This function estimates maximum likelihood models (e.g., Poisson or Logit) with non-linear in parameters right-hand-sides and is efficient to handle any number of fixed effects. If you do not use non-linear in parameters right-hand-side, use femlm or feglm instead (their design is simpler).
feNmlm(
fml,
data,
family = c("poisson", "negbin", "logit", "gaussian"),
NL.fml,
fixef,
fixef.rm = "perfect",
NL.start,
lower,
upper,
NL.start.init,
offset,
subset,
split,
fsplit,
cluster,
se,
dof,
panel.id,
start = 0,
jacobian.method = "simple",
useHessian = TRUE,
hessian.args = NULL,
opt.control = list(),
nthreads = getFixest_nthreads(),
lean = FALSE,
verbose = 0,
theta.init,
fixef.tol = 1e-05,
fixef.iter = 10000,
deriv.tol = 1e-04,
deriv.iter = 1000,
warn = TRUE,
notes = getFixest_notes(),
combine.quick,
mem.clean = FALSE,
only.env = FALSE,
env,
...
)A formula. This formula gives the linear formula to be estimated (it is similar to a lm formula), for example: fml = z~x+y. To include fixed-effects variables, insert them in this formula using a pipe (e.g. fml = z~x+y|fixef_1+fixef_2). To include a non-linear in parameters element, you must use the argment NL.fml. 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). This leads to 6 estimation fml = c(y1, y2) ~ x1 + cw0(x2, x3). See details.
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.
Character scalar. It should provide the family. The possible values are "poisson" (Poisson model with log-link, the default), "negbin" (Negative Binomial model with log-link), "logit" (LOGIT model with log-link), "gaussian" (Gaussian model).
A formula. If provided, this formula represents the non-linear part of the right hand side (RHS). Note that contrary to the fml argument, the coefficients must explicitly appear in this formula. For instance, it can be ~a*log(b*x + c*x^3), where a, b, and c are the coefficients to be estimated. Note that only the RHS of the formula is to be provided, and NOT the left hand side.
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.
(For NL models only) A list of starting values for the non-linear parameters. ALL the parameters are to be named and given a staring value. Example: NL.start=list(a=1,b=5,c=0). Though, there is an exception: if all parameters are to be given the same starting value, you can use a numeric scalar.
(For NL models only) A list. The lower bound for each of the non-linear parameters that requires one. Example: lower=list(b=0,c=0). Beware, if the estimated parameter is at his lower bound, then asymptotic theory cannot be applied and the standard-error of the parameter cannot be estimated because the gradient will not be null. In other words, when at its upper/lower bound, the parameter is considered as 'fixed'.
(For NL models only) A list. The upper bound for each of the non-linear parameters that requires one. Example: upper=list(a=10,c=50). Beware, if the estimated parameter is at his upper bound, then asymptotic theory cannot be applied and the standard-error of the parameter cannot be estimated because the gradient will not be null. In other words, when at its upper/lower bound, the parameter is considered as 'fixed'.
(For NL models only) Numeric scalar. If the argument NL.start is not provided, or only partially filled (i.e. there remain non-linear parameters with no starting value), then the starting value of all remaining non-linear parameters is set to NL.start.init.
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 clusters in the estimation, you could further use cluster = 1:2 or leave it blank with se = "twoway" (assuming var1 [resp. var2] was the 1st [res. 2nd] cluster). 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 = "standard". Note that this argument can be implicitly deduced from the argument cluster.
An object of class dof.type obtained with the function dof. Represents how the degree of freedom correction should be done.You must use the function dof for this argument. The arguments and defaults of the function dof 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 dof 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).
Starting values for the coefficients in the linear part (for the non-linear part, use NL.start). Can be: i) a numeric of length 1 (e.g. start = 0, the default), 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).
(For NL models only) Character scalar. Provides the method used to numerically compute the Jacobian of the non-linear part. Can be either "simple" or "Richardson". Default is "simple". See the help of jacobian for more information.
Logical. Should the Hessian be computed in the optimization stage? Default is TRUE.
List of arguments to be passed to function genD. Defaults is missing. Only used with the presence of NL.fml.
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, default is 0. It represents the level of information that should be reported during the optimisation process. If verbose=0: nothing is reported. If verbose=1: the value of the coefficients and the likelihood are reported. If verbose=2: 1 + information on the computing time of the null model, the fixed-effects coefficients and the hessian are reported.
Positive numeric scalar. The starting value of the dispersion parameter if family="negbin". By default, the algorithm uses as a starting value the theta obtained from the model with only the intercept.
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$double.eps. Note that this parameter is dynamically controlled by the algorithm.
Maximum number of iterations in fixed-effects algorithm (only in use for 2+ fixed-effects). Default is 10000.
Precision used to obtain the fixed-effects derivatives. Defaults to 1e-4. It corresponds to the maximum absolute difference allowed between two coefficients of successive iterations. Argument deriv.tol cannot be lower than 10000*.Machine$double.eps.
Maximum number of iterations in the algorithm to obtain the derivative of the fixed-effects (only in use for 2+ fixed-effects). Default is 1000.
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 only 0 (or 0/1) outcomes in a fixed-effect setup (in Poisson/Neg. Bin./Logit models). 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 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.
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).
The named vector of coefficients.
The table of the coefficients with their standard errors, z-values and p-values.
The loglikelihood.
Number of iterations of the algorithm.
The number of observations.
The number of parameters of the model.
The call.
The linear formula of the call.
A list containing different parts of the formula. Always contain the linear formula. Then, if relevant: fixef: the fixed-effects; NL: the non linear part of the formula.
Log-likelihood of the null model (i.e. with the intercept only).
The adjusted pseudo R2.
The convergence message from the optimization procedures.
Squared correlation between the dependent variable and the expected predictor (i.e. fitted.values) obtained by the estimation.
The Hessian of the parameters.
The fitted values are the expected value of the dependent variable for the fitted model: that is \(E(Y|X)\).
The variance-covariance matrix of the parameters.
The standard-error of the parameters.
The matrix of the scores (first derivative for each observation).
The ML family that was used for the estimation.
The difference between the dependent variable and the expected predictor.
The sum of the fixed-effects for each observation.
The offset formula.
The nonlinear formula of the call.
Whether the coefficients were upper or lower bounded. -- This can only be the case when a non-linear formula is included and the arguments 'lower' or 'upper' are provided.
The logical vector that gives for each coefficient whether it was bounded or not. This can only be the case when a non-linear formula is included and the arguments 'lower' or 'upper' are provided.
The names of each fixed-effect dimension.
The list (of length the number of fixed-effects) of the fixed-effects identifiers for each observation.
The size of each fixed-effect (i.e. the number of unique identifierfor each fixed-effect dimension).
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 row numbers of the observations that were removed. Also reports the NA observations that were removed.
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.
In the case of a negative binomial estimation: the overdispersion parameter.
@seealso 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, feglm, fepois, fenegbin.
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.
You can interact a numeric variable with a "factor-like" variable by using interact(var, fe, ref), where fe is the variable to be interacted with and the argument ref is a value of fe taken as a reference (optional). Instead of using the function interact, you can use the alias i(var, fe, ref).
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 (using the syntax fe[[var]], as explained in the section “Varying slopes”).
The function interact has in fact more arguments, please see details in its associated help page.
Standard-errors can be computed in different ways, you can use the arguments se and dof 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_se and setFixest_dof to permanently set the way the standard-errors are computed.
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.
To include mutliple 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 associated to 4 short aliases. These are a) stepwise, stepwise0, cstepwise, cstepwise0, and b) sw, sw0, csw, csw0. 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 wihtout 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).
Fixed-effects cannot be included in a stepwise fashion: they are there or not and stay the same for all estimations.
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.
This function estimates maximum likelihood models where the conditional expectations are as follows:
Gaussian likelihood: $$E(Y|X)=X\beta$$ Poisson and Negative Binomial likelihoods: $$E(Y|X)=\exp(X\beta)$$ where in the Negative Binomial there is the parameter \(\theta\) used to model the variance as \(\mu+\mu^2/\theta\), with \(\mu\) the conditional expectation. Logit likelihood: $$E(Y|X)=\frac{\exp(X\beta)}{1+\exp(X\beta)}$$
When there are one or more fixed-effects, the conditional expectation can be written as: $$E(Y|X) = h(X\beta+\sum_{k}\sum_{m}\gamma_{m}^{k}\times C_{im}^{k}),$$ where \(h(.)\) is the function corresponding to the likelihood function as shown before. \(C^k\) is the matrix associated to fixed-effect dimension \(k\) such that \(C^k_{im}\) is equal to 1 if observation \(i\) is of category \(m\) in the fixed-effect dimension \(k\) and 0 otherwise.
When there are non linear in parameters functions, we can schematically split the set of regressors in two:
$$f(X,\beta)=X^1\beta^1 + g(X^2,\beta^2)$$
with first a linear term and then a non linear part expressed by the function g. That is, we add a non-linear term to the linear terms (which are \(X*beta\) and the fixed-effects coefficients). It is always better (more efficient) to put into the argument NL.fml only the non-linear in parameter terms, and add all linear terms in the fml argument.
To estimate only a non-linear formula without even the intercept, you must exclude the intercept from the linear formula by using, e.g., fml = z~0.
The over-dispersion parameter of the Negative Binomial family, theta, is capped at 10,000. If theta reaches this high value, it means that there is no overdispersion.
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
On the unconditionnal Negative Binomial model:
Allison, Paul D and Waterman, Richard P, 2002, "Fixed-Effects Negative Binomial Regression Models", Sociological Methodology 32(1) pp. 247--265
# NOT RUN {
# This section covers only non-linear in parameters examples
# For linear relationships: use femlm or feglm instead
# Generating data for a simple example
set.seed(1)
n = 100
x = rnorm(n, 1, 5)**2
y = rnorm(n, -1, 5)**2
z1 = rpois(n, x*y) + rpois(n, 2)
base = data.frame(x, y, z1)
# Estimating a 'linear' relation:
est1_L = femlm(z1 ~ log(x) + log(y), base)
# Estimating the same 'linear' relation using a 'non-linear' call
est1_NL = feNmlm(z1 ~ 1, base, NL.fml = ~a*log(x)+b*log(y), NL.start = list(a=0, b=0))
# we compare the estimates with the function esttable (they are identical)
etable(est1_L, est1_NL)
# Now generating a non-linear relation (E(z2) = x + y + 1):
z2 = rpois(n, x + y) + rpois(n, 1)
base$z2 = z2
# Estimation using this non-linear form
est2_NL = feNmlm(z2 ~ 0, base, NL.fml = ~log(a*x + b*y),
NL.start = 2, lower = list(a=0, b=0))
# we can't estimate this relation linearily
# => closest we can do:
est2_L = femlm(z2 ~ log(x) + log(y), base)
# Difference between the two models:
etable(est2_L, est2_NL)
# Plotting the fits:
plot(x, z2, pch = 18)
points(x, fitted(est2_L), col = 2, pch = 1)
points(x, fitted(est2_NL), col = 4, pch = 2)
# }
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