Spatial gravity model for international trade: a panel analysis among CIS member

Spatial gravity model for international trade: a panel
analysis among CIS member states
Model Setting and methods
The first theoretical – based gravity model derived by Anderson (1979) results from the equality of supply of export and import demand in the case of Constant Elasticity of Substitution (CES) expenditure function and Constant Elasticity of Trade (CET) profit equation is generically represented by the following equation:
Y_ij＝(X_i^(β^o )*E_j^(β^d ))/(d_ij^θ ) (1)
This is the simplified form of the more inclusive model in equation 2, if P_j^(1-σ) and Π_i are equal.
Y_ij＝(X_i^(β^o )*E_j^(β^d ))/(d_ij^θ*P_j^(1-σ)*Π_i ) (2)

where Y_ij is the trade flow from country i to country j, X_i and E_j represent the economic size of the origin country and destination country, respectively; d_ij is the distance between the two countries (as a proxy of the transport cost), Π_i measures the market openness for the origin country (exporting goods to other countries) and P_j the market openness of the destination country (importing goods from other countries). Furthermore, β′s and θ′s are the model parameters to be estimated.
A wide variety of model had been specified in the spatial economic literature which can be adapted for the analysis of international trade with the use of the gravity model. The main models, proposed by Anselin (1988) for cross sectional specification are called SAR (spatial autoregressive model) and SEM (spatial error model). It is possible to derive each of those econometric models starting from an economic motivation. The motivation for the SAR model comes from viewing spatial dependence as a long-run equilibrium of an underlying spatio-temporal process; the motivation for the SEM model asserts that omitted variables that exhibit spatial dependence led to a model with spatial lags of both the explanatory and the dependent variable.
Referring to the first motivation, we can think to the spatial dependence as based on a time-lag relationship describing a diffusion process over space. SAR models contain spatial lags of the dependent variable. The second econometric motivation leads us to a model with a spatial lag of both the explanatory and the dependent variables. However, if the included and excluded explanatory variables are not correlated, a spatial error model (SEM) emerges.
Taking into account the spatial dependence (or, in other words, to represent in formula the spatial lags) consists on defining the structure of the spatial components, that involves the weights matrix. In a typical cross-sectional model with n countries and one observation per country, the spatial weights matrix that combines neighbors, called W, is a n-by-n dimension, non-negative and sparse. For the elements of this matrix hold that w_ij˃0, and, by convention, w_ij＝0, to prevent an observation from being defined as a neighbor to itself.
Given an origin-destination organization of the data, as flow data, we should define a weight matrix of order n^2 by n^2 that combines each couple of regions having neighboring origin country or neighboring destination country. Given the above defined W, we can define the n^2 by n^2 matrix W_od as the Kronecker product of W with itself, which keep into account the interaction between origin and destination neighbors. We can now define the SAR (equation 3) and the SEM (equation 4) model derivation of the gravity equation:
Y＝ I_n⊗ I_t⊗α_i+I_n⊗ I_t⊗α_j+I_n⊗ I_t⊗α_t+X_o β_o+X_d β_d+D_γ+ρ(W_od⊗I_t )Y+ε (3)
Y＝ I_n⊗ I_t⊗α_i+I_n⊗ I_t⊗α_j+I_n⊗ I_t⊗α_t+X_o β_o+X_d β_d+D_γ+λ(W_od⊗I_t )ε+u (4)
where Y is vector of dimension n*n*t, I_T is a vector of ones of dimension T and I_n is vector of ones dimension n. α_i and α_j are the vector of order n that represent the origin and destination fixed individual effect, and α_t is the T dimension time counterpart. Moreover, X_o and X_d are the matrices of order n*n*t by k, that contain respectively the n*n*t explanatory variables of the origin and destination countries. β_o and β_d are the k by 1 vector of coefficients for the explanatory variables. γ is the scalar coefficients relative to D, that is the n*n*t by h matrix with the h variables (like distance) relative to the interaction between the origin and destination countries. Moreover, ε in equation 3 is the n*n*t vector of the residuals, that is assumed to be a stochastic i.i.d. variable with zero mean and common variance σ_ε^2. ρ is the coefficient for the spatial dependence captured by the spatially lagged dependent variable, W_od is defined above and I_t is a t by t identity matrix that account for the time index. The term u in equation 4 is i.i.d with zero mean and σ_u^2 variance. The symbol ⊗ is Kronecker product. Here, we assume a simplification of the model in 5, derived from filtering the dependent variable from spatial components, that would lead to the following model (as defined in LeSage, Pace, 2008):
Y＝ I_n⊗ I_t⊗α_i+I_n⊗ I_t⊗α_j+I_n⊗ I_t⊗α_t+X_o β_o+X_d β_d+D_γ+〖 ρ〗_1 (W_o⊗I_t )Y+〖 ρ〗_2 (W_d⊗I_t )+ρ(W_od⊗I_t )Y+ ε (5)
The economic motivations that lead to the two different econometric models could be, anyway, complementary, so, we can jointly take these into account. The model that in this case emerges is a SARAR model with spatial lags of both kinds.
Y＝ I_n⊗ I_t⊗α_i+I_n⊗ I_t⊗α_j+I_n⊗ I_t⊗α_t+X_o β_o+X_d β_d+D_γ+ρ(W_od⊗I_t )Y+λ(W_od⊗I_t )Yε+u (6)
Defined the model, we must choose the best estimation method to end up with correct coefficients. Every spatial econometric model suffers from endogeneity problems, since the spatially lagged component is naturally correlated with the dependent variable itself.
To sum up, our empirical model takes the following form:
〖Export〗_ijt＝α_ij+α_t+ β_1^o 〖Pop〗_it^o+β_2^o 〖GDP〗_it^o+β_4^o 〖CISFTA 〗_it^o+β_5^o 〖EAEU 〗_it^o+β_1^d 〖Pop〗_jt^d+β_2^d 〖GDP〗_jt^d+β_4^d 〖CISFTA 〗_jt^d+β_5^d 〖EAEU 〗_jt^d+β_6 〖Migrat〗_ijt+ψ_1^od 〖Contig〗_ij+ψ_2^od 〖Comlang〗_ij+ψ_3^od 〖Comcur〗_ij+ψ_4^od 〖Dist〗_ij+u_ijt (7)
where 〖Export〗_ijt are exports of origin and destination countries, 〖Pop〗_it^o and 〖Pop〗_jt^d are the population of origin and destination countries, 〖GDP〗_it^o and 〖GDP〗_jt^d are the GDP of origin and destination countries, 〖CISFTA 〗_it^o and 〖CISFTA 〗_jt^d are set of dummies to identify the belongs to the CIS free trade agreement for origin and destination countries; 〖EAEU 〗_it^o and 〖EAEU 〗_it^d are set of dummies to identify the belongs to the Eurasian Economic Union, 〖Contig〗_ij are set of dummies which identify if the couple have common
borders, 〖Comlang〗_ij are set of dummies which identify if the couple speak the same language, 〖Comcur〗_ij are a set of dummies which identify if the couple have the same currency, 〖Migrat〗_ijt represents the Stock of immigrants from country i to country j and 〖Dist〗_ij represents the geographical distance between the couple of countries,
We want to model the country-to-country trade flow over the period from 1991 to 2020 for the CIS. The sample was restricted to the 9 member states of the CIS for T＝30 years, resulting in a n*n*T ＝ 2430 observations.
Table 1: List of CIS members
Armenia Moldova
Azerbaijan Russia
Belarus Tajikistan
Kazakhstan Ukraine
Kyrgyzstan Uzbekistan
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