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,
vcov,
fixef,
fixef.rm = "perfect",
NL.start,
lower,
upper,
NL.start.init,
offset,
subset,
split,
fsplit,
cluster,
se,
ssc,
panel.id,
start = 0,
jacobian.method = "simple",
useHessian = TRUE,
hessian.args = NULL,
opt.control = list(),
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,
...
)

## Arguments

fml 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. 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. 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. 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. 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 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). 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. List of elements to be passed to the optimization method nlminb. See the help page of nlminb for more information. 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.

## 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).

coefficients

The named vector of coefficients.

coeftable

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

loglik

The loglikelihood.

iterations

Number of iterations of the algorithm.

nobs

The number of observations.

nparams

The number of parameters of the model.

call

The call.

fml

The linear formula of the call.

fml_all

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.

ll_null

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

pseudo_r2

message

The convergence message from the optimization procedures.

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.

fitted.values

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

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).

family

The ML family that was used for the estimation.

residuals

The difference between the dependent variable and the expected predictor.

sumFE

The sum of the fixed-effects for each observation.

offset

The offset formula.

NL.fml

The nonlinear formula of the call.

bounds

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.

isBounded

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.

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).

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

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.

theta

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.

## Details

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.

## 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.

## 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.

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.

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

Laurent Berge

## Examples


# 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)
#>                             est1_L            est1_NL
#> Dependent Var.:                 z1                 z1
#>
#> (Intercept)       0.0632* (0.0298)   0.0632* (0.0298)
#> log(x)          0.9945*** (0.0051)
#> log(y)          0.9901*** (0.0058)
#> a                                  0.9945*** (0.0051)
#> b                                  0.9901*** (0.0058)
#> _______________ __________________ __________________
#> S.E. type                      IID                IID
#> Observations                   100                100
#> Squared Cor.               0.99957            0.99957
#> Pseudo R2                  0.99338            0.99338
#> BIC                         797.47             797.47
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# 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)
#>                             est2_L           est2_NL
#> Dependent Var.:                 z2                z2
#>
#> (Intercept)      2.824*** (0.0386)
#> log(x)          0.2695*** (0.0094)
#> log(y)          0.1774*** (0.0087)
#> a                                  1.057*** (0.0265)
#> b                                  1.005*** (0.0248)
#> _______________ __________________ _________________
#> S.E. type                      IID               IID
#> Observations                   100               100
#> Squared Cor.               0.46596           0.96247
#> Pseudo R2                  0.32078           0.81626
#> BIC                        2,545.9            694.19
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# 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) 