Multilateral and Bilateral Trade Agreements

In most traditional network representations of the ITN, the trade volume is represented by weighted and directed links that connect two industrial sectors or, at a coarser resolution, two countries. Here, we assess the interconnectedness between two national economies with a newly developed framework that is based upon the interpretation of the ITN as a flow network. Generally, flow networks encode the probabilities of a random walker to move from one node to another. Thus, the ITN as a flow network represents the probability that a unit good follows certain paths through different industrial sectors down the supply chain. This probabilistic approach becomes necessary as individual supply chains cannot be traced from existing data.

In this work, we utilize the idea of flow networks to define the trade interconnectedness (TI) between two countries based upon the input and output dependency measures $p_{i j}^{\bullet}$ originally used in Distributions of positive correlations in sectoral value added growth in the global economic network,

$p_{i j}^{\text {out }}=\frac{w_{i j}}{\sum_{t} w_{l l}} \text { and } p_{j i}^{i n}=\frac{w_{j t}}{\sum_{t} w_{i t}}$

Here, $w_{i j}$ describes the aggregated monetary value (in nominal US $) of all goods that have been sold in 1 year from sector $i$ to another sector $j$. The values of $p_{i j}^{\text {out }}\left(p_{j i}^{i n}\right)$ can be interpreted as the empirical probability of a unit good (respectively, of a certain monetary unit) to follow the corresponding edge in the ITN from $i$ to $j$ as a random walker. 

With the matrices $\left(P_{o u t}\right)_{i j}=p_{i j}^{\text {out }}$, the probability that a unit good follows a path of length $\alpha$ from sector $i$ to sector $j$ is given by $\left(P_{o u t}^{\alpha}\right)_{i j} .$ Analogously, $\left(P_{i n}^{\alpha}\right)_{j i}$ measures the flow of the associated monetary units. To measure how likely it is for a random walker on the ITN to start from a sector in one country and eventually end in another country, we define the trade interconnectedness $(T I)$ between two countries $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ as

$T I^{*}\left(\mathcal{C}_{1}, \mathcal{C}_{2}\right)=\frac{1}{\left|\mathcal{C}_{1}\right| \cdot\left|\mathcal{C}_{2}\right|} \sum_{i \in \mathcal{C}_{1}}\left(\sum_{\alpha=1}^{\alpha_{m a x}}\left(P_{\bullet}^{\alpha}\right)_{i j}\right)$

with $\mathcal{C}_{c}$ denoting the subset of all sectors $i$ that belong to one country $c .$ We refer to $T I^{\text {out }}\left(\mathcal{C}_{1}, \mathcal{C}_{2}\right)$ as the output TI of $\mathcal{C}_{1}$ to $\mathcal{C}_{2}$, which can be interpreted as the relative importance of $\mathcal{C}_{2}$ in the role of a consumer for $\mathcal{C}_{1}$. The relative importance of $\mathcal{C}_{1}$ in the role of a supplier for $\mathcal{C}_{2}$ is analogously quantified by the input $Ti$ of $\mathcal{C}_{2}$ to $\mathcal{C}_{1}, T I^{\text {in }}\left(\mathcal{C}_{1}, \mathcal{C}_{2}\right)$. In Equation (2), $\alpha_{\max }$ describes the maximal path length (in terms of national economic sectors) of the random walker that is to be considered. We find that a reasonable choice of $\alpha_{\max }$ is twice the average path length between the two subgraphs of the ITN spanned by the national economies of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. A more detailed discussion of this choice and a sensitivity analysis of the results with respect to different values of $\alpha_{\max }$ will be presented in section 4.

As formalized in Equation (2), the dependency measures allow for the definition of the output TI which describes the probability of a unit good that is supplied from $\mathcal{C}_{1}$ to end in $\mathcal{C}_{2}$. The input TI describes this probability for a flow of successive payments. In Figure 1 , we schematically illustrate the paths that contribute to the TI using an exemplary network of trade between China (CHN) and Vietnam (VNM). The output TI of China to Vietnam, $TI^out$ (CHN, VNM), accounts for the paths of goods that originate in China and end in Vietnam (see Figure 1A). On the other hand, the input TI of China to Vietnam, $T I^{i n}(\mathrm{VNM}, \mathrm{CHN})$, takes the paths of the monetary flows into account (see Figure 1B). Here, the paths are defined in the opposite direction, as the payment flows opposite to the supply of materials, goods or services in the trade network. The definition of the TI is not symmetric: the corresponding paths of this exemplary network that contribute to the TIs of Vietnam to China are illustrated in Figure 1C for $T I^{\text {out }}$ (VNM, CHN) and Figure 1D for $T I^{\text {in }}$(CHN, VNM), respectively. Notice that we do not consider paths that traverse a third country in the definition of $T I^{*}$

BTA Impact Index

To quantify the impact of a BTA on the TI between the involved countries, we define the BTA impact index $\Pi$ • that takes both the level and local trend properties of the time series of $T I$ into account. Thus, the investigation of the BTA impact index allows for a comparison between the impacts of individual BTAs. In contrast, more traditional methods such as as a difference-in-differences approach would only assess the impact of BTAs compared to country pairs without agreement.

Firstly, we investigate if the mean level of TI has changed markedly after the date of entry into force $t_{f}$ of a specific trade agreement. For this purpose, we consider the annual TI values during a 5 -year interval before the agreement's implementation $\mathcal{I}_{p}=\left[T I_{t_{f}-5}^{\bullet}, . ., T I_{t_{f}-1}^{\bullet}\right]$ and a 5 -year interval after the implementation $\mathcal{I}_{s}=\left[T I_{t_{f}+1}^{\bullet}, \ldots, T I_{t_{f}+5}^{\bullet}\right]$ and define a corresponding score as

$z:=\frac{\mu\left(\tau_{s}\right)-\mu\left(\tau_{p}\right)}{\sigma\left(\mathcal{I}_{p}\right)}$

Here, $\mu(\cdot)$ and $\sigma(\cdot)$ represent the mean value and standard deviation of annual TI values within the respective periods. The score $z$ relates the TI values after $t_{f}$ with the previous levels of the variable. Since the $T I$ ' most commonly do not follow a Gaussian distribution, we will utilize a coarse classification of the explicit values of $z$ defined by Equation (3) in the definition of $\Pi^{*}$, as it will be described below, instead of considering the precise value of $z$. In general, a more sophisticated approach to assess potential changes in the level of a random variable would include an analysis of variance (ANOVA), most likely via the Mann-Whitney $U$ test. However, the small sample size of $T I^{*}$ prevents a meaningful interpretation of the test results in this case, which is why we refrain from performing such explicit statistical significance testing at this point.

Secondly, we are interested in the evolution of the annual TI values after the date of entry into force of an agreement. and therefore statistically characterize their trend during the interval $\left[T I_{t_{f}}^{\bullet}, \ldots, T I_{t_{f}+5}^{\bullet}\right]$ (including the year of BTA implementation and the five following years). To assess this trend, we consider two possible models: In the first model, we perform a simple linear regression

$y_{l}(t)=\beta_{0}+\beta_{1} t+\epsilon_{l}(t)$

with the parameters $\beta_{i}(i=0,1)$ and an independent and identically distributed Gaussian error $\epsilon_{l}(t)$ Alternatively, in order to better recognize oscillating or saturating behavior of the time series during the considered 6-year period, we additionally perform a two-segment piecewise linear regression. The form of this segmented linear model is

$y_{s}(t)=\gamma_{0}+\gamma_{1} t+\gamma_{2}(t-\psi) \theta(t-\psi)+\epsilon_{s}(t)$

with the Heaviside function $\theta$, the (unknown) break-point $\psi$, trend parameters $\gamma_{i}(i=1,2)$ and a Gaussian error term $\epsilon_{s}(t)$ as in the linear model. In contrast to the linear regression, the model in Equation (5) can also account for one local extreme value during the investigated time period, which would be represented by a change in the signs of the slopes between the two segments. More complex regression models that exhibit multiple breakpoints cannot be reliably applied due to the coarse (annual) resolution of the considered data. Therefore, we do not consider such more general models, emphasizing that we are only interested in the sign and statistical relevance of short-term (multi-annual) trends after BTA implementation rather than exact functional descriptions of the shape of these trends or explicit quantitative estimates thereof. Since the segmented model contains two additional parameters as compared to the linear regression model, we perform a model selection based upon the Akaike Information Criterion (AIC) to avoid overfitting by the statistical model with a higher number of degrees of freedom.

If the simple linear model is selected by the AIC criterion, we assess the relevance of the trend identified by the linear regression model and categorize it as relevant and positive (+), relative and negative (-) or not relevant (o). Here, we consider the trend of the linear regression model as relevant, if the estimated variance of the error $\widehat{\sigma}_{\epsilon}$ in $y_{l}$ is smaller than the difference $\Delta \hat{y}_{l}:=\left|\hat{y}_{l}\left(t_{f}\right)-\hat{y}_{l}\left(t_{f}+5\right)\right|$ of the values at the margins of the regression period. If $\widehat{\sigma}_{\epsilon}>\Delta \hat{y}_{l}$, we do not consider the estimated slope of the linear model to be relevant and categorize the trend as (o).

In the case of the segmented model, the considered time series is too short for a similar relevance assessment. Accordingly, if that model is preferred, the additional breakpoint improves the AIC score as compared to the linear regression model. We then consider the slopes of the two segments as relevant. Thus, any pairwise combination between $(+)$ and $(-)$ is possible for the segmented model. Combining both the trend properties and score parameter $z$ of the time series of TI values, we finally define the impact index of a BTA as follows:

\begin{equation} \Pi^{*} (\mathcal{C}_{1}, \mathcal{C}_{2}) = \begin{cases}1 & \text { if } z>1 \text { and }(+\mid++) \\ 0.5 & \text { if } 0 < z < 1 \text { and }(+\mid++) \\ -0.5 & \text { if }-1 < z < 0 \text { and }(-\mid--) \\ -1 & \text { if }-1 < z \text { and }(-\mid--) \\ 0 & \text { else }\end{cases}\end{equation} 

As for the TI, we refer to $\Pi^{\text {out }}\left(\mathcal{C}_{1}, \mathcal{C}_{2}\right)$ as the output BTA impact index of $\mathcal{C}_{1}$ to $\mathcal{C}_{2}\left(\Pi^{i n}\left(\mathcal{C}_{1}, \mathcal{C}_{2}\right)\right.$ as the input BTA impact index of $\mathcal{C}_{2}$ to $\left.\mathcal{C}_{1}\right)$. The average BTA impact indices $\Pi^{\text {out }}$ and $\Pi^{\text {in }}$ of a country $\mathcal{C}_{c}$ are defined as the average $\Pi^{\text {out }}\left(\mathcal{C}_{c}, \bullet\right)$ and $\Pi^{i n}\left(\bullet, \mathcal{C}_{c}\right)$ for the export and import linkages, respectively, taken over all countries that have negotiated a BTA with $\mathcal{C}_{c}$.