summary.fixest`

to define how to compute them. By default, in the presence
of fixed-effects, standard-errors are automatically clustered.

The following vignette: On standard-errors describes in details how the standard-errors are computed in
`fixest`

and how you can replicate standard-errors from other software.

You can use the functions `setFixest_vcov`

and `setFixest_ssc`

to
permanently set the way the standard-errors are computed.

## Multiple estimations

Multiple estimations can be performed at once, they just have to be specified in the formula.
Multiple estimations yield a `fixest_multi`

object which is ‘kind of’ a list of
all the results but includes specific methods to access the results in a handy way.
Please have a look at the dedicated vignette:
Multiple estimations.

To include multiple dependent variables, wrap them in `c()`

(`list()`

also works).
For instance `fml = c(y1, y2) ~ x1`

would estimate the model `fml = y1 ~ x1`

and
then the model `fml = y2 ~ x1`

.

To include multiple independent variables, you need to use the stepwise functions.
There are 4 stepwise functions: `sw`

, `sw0`

, `csw`

, `csw0`

, and `mvsw`

. Of course `sw`

stands for stepwise, and `csw`

for cumulative stepwise. Finally `mvsw`

is a bit special,
it stands for multiverse stepwise. Let's explain that.
Assume you have the following formula: `fml = y ~ x1 + sw(x2, x3)`

.
The stepwise function `sw`

will estimate the following two models: `y ~ x1 + x2`

and
`y ~ x1 + x3`

. That is, each element in `sw()`

is sequentially, and separately,
added to the formula. Would have you used `sw0`

in lieu of `sw`

, then the model
`y ~ x1`

would also have been estimated. The `0`

in the name means that the model
without any stepwise element also needs to be estimated.
The prefix `c`

means cumulative: each stepwise element is added to the next. That is,
`fml = y ~ x1 + csw(x2, x3)`

would lead to the following models `y ~ x1 + x2`

and
`y ~ x1 + x2 + x3`

. The `0`

has the same meaning and would also lead to the model without
the stepwise elements to be estimated: in other words, `fml = y ~ x1 + csw0(x2, x3)`

leads to the following three models: `y ~ x1`

, `y ~ x1 + x2`

and `y ~ x1 + x2 + x3`

.
Finally `mvsw`

will add, in a stepwise fashion all possible combinations of the variables
in its arguments. For example `mvsw(x1, x2, x3)`

is equivalent to
`sw0(x1, x2, x3, x1 + x2, x1 + x3, x2 + x3, x1 + x2 + x3)`

. The number of models
to estimate grows at a factorial rate: so be cautious!

Multiple independent variables can be combined with multiple dependent variables, as in
`fml = c(y1, y2) ~ cw(x1, x2, x3)`

which would lead to 6 estimations. Multiple
estimations can also be combined to split samples (with the arguments `split`

, `fsplit`

).

You can also add fixed-effects in a stepwise fashion. Note that you cannot perform
stepwise estimations on the IV part of the formula (`feols`

only).

If NAs are present in the sample, to avoid too many messages, only NA removal concerning the variables common to all estimations is reported.

A note on performance. The feature of multiple estimations has been highly optimized for
`feols`

, in particular in the presence of fixed-effects. It is faster to estimate
multiple models using the formula rather than with a loop. For non-`feols`

models using
the formula is roughly similar to using a loop performance-wise.

## Argument sliding

When the data set has been set up globally using
`setFixest_estimation`

`(data = data_set)`

, the argument `vcov`

can be used implicitly.
This means that calls such as `feols(y ~ x, "HC1")`

, or `feols(y ~ x, ~id)`

, are valid:
i) the data is automatically deduced from the global settings, and ii) the `vcov`

is deduced to be the second argument.

## Piping

Although the argument 'data' is placed in second position, the data can be piped to the
estimation functions. For example, with R >= 4.1, `mtcars |> feols(mpg ~ cyl)`

works as
`feols(mpg ~ cyl, mtcars)`

.

## Tricks to estimate multiple LHS

To use multiple dependent variables in `fixest`

estimations, you need to include them
in a vector: like in `c(y1, y2, y3)`

.

First, if names are stored in a vector, they can readily be inserted in a formula to
perform multiple estimations using the dot square bracket operator. For instance if
`my_lhs = c("y1", "y2")`

, calling `fixest`

with, say `feols(.[my_lhs] ~ x1, etc)`

is
equivalent to using `feols(c(y1, y2) ~ x1, etc)`

. Beware that this is a special feature
unique to the *left-hand-side* of `fixest`

estimations (the default behavior of the DSB
operator is to aggregate with sums, see `xpd`

).

Second, you can use a regular expression to grep the left-hand-sides on the fly. When the
`..("regex")`

feature is used naked on the LHS, the variables grepped are inserted into
`c()`

. For example `..("Pe") ~ Sepal.Length, iris`

is equivalent to
`c(Petal.Length, Petal.Width) ~ Sepal.Length, iris`

. Beware that this is a
special feature unique to the *left-hand-side* of `fixest`

estimations
(the default behavior of `..("regex")`

is to aggregate with sums, see `xpd`

).

## Dot square bracket operator in formulas

In a formula, the dot square bracket (DSB) operator can: i) create manifold variables at once, or ii) capture values from the current environment and put them verbatim in the formula.

Say you want to include the variables `x1`

to `x3`

in your formula. You can use
`xpd(y ~ x.[1:3])`

and you'll get `y ~ x1 + x2 + x3`

.

To summon values from the environment, simply put the variable in square brackets. For example:
`for(i in 1:3) xpd(y.[i] ~ x)`

will create the formulas `y1 ~ x`

to `y3 ~ x`

depending on the
value of `i`

.

You can include a full variable from the environment in the same way:
`for(y in c("a", "b")) xpd(.[y] ~ x)`

will create the two formulas `a ~ x`

and `b ~ x`

.

The DSB can even be used within variable names, but then the variable must be nested in
character form. For example `y ~ .["x.[1:2]_sq"]`

will create `y ~ x1_sq + x2_sq`

. Using the
character form is important to avoid a formula parsing error. Double quotes must be used. Note
that the character string that is nested will be parsed with the function `dsb`

, and thus it
will return a vector.

By default, the DSB operator expands vectors into sums. You can add a comma, like in `.[, x]`

,
to expand with commas--the content can then be used within functions. For instance:
`c(x.[, 1:2])`

will create `c(x1, x2)`

(and *not* `c(x1 + x2)`

).

In all `fixest`

estimations, this special parsing is enabled, so you don't need to use `xpd`

.

One-sided formulas can be expanded with the DSB operator: let `x = ~sepal + petal`

, then
`xpd(y ~ .[x])`

leads to `color ~ sepal + petal`

.

You can even use multiple square brackets within a single variable, but then the use of nesting
is required. For example, the following `xpd(y ~ .[".[letters[1:2]]_.[1:2]"])`

will create
`y ~ a_1 + b_2`

. Remember that the nested character string is parsed with `dsb`

,
which explains this behavior.

When the element to be expanded i) is equal to the empty string or, ii) is of length 0, it is
replaced with a neutral element, namely `1`

. For example, `x = "" ; xpd(y ~ .[x])`

leads to
`y ~ 1`

.

## References

Berge, Laurent, 2018, "Efficient estimation of maximum likelihood models with multiple fixed-effects: the R package FENmlm." CREA Discussion Papers, 13 ().

For models with multiple fixed-effects:

Gaure, Simen, 2013, "OLS with multiple high dimensional category variables", Computational Statistics & Data Analysis 66 pp. 8--18

## See also

See also `summary.fixest`

to see the results with the appropriate standard-errors,
`fixef.fixest`

to extract the fixed-effects coefficients, and the function `etable`

to visualize the results of multiple estimations.
And other estimation methods: `feols`

, `femlm`

, `fenegbin`

, `feNmlm`

.

## Examples

```
# Poisson estimation
res = feglm(Sepal.Length ~ Sepal.Width + Petal.Length | Species, iris, "poisson")
# You could also use fepois
res_pois = fepois(Sepal.Length ~ Sepal.Width + Petal.Length | Species, iris)
# With the fit method:
res_fit = feglm.fit(iris$Sepal.Length, iris[, 2:3], iris$Species, "poisson")
# All results are identical:
etable(res, res_pois, res_fit)
#> res res_pois res_fit
#> Dependent Var.: Sepal.Length Sepal.Length iris$Sepal.Length
#>
#> Sepal.Width 0.0778* (0.0339) 0.0778* (0.0339) 0.0778* (0.0339)
#> Petal.Length 0.1221*** (0.0150) 0.1221*** (0.0150) 0.1221*** (0.0150)
#> Fixed-Effects: ------------------ ------------------ ------------------
#> Species Yes Yes No
#> iris$Species No No Yes
#> _______________ __________________ __________________ __________________
#> S.E.: Clustered by: Species by: Species by: iris$Species
#> Observations 150 150 150
#> Squared Cor. 0.86314 0.86314 0.86314
#> Pseudo R2 0.02676 0.02676 0.02676
#> BIC 570.76 570.76 570.76
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Note that you have many more examples in feols
#
# Multiple estimations:
#
# 6 estimations
est_mult = fepois(c(Ozone, Solar.R) ~ Wind + Temp + csw0(Wind:Temp, Day), airquality)
# We can display the results for the first lhs:
etable(est_mult[lhs = 1])
#> est_mult[lhs = 1].1 est_mult[lhs = ..2 est_mult[lhs = ..3
#> Dependent Var.: Ozone Ozone Ozone
#>
#> Constant 0.5334** (0.1943) 0.8890* (0.4430) 0.8028. (0.4441)
#> Wind -0.0761*** (0.0052) -0.1151** (0.0439) -0.1152** (0.0440)
#> Temp 0.0483*** (0.0020) 0.0441*** (0.0051) 0.0444*** (0.0051)
#> Wind x Temp 0.0005 (0.0005) 0.0005 (0.0005)
#> Day 0.0051*** (0.0015)
#> _______________ ___________________ __________________ __________________
#> S.E. type IID IID IID
#> Observations 116 116 116
#> Squared Cor. 0.62615 0.63129 0.63379
#> Pseudo R2 0.53377 0.53402 0.53759
#> BIC 1,564.7 1,568.7 1,561.5
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# And now the second (access can be made by name)
etable(est_mult[lhs = "Solar.R"])
#> est_mult[lhs = ..1 est_mult[lhs = ..2 est_mult[lhs = "..3
#> Dependent Var.: Solar.R Solar.R Solar.R
#>
#> Constant 3.777*** (0.0714) 4.343*** (0.1689) 4.529*** (0.1708)
#> Wind 0.0119*** (0.0020) -0.0415** (0.0146) -0.0433** (0.0147)
#> Temp 0.0169*** (0.0008) 0.0098*** (0.0021) 0.0088*** (0.0021)
#> Wind x Temp 0.0007*** (0.0002) 0.0007*** (0.0002)
#> Day -0.0061*** (0.0007)
#> _______________ __________________ __________________ ___________________
#> S.E. type IID IID IID
#> Observations 146 146 146
#> Squared Cor. 0.08076 0.08166 0.08983
#> Pseudo R2 0.06045 0.06207 0.07100
#> BIC 8,192.6 8,183.5 8,110.7
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Now we focus on the two last right hand sides
# (note that .N can be used to specify the last item)
etable(est_mult[rhs = 2:.N])
#> est_mult[rhs = ..1 est_mult[rhs = ..2 est_mult[rhs = ..3
#> Dependent Var.: Ozone Solar.R Ozone
#>
#> Constant 0.8890* (0.4430) 4.343*** (0.1689) 0.8028. (0.4441)
#> Wind -0.1151** (0.0439) -0.0415** (0.0146) -0.1152** (0.0440)
#> Temp 0.0441*** (0.0051) 0.0098*** (0.0021) 0.0444*** (0.0051)
#> Wind x Temp 0.0005 (0.0005) 0.0007*** (0.0002) 0.0005 (0.0005)
#> Day 0.0051*** (0.0015)
#> _______________ __________________ __________________ __________________
#> S.E. type IID IID IID
#> Observations 116 146 116
#> Squared Cor. 0.63129 0.08166 0.63379
#> Pseudo R2 0.53402 0.06207 0.53759
#> BIC 1,568.7 8,183.5 1,561.5
#>
#> est_mult[rhs = 2..4
#> Dependent Var.: Solar.R
#>
#> Constant 4.529*** (0.1708)
#> Wind -0.0433** (0.0147)
#> Temp 0.0088*** (0.0021)
#> Wind x Temp 0.0007*** (0.0002)
#> Day -0.0061*** (0.0007)
#> _______________ ___________________
#> S.E. type IID
#> Observations 146
#> Squared Cor. 0.08983
#> Pseudo R2 0.07100
#> BIC 8,110.7
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Combining with split
est_split = fepois(c(Ozone, Solar.R) ~ sw(poly(Wind, 2), poly(Temp, 2)),
airquality, split = ~ Month)
# You can display everything at once with the print method
est_split
#> Standard-errors: IID
#>
#>
#> # SAMPLE: 5
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.30403 0.042415 77.89732 < 2.2e-16 ***
#> poly(Wind, 2)1 -6.41466 0.617300 -10.39149 < 2.2e-16 ***
#> poly(Wind, 2)2 3.07584 0.570194 5.39437 6.8765e-08 ***
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 4.05816 0.109394 37.09674 < 2.2e-16 ***
#> poly(Temp, 2)1 11.28467 1.448123 7.79261 6.5637e-15 ***
#> poly(Temp, 2)2 2.50568 1.171805 2.13830 3.2492e-02 *
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.207784 0.018194 286.23787 < 2.2e-16 ***
#> poly(Wind, 2)1 -0.569626 0.271696 -2.09656 3.6033e-02 *
#> poly(Wind, 2)2 -1.834732 0.250895 -7.31274 2.6176e-13 ***
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.10262 0.052341 97.48802 < 2.2e-16 ***
#> poly(Temp, 2)1 -2.63000 0.635659 -4.13744 3.512e-05 ***
#> poly(Temp, 2)2 -5.59769 0.432368 -12.94657 < 2.2e-16 ***
#>
#>
#> # SAMPLE: 6
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.07100 0.124756 24.61599 < 2.2e-16 ***
#> poly(Wind, 2)1 6.09901 1.922777 3.17198 0.001514 **
#> poly(Wind, 2)2 -2.28442 1.031893 -2.21381 0.026842 *
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.41843 0.080824 42.29472 < 2.2e-16 ***
#> poly(Temp, 2)1 3.26499 1.293450 2.52425 0.01159456 *
#> poly(Temp, 2)2 5.66897 1.562592 3.62792 0.00028571 ***
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.218537 0.014191 367.74415 < 2.2e-16 ***
#> poly(Wind, 2)1 2.220448 0.177688 12.49633 < 2.2e-16 ***
#> poly(Wind, 2)2 -0.624761 0.139873 -4.46664 7.9458e-06 ***
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.200307 0.019697 264.014030 < 2.2e-16 ***
#> poly(Temp, 2)1 3.286120 0.364969 9.003834 < 2.2e-16 ***
#> poly(Temp, 2)2 0.155763 0.322767 0.482586 0.62939
#>
#>
#> # SAMPLE: 7
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.778200 0.043180 87.49820 < 2.2e-16 ***
#> poly(Wind, 2)1 -6.722474 0.703358 -9.55768 < 2.2e-16 ***
#> poly(Wind, 2)2 -0.959048 0.681576 -1.40710 0.1594
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 2.59501 0.190413 13.62828 < 2.2e-16 ***
#> poly(Temp, 2)1 24.89459 3.062021 8.13011 4.2888e-16 ***
#> poly(Temp, 2)2 -8.57801 1.768706 -4.84988 1.2354e-06 ***
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.360516 0.014178 378.099115 < 2.2e-16 ***
#> poly(Wind, 2)1 -0.682752 0.216716 -3.150439 0.0016303 **
#> poly(Wind, 2)2 0.013888 0.277451 0.050054 0.9600792
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.10937 0.056186 90.93638 < 2.2e-16 ***
#> poly(Temp, 2)1 4.82809 0.928155 5.20181 1.9735e-07 ***
#> poly(Temp, 2)2 -1.11475 0.615860 -1.81007 7.0285e-02 .
#>
#>
#> # SAMPLE: 8
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.755844 0.041522 90.453699 < 2.2e-16 ***
#> poly(Wind, 2)1 -6.956477 0.697914 -9.967528 < 2.2e-16 ***
#> poly(Wind, 2)2 -0.344133 0.486153 -0.707869 0.47903
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.14340 0.097610 32.20355 < 2.2e-16 ***
#> poly(Temp, 2)1 17.29925 1.746264 9.90643 < 2.2e-16 ***
#> poly(Temp, 2)2 -5.78006 0.896709 -6.44586 1.1495e-10 ***
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.134474 0.016224 316.48225 < 2.2e-16 ***
#> poly(Wind, 2)1 -0.459093 0.272152 -1.68690 0.0916227 .
#> poly(Wind, 2)2 0.683835 0.246613 2.77291 0.0055558 **
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 4.926595 0.041382 119.05115 < 2.2e-16 ***
#> poly(Temp, 2)1 4.222008 0.816701 5.16959 2.3461e-07 ***
#> poly(Temp, 2)2 -0.794082 0.462440 -1.71716 8.5950e-02 .
#>
#>
#> # SAMPLE: 9
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.36418 0.036953 91.04039 < 2.2e-16 ***
#> poly(Wind, 2)1 -4.04407 0.492839 -8.20566 2.2933e-16 ***
#> poly(Wind, 2)2 2.98908 0.470324 6.35536 2.0794e-10 ***
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.36946 0.038292 87.99344 < 2.2e-16 ***
#> poly(Temp, 2)1 6.10018 0.690949 8.82870 < 2.2e-16 ***
#> poly(Temp, 2)2 2.25482 0.627045 3.59595 0.00032321 ***
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.122995 0.014639 349.966703 < 2.2e-16 ***
#> poly(Wind, 2)1 -0.564257 0.193672 -2.913473 0.0035743 **
#> poly(Wind, 2)2 0.144413 0.224957 0.641955 0.5209022
#> ---
#> Expl. vars.: poly(Temp, 2)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 5.111358 0.015302 334.02914 < 2.2e-16 ***
#> poly(Temp, 2)1 1.286054 0.251169 5.12026 3.0511e-07 ***
#> poly(Temp, 2)2 -0.928769 0.260074 -3.57117 3.5539e-04 ***
# Different way of displaying the results with "compact"
summary(est_split, "compact")
#> sample lhs rhs (Intercept) poly(Wind, 2)1
#> 1 5 Ozone poly(Wind, 2) 3.30*** (0.042) -6.41*** (0.617)
#> 2 5 Ozone poly(Temp, 2) 4.06*** (0.109)
#> 3 5 Solar.R poly(Wind, 2) 5.21*** (0.018) -0.570* (0.272)
#> 4 5 Solar.R poly(Temp, 2) 5.10*** (0.052)
#> 5 6 Ozone poly(Wind, 2) 3.07*** (0.125) 6.10** (1.92)
#> 6 6 Ozone poly(Temp, 2) 3.42*** (0.081)
#> 7 6 Solar.R poly(Wind, 2) 5.22*** (0.014) 2.22*** (0.178)
#> 8 6 Solar.R poly(Temp, 2) 5.20*** (0.020)
#> 9 7 Ozone poly(Wind, 2) 3.78*** (0.043) -6.72*** (0.703)
#> 10 7 Ozone poly(Temp, 2) 2.60*** (0.190)
#> 11 7 Solar.R poly(Wind, 2) 5.36*** (0.014) -0.683** (0.217)
#> 12 7 Solar.R poly(Temp, 2) 5.11*** (0.056)
#> 13 8 Ozone poly(Wind, 2) 3.76*** (0.042) -6.96*** (0.698)
#> 14 8 Ozone poly(Temp, 2) 3.14*** (0.098)
#> 15 8 Solar.R poly(Wind, 2) 5.13*** (0.016) -0.459. (0.272)
#> 16 8 Solar.R poly(Temp, 2) 4.93*** (0.041)
#> 17 9 Ozone poly(Wind, 2) 3.36*** (0.037) -4.04*** (0.493)
#> 18 9 Ozone poly(Temp, 2) 3.37*** (0.038)
#> 19 9 Solar.R poly(Wind, 2) 5.12*** (0.015) -0.564** (0.194)
#> 20 9 Solar.R poly(Temp, 2) 5.11*** (0.015)
#> poly(Wind, 2)2 poly(Temp, 2)1 poly(Temp, 2)2
#> 1 3.08*** (0.570)
#> 2 11.3*** (1.45) 2.51* (1.17)
#> 3 -1.83*** (0.251)
#> 4 -2.63*** (0.636) -5.60*** (0.432)
#> 5 -2.28* (1.03)
#> 6 3.26* (1.29) 5.67*** (1.56)
#> 7 -0.625*** (0.140)
#> 8 3.29*** (0.365) 0.156 (0.323)
#> 9 -0.959 (0.682)
#> 10 24.9*** (3.06) -8.58*** (1.77)
#> 11 0.014 (0.277)
#> 12 4.83*** (0.928) -1.11. (0.616)
#> 13 -0.344 (0.486)
#> 14 17.3*** (1.75) -5.78*** (0.897)
#> 15 0.684** (0.247)
#> 16 4.22*** (0.817) -0.794. (0.462)
#> 17 2.99*** (0.470)
#> 18 6.10*** (0.691) 2.25*** (0.627)
#> 19 0.144 (0.225)
#> 20 1.29*** (0.251) -0.929*** (0.260)
# You can still select which sample/LHS/RHS to display
est_split[sample = 1:2, lhs = 1, rhs = 1]
#> Standard-errors: IID
#> Dep. var.: Ozone
#> Expl. vars.: poly(Wind, 2)
#> Sample: 5
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.30403 0.042415 77.89732 < 2.2e-16 ***
#> poly(Wind, 2)1 -6.41466 0.617300 -10.39149 < 2.2e-16 ***
#> poly(Wind, 2)2 3.07584 0.570194 5.39437 6.8765e-08 ***
#> ---
#> Sample: 6
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 3.07100 0.124756 24.61599 < 2.2e-16 ***
#> poly(Wind, 2)1 6.09901 1.922777 3.17198 0.001514 **
#> poly(Wind, 2)2 -2.28442 1.031893 -2.21381 0.026842 *
```