penppml
is an R package that enables users to fit penalized Poisson Pseudo Maximum Likelihood (PPML) regressions with highdimensional fixed effects (HDFE). Supported penalties in the current version are ridge and lasso. The original application that motivated the development of penppml
was the estimation of threeway gravity models of trade with a large number of PTA provision dummies (Breinlich, Corradi, Rocha, Ruta, Santos Silva and Zylkin, 2021).
To install and load penppml
, you can use the following commands:
install.packages(penppml)
library(penppml)
The penppml
package features the trade
data set, which integrates panel data on bilateral trade flows with information about specific provisions in trade agreements for 220 exporters and 270 importers. The provisions included in the package are a subset of 16 out of 305 “essential” provisions featured in the full data set. More information about the data set and the variables is included in the corresponding help file, accessible via ?trade
.
exp  imp  time  export  id  agreement  ad_prov_14  cp_prov_23  tbt_prov_07  tbt_prov_33 

ABW  ANT  1988  13087119  0  0  0  0  0  
ABW  ANT  1996  2371576  0  0  0  0  0  
ABW  ARG  1988  3351  0  0  0  0  0  
ABW  ARG  2012  50642  0  0  0  0  0  
ABW  ATG  1988  558  0  0  0  0  0  
ABW  ATG  1996  132070  0  0  0  0  0  
ABW  AUS  1996  1129  0  0  0  0  0  
ABW  BHS  1988  1117  0  0  0  0  0  
ABW  BHS  1996  587  0  0  0  0  0  
ABW  BHS  2012  830249  0  0  0  0  0 
Along with the trade
data set, the package includes an auxiliary data frame, countries
, which contains basic information about the country ISO codes included in the main data set.
iso  name  region  sub.region 

AFG  Afghanistan  Asia  Southern Asia 
ALA  Islands  Europe  Northern Europe 
ALB  Albania  Europe  Southern Europe 
DZA  Algeria  Africa  Northern Africa 
ASM  American Samoa  Oceania  Polynesia 
AND  Andorra  Europe  Southern Europe 
AGO  Angola  Africa  SubSaharan Africa 
AIA  Anguilla  Americas  Latin America and the Caribbean 
ATA  Antarctica  
ATG  Antigua and Barbuda  Americas  Latin America and the Caribbean 
This enables users to easily filter by region or subregion. For instance, if we want to restrict our analysis to countries in the Americas, we can do the following:
< countries$iso[countries$region %in% c("Americas")]
selected < trade[(trade$exp %in% selected) & (trade$imp %in% selected), (5:6)] # We remove columns 5 and
trade2 # 6 because these variables are not needed in our regressions.
We will show the capabilities of this package using the filtered trade2
data frame.
The package enables users to run unpenalized PPML regressions with HDFE, using the hdfeppml
function as follows:
< hdfeppml(data = trade2,
reg1 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")))
#> User did not specify independent variables. By default, the function takes all variables
#> not included in 'dep' or 'fixed' as regressors.
As we can see, the function is designed to be dataframefriendly: the user sets a data frame of reference in the data
argument and then picks the dependent, independent and fixed effect variables either by name or by column number within the provided data frame. Also, note the following points about the syntax:
The fixed
argument can take either a single vector (just like the dep
argument; each element will be used as a separate fixed effect) or a list of vectors. The latter option is useful in cases where the desired fixed effect is the result of the interaction of two or more variables, as in the gravity model of trade. In those cases, you can specify any number of separate fixed effects in distinct elements of the list and, inside each element, which variables in the data set you want to interact.
As explained in the note, if indep
is empty, the function uses all remaining columns in the data frame by default.
Internally, the function will do the necessary transformations of the columns into vectors and matrices, as needed by the algorithm.
Alternatively, more advanced users who prefer to handle data transformations themselves can use the internal function hdfeppml_int
. For more information and examples on this issue, run ?hdfeppml
or ?hdfeppml_int
. Note also that this applies to all of the main functions in our package: both the dataframefriendly wrapper and the internal function are available for use.
The results of our PPML model are:
< data.frame(prov = rownames(reg1$coefficients), b = reg1$coefficients, se = 0)
results $se[!is.na(reg1$coefficients)] < reg1$se
results results


(Note that the functions is automatically dropping perfectly collinear variables from the estimation algorithm and reporting NA
as the coefficient.)
The mlfitppml
function is a flexible tool for computing penalized PPML regressions with HDFE. For instance, if we want to fit a PPML regression with a lasso penalty for several values of the penalty parameter (lambda) at once, we can run:
< c(0.05, 0.025, 0.01, 0.0075, 0.005, 0.0025, 0.001, 0.00075, 0.0005, 0.00025, 0.0001, 0)
lambdas
< mlfitppml(data = trade2,
reg2 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")),
penalty = "lasso",
lambdas = lambdas)
The function returns a list that contains two sets of coefficients for each value of lambda: the penalized coefficients (in the beta_pre
element) and the postpenalty or unpenalized coefficients (in the beta
element; these are calculated by estimating a postpenalty regression with just the selected variables). We can plot the regularization path of the penalized coefficients as follows:
< as.data.frame(reg2$beta_pre)
results names(results) < lambdas
$provision < row.names(results)
results< reshape2::melt(results, id.vars = "provision", variable.name = "lambda",
results value.name = "coefficient")
$lambda < as.numeric(as.character(results$lambda))
results
::ggplot(data = results, mapping = ggplot2::aes(x = lambda, y = coefficient, col = provision)) +
ggplot2::geom_line(show.legend = FALSE) +
ggplot2::scale_x_reverse(expand = ggplot2::expansion(add = c(0, 0.015))) +
ggplot2::theme_classic() +
ggplot2::geom_dl(ggplot2::aes(label = provision),
directlabelsmethod = list(directlabels::dl.trans(x = x + 0.5), "last.bumpup")) +
::labs(x = "Penalty parameter (lambda)", y = "Coefficient",
ggplot2title = "Figure 1: Regularization path for lasso")
If the user wishes to obtain the penalized estimates for a single value of lambda, they can either use mlfitppml
as described above (just setting lambdas == x
, where x
is a number) or use the penhdfeppml
function, upon which mlfitppml
is built, directly. For instance:
< penhdfeppml(data = trade2,
reg3 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")),
penalty = "lasso",
lambda = 0.005)
We can easily check that the penalized coefficient estimates of penhdfeppml
and mlfitppml
are equal for a lambda value of 0.005 (within a numerical tolerance):
all.equal(as.vector(reg3$beta[!is.na(reg3$beta)]), as.vector(reg2$beta_pre[, 5]), tol = 1e05)
#> [1] TRUE
For more details on penhdfeppml
, run ?penhdfeppml
.
mlfitppml
also allows user to use the ridge penalty in their PPML regressions. Simply run:
< seq(0.0001, 0, length.out = 10)
lambdas
< mlfitppml(data = trade2,
reg4 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")),
penalty = "ridge",
lambdas = lambdas)
Note that this feature is still in development and may contain bugs.
The mlfitppml
function enables users to carry out crossvalidation of their models via the xval
and IDs
arguments. When xval
is set to TRUE
, the function performs crossvalidation using a userprovided vector of IDs. For instance, if we want to do kfold cross validation with k = 20, splitting the data set by agreement (not by observation), we can do the following:
< unique(trade[(trade$exp %in% selected) & (trade$imp %in% selected), 5])
id < 20
nfolds < data.frame(id = id, fold = sample(1:nfolds, size = length(id), replace = TRUE))
unique_ids
< merge(trade[(trade$exp %in% selected) & (trade$imp %in% selected), 5, drop = FALSE],
cross_ids by = "id", all.x = TRUE) unique_ids,
Now we activate the crossvalidation option in mlfitppml
and input the ID vector:
< mlfitppml(data = trade2,
reg5 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")),
penalty = "lasso",
lambdas = c(seq(0.5, 0.1, by = 0.1), 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001, 0),
xval = TRUE,
IDs = cross_ids$fold)
Now the function also returns a rmse
element that includes the crossvalidation results (the average RMSE or root mean squared error for each value of lambda). Users can employ this tool to choose the value of the penalty parameter that minimizes the RMSE:
$rmse reg5
Penalty (lambda)  RMSE 

5e01  108.5813 
4e01  108.5813 
3e01  108.5813 
2e01  108.5813 
1e01  108.5813 
5e02  108.5813 
1e02  108.5750 
5e03  108.5743 
1e03  108.5740 
5e04  108.5739 
1e04  108.5739 
0e+00  108.5739 
In this case, the RMSE criterion suggest that the penalty should be set at 0. In other words, the results in Table 3 are the ones that minimize the mean squared error of the regression, according to 20fold crossvalidation.
This package also enables the use of the plugin lasso method, incorporating coefficientspecific penalty weights calculated automatically by the package. The most convenient way to do this is to set method = "plugin"
in mlfitppml
. Note that the plugin algorithm requires a clustering variable  we are using the interaction of the exporter and importer variables in the trade
data set:
< mlfitppml(data = trade2,
reg6 dep = "export",
fixed = list(c("exp", "time"),
c("imp", "time"),
c("exp", "imp")),
penalty = "lasso",
method = "plugin",
cluster = c("exp", "imp"))
< data.frame(prov = rownames(reg6$beta), b_pre = reg6$beta_pre, b = reg6$beta, se = 0)
results $se[!is.na(reg6$beta)] < reg6$ses
results results


We can see that the plugin lasso is very strict: only two provisions have nonzero coefficients when the penalties are set at the level suggested by the plugin algorithm. Compare this to the crossvalidation results, which didn’t remove any of the provisions.
This package also allows users to implement the twostep lasso described in Breinlich et al. (2021). This method consists in, first, running a plugin lasso estimation and second, running individual lasso regressions on each of the variables selected in the first stage. The iceberg
function is designed precisely with this second step in mind. It takes a vector of dependent variables and returns a lasso regression for each one of them:
< iceberg(data = trade2[, (1:4)],
iceberg_results dep = results$prov[results$b != 0],
selectobs = (trade2$time == "2016"))
Currently, the function returns a matrix with coefficient estimates for each of the selected provisions. Support for standard errors (including clustered) is in development as of the current version of the package. The iceberg lasso coefficients are:
iceberg_results
tf_prov_45  inv_prov_22  

ad_prov_14  0.0000  0.0000 
cp_prov_23  0.5912  0.0000 
tbt_prov_07  0.0961  0.0000 
tbt_prov_33  0.0000  0.0000 
tf_prov_41  0.2144  0.0000 
ser_prov_47  0.0000  0.5083 
et_prov_38  0.0000  0.3315 
ipr_prov_44  0.0000  0.0000 
env_prov_18  0.0000  0.0000 
ipr_prov_15  0.0000  0.0000 
moc_prov_21  0.0000  0.0000 
ste_prov_30  0.0580  0.0668 
lm_prov_10  0.0000  0.0000 
cp_prov_26  0.0334  0.0040 
Since PPML coefficients can’t be easily interpreted, you may find it useful to see the raw correlations between the variables in the iceberg lasso step:
< cor(trade2[, results$prov])
provcorr < provcorr[, results$prov[results$b != 0]]) (provcorr
tf_prov_45  inv_prov_22  

ad_prov_14  0.1580  0.2260 
cp_prov_23  0.9691  0.1612 
tbt_prov_07  0.7023  0.3039 
tbt_prov_33  0.1813  0.1318 
tf_prov_41  0.3306  0.0285 
tf_prov_45  1.0000  0.1469 
ser_prov_47  0.0605  0.6679 
inv_prov_22  0.1469  1.0000 
et_prov_38  0.0771  0.5961 
env_prov_18  0.0065  0.0752 
moc_prov_21  0.0784  0.6007 
ste_prov_30  0.7716  0.5569 
lm_prov_10  0.0231  0.0752 
cp_prov_26  0.7975  0.2416 
Breinlich, H., Corradi, V., Rocha, N., Ruta, M., Santos Silva, J.M.C. and T. Zylkin, T. (2021). “Machine Learning in International Trade Research: Evaluating the Impact of Trade Agreements”, Policy Research Working Paper; No. 9629. World Bank, Washington, DC.
Correia, S., P. Guimaraes and T. Zylkin (2020). “Fast Poisson estimation with high dimensional fixed effects”, STATA Journal, 20, 90115.
Gaure, S (2013). “OLS with multiple high dimensional category variables”, Computational Statistics & Data Analysis, 66, 818.
Friedman, J., T. Hastie, and R. Tibshirani (2010). “Regularization paths for generalized linear models via coordinate descent”, Journal of Statistical Software, 33, 122.
Belloni, A., V. Chernozhukov, C. Hansen and D. Kozbur (2016). “Inference in high dimensional panel models with an application to gun control”, Journal of Business & Economic Statistics, 34, 590605.