2 The Model

Building on the random utility framework (McFadden Reference McFadden1980), we model an actor $i=1,\ldots ,N$ as having net utility from forming a tie with another actor $j$ : $z_{ij}=\unicode[STIX]{x1D707}_{ij}+\unicode[STIX]{x1D716}_{ij}$ with $\unicode[STIX]{x1D707}_{ij}$ representing the systematic component (that depends on observables) and $\unicode[STIX]{x1D716}_{ij}$ representing the stochastic error. To account for interdependencies in actors’ utilities from having ties, we use the “latent space” approach (Hoff Reference Hoff2005) and model these utilities as follows:

(1) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}z_{ij}\\ z_{ji}\end{array}\right)\sim {\mathcal{N}}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D707}_{ij}+a_{i}+b_{j}+\boldsymbol{u}_{i}^{\prime }\boldsymbol{v}_{j}\\ \unicode[STIX]{x1D707}_{ji}+a_{j}+b_{i}+\boldsymbol{u}_{j}^{\prime }\boldsymbol{v}_{i}\end{array},\left[\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70E}^{2} & \unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}^{2}\\ \unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}^{2} & \unicode[STIX]{x1D70E}^{2}\end{array}\right]\right).\end{eqnarray}$$

The correlation, $\unicode[STIX]{x1D70C}$ , captures the “reciprocity” between the utilities that actors derive from tie formation. Parameters $a_{i}$ and $b_{j}$ are sender- and receiver-specific random effects, respectively, and they capture second-order network dependencies. The vectors $\boldsymbol{u}_{i}$ and $\boldsymbol{v}_{i}$ represent the location of actor $i$ in the latent space of “senders” and “receivers,” respectively. These random effects capture higher-order dependencies in network ties: $i$ derives a large utility from forming a tie with $j$ , if $i$ ’s location in the latent space of “senders” $\boldsymbol{u}_{i}$ is close to $j$ ’s location in the latent space of “receivers” $\boldsymbol{u}_{j}$ (so that the cross product $\boldsymbol{u}_{i}^{\prime }\boldsymbol{v}_{j}$ is large).Footnote ² We express the systematic components of actors’ utilities as linear functions of predictors:

(2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}_{ij}=\unicode[STIX]{x1D737}^{(s)}\boldsymbol{x}_{i}^{(s)}+\unicode[STIX]{x1D737}^{(r)}\boldsymbol{x}_{j}^{(r)}+\unicode[STIX]{x1D737}^{(d)}\boldsymbol{x}_{ij}^{(d)}, & \displaystyle\end{eqnarray}$$

(3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}_{ji}=\unicode[STIX]{x1D737}^{(s)}\boldsymbol{x}_{j}^{(s)}+\unicode[STIX]{x1D737}^{(r)}\boldsymbol{x}_{i}^{(r)}+\unicode[STIX]{x1D737}^{(d)}\boldsymbol{x}_{ji}^{(d)}. & \displaystyle\end{eqnarray}$$

A researcher cannot directly observe net utility (the $z$ ’s). We only observe agents’ behaviors, in this case undirected bilateral ties. A directional tie $i\rightarrow j$ is formed if and only if $i$ ’s net gain from doing so is strictly positive, $z_{ij}>0$ . Accordingly, the bilateral tie $i\leftrightarrow j$ is formed if and only if both actors derive a net positive payoff from having a tie so that $z_{ij}>0$ and $z_{ji}>0$ . A researcher observes an undirected (bilateral) tie $y_{ij}=y_{ji}=\{0,1\}$ arising from the following data generating process:

(4) $$\begin{eqnarray}y_{ij}=y_{ji}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }z_{ij}>0\text{ and }z_{ji}>0,\\ 0\quad & \text{else}.\end{array}\right.\end{eqnarray}$$

Under a standard identifying restriction $\unicode[STIX]{x1D70E}^{2}=1$ , the model is a partially observable probit regression (Poirier Reference Poirier1980), augmented with random effects to capture unobserved heterogeneity and interdyadic dependencies. Vectors $\boldsymbol{x}_{i}^{(s)}$ and $\boldsymbol{x}_{i}^{(r)}$ represent the sender-specific and receiver-specific covariates, respectively. A model for the directional link $i\rightarrow j$ (Equation (2)), uses variables $\boldsymbol{x}_{i}$ as sender-specific predictors, but these same predictors become receiver-specific in the model for the directional link $j\rightarrow i$ (Equation (3)). Vector $\boldsymbol{x}_{ij}^{(d)}$ contains dyad-specific variables. These dyad-specific variables might be symmetric, $x_{ij}=x_{ji}$ (e.g., distance between countries), or not (e.g., export–import).

A partially observed probit model requires at least one of the following identifying restrictions: (a) regression Equations (2) and (3) must have the same parameters and/or (b) one equation contains a predictor not included in another equation (Poirier Reference Poirier1980). If the dyadic predictors are asymmetric, $\boldsymbol{x}_{ij}\neq \boldsymbol{x}_{ji}$ , then condition (2) is satisfied. Furthermore, regression Equations (2) and (3) have the same parameters, and so condition (1) holds as well; thus, the above model is parametrically identified. However, we impose an additional restriction that $\unicode[STIX]{x1D70C}=0$ . While this restriction is not required for parametric identification, Rajbhandari (Reference Rajbhandari, Jeliazkov and Yang2014) showed that finite sample estimates of $\unicode[STIX]{x1D70C}$ are sensitive to the starting values and generally cannot be treated as reliable.Footnote ³

We estimate the model in a Bayesian framework using Markov chain Monte Carlo. In the supplemental materials give a more detailed exposition of the model, prior assumptions, the sampling algorithm. We also provide results from a simulation study demonstrating that the model successfully recovers true parameter values.

3 Application: Bilateral Investment Treaties

We apply the P-GBME model to the network of bilateral investment treaties (BITs) from 1990 to 2012 using the standard United Nations BIT database. There is a vibrant debate on whether BITs boost FDI (Jandhyala, Henisz, and Mansfield Reference Jandhyala, Henisz and Mansfield2011; Simmons Reference Simmons2014; Minhas Reference Minhas2016), but a proper resolution of this debate requires a convincing empirical model of treaty formation (Rosendorff and Shin Reference Rosendorff and Shin2012). Partial observability is one key challenge in building such model: the observed network of signed bilateral treaties is symmetric, while the underlying preferences for these treaties are asymmetric.

We fit the P-GBME model separately for each year of data using a suite of covariates that closely follows the existing empirical literature (see supplementary materials).Footnote ⁴ Our model improves on the previous literature by accounting for both network interdependence and partial observability.

3.2 Regression Parameters

Existing models, including GBME, estimate a single coefficient for each predictor when the outcome is represented as a symmetric matrix. This assumes away the possibility that the same factor differentially “affects” $i$ ’s demand for a treaty with $j$ and $i$ ’s attractiveness to $j$ . The P-GBME recovers directed sender and receiver effects for node-level covariates. For instance, our estimates suggest that, countries with faster growth in GDP per capita were no more inclined to sign BITs with others (sender effect). But high-growth countries were more attractive BIT partners to others (receiver effect). Supplementary materials describes regression parameter estimates in detail.

Figure 1.

Each panels displays the posterior mean probability that China demands and will be demanded as a BIT partner. Country labels in black designate those that had formed a BIT with China by the specified year.

Table 1.

Predictive performance in BIT data: area under the receiver operating characteristic curve (ROC) and area under the precision recall curve (PR).

3.3 The Structure of Latent Treaty Preferences

The P-GBME model allows us to extract “latent preferences” for treaty formation—the estimated probability that country $i$ demands a treaty from $j$ , and vice versa. Figure 1 displays the mean posterior predicted probabilities relevant to China in 1995 and 2010. The horizontal axis represents a country’s attractiveness to China as a BIT partner and the vertical axis is China’s attractiveness to that country. Countries above the diagonal line find China a more attractive BIT partner than China finds them, and vice versa. Heavier color identifies observed BITs.

The plots reveal how China’s position in the BIT network changed over time. In 1995, China was moving aggressively to demand BITs around the world, forming ties that the model views as relatively unlikely. By 2010, China had many more BITs in place and is more likely to be a treaty target of the remaining countries, an indication of China’s expanded role in the global economy.

Figure 2 illustrates a different use of the P-GBME model. It shows the predicted probabilities that the USA is demanded (left) and demands (right) a BIT in 2010. Observe that the model predicts Peru, Mexico, and Chile—the top 3 non-BITs—demand a BIT with the USA at probabilities close to one, and are also likely BIT targets of the USA with probabilities of about $0.5$ . Closer inspection of UN treaty data reveals that, by 2010, all three had signed other agreements with the US that contain provisions functionally equivalent to BITs; these agreements do not appear in the BIT dataset commonly used in the literature.Footnote ⁵ The P-GBME model nevertheless highlights these “hidden” agreements as dyads likely to have a BIT. This suggests that researchers studying BITs need to carefully examine the dataset they employ and perhaps expand the set of treaties considered relevant.

Figure 2.

The posterior probability a country demands a BIT (left) and is demanded (right) with the USA in 2010 (the top and bottom decile of countries). P-GBME is good at separating likely from unlikely treaties as well as identifying “hidden agreements.”