8  OLS Gravity

8.1 Purpose

This chapter develops the baseline log-linear gravity model used as the first econometric step in the Post-Soviet replication. It explains the theory, the R workflow, the Python workflow, and the interpretation of coefficients.

8.2 Post-Soviet variables

The running model uses:

Role Variable
Trade flow flow
Exporter GDP gdp_o
Importer GDP gdp_d
Weighted distance distw
Common official language comlang_off
Contiguity contig
Joint WTO status wto_joint
Joint EU status EU_joint
Joint EAEU status EAEU_joint
Year year

No numerical result should be reported until the replication dataset is loaded and the model is estimated.

8.3 Theory

The gravity model starts from a stable empirical regularity: larger economies trade more, and more distant economies trade less. In the Post-Soviet case, the model asks whether institutional relationships such as WTO, EU, or EAEU joint status are associated with bilateral trade after controlling for economic mass and basic trade costs.

The multiplicative gravity idea is:

\[ Flow_{ijt} = A GDP_{it}^{\beta_1} GDP_{jt}^{\beta_2} Distw_{ij}^{\beta_3} \exp(\gamma Z_{ijt}) \eta_{ijt} \]

where \(Z_{ijt}\) includes comlang_off, contig, wto_joint, EU_joint, and EAEU_joint.

8.4 Equation

The baseline log-linear model is:

\[ \begin{aligned} \log(flow_{ijt}) &= \beta_0 + \beta_1 \log(gdp\_o_{it}) + \beta_2 \log(gdp\_d_{jt}) \\ &\quad + \beta_3 \log(distw_{ij}) + \gamma_1 comlang\_off_{ij} + \gamma_2 contig_{ij} \\ &\quad + \gamma_3 wto\_joint_{ijt} + \gamma_4 EU\_joint_{ijt} + \gamma_5 EAEU\_joint_{ijt} \\ &\quad + \delta_t + \varepsilon_{ijt} \end{aligned} \]

The year effect \(\delta_t\) controls for common shocks in a given year.

8.5 Why OLS exists

OLS is the simplest way to estimate the log-linear gravity equation. It is useful for teaching, replication diagnostics, and comparison with older gravity workflows. It also gives direct elasticity interpretations for logged variables.

8.6 Advantages

  • Transparent and easy to reproduce.
  • Useful for checking signs and magnitudes.
  • Coefficients on logged variables are elasticities.
  • Dummy variables can be converted into approximate percent differences.
  • R and Python workflows are easy to compare.

8.7 Limitations

  • log(flow) is undefined when flow == 0.
  • Dropping zero flows can change the sample.
  • Log-linear OLS can be biased under heteroskedasticity.
  • OLS is not usually the preferred final estimator for modern gravity when zeros and multiplicative errors matter.
WarningZero trade flows

OLS with log(flow) uses positive trade flows only. Do not silently drop zero flows without reporting how many observations are removed.

8.8 R implementation

The R workflow usually creates logged variables, filters positive trade flows, and estimates a formula model.

library(dplyr)
library(fixest)

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

ols_df <- df %>%
  filter(flow > 0) %>%
  mutate(
    log_flow = log(flow),
    log_gdp_o = log(gdp_o),
    log_gdp_d = log(gdp_d),
    log_distw = log(distw)
  )

ols_r <- feols(
  log_flow ~ log_gdp_o + log_gdp_d + log_distw +
    comlang_off + contig + wto_joint + EU_joint + EAEU_joint |
    year,
  data = ols_df,
  vcov = "hetero"
)

summary(ols_r)

The R code above is a workflow template. Students should match the exact sample and variable definitions used in the Post-Soviet replication before comparing estimates.

8.9 Python implementation

The Python workflow uses the same sample logic and model variables.

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf
df = pd.read_csv("data/gravity_clean.csv")

ols_df = df.loc[df["flow"] > 0].copy()
ols_df["log_flow"] = np.log(ols_df["flow"])
ols_df["log_gdp_o"] = np.log(ols_df["gdp_o"])
ols_df["log_gdp_d"] = np.log(ols_df["gdp_d"])
ols_df["log_distw"] = np.log(ols_df["distw"])
ols_formula = (
    "log_flow ~ log_gdp_o + log_gdp_d + log_distw + "
    "comlang_off + contig + wto_joint + EU_joint + EAEU_joint + C(year)"
)

ols_py = smf.ols(ols_formula, data=ols_df).fit(cov_type="HC1")

print(ols_py.summary())

8.10 Coefficient interpretation

For logged continuous variables:

  • log_gdp_o: exporter GDP elasticity.
  • log_gdp_d: importer GDP elasticity.
  • log_distw: distance elasticity.

If the coefficient on log_distw is \(\beta_3\), a 1 percent increase in weighted distance is associated with approximately \(\beta_3\) percent change in trade, conditional on the model.

For dummy variables such as wto_joint, EU_joint, and EAEU_joint, use:

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

for variable in ["wto_joint", "EU_joint", "EAEU_joint", "comlang_off", "contig"]:
    if variable in ols_py.params.index:
        beta = ols_py.params[variable]
        percent = 100 * (np.exp(beta) - 1)
        print(variable, percent)

Do not interpret these transformed coefficients as causal effects unless the research design supports a causal claim.

8.11 Replication checks

Before moving to fixed effects, record:

  • number of rows in the raw dataset;
  • number of rows with flow > 0;
  • variable definitions used for logs;
  • whether year effects are included;
  • whether robust standard errors are used;
  • whether the model matches the Post-Soviet manuscript specification.

8.12 Research output

Produce an OLS replication table shell and a short note explaining the positive-flow sample, the model equation, and the interpretation of the institutional coefficients.