# femlm: Fixed effects maximum likelihood models In FENmlm: Fixed Effects Nonlinear Maximum Likelihood Models

## Description

This function estimates maximum likelihood models (e.g., Poisson or Logit) and is efficient to handle any number of fixed effects (i.e. cluster variables). It further allows for nonlinear in parameters right hand sides.

## Usage

 1 2 3 4 5 femlm(fml, data, family = c("poisson", "negbin", "logit", "gaussian"), NL.fml, cluster, useAcc = TRUE, start, lower, upper, env, start.init, offset, nl.gradient, linear.start = 0, jacobian.method = c("simple", "Richardson"), useHessian = TRUE, opt.control = list(), cores = 1, debug = FALSE, theta.init, precision.cluster, ...) 

## 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 cluster variables, you can 1) either insert them in this formula using a pipe (e.g. fml = z~x+y|cluster1+cluster2), or 2) either use the argment cluster. You can add a non-linear element in this formula by using the argment NL.fml. If you want to estimate only a non-linear formula without even the intercept, you can use fml = z~0 in combination with NL.fml. data A data.frame containing the necessary variables to run the model. The variables of the non-linear right hand side of the formula are identified with this data.frame names. Note that no NA is allowed in the variables to be used in the estimation. family 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). NL.fml 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 explicitely 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. cluster Character vector. The name/s of a/some variable/s within the dataset to be used as clusters. These variables should contain the identifier of each observation (e.g., think of it as a panel identifier). useAcc Default is TRUE. Whether an acceleration algorithm (Irons and Tuck iterations) should be used to otbain the cluster coefficients when there are two clusters. start A list. Starting values for the non-linear parameters. ALL the parameters are to be named and given a staring value. Example: 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 the argument start.init. lower 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'. upper 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'. env An environment. You can provide an environement in which the non-linear part will be evaluated. (May be useful for some particular non-linear functions.) start.init Numeric scalar. If the argument 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 start.init. offset A formula. An offset can be added to the estimation. It should be a formula of the form (for example) ~0.5*x**2. This offset is linearily added to the elements of the main formula 'fml'. Note that when using the argument 'NL.fml', you can directly add the offset there. nl.gradient A formula. The user can prodide a function that computes the gradient of the non-linear part. The formula should be of the form ~f0(a1,x1,a2,a2). The important point is that it should be able to be evaluated by: eval(nl.gradient[[2]], env) where env is the working environment of the algorithm (which contains all variables and parameters). The function should return a list or a data.frame whose names are the non-linear parameters. linear.start Numeric named vector. The starting values of the linear part. Note that you can jacobian.method 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. useHessian Logical. Should the Hessian be computed in the optimization stage? Default is TRUE. opt.control List of elements to be passed to the optimization method nlminb. cores Integer, default is 1. Number of threads to be used (accelerates the algorithm via the use of openMP routines). This is particularly efficient for the negative binomial and logit models, less so for Gaussian and Poisson likelihoods (unless for large datasets). debug Logical. If TRUE then the log-likelihood as well as all parameters are printed at each iteration. Default is FALSE. theta.init 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.cluster Precision used to obtain the fixed-effects (ie cluster coefficients). Defaults to 1e-5. It corresponds to the maximum absolute difference allowed between two iterations. ... Not currently used.

## 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) = exp(X*beta) / (1 + exp(X*beta))

When there are one or more clusters, the conditional expectation can be written as:

E(Y|X) = h(Xβ+∑_{k}∑_{m}γ_{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 cluster k such that C^k_{im} is equal to 1 if observation i is of category m in cluster k and 0 otherwise.

When there are non linear in parameters functions, we can schematically split the set of regressors in two:

f(X,β)=X^1β^1 + g(X^2,β^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 cluster 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.

## Value

An femlm object.

 coef The 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. n The number of observations. k The number of parameters of the model. call The call. NL.fml The nonlinear formula of the call. linear.formula The linear formula of the call. ll_null Log-likelihood of the null model (i.e. with the intercept only). pseudo_r2 The adjusted pseudo R2. naive.r2 The R2 as if the expected predictor was the linear predictor in OLS. message The convergence message from the optimization procedures. sq.cor Squared correlation between the dependent variable and its expected value as given by the optimization. hessian The Hessian of the parameters. expected.predictor The expected predictor is the expected value of the dependent variable. cov.unscaled The variance-covariance matrix of the parameters. 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. 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. resids The difference between the dependent variable and the expected predictor. dummies The sum of the cluster coefficients for each observation. clusterNames The names of each cluster. id_dummies The list (of length the number of clusters) of the cluser identifiers for each observation. clusterSize The size of each cluster. obsRemoved In the case there were clusters and some observations were removed because of only 0/1 outcome wirhin a cluster, it gives the row numbers of the observations that were removed. clusterRemoved In the case there were clusters and some observations were removed because of only 0/1 outcome wirhin a cluster, it gives the list (for each cluster) of the clustr identifiers that were removed. theta In the case of a negative binomial estimation: the overdispersion parameter.

Laurent Berge

## References

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

See also summary.femlm to see the results with the appropriate standard-errors, getFE to extract the cluster coefficients, and the functions res2table and res2tex to visualize the results of multiple estimations.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 # # Linear examples # # Load trade data data(trade) # We estimate the effect of distance on trade => we account for 3 cluster effects # 1) Poisson estimation est_pois = femlm(Euros ~ log(dist_km)|Origin+Destination+Product, trade) # alternative formulation giving the same results: # est_pois = femlm(Euros ~ log(dist_km), trade, cluster = c("Origin", "Destination", "Product")) # 2) Log-Log Gaussian estimation est_gaus = femlm(log(Euros+1) ~ log(dist_km)|Origin+Destination+Product, trade, family="gaussian") # 3) Negative Binomial estimation est_nb = femlm(Euros ~ log(dist_km)|Origin+Destination+Product, trade, family="negbin") # Comparison of the results using the function res2table res2table(est_pois, est_gaus, est_nb) # Now using two way clustered standard-errors res2table(est_pois, est_gaus, est_nb, se = "twoway") # Comparing different types of standard errors sum_white = summary(est_pois, se = "white") sum_oneway = summary(est_pois, se = "cluster") sum_twoway = summary(est_pois, se = "twoway") sum_threeway = summary(est_pois, se = "threeway") res2table(sum_white, sum_oneway, sum_twoway, sum_threeway) # # Non-linear examples # # Generating data for a simple example n = 100 x = rnorm(n, 1, 5)**2 y = rnorm(n, -1, 5)**2 z = rpois(n, x*y) + rpois(n, 2) base = data.frame(x, y, z) # Comparing the results of a 'linear' function using a 'non-linear' call est0L = femlm(z~log(x)+log(y), base) est0NL = femlm(z~1, base, NL.fml = ~a*log(x)+b*log(y), start = list(a=0, b=0)) # we compare the estimates with the function res2table res2table(est0L, est0NL) # Generating a non-linear relation z2 = rpois(n, x + y) + rpois(n, 1) base\$z2 = z2 # Using a non-linear form est1NL = femlm(z2~0, base, NL.fml = ~log(a*x + b*y), start = list(a=1, b=2), lower = list(a=0, b=0)) # we can't estimate this relation linearily # => closest we can do: est1L = femlm(z2~log(x)+log(y), base) res2table(est1L, est1NL) # Using a custom Jacobian for the function log(a*x + b*y) myGrad = function(a,x,b,y){ # Custom Jacobian s = a*x+b*y data.frame(a = x/s, b = y/s) } est1NL_grad = femlm(z2~0, base, NL.fml = ~log(a*x + b*y), start = list(a=1,b=2), nl.gradient = ~myGrad(a,x,b,y))