Estimates OLS with any number of fixed-effects.

feols(
fml,
data,
vcov,
weights,
offset,
subset,
split,
fsplit,
cluster,
se,
ssc,
panel.id,
fixef,
fixef.rm = "none",
fixef.tol = 1e-06,
fixef.iter = 10000,
collin.tol = 1e-10,
lean = FALSE,
verbose = 0,
warn = TRUE,
notes = getFixest_notes(),
combine.quick,
demeaned = FALSE,
mem.clean = FALSE,
only.env = FALSE,
env,
...
)

feols.fit(
y,
X,
fixef_df,
vcov,
offset,
split,
fsplit,
cluster,
se,
ssc,
weights,
subset,
fixef.rm = "perfect",
fixef.tol = 1e-06,
fixef.iter = 10000,
collin.tol = 1e-10,
lean = FALSE,
warn = TRUE,
notes = getFixest_notes(),
mem.clean = FALSE,
verbose = 0,
only.env = FALSE,
env,
...
)

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 | fe_1+fe_2. You can combine two fixed-effects with ^: e.g. fml = z~x+y|fe_1^fe_2, see details. You can also use variables with varying slopes using square brackets: e.g. in fml = z~y|fe_1[x] + fe_2, see details. To add IVs, insert the endogenous vars./instruments after a pipe, like in y ~ x | c(x_endo1, x_endo2) ~ x_inst1 + x_inst2. Note that it should always be the last element, see details. 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). 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. 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. 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. 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'. 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. 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. 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. This argument is the same as split but also includes the full sample as the first estimation. 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 [res. 2nd] fixed-effect). You can interact two variables using ^ with the following syntax: cluster = ~var1^var2 or cluster = "var1^var2". 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. 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 = "conventional", t.df = "conventional", fixef.force_exact=FALSE). See the help of the function ssc for details. 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). 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. 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. The meaning of "both" and "none" is direct. Precision used to obtain the fixed-effects. Defaults to 1e-5. It corresponds to the maximum absolute difference allowed between two coefficients of successive iterations. Argument fixef.tol cannot be lower than 10000*.Machine$double.eps. Note that this parameter is dynamically controlled by the algorithm. Maximum number of iterations in fixed-effects algorithm (only in use for 2+ fixed-effects). Default is 10000. 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). 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. 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. Integer. Higher values give more information. In particular, it can detail the number of iterations in the demeaning algorithm (the first number is the left-hand-side, the other numbers are the right-hand-side variables). Logical, default is TRUE. Whether warnings should be displayed (concerns warnings relating to convergence state). Logical. By default, two notes are displayed: when NAs are removed (to show additional information) and when some observations are removed because of collinearity. To avoid displaying these messages, you can set notes = FALSE. You can remove these messages permanently by using setFixest_notes(FALSE). 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. Logical, default is FALSE. Only used in the presence of fixed-effects: should the centered variables be returned? If TRUE, it creates the items y_demeaned and X_demeaned. 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. (Advanced users.) Logical, default is FALSE. If TRUE, then only the environment used to make the estimation is returned. (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. Numeric vector/matrix/data.frame of the dependent variable(s). Multiple dependent variables will return a fixest_multi object. Numeric matrix of the regressors. 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. fml_all A list containing different parts of the formula. Always contain the linear formula. Then depending on the cases: fixef: the fixed-effects, iv: the IV part of the formula. 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 identifierfor each fixed-effect dimension). coefficients The named vector of estimated coefficients. multicol Logical, if multicollinearity was found. coeftable The table of the coefficients with their standard errors, z-values and p-values. loglik The loglikelihood. ssr_null Sum of the squared residuals of the null model (containing only with the intercept). ssr_fe_only Sum of the squared residuals of the model estimated with fixed-effects only. ll_null The log-likelihood of the null model (containing only with the intercept). ll_fe_only The log-likelihood of the model estimated with fixed-effects only. fitted.values The fitted values. 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. obs_selection (When relevant.) List containing vectors of integers. It represents the sequential selection of observation vis a vis the original data set. 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. collin.min_norm The minimal diagonal value of the Cholesky decomposition. Small values indicate possible presence collinearity. y_demeaned Only when demeaned = TRUE: the centered dependent variable. X_demeaned Only when demeaned = TRUE: the centered explanatory variable. ## Details 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. Doing either of the two will give you acceess to the lagging functions l, 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 and offers a special representation of the interacted coefficients in the function coefplot. 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. ## Instrumental variables To estimate two stage least square regressions, insert the relationship between the endogenous regressor(s) and the instruments in a formula, after a pipe. For example, fml = y ~ x1 | x_endo ~ x_inst will use the variables x1 and x_inst in the first stage to explain x_endo. Then will use the fitted value of x_endo (which will be named fit_x_endo) and x1 to explain y. To include several endogenous regressors, just use "+", like in: fml = y ~ x1 | x_endo1 + x_end2 ~ x_inst1 + x_inst2. Of course you can still add the fixed-effects, but the IV formula must always come last, like in fml = y ~ x1 | fe1 + fe2 | x_endo ~ x_inst. If you want to estimate a model without exogenous variables, use "1" as a placeholder: e.g. fml = y ~ 1 | x_endo + x_inst. By default, the second stage regression is returned. You can access the first stage(s) regressions either directly in the slot iv_first_stage (not recommended), or using the argument stage = 1 from the function summary.fixest. For example summary(iv_est, stage = 1) will give the first stage(s). Note that using summary you can display both the second and first stages at the same time using, e.g., stage = 1:2 (using 2:1 would reverse the order). ## 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. Of course sw stands for stepwise, and csw for cumulative 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. Finally, 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. 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). ## 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. In all fixest estimations, this special parsing is enabled, so you don't need to use xpd. Limitations: the use of multiple square brackets within a single variable is not implemented. For example, the following will not work xpd(y ~ ..x, ..x = x.[1:3]_.[1:3]). Berge, Laurent, 2018, "Efficient estimation of maximum likelihood models with multiple fixed-effects: the R package FENmlm." CREA Discussion Papers, 13 (https://wwwen.uni.lu/content/download/110162/1299525/file/2018_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. For plotting coefficients: see coefplot. And other estimation methods: femlm, feglm, fepois, fenegbin, feNmlm. ## Author Laurent Berge ## Examples  # # Basic estimation # res = feols(Sepal.Length ~ Sepal.Width + Petal.Length, iris) # You can specify clustered standard-errors in summary: summary(res, cluster = ~Species) #> OLS estimation, Dep. Var.: Sepal.Length #> Observations: 150 #> Standard-errors: Clustered (Species) #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 2.249140 0.162626 13.8302 0.00518747 ** #> Sepal.Width 0.595525 0.051733 11.5115 0.00746202 ** #> Petal.Length 0.471920 0.006873 68.6673 0.00021201 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> RMSE: 0.329937 Adj. R2: 0.838003 # # Just one set of fixed-effects: # res = feols(Sepal.Length ~ Sepal.Width + Petal.Length | Species, iris) # By default, the SEs are clustered according to the first fixed-effect summary(res) #> OLS estimation, Dep. Var.: Sepal.Length #> Observations: 150 #> Fixed-effects: Species: 3 #> Standard-errors: Clustered (Species) #> Estimate Std. Error t value Pr(>|t|) #> Sepal.Width 0.432217 0.161308 2.67945 0.115623 #> Petal.Length 0.775629 0.126546 6.12925 0.025601 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> RMSE: 0.305129 Adj. R2: 0.859538 #> Within R2: 0.641507 # # Varying slopes: # res = feols(Sepal.Length ~ Petal.Length | Species[Sepal.Width], iris) summary(res) #> OLS estimation, Dep. Var.: Sepal.Length #> Observations: 150 #> Fixed-effects: Species: 3 #> Varying slopes: Sepal.Width (Species: 3) #> Standard-errors: Clustered (Species) #> Estimate Std. Error t value Pr(>|t|) #> Petal.Length 0.822045 0.10061 8.1706 0.014651 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> RMSE: 0.298903 Adj. R2: 0.863326 #> Within R2: 0.518738 # # Combining the FEs: # base = iris base$fe_2 = rep(1:10, 15)
res_comb = feols(Sepal.Length ~ Petal.Length | Species^fe_2, base)
summary(res_comb)
#> OLS estimation, Dep. Var.: Sepal.Length
#> Observations: 150
#> Fixed-effects: Species^fe_2: 30
#> Standard-errors: Clustered (Species^fe_2)
#>              Estimate Std. Error t value   Pr(>|t|)
#> Petal.Length 0.875613   0.062395 14.0334 1.8483e-14 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 0.309865     Adj. R2: 0.823494
#>                  Within R2: 0.548794fixef(res_comb)[]
#>      setosa_1     setosa_10      setosa_2      setosa_3      setosa_4
#>      3.826581      3.646581      3.661605      3.601703      3.621605
#>      setosa_5      setosa_6      setosa_7      setosa_8      setosa_9
#>      3.736532      3.884093      3.859118      3.699118      3.721605
#>  versicolor_1 versicolor_10  versicolor_2  versicolor_3  versicolor_4
#>      2.067402      2.142623      2.322426      2.359841      1.922328
#>  versicolor_5  versicolor_6  versicolor_7  versicolor_8  versicolor_9
#>      2.187402      2.287304      2.209694      2.284914      2.274963
#>   virginica_1  virginica_10   virginica_2   virginica_3   virginica_4
#>      1.769007      1.841642      1.826716      1.716471      1.639252
#>   virginica_5   virginica_6   virginica_7   virginica_8   virginica_9
#>      1.474081      2.006422      1.591887      1.791495      1.609007
#
#

data(base_did)
# We need to set up the panel with the arg. panel.id
est1 = feols(y ~ l(x1, 0:1), base_did, panel.id = ~id+period)
#> NOTE: 108 observations removed because of NA values (RHS: 108).est2 = feols(f(y) ~ l(x1, -1:1), base_did, panel.id = ~id+period)
#> NOTE: 216 observations removed because of NA values (LHS: 108, RHS: 216).etable(est1, est2, order = "f", drop="Int")
#>                               est1               est2
#> Dependent Var.:                  y             f(y,1)
#>
#> f(x1,1)                            0.9940*** (0.0542)
#> x1              0.9948*** (0.0487)    0.0081 (0.0592)
#> l(x1,1)            0.0410 (0.0558)    0.0157 (0.0640)
#> _______________ __________________ __________________
#> S.E.: Clustered             by: id             by: id
#> Observations                   972                864
#> R2                         0.26558            0.25697
#
# Using interactions:
#

data(base_did)
# We interact the variable 'period' with the variable 'treat'
est_did = feols(y ~ x1 + i(period, treat, 5) | id+period, base_did)

# Now we can plot the result of the interaction with coefplot
coefplot(est_did) # You have many more example in coefplot help

#
# Instrumental variables
#

# To estimate Two stage least squares,
# insert a formula describing the endo. vars./instr. relation after a pipe:

base = iris
names(base) = c("y", "x1", "x2", "x3", "fe1")
base$x_inst1 = 0.2 * base$x1 + 0.7 * base$x2 + rpois(150, 2) base$x_inst2 = 0.2 * base$x2 + 0.7 * base$x3 + rpois(150, 3)
base$x_endo1 = 0.5 * base$y + 0.5 * base$x3 + rnorm(150, sd = 2) base$x_endo2 = 1.5 * base$y + 0.5 * base$x3 + 3 * base\$x_inst1 + rnorm(150, sd = 5)

# Using 2 controls, 1 endogenous var. and 1 instrument
res_iv = feols(y ~ x1 + x2 | x_endo1 ~ x_inst1, base)

# The second stage is the default
summary(res_iv)
#> TSLS estimation, Dep. Var.: y, Endo.: x_endo1, Instr.: x_inst1
#> Second stage: Dep. Var.: y
#> Observations: 150
#> Standard-errors: IID
#>              Estimate Std. Error   t value Pr(>|t|)
#> (Intercept) -5.444083   82.90731 -0.065665  0.94773
#> fit_x_endo1  4.156853   44.67892  0.093038  0.92600
#> x1           0.533483    1.80938  0.294843  0.76853
#> x2          -1.358091   19.67381 -0.069030  0.94506
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 7.97755   Adj. R2: -94.4
#> F-test (1st stage), x_endo1: stat = 0.008704, p = 0.925795, on 1 and 146 DoF.
#>                  Wu-Hausman: stat = 5.23866 , p = 0.023535, on 1 and 145 DoF.
# To show the first stage:
summary(res_iv, stage = 1)
#> TSLS estimation, Dep. Var.: x_endo1, Endo.: x_endo1, Instr.: x_inst1
#> First stage: Dep. Var.: x_endo1
#> Observations: 150
#> Standard-errors: IID
#>              Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)  1.867844   1.454871  1.283856 0.20122622
#> x_inst1     -0.011136   0.119357 -0.093298 0.92579498
#> x1           0.018677   0.405512  0.046059 0.96332636
#> x2           0.447831   0.128635  3.481404 0.00065818 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1.9138   Adj. R2: 0.122258
#> F-test (1st stage): stat = 0.008704, p = 0.925795, on 1 and 146 DoF.
# To show both the first and second stages:
summary(res_iv, stage = 1:2)
#> IV: First stage: x_endo1
#> TSLS estimation, Dep. Var.: x_endo1, Endo.: x_endo1, Instr.: x_inst1
#> First stage: Dep. Var.: x_endo1
#> Observations: 150
#> Standard-errors: IID
#>              Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)  1.867844   1.454871  1.283856 0.20122622
#> x_inst1     -0.011136   0.119357 -0.093298 0.92579498
#> x1           0.018677   0.405512  0.046059 0.96332636
#> x2           0.447831   0.128635  3.481404 0.00065818 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1.9138   Adj. R2: 0.122258
#> F-test (1st stage): stat = 0.008704, p = 0.925795, on 1 and 146 DoF.
#>
#> IV: Second stage
#> TSLS estimation, Dep. Var.: y, Endo.: x_endo1, Instr.: x_inst1
#> Second stage: Dep. Var.: y
#> Observations: 150
#> Standard-errors: IID
#>              Estimate Std. Error   t value Pr(>|t|)
#> (Intercept) -5.444083   82.90731 -0.065665  0.94773
#> fit_x_endo1  4.156853   44.67892  0.093038  0.92600
#> x1           0.533483    1.80938  0.294843  0.76853
#> x2          -1.358091   19.67381 -0.069030  0.94506
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 7.97755   Adj. R2: -94.4
#> F-test (1st stage), x_endo1: stat = 0.008704, p = 0.925795, on 1 and 146 DoF.
#>                  Wu-Hausman: stat = 5.23866 , p = 0.023535, on 1 and 145 DoF.
# Adding a fixed-effect => IV formula always last!
res_iv_fe = feols(y ~ x1 + x2 | fe1 | x_endo1 ~ x_inst1, base)

# With two endogenous regressors
res_iv2 = feols(y ~ x1 + x2 | x_endo1 + x_endo2 ~ x_inst1 + x_inst2, base)

# Now there's two first stages => a fixest_multi object is returned
sum_res_iv2 = summary(res_iv2, stage = 1)

# You can navigate through it by subsetting:
sum_res_iv2[iv = 1]
#> TSLS estimation, Dep. Var.: x_endo1, Endo.: x_endo1, x_endo2, Instr.: x_inst1, x_inst2
#> First stage: Dep. Var.: x_endo1
#> Observations: 150
#> Standard-errors: IID
#>              Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)  1.850720   1.516307  1.220544 0.22423981
#> x_inst1     -0.011391   0.119922 -0.094984 0.92445853
#> x_inst2      0.003804   0.091032  0.041784 0.96672840
#> x1           0.020822   0.410131  0.050770 0.95957842
#> x2           0.446540   0.132727  3.364349 0.00098184 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1.91379   Adj. R2: 0.116216
#> F-test (1st stage): stat = 0.005195, p = 0.994818, on 2 and 145 DoF.
# The stage argument also works in etable:
etable(res_iv, res_iv_fe, res_iv2, order = "endo")
#>                         res_iv       res_iv_fe          res_iv2
#> Dependent Var.:              y               y                y
#>
#> x_endo1          4.157 (44.68)   3.414 (15.83)   -1.912 (56.41)
#> x_endo2                                        -0.0206 (0.2092)
#> (Intercept)     -5.444 (82.91)                    5.959 (106.1)
#> x1              0.5335 (1.809) 0.6988 (0.9094)   0.6664 (1.476)
#> x2              -1.358 (19.67) -0.6848 (6.072)    1.368 (25.36)
#> Fixed-Effects:  -------------- --------------- ----------------
#> fe1                         No             Yes               No
#> _______________ ______________ _______________ ________________
#> S.E. type                  IID         by: fe1              IID
#> Observations               150             150              150
#> R2                     -92.436         -61.946          -18.724
#> Within R2                   --         -164.08               --
etable(res_iv, res_iv_fe, res_iv2, stage = 1:2, order = c("endo", "inst"),
group = list(control = "!endo|inst"))
#>                         res_iv.1      res_iv.2      res_iv_fe.1   res_iv_fe.2
#> IV stages                  First        Second            First        Second
#> Dependent Var.:          x_endo1             y          x_endo1             y
#>
#> x_endo1                          4.157 (44.68)                  3.414 (15.83)
#> x_endo2
#> x_inst1         -0.0111 (0.1194)               -0.0127 (0.0508)
#> x_inst2
#> control                      Yes           Yes              Yes           Yes
#> Fixed-Effects:  ---------------- ------------- ---------------- -------------
#> fe1                           No            No              Yes           Yes
#> _______________ ________________ _____________ ________________ _____________
#> S.E. type                    IID           IID          by: fe1       by: fe1
#> Observations                 150           150              150           150
#> R2                       0.13993       -92.436          0.14157       -61.946
#> Within R2                     --            --          0.00829       -164.08
#>                        res_iv2.1         res_iv2.2        res_iv2.3
#> IV stages                  First             First           Second
#> Dependent Var.:          x_endo1           x_endo2                y
#>
#> x_endo1                                              -1.912 (56.41)
#> x_endo2                                            -0.0206 (0.2092)
#> x_inst1         -0.0114 (0.1199) 3.295*** (0.3310)
#> x_inst2          0.0038 (0.0910)  -0.2070 (0.2513)
#> control                      Yes               Yes              Yes
#> Fixed-Effects:  ---------------- ----------------- ----------------
#> fe1                           No                No               No
#> _______________ ________________ _________________ ________________
#> S.E. type                    IID               IID              IID
#> Observations                 150               150              150
#> R2                       0.13994           0.57761          -18.724
#> Within R2                     --                --               --
#
# Multiple estimations:
#

# 6 estimations
est_mult = feols(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])
#>                            model 1             model 2             model 3
#> Dependent Var.:              Ozone               Ozone               Ozone
#>
#> (Intercept)       -71.03** (23.58)   -248.5*** (48.14)   -257.7*** (48.45)
#> Wind            -3.055*** (0.6633)    14.34*** (4.239)    14.58*** (4.228)
#> Temp             1.840*** (0.2500)   4.076*** (0.5875)   4.136*** (0.5871)
#> Wind x Temp                        -0.2239*** (0.0540) -0.2273*** (0.0539)
#> Day                                                        0.2940 (0.2185)
#> _______________ __________________ ___________________ ___________________
#> S.E. type                      IID                 IID                 IID
#> Observations                   116                 116                 116
#> R2                         0.56871             0.62613             0.63213
#> Adj. R2                    0.56108             0.61611             0.61887
# And now the second (access can be made by name)
etable(est_mult[lhs = "Solar.R"])
#>                           model 1         model 2         model 3
#> Dependent Var.:           Solar.R         Solar.R         Solar.R
#>
#> (Intercept)        -76.36 (82.00)   10.01 (189.6)   45.19 (190.6)
#> Wind                2.211 (2.308)  -5.893 (16.20)  -6.214 (16.15)
#> Temp            3.075*** (0.8778)   1.982 (2.333)   1.788 (2.329)
#> Wind x Temp                       0.1044 (0.2064) 0.1069 (0.2057)
#> Day                                               -1.161 (0.8277)
#> _______________ _________________ _______________ _______________
#> S.E. type                     IID             IID             IID
#> Observations                  146             146             146
#> R2                        0.08198         0.08363         0.09624
#> Adj. R2                   0.06914         0.06427         0.07061
# 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])
#>                             model 1         model 2             model 3
#> Dependent Var.:               Ozone         Solar.R               Ozone
#>
#> (Intercept)       -248.5*** (48.14)   10.01 (189.6)   -257.7*** (48.45)
#> Wind               14.34*** (4.239)  -5.893 (16.20)    14.58*** (4.228)
#> Temp              4.076*** (0.5875)   1.982 (2.333)   4.136*** (0.5871)
#> Wind x Temp     -0.2239*** (0.0540) 0.1044 (0.2064) -0.2273*** (0.0539)
#> Day                                                     0.2940 (0.2185)
#> _______________ ___________________ _______________ ___________________
#> S.E. type                       IID             IID                 IID
#> Observations                    116             146                 116
#> R2                          0.62613         0.08363             0.63213
#> Adj. R2                     0.61611         0.06427             0.61887
#>                         model 4
#> Dependent Var.:         Solar.R
#>
#> (Intercept)       45.19 (190.6)
#> Wind             -6.214 (16.15)
#> Temp              1.788 (2.329)
#> Wind x Temp     0.1069 (0.2057)
#> Day             -1.161 (0.8277)
#> _______________ _______________
#> S.E. type                   IID
#> Observations                146
#> R2                      0.09624
# Combining with split
est_split = feols(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  t value   Pr(>|t|)
#> (Intercept)      29.8505    4.62478  6.45448 1.3810e-06 ***
#> poly(Wind, 2)1 -180.1531   68.14588 -2.64364 1.4516e-02 *
#> poly(Wind, 2)2   93.3187   55.68940  1.67570 1.0734e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error t value   Pr(>|t|)
#> (Intercept)     60.8847    13.3314 4.56701 0.00013696 ***
#> poly(Temp, 2)1 436.0792   162.6142 2.68168 0.01332077 *
#> poly(Temp, 2)2 156.5738   107.6842 1.45401 0.15945485
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#>                Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)     185.975    27.2165  6.833171 4.5627e-07 ***
#> poly(Wind, 2)1 -101.509   403.9730 -0.251277 8.0374e-01
#> poly(Wind, 2)2 -248.609   327.3103 -0.759552 4.5492e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error   t value Pr(>|t|)
#> (Intercept)     184.737    72.5274  2.547133  0.01769 *
#> poly(Temp, 2)1 -306.202   881.2266 -0.347473  0.73126
#> poly(Temp, 2)2 -822.991   547.8534 -1.502211  0.14609
#>
#>
#> # SAMPLE: 6
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#>                Estimate Std. Error   t value Pr(>|t|)
#> (Intercept)     21.8206    9.48902  2.299565 0.061137 .
#> poly(Wind, 2)1 166.9646  161.90174  1.031271 0.342179
#> poly(Wind, 2)2 -60.6305   92.54148 -0.655171 0.536652
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error t value   Pr(>|t|)
#> (Intercept)     34.2959    4.15309 8.25794 0.00017054 ***
#> poly(Temp, 2)1 114.9172   62.20576 1.84737 0.11420526
#> poly(Temp, 2)2 206.4879   79.99940 2.58112 0.04170670 *
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#>                Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)    187.7785    16.4660 11.404021 7.8882e-12 ***
#> poly(Wind, 2)1 389.5634   194.8959  1.998828 5.5787e-02 .
#> poly(Wind, 2)2 -72.8180   156.4046 -0.465574 6.4525e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error  t value   Pr(>|t|)
#> (Intercept)     188.337    21.4096 8.796855 2.0604e-09 ***
#> poly(Temp, 2)1  575.587   374.3715 1.537475 1.3581e-01
#> poly(Temp, 2)2  133.702   363.9611 0.367352 7.1622e-01
#>
#>
#> # SAMPLE: 7
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#>                 Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)      49.8115    5.51938  9.024837 5.1064e-09 ***
#> poly(Wind, 2)1 -280.6407   84.25343 -3.330912 2.9055e-03 **
#> poly(Wind, 2)2   58.0027  107.62694  0.538924 5.9512e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error  t value Pr(>|t|)
#> (Intercept)     29.2107    16.0689 1.817844 0.082138 .
#> poly(Temp, 2)1 584.0963   265.9188 2.196521 0.038399 *
#> poly(Temp, 2)2  14.1887   198.9617 0.071314 0.943764
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#>                  Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)     213.16197    16.3352 13.049206 2.0111e-13 ***
#> poly(Wind, 2)1 -144.08420   246.7578 -0.583909 5.6396e-01
#> poly(Wind, 2)2    8.70905   324.8247  0.026812 9.7880e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error   t value  Pr(>|t|)
#> (Intercept)     172.303    54.6087  3.155239 0.0038125 **
#> poly(Temp, 2)1  835.016   904.9731  0.922697 0.3640481
#> poly(Temp, 2)2 -107.276   645.9680 -0.166070 0.8692954
#>
#>
#> # SAMPLE: 8
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#>                 Estimate Std. Error  t value   Pr(>|t|)
#> (Intercept)      49.7316    6.14979  8.08670 3.5632e-08 ***
#> poly(Wind, 2)1 -289.2515   97.10974 -2.97860 6.7192e-03 **
#> poly(Wind, 2)2  116.2989   90.00343  1.29216 2.0913e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                 Estimate Std. Error   t value Pr(>|t|)
#> (Intercept)      26.1289    16.6482  1.569471 0.130195
#> poly(Temp, 2)1  678.1406   325.9933  2.080228 0.048834 *
#> poly(Temp, 2)2 -162.1223   188.3344 -0.860822 0.398222
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#>                Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)    170.3429    15.8361 10.756599 7.2127e-11 ***
#> poly(Wind, 2)1 -78.9215   265.2646 -0.297520 7.6853e-01
#> poly(Wind, 2)2 130.2447   250.8995  0.519111 6.0825e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error   t value  Pr(>|t|)
#> (Intercept)    141.7373    34.9269  4.058112 0.0004267 ***
#> poly(Temp, 2)1 610.4434   708.8908  0.861125 0.3973535
#> poly(Temp, 2)2 -62.8161   420.0627 -0.149540 0.8823275
#>
#>
#> # SAMPLE: 9
#>
#>
#> ### Dep. var.: Ozone
#>
#> Expl. vars.: poly(Wind, 2)
#>                 Estimate Std. Error  t value   Pr(>|t|)
#> (Intercept)      32.4899    3.00730 10.80367 4.1394e-11 ***
#> poly(Wind, 2)1 -151.1122   39.26645 -3.84838 6.9340e-04 ***
#> poly(Wind, 2)2  166.7517   45.93576  3.63011 1.2171e-03 **
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error  t value   Pr(>|t|)
#> (Intercept)     35.7733    1.92538 18.57989  < 2.2e-16 ***
#> poly(Temp, 2)1 198.9668   30.21319  6.58543 5.5334e-07 ***
#> poly(Temp, 2)2 160.4793   32.36259  4.95879 3.7465e-05 ***
#>
#> ### Dep. var.: Solar.R
#>
#> Expl. vars.: poly(Wind, 2)
#>                Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)    168.0471    14.9433 11.245665 1.0799e-11 ***
#> poly(Wind, 2)1 -94.9627   195.7141 -0.485211 6.3144e-01
#> poly(Wind, 2)2  27.9996   231.6709  0.120859 9.0470e-01
#> ---
#> Expl. vars.: poly(Temp, 2)
#>                Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)     166.482    15.3173 10.868939 2.3073e-11 ***
#> poly(Temp, 2)1  204.784   241.8651  0.846686 4.0461e-01
#> poly(Temp, 2)2 -146.653   256.4108 -0.571947 5.7209e-01
# Different way of displaying the results with "compact"
summary(est_split, "compact")
#>    sample     lhs           rhs    (Intercept) poly(Wind, 2)1 poly(Wind, 2)2
#> 1       5 Ozone   poly(Wind, 2) 29.9*** (4.62)   -180* (68.1)    93.3 (55.7)
#> 2       5 Ozone   poly(Temp, 2) 60.9*** (13.3)
#> 3       5 Solar.R poly(Wind, 2)  186*** (27.2)    -102 (404)     -249 (327)
#> 4       5 Solar.R poly(Temp, 2)    185* (72.5)
#> 5       6 Ozone   poly(Wind, 2)   21.8. (9.49)     167 (162)    -60.6 (92.5)
#> 6       6 Ozone   poly(Temp, 2) 34.3*** (4.15)
#> 7       6 Solar.R poly(Wind, 2)  188*** (16.5)    390. (195)    -72.8 (156)
#> 8       6 Solar.R poly(Temp, 2)  188*** (21.4)
#> 9       7 Ozone   poly(Wind, 2) 49.8*** (5.52)  -281** (84.3)      58 (108)
#> 10      7 Ozone   poly(Temp, 2)   29.2. (16.1)
#> 11      7 Solar.R poly(Wind, 2)  213*** (16.3)    -144 (247)     8.71 (325)
#> 12      7 Solar.R poly(Temp, 2)   172** (54.6)
#> 13      8 Ozone   poly(Wind, 2) 49.7*** (6.15)  -289** (97.1)     116 (90)
#> 14      8 Ozone   poly(Temp, 2)    26.1 (16.6)
#> 15      8 Solar.R poly(Wind, 2)  170*** (15.8)   -78.9 (265)      130 (251)
#> 16      8 Solar.R poly(Temp, 2)  142*** (34.9)
#> 17      9 Ozone   poly(Wind, 2) 32.5*** (3.01) -151*** (39.3)   167** (45.9)
#> 18      9 Ozone   poly(Temp, 2) 35.8*** (1.93)
#> 19      9 Solar.R poly(Wind, 2)  168*** (14.9)     -95 (196)       28 (232)
#> 20      9 Solar.R poly(Temp, 2)  166*** (15.3)
#>    poly(Temp, 2)1 poly(Temp, 2)2
#> 1
#> 2     436* (163)      157 (108)
#> 3
#> 4     -306 (881)     -823 (548)
#> 5
#> 6      115 (62.2)    206* (80)
#> 7
#> 8      576 (374)      134 (364)
#> 9
#> 10    584* (266)     14.2 (199)
#> 11
#> 12     835 (905)     -107 (646)
#> 13
#> 14    678* (326)     -162 (188)
#> 15
#> 16     610 (709)    -62.8 (420)
#> 17
#> 18  199*** (30.2)  160*** (32.4)
#> 19
#> 20     205 (242)     -147 (256)
# 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  t value   Pr(>|t|)
#> (Intercept)      29.8505    4.62478  6.45448 1.3810e-06 ***
#> poly(Wind, 2)1 -180.1531   68.14588 -2.64364 1.4516e-02 *
#> poly(Wind, 2)2   93.3187   55.68940  1.67570 1.0734e-01
#> ---
#> Sample: 6
#>                Estimate Std. Error   t value Pr(>|t|)
#> (Intercept)     21.8206    9.48902  2.299565 0.061137 .
#> poly(Wind, 2)1 166.9646  161.90174  1.031271 0.342179
#> poly(Wind, 2)2 -60.6305   92.54148 -0.655171 0.536652

#
# Argument sliding
#

# When the data set is set up globally, you can use the vcov argument implicitly

base = iris
names(base) = c("y", "x1", "x2", "x3", "species")

no_sliding = feols(y ~ x1 + x2, base, ~species)

# With sliding
setFixest_estimation(data = base)

# ~species is implicitly deduced to be equal to 'vcov'
sliding = feols(y ~ x1 + x2, ~species)

etable(no_sliding, sliding)
#>                         no_sliding            sliding
#> Dependent Var.:                  y                  y
#>
#> (Intercept)       2.249** (0.1626)   2.249** (0.1626)
#> x1               0.5955** (0.0517)  0.5955** (0.0517)
#> x2              0.4719*** (0.0069) 0.4719*** (0.0069)
#> _______________ __________________ __________________
#> S.E.: Clustered        by: species        by: species
#> Observations                   150                150
#> R2                         0.84018            0.84018