Appendix C — Formula Sheet

C.1 Purpose

This appendix collects the formulas used in the Post-Soviet replication and in the final region-specific gravity paper. Use it to keep notation consistent across notebooks, tables, and the methodology section.

C.2 Notation

Symbol	Meaning
\(i\)	Exporter
\(j\)	Importer
\(t\)	Year
\(X_{ijt}\)	Bilateral exports or trade flow
\(Y_{it}\)	Exporter economic size
\(Y_{jt}\)	Importer economic size
\(D_{ij}\)	Bilateral distance or trade-cost proxy
\(Z_{ijt}\)	Additional bilateral and institutional controls
\(\alpha_i\), \(\delta_j\), \(\lambda_t\)	Exporter, importer, and year fixed effects
\(\mu_{ij}\)	Country-pair fixed effect
\(\alpha_{it}\), \(\delta_{jt}\)	Exporter-year and importer-year fixed effects

In the Post-Soviet project, \(X_{ijt}\) is measured by \(flow\), economic size by \(gdp\_o\) and \(gdp\_d\), distance by \(distw\), and institutional variables by \(wto\_joint\), \(EU\_joint\), and \(EAEU\_joint\).

C.3 Basic Gravity

The intuitive gravity equation is:

\[ X_{ij} = G Y_i^{\beta_1} Y_j^{\beta_2} D_{ij}^{\beta_3} \]

If distance is expected to reduce trade, \(\beta_3\) should be negative. Some textbooks write distance in the denominator; both forms express the same intuition when the distance coefficient is signed correctly.

C.4 Log-Linear Gravity

The baseline empirical model is:

\[ \begin{aligned} \log X_{ijt} &= \beta_0 + \beta_1 \log Y_{it} + \beta_2 \log Y_{jt} \\ &\quad + \beta_3 \log D_{ij} + \gamma Z_{ijt} + \varepsilon_{ijt} \end{aligned} \]

This model is easy to estimate with OLS and easy to interpret. It should be treated as a teaching baseline, not the final word, because it excludes zero trade flows and can be biased under heteroskedasticity.

C.5 Post-Soviet OLS Specification

The course replication begins with:

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

Logged GDP coefficients are elasticity-like. Dummy coefficients should be converted to percent effects using the formula below.

C.6 Structural Gravity

A compact structural gravity expression is:

\[ \begin{aligned} X_{ij} &= \frac{Y_i E_j}{Y} \left( \frac{\tau_{ij}}{\Pi_i P_j} \right)^{1-\sigma} \end{aligned} \]

Here, \(\tau_{ij}\) is bilateral trade cost, \(\Pi_i\) is outward multilateral resistance, \(P_j\) is inward multilateral resistance, and \(\sigma\) is the elasticity of substitution. The key implication is practical: bilateral trade depends on relative trade costs, not only the direct cost between two countries.

C.7 Fixed-Effects Gravity

A standard fixed-effects version is:

\[ \begin{aligned} \log X_{ijt} &= \beta_1 \log D_{ij} + \gamma Z_{ijt} + \alpha_i + \delta_j + \lambda_t + \varepsilon_{ijt} \end{aligned} \]

Exporter, importer, and year fixed effects absorb broad country and time differences. They make coefficient interpretation more conditional.

C.8 Pair Fixed Effects

Pair fixed effects compare a country pair with itself over time:

\[ \begin{aligned} \log X_{ijt} &= \gamma_1 wto\_joint_{ijt} + \gamma_2 EU\_joint_{ijt} + \gamma_3 EAEU\_joint_{ijt} \\ &\quad + \mu_{ij} + \lambda_t + \varepsilon_{ijt} \end{aligned} \]

Time-invariant variables such as \(distw\), \(comlang\_off\), and \(contig\) are absorbed by \(\mu_{ij}\) and cannot be estimated separately.

C.9 Structural Fixed Effects

A saturated structural gravity specification often uses exporter-year and importer-year fixed effects:

\[ \begin{aligned} E[X_{ijt} \mid Z_{ijt}] &= \exp\left( \gamma Z_{ijt} + \mu_{ij} + \alpha_{it} + \delta_{jt} \right) \end{aligned} \]

This structure absorbs exporter-year supply conditions, importer-year demand conditions, and time-varying multilateral resistance. GDP variables are usually absorbed.

C.10 Double Demeaning

For a variable \(x_{ij}\) with exporter and importer groups:

\[ \tilde{x}_{ij} = x_{ij} - \bar{x}_{i\cdot} - \bar{x}_{\cdot j} + \bar{x} \]

The transformed variable removes exporter and importer means. Double demeaning is useful for teaching fixed-effects logic because it shows how group averages are removed before estimation.

C.11 BVU

Bonus Vetus OLS uses normalized gravity logic to approximate structural trade-cost effects. A simple teaching version uses trade intensity:

\[ \begin{aligned} \log TI_{ijt} &= \log(flow_{ijt}) - \log(gdp\_o_{it}) - \log(gdp\_d_{jt}) \end{aligned} \]

Then estimate:

\[ \begin{aligned} \log TI_{ijt} &= \beta_0 + \beta_1 \log(distw_{ij}) + \gamma Z_{ijt} + \varepsilon_{ijt} \end{aligned} \]

This is a teaching approximation. Publication work should verify the exact Bonus Vetus transformation against the intended reference implementation.

C.12 BVW

BVW extends the normalization logic by using GDP-weighted averages to approximate multilateral resistance:

\[ \begin{aligned} x^{BVW}_{ijt} &= x_{ijt} - \sum_j s_{jt} x_{ijt} - \sum_i s_{it} x_{ijt} \\ &\quad + \sum_i \sum_j s_{it}s_{jt} x_{ijt} \end{aligned} \]

Here, \(s_{it}\) and \(s_{jt}\) are GDP shares. The exact implementation is more involved than this compact expression, so students should treat classroom BVW code as a transparent learning tool.

C.13 PPML

PPML estimates trade in levels:

\[ \begin{aligned} E[X_{ijt} \mid Z_{ijt}] &= \exp\left( \beta_0 + \beta_1 \log D_{ij} + \gamma Z_{ijt} \right) \end{aligned} \]

PPML can include valid zero trade flows and is robust to important heteroskedasticity problems in log-linear gravity models. It is estimated with a Poisson conditional mean but does not require trade to be count data.

C.14 GPML

GPML uses a Gamma conditional mean with a log link:

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

GPML is a positive-flow estimator. It is useful as a robustness check, not as a replacement for explaining zero-flow treatment.

C.15 NBPML

NBPML keeps the multiplicative conditional mean but allows a negative-binomial variance structure:

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

Use NBPML as a robustness estimator when overdispersion is substantively important and convergence is reliable.

C.16 Dummy Interpretation

For a dummy coefficient \(\hat{\gamma}\) in a log-link model:

\[ \% \Delta X = 100 \times \left[ \exp(\hat{\gamma}) - 1 \right] \]

Example interpretation template: joint membership in [institution] is associated with a conditional trade difference of \(100 \times [\exp(\hat{\gamma}) - 1]\) percent, holding the model controls fixed.

C.17 Elasticity Interpretation

When both dependent and independent variables are logged:

\[ \frac{\partial \log X}{\partial \log D} = \beta_D \]

A 1 percent increase in distance is associated with an approximate \(\beta_D\) percent change in trade. Because distance coefficients are usually negative, larger distance is expected to reduce trade.

C.18 Reporting Rules

State the dependent variable exactly.
State whether zero flows are included.
State which fixed effects are absorbed.
Do not report absorbed variables as if they were omitted accidentally.
Convert dummy coefficients with \(100 \times [\exp(\hat{\gamma}) - 1]\).
Use the same notation in tables, equations, and text.