11 PPML

11.1 Purpose

This chapter explains zero-inclusive and multiplicative gravity estimators for the Post-Soviet replication. It covers PPML, GPML, NBPML, and structural PPML with exporter-year and importer-year fixed effects.

11.2 Theory

Modern gravity models are multiplicative. Estimating a log-linear model with OLS changes the model and drops zero trade flows. PML estimators keep flow in levels and specify a conditional mean.

The general conditional-mean form is:

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp(X_{ijt}\beta) \end{aligned} \]

where \(X_{ijt}\) includes log(distw), comlang_off, contig, wto_joint, EU_joint, EAEU_joint, year, and fixed effects when required.

11.3 Why PML estimators exist

PML estimators exist because trade data are skewed, heteroskedastic, and often include zero flows. They allow students to estimate gravity models without transforming the dependent variable into log(flow).

11.4 Shared setup

import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf

df = pd.read_csv("data/gravity_clean.csv")

df = df.loc[df["flow"].notna()].copy()
df["log_gdp_o"] = np.log(df["gdp_o"])
df["log_gdp_d"] = np.log(df["gdp_d"])
df["log_distw"] = np.log(df["distw"])

The examples below do not fabricate fitted values or results. They show how the Post-Soviet variables enter each estimator.

11.5 PPML

11.5.1 Theory

PPML estimates the exponential conditional mean of trade in levels. It is widely used in gravity because it can keep valid zero trade flows and is robust to important heteroskedasticity concerns.

11.5.2 Equation

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp\left( \beta_0 + \beta_1 \log(gdp\_o_{it}) + \beta_2 \log(gdp\_d_{jt}) \right.\\ &\quad \left. + \beta_3 \log(distw_{ij}) + \gamma Z_{ijt} + \delta_t \right) \end{aligned} \]

11.5.3 Advantages

Keeps flow == 0 when zeros are valid observations.
Estimates the multiplicative gravity model directly.
Coefficients can be interpreted through semi-elasticities.
Works naturally with fixed effects in structural gravity.

11.5.4 Limitations

Can be computationally heavy with many fixed effects.
Sensitive to separation and convergence issues.
Requires careful checking of zeros and missing values.
Standard statsmodels dummy expansion may not scale to very large panels.

11.5.5 R implementation

library(fixest)

df <- read.csv("data/gravity_clean.csv")

ppml_r <- fepois(
  flow ~ log(distw) + comlang_off + contig +
    wto_joint + EU_joint + EAEU_joint |
    year,
  data = df,
  vcov = "hetero"
)

summary(ppml_r)

11.5.6 Python implementation

ppml_formula = (
    "flow ~ log_gdp_o + log_gdp_d + log_distw + "
    "comlang_off + contig + wto_joint + EU_joint + EAEU_joint + C(year)"
)

ppml_py = smf.glm(
    formula=ppml_formula,
    data=df,
    family=sm.families.Poisson(),
).fit(cov_type="HC1")

11.5.7 Interpretation

For continuous logged variables, coefficients are elasticity-like. For dummy variables:

\[ 100 \times \left(\exp(\gamma) - 1\right) \]

is the conditional percent difference associated with the dummy.

11.6 GPML

11.6.1 Theory

Gamma PML estimates a positive continuous dependent variable with a log link. It is sometimes used as an alternative multiplicative estimator when the dependent variable is strictly positive.

11.6.2 Equation

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp(X_{ijt}\beta), \quad flow_{ijt} > 0 \end{aligned} \]

11.6.3 Why it exists

GPML exists as an alternative quasi-likelihood for positive continuous outcomes. In gravity work, it can be compared with PPML when the sample excludes zero flows.

11.6.4 Advantages

Models positive continuous trade values in levels.
Uses a log link, preserving multiplicative interpretation.
Useful as a robustness comparison.

11.6.5 Limitations

Cannot use zero trade flows directly.
Not the default modern gravity estimator.
Results can differ from PPML because the variance assumption differs.

11.6.6 R implementation

gpml_df <- subset(df, flow > 0)

gpml_r <- glm(
  flow ~ log(gdp_o) + log(gdp_d) + log(distw) +
    comlang_off + contig + wto_joint + EU_joint + EAEU_joint +
    factor(year),
  data = gpml_df,
  family = Gamma(link = "log")
)

11.6.7 Python implementation

gpml_df = df.loc[df["flow"] > 0].copy()

gpml_formula = (
    "flow ~ log_gdp_o + log_gdp_d + log_distw + "
    "comlang_off + contig + wto_joint + EU_joint + EAEU_joint + C(year)"
)

gpml_py = smf.glm(
    formula=gpml_formula,
    data=gpml_df,
    family=sm.families.Gamma(link=sm.families.links.Log()),
).fit(cov_type="HC1")

11.6.8 Interpretation

Interpret dummy and continuous coefficients using the same exponential logic as PPML, but state that GPML uses a positive-flow sample.

11.7 NBPML

11.7.1 Theory

Negative binomial PML is another multiplicative estimator. It allows overdispersion relative to Poisson. In trade applications, it can be used as a robustness estimator when variance appears much larger than the mean.

11.7.2 Equation

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp(X_{ijt}\beta) \end{aligned} \]

with a negative-binomial variance structure.

11.7.3 Why it exists

NBPML exists to relax the strict Poisson variance relationship while retaining a log-link conditional mean.

11.7.4 Advantages

Allows overdispersion.
Keeps level-based interpretation.
Can include zero flows.
Useful as a robustness comparison.

11.7.5 Limitations

Can be harder to estimate and interpret.
May be sensitive to convergence and dispersion assumptions.
Not a substitute for thinking carefully about zeros, fixed effects, and model design.

11.7.6 R implementation

library(MASS)

nbpml_r <- glm.nb(
  flow ~ log(gdp_o) + log(gdp_d) + log(distw) +
    comlang_off + contig + wto_joint + EU_joint + EAEU_joint +
    factor(year),
  data = df
)

11.7.7 Python implementation

nbpml_formula = (
    "flow ~ log_gdp_o + log_gdp_d + log_distw + "
    "comlang_off + contig + wto_joint + EU_joint + EAEU_joint + C(year)"
)

nbpml_py = smf.glm(
    formula=nbpml_formula,
    data=df,
    family=sm.families.NegativeBinomial(),
).fit(cov_type="HC1")

11.7.8 Interpretation

Interpret coefficients through the exponential conditional mean. Report NBPML as a robustness estimator unless the research design gives a specific reason to prefer it.

11.8 Zero-inclusive PPML

11.8.1 Theory

Zero-inclusive PPML keeps observations where flow == 0 as long as they are true observed zero trade flows rather than missing or miscoded values.

11.8.2 Equation

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp(X_{ijt}\beta), \quad flow_{ijt} \ge 0 \end{aligned} \]

11.8.3 Why it exists

Zero-inclusive PPML exists because zero trade flows are economically meaningful in many gravity panels. Dropping them can change the research question.

11.8.4 R implementation

zero_ppml_r <- fepois(
  flow ~ log(distw) + comlang_off + contig +
    wto_joint + EU_joint + EAEU_joint |
    year,
  data = df,
  vcov = "hetero"
)

11.8.5 Python implementation

zero_ppml_df = df.loc[df["flow"] >= 0].copy()

zero_ppml_py = smf.glm(
    formula=ppml_formula,
    data=zero_ppml_df,
    family=sm.families.Poisson(),
).fit(cov_type="HC1")

11.8.6 Interpretation

State clearly whether zeros are included and how many observations enter the model. The key comparison is not only coefficient size but also sample definition.

11.9 Structural PPML with exporter-year and importer-year fixed effects

11.9.1 Theory

Structural PPML absorbs exporter-year and importer-year conditions. This is close to modern structural gravity practice because it controls for time-varying exporter supply, importer demand, and multilateral resistance.

11.9.2 Equation

\[ \begin{aligned} E[flow_{ijt} \mid X_{ijt}] &= \exp\left( \gamma_1 wto\_joint_{ijt} + \gamma_2 EU\_joint_{ijt} + \gamma_3 EAEU\_joint_{ijt} \right.\\ &\quad \left. + \mu_{ij} + \alpha_{it} + \delta_{jt} \right) \end{aligned} \]

where \(\mu_{ij}\) is a pair fixed effect, \(\alpha_{it}\) is an exporter-year fixed effect, and \(\delta_{jt}\) is an importer-year fixed effect.

11.9.3 Why it exists

This model exists to compare institutional changes within pairs while absorbing time-varying exporter and importer conditions.

11.9.4 Advantages

Closely connected to structural gravity.
Keeps zero flows through PPML.
Controls for rich exporter-year and importer-year heterogeneity.
Focuses interpretation on institutional variables that vary over time.

11.9.5 Limitations

Requires exporter and importer identifiers.
Absorbs gdp_o, gdp_d, and many country-year variables.
Can be computationally expensive in ordinary Python dummy expansion.
May require specialized estimators for large panels.

11.9.6 R implementation

df$pair_id <- paste(df$iso3_o, df$iso3_d, sep = "_")

structural_ppml_r <- fepois(
  flow ~ wto_joint + EU_joint + EAEU_joint |
    pair_id + iso3_o^year + iso3_d^year,
  data = df,
  vcov = "hetero"
)

11.9.7 Python implementation

The transparent statsmodels version is useful for teaching but may be slow.

df["pair_id"] = df["iso3_o"] + "_" + df["iso3_d"]

structural_ppml_formula = (
    "flow ~ wto_joint + EU_joint + EAEU_joint + "
    "C(pair_id) + C(iso3_o):C(year) + C(iso3_d):C(year)"
)

structural_ppml_py = smf.glm(
    formula=structural_ppml_formula,
    data=df,
    family=sm.families.Poisson(),
).fit(cov_type="HC1")

11.9.8 Interpretation

In this saturated model, the institutional coefficients are identified from within-pair changes after controlling for exporter-year and importer-year shocks. Do not interpret gdp_o, gdp_d, or distw in this model, because they are absorbed or excluded by design.

11.10 Estimator comparison table

Students should create a comparison table with no invented results:

Estimator	Dependent variable	Zero flows	Key institutional variables	Main caution
OLS	`log(flow)`	Dropped	`wto_joint`, `EU_joint`, `EAEU_joint`	Positive-flow sample only
PPML	`flow`	Included if valid	`wto_joint`, `EU_joint`, `EAEU_joint`	Check convergence and separation
GPML	`flow`	Dropped	`wto_joint`, `EU_joint`, `EAEU_joint`	Positive-flow sample only
NBPML	`flow`	Included if valid	`wto_joint`, `EU_joint`, `EAEU_joint`	Dispersion assumptions
Structural PPML	`flow`	Included if valid	`wto_joint`, `EU_joint`, `EAEU_joint`	High-dimensional fixed effects

11.11 Research output

Prepare a PML replication plan that specifies which observations enter PPML, GPML, NBPML, and structural PPML; which fixed effects are included; and how institutional coefficients will be interpreted.