# Penalized PPML Regression with penppml

# 1 Introduction

`penppml` is an R package that enables users to fit penalized Poisson Pseudo Maximum Likelihood (PPML) regressions with high-dimensional 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 three-way 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)

# 2 Data Sets

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`.

Table 1: International Trade Data Set
exp | imp | time | export | id | agreement | ad_prov_14 | cp_prov_23 | tbt_prov_07 | tbt_prov_33
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
ABW | AIA | 1988 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | AIA | 1996 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | AIA | 2012 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | AIA | 2016 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | ANT | 1988 | 13087119 | 0 |  | 0 | 0 | 0 | 0
ABW | ANT | 1996 | 2371576 | 0 |  | 0 | 0 | 0 | 0
ABW | ARE | 1988 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | ARE | 1996 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | ARE | 2012 | 0 | 0 |  | 0 | 0 | 0 | 0
ABW | ARE | 2016 | 0 | 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.

Table 2: Country 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 | Sub-Saharan 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:

    selected <- countries$iso[countries$region %in% c("Americas")]
    trade2 <- trade[(trade$exp %in% selected) & (trade$imp %in% selected), -(5:6)] # We remove columns 5 and
    # 6 because these variables are not needed in our regressions.

We will show the capabilities of this package using the filtered `trade2` data frame.

# 3 Unpenalized PPML estimation with HDFE

The package enables users to run unpenalized PPML regressions with HDFE, using the `hdfeppml` function as follows:

    reg1 <- hdfeppml(data = trade2,
                     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 data-frame-friendly: 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 data-frame-friendly wrapper and the internal function are available for use.

The results of our PPML model are:

    results <- data.frame(prov = rownames(reg1$coefficients), b = reg1$coefficients, se = 0)
    results$se[!is.na(reg1$coefficients)] <- reg1$se
    results

Table 3: Unpenalized PPML results
Provision | Coefficient | SE
--- | --- | ---
ad_prov_14 | 0.0392 | 0.3253
cp_prov_23 | -0.0317 | 0.0514
tbt_prov_07 | 0.3924 | 0.3171
tbt_prov_33 | -0.0484 | 0.3303
tf_prov_41 | 0.6611 | 0.1829
tf_prov_45 | 0.1357 | 0.0618
env_prov_18 | -0.2564 | 0.1340
et_prov_38 | -0.0104 | 0.1276
Provision | Coefficient | SE
--- | --- | ---
inv_prov_22 | 0.3549 | 0.0617
ipr_prov_15 | NA | 0.0000
ipr_prov_44 | NA | 0.0000
lm_prov_10 | -0.5864 | 0.1742
moc_prov_21 | 0.0930 | 0.1492
ser_prov_47 | -0.2656 | 0.1132
ste_prov_30 | 0.0263 | 0.0636
cp_prov_26 | -0.1628 | 0.0854

(Note that the functions is automatically dropping perfectly collinear variables from the estimation algorithm and reporting `NA` as the coefficient.)

# 4 Penalized PPML estimation with HDFE

## 4.1 Lasso regression

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:

    lambdas <- c(0.05, 0.025, 0.01, 0.0075, 0.005, 0.0025, 0.001, 0.00075, 0.0005, 0.00025, 0.0001, 0)

    reg2 <- mlfitppml(data = trade2,
                      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 post-penalty or unpenalized coefficients (in the `beta` element; these are calculated by estimating a post-penalty regression with just the selected variables). We can plot the regularization path of the penalized coefficients as follows:

    results <- as.data.frame(reg2$beta_pre)
    names(results) <- lambdas
    results$provision <- row.names(results)
    results <- reshape2::melt(results, id.vars = "provision", variable.name = "lambda",
                              value.name = "coefficient")
    results$lambda <- as.numeric(as.character(results$lambda))

    ggplot2::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() +
      directlabels::geom_dl(ggplot2::aes(label = provision),
                            method = list(directlabels::dl.trans(x = x + 0.5), "last.bumpup")) +
      ggplot2::labs(x = "Penalty parameter (lambda)", y = "Coefficient",
                    title = "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:

    reg3 <- penhdfeppml(data = trade2,
                      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 = 1e-05)
    #> [1] TRUE

For more details on `penhdfeppml`, run `?penhdfeppml`.

## 4.2 Ridge regression (in development)

`mlfitppml` also allows user to use the ridge penalty in their PPML regressions. Simply run:

    lambdas <- seq(0.0001, 0, length.out = 10)

    reg4 <- mlfitppml(data = trade2,
                      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.

## 4.3 Penalty selection

### 4.3.1 Cross-validation

The `mlfitppml` function enables users to carry out cross-validation of their models via the `xval` and `IDs` arguments. When `xval` is set to `TRUE`, the function performs cross-validation using a user-provided vector of IDs. For instance, if we want to do k-fold cross validation with k = 20, splitting the data set by agreement (not by observation), we can do the following:

    id <- unique(trade[(trade$exp %in% selected) & (trade$imp %in% selected), 5])
    nfolds <- 20
    unique_ids <- data.frame(id = id, fold = sample(1:nfolds, size = length(id), replace = TRUE))

    cross_ids <- merge(trade[(trade$exp %in% selected) & (trade$imp %in% selected), 5, drop = FALSE],
                       unique_ids, by = "id", all.x = TRUE)

Now we activate the cross-validation option in `mlfitppml` and input the ID vector:

    reg5 <- mlfitppml(data = trade2,
                      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 cross-validation 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:

    reg5$rmse

Table 4: Cross-validation results
Penalty (lambda) | RMSE
--- | ---
5e-01 | 108.5813
4e-01 | 108.5813
3e-01 | 108.5813
2e-01 | 108.5813
1e-01 | 108.5813
5e-02 | 108.5813
1e-02 | 108.5750
5e-03 | 108.5743
1e-03 | 108.5740
5e-04 | 108.5739
1e-04 | 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 20-fold cross-validation.

### 4.3.2 Plugin lasso

This package also enables the use of the plugin lasso method, incorporating coefficient-specific 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:

    reg6 <- mlfitppml(data = trade2,
                      dep = "export",
                      fixed = list(c("exp", "time"),
                                  c("imp", "time"),
                                  c("exp", "imp")),
                      penalty = "lasso",
                      method = "plugin",
                      cluster = c("exp", "imp"))

    results <- 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

Table 5: Plugin Lasso results
Provision | Lasso Coefficient | Post-Lasso Coefficient | SE
--- | --- | --- | ---
ad_prov_14 | 0.0205 | 0.4593 | 0.1397
cp_prov_23 | 0.0000 | 0.0000 | NA
tbt_prov_07 | 0.0000 | 0.0000 | NA
tbt_prov_33 | 0.0000 | 0.0000 | NA
tf_prov_41 | 0.0000 | 0.0000 | NA
tf_prov_45 | 0.1403 | 0.2362 | 0.0760
env_prov_18 | 0.0000 | 0.0000 | NA
Provision | Lasso Coefficient | Post-Lasso Coefficient | SE
--- | --- | --- | ---
et_prov_38 | 0.0000 | 0.000 | NA
inv_prov_22 | 0.0082 | 0.161 | 0.0616
lm_prov_10 | 0.0000 | 0.000 | NA
moc_prov_21 | 0.0000 | 0.000 | NA
ser_prov_47 | 0.0000 | 0.000 | NA
ste_prov_30 | 0.0000 | 0.000 | NA
cp_prov_26 | 0.0000 | 0.000 | NA

We can see that the plugin lasso is very strict: only two provisions have non-zero coefficients when the penalties are set at the level suggested by the plugin algorithm. Compare this to the cross-validation results, which didn’t remove any of the provisions.

### 4.3.3 Iceberg lasso (in development)

This package also allows users to implement the two-step 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_results <- iceberg(data = trade2[, -(1:4)],
                               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

Table 6: Iceberg Lasso coefficients
| ad_prov_14 | tf_prov_45 | inv_prov_22
--- | --- | --- | ---
cp_prov_23 | 0.0000 | 0.6143 | 0.0000
tbt_prov_07 | 0.0000 | 0.1002 | 0.0000
tbt_prov_33 | -0.0042 | 0.0000 | 0.0000
tf_prov_41 | 0.0010 | 0.1793 | 0.0000
env_prov_18 | 0.0000 | 0.0000 | 0.0000
et_prov_38 | 0.0000 | 0.0000 | 0.3399
ipr_prov_15 | 0.0000 | 0.0000 | 0.0000
ipr_prov_44 | 0.0000 | 0.0000 | 0.0000
lm_prov_10 | 0.0000 | 0.0000 | 0.0000
moc_prov_21 | 0.0000 | 0.0000 | 0.0000
ser_prov_47 | 0.0000 | 0.0000 | 0.5074
ste_prov_30 | 0.0000 | 0.0489 | 0.0729
cp_prov_26 | -0.0047 | 0.0491 | -0.0191

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:

    provcorr <- cor(trade2[, results$prov])
    (provcorr <- provcorr[, results$prov[results$b != 0]])

Table 7: Iceberg Lasso correlations
| ad_prov_14 | tf_prov_45 | inv_prov_22
--- | --- | --- | ---
ad_prov_14 | 1.0000 | 0.1471 | 0.2265
cp_prov_23 | 0.1466 | 0.9744 | 0.1593
tbt_prov_07 | 0.1096 | 0.7372 | 0.3075
tbt_prov_33 | -0.0121 | 0.1790 | 0.1497
tf_prov_41 | -0.0044 | 0.3072 | -0.0196
tf_prov_45 | 0.1471 | 1.0000 | 0.1464
env_prov_18 | -0.0007 | -0.0048 | 0.0754
et_prov_38 | 0.3522 | 0.0792 | 0.5964
inv_prov_22 | 0.2265 | 0.1464 | 1.0000
lm_prov_10 | -0.0007 | 0.0221 | 0.0754
moc_prov_21 | 0.3547 | 0.0804 | 0.6010
ser_prov_47 | 0.3154 | 0.0652 | 0.6676
ste_prov_30 | 0.1349 | 0.8065 | 0.5273
cp_prov_26 | -0.0149 | 0.8290 | 0.2387

# 5 References

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, 90-115.

Gaure, S (2013). “OLS with multiple high dimensional category variables”, Computational Statistics & Data Analysis, 66, 8-18.

Friedman, J., T. Hastie, and R. Tibshirani (2010). “Regularization paths for generalized linear models via coordinate descent”, Journal of Statistical Software, 33, 1-22.

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, 590-605.