A. Macroeconomic Indicators and Preprocessing

Our predictor set combines macroeconomic indicators from the “Revisions Analysis Dataset—Infra-annual Indicators” of the OECD Main Economic Indicators (MEI) database with sentiment indicators from the OECD Key Short-Term Economic Indicators data set, country-specific economic policy uncertainty indexes developed by Baker, Bloom, and Davis (Reference Baker, Bloom and Davis2016),Footnote ³ and financial variables from Bloomberg. The predictor block covers real-sector indicators, fiscal and inflation measures, financial variables (interest-rate spreads, stock indices, USD exchange rates), and global factors such as the VIX and the global economic activity index of Kilian (Reference Kilian2009), incorporating the correction discussed in Kilian (Reference Kilian2019).

Two small variable choices are worth noting. The OECD Composite Leading Indicator (CLI) was discontinued in the OECD Revisions Database after 2022; we replace it with two sentiment-based indicators from the OECD Key Short-Term Economic Indicators data set, the business confidence indicator (BCI) and the consumer confidence indicator (CCI), both available over the full sample. The EPU index of Baker et al. (Reference Baker, Bloom and Davis2016) is not available for all countries in our panel; we approximate missing values using the EPU series of a neighboring country with close economic and geographical ties (Sweden for Finland, Spain for Portugal, and so on). Table A1 in Appendix B lists each variable together with its source and transformation code, and the average correlation structure is reported in Section B of the Supplementary Material.

We align our predictor transformations with the FRED-MD convention of McCracken and Ng (Reference McCracken and Ng2016) to ensure that the predictor block is stationary by construction. Each indicator is mapped to one of four transformation codes (tcode). Let $ {x}_t $ denote the original series and $ {\tilde{x}}_t $ the transformed predictor; we use

(5) $$ {\tilde{x}}_t=\left\{\begin{array}{ll}{x}_t& \left(\mathtt{tcode}=1,\mathrm{level}\right)\\ {}{x}_t-{x}_{t-p}& \left(\mathtt{tcode}=2,\mathrm{first}\ \mathrm{difference}\right)\\ {}\log \left({x}_t\right)-\log \left({x}_{t-p}\right)& \left(\mathtt{tcode}=5,\log\;\mathrm{first}\ \mathrm{difference}\right)\\ {}\left({x}_t/{x}_{t-p}-1\right)\times 100& \left(\mathtt{tcode}=7,\mathrm{percentage}\ \mathrm{change}\right)\end{array}\right. $$

with $ p=12 $ for the monthly series and $ p=4 $ for the quarterly series, so that differences and growth rates are measured year-over-year. The variable-by-variable mapping is reported in Table A1 in Appendix B. GDP and its real-activity components enter as year-over-year percentage changes; price and activity indices as year-over-year log differences; unemployment and short-term interest rates as year-over-year first differences; and scale-free ratios, sentiment indicators, and global financial factors in levels.

The stationarity of the target variable—the log CDS spread—is a separate question. Panel unit-root tests (Pesaran (Reference Pesaran2007)) and stationarity tests (Kwiatkowski, Phillips, Schmidt, and Shin (Reference Kwiatkowski, Phillips, Schmidt and Shin1992), Hadri (Reference Hadri2000)) reject their respective nulls for all four CDS maturities at the 5% level or better; the combined pattern points to high persistence of log CDS spreads with slowly drifting levels, consistent with the characterization of sovereign credit spreads in Longstaff et al. (Reference Longstaff, Pan, Pedersen and Singleton2011) and Augustin (Reference Augustin2018). The transformed predictor block is stationary by construction under the FRED-MD transformations. The forecasting results reported in Section IV are unlikely to be affected by spurious correlations. Detailed test results are reported in Section B of the Supplementary Material.

B. Forecasting Methods and Evaluation Protocol

We compare eight linear specifications against five nonlinear specifications in a unified out-of-sample protocol. The linear models implement the log-linear benchmark of equation (4); the three nonlinear families relax it through distinct representational mechanisms—recursive partitioning of the feature space, kernel-based feature expansions, and hierarchical nonlinear transformations.

Every method receives the same set of macroeconomic indicators $ {\mathbf{x}}_{i,t-1} $ and four CDS maturity dummies, with the maturity dummies entering as interactions in the linear specifications and as ordinary predictors in the nonlinear specifications. For every method that either shrinks coefficients toward 0 or trains on cross-sectionally demeaned targets, we subtract the cross-country mean at each time and maturity before estimation, and recenter the maturity-specific means at the forecast stage, thereby preventing shrinkage or demeaning from contaminating the maturity-interaction structure.

1. Linear Specifications

Following Balduzzi et al. (Reference Balduzzi, Savona and Alessi2023), we adopt a “level + slope” form that interacts the real-time macroeconomic indicators $ {\mathbf{x}}_{i,t-1} $ (plus an intercept) with the CDS maturity $ j $ ,

(6) $$ {s}_{ij,t}={\mathbf{x}}_{i,t-1}^{\prime}\beta +\sum \limits_{j\in \mathcal{J}\backslash \left\{3\right\}}{\left({\mathbf{x}}_{i,t-1}\times {m}_j\right)}^{\prime }{\delta}_j+{\varepsilon}_{ij,t}, $$

where $ {m}_j $ is a dummy variable that takes the value 1 if the contract has a $ j $ -year maturity, and 0 otherwise, $ \mathcal{J}=\left\{\mathrm{3,5,7,10}\right\} $ , and the 3-year CDS is the reference group. On top of this common form, we consider three strategies for managing the high dimensionality of the predictor block in short rolling windows: coefficient regularization (Ridge, Lasso, Elastic Net; Hsiang (Reference Hsiang1975), Tibshirani (Reference Tibshirani1996), and Zou and Hastie (Reference Zou and Hastie2005)), forecast combination via Complete Subset Regression (CSR; Elliott, Gargano, and Timmermann (Reference Elliott, Gargano and Timmermann2013)) and its equal-weighted special case (EWC; Rapach, Strauss, and Zhou (Reference Rapach, Strauss and Zhou2010)), and latent-factor dimension reduction via Principal Component Regression (PCR; Ludvigson and Ng (Reference Ludvigson and Ng2009), Stock and Watson (Reference Stock, Watson, Clements and Hendry2011), and Kelly and Pruitt (Reference Kelly and Pruitt2013)) and Partial Least Squares (PLS). Implementation details are in Section A of the Supplementary Material. For robustness, we also estimate “level-only” variants of each linear specification that omit the maturity-interaction terms in equation (6), and report the corresponding results in Section C of the Supplementary Material.

2. Tree-Based Methods

The first nonlinear family is based on recursive partitioning of the feature space (e.g., Breiman, Friedman, Stone, and Olshen (Reference Breiman, Friedman, Stone and Olshen1984), Döpke, Fritsche, and Pierdzioch (Reference Döpke, Fritsche and Pierdzioch2017), and Medeiros, Vasconcelos, Veiga, and Zilberman (Reference Medeiros, Vasconcelos, Veiga and Zilberman2021)). At each node, the feature space is split on a predictor $ {x}_{i,t-1,p} $ (the $ p $ -th component of the state vector $ {\mathbf{x}}_{i,t-1} $ ) and a cutoff value $ c $ , chosen jointly to minimize the within-region sum of squared errors of the target $ {s}_{ij,t} $ ,

(7) $$ \mathrm{\ell}=\sum \limits_{\left(i,j,t\right)\in {I}_{<}}{\left({s}_{ij,t}-{\overline{s}}_{I_{<}}\right)}^2+\sum \limits_{\left(i,j,t\right)\in {I}_{>}}{\left({s}_{ij,t}-{\overline{s}}_{I_{>}}\right)}^2, $$

with $ {I}_{<}=\left\{\left(i,j,t\right)|{x}_{i,t-1,p}\le c\right\} $ , $ {I}_{>}=\left\{\left(i,j,t\right)|{x}_{i,t-1,p}>c\right\} $ , and leaf predictions equal to the mean target value in each region. We consider three variants that control a fully grown tree’s tendency to overfit. A pruned regression tree penalizes leaf count directly. A random forest (Breiman (Reference Breiman2001)) averages over an ensemble of trees trained on bootstrap samples, with a random subset of predictors evaluated at each split. XGBoost (Chen and Guestrin (Reference Chen and Guestrin2016)) trains a sequence of trees in which each new tree is fit to the gradient of a regularized loss function, yielding an additive ensemble:

(8) $$ {\hat{s}}_{ij,t}=\sum \limits_{k=1}^K{f}_k\left({\mathbf{x}}_{i,t-1}\right),\hskip1em {f}_k\in \mathrm{\mathcal{F}}, $$

with $ \mathrm{\mathcal{F}} $ the space of regression trees. Implementation details for all these tree-based methods are in Section A of the Supplementary Material.

A. Time-Varying Predictability

We begin with the yearly out-of-sample $ {R}^2 $ , which we compute as the average of $ {R}_{t, oos}^2 $ within each calendar year. Figure 2 reports the yearly $ {R}^2 $ on the Global panel, 2010–2024, across two benchmarks (rows) and two forecasting horizons (columns).Footnote ⁵ The heatmap colors each yearly cell by its $ {R}^2 $ value, with blue shading for positive values, red shading for negative values, and white at 0; cells with $ \mid {R}^2\mid \ge 0.7 $ are annotated.

FIGURE 2Yearly $R_{\mathit{OOS}}^2$ $ {R}_{OOS}^2 $ ROOS 2 Across Benchmarks and Horizons, Global Panel

Figure 2 reports the annual average of $ {R}_{t, oos}^2 $ on the Global panel of 25 countries. Graphs A and B use the benchmark $ {\overline{s}}_{\cdot j} $ (log CDS averaged across countries); Graphs C and D use $ {\overline{s}}_{i\cdot } $ (averaged across maturities). Columns: 1-month (Graphs A and C) and 3-month (Graphs B and D) testing horizons. Cells are colored by $ {R}^2 $ using a diverging scale centered at 0; values with $ \mid {R}^2\mid \ge 0.7 $ are annotated.

FIGURE 2 Long description

A four-panel grid of heat maps showing yearly out-of-sample R super 2 values.

Common Structure:

* Y-axis: Eight models listed as X G B, R F, S V R, N N, E W C, C S R, P C R, and E Net.

* X-axis: Years from 2010 to 2024.

* Color Scale: A diverging vertical bar on the right ranging from 1.00 (dark blue) through 0.00 (white) to -1.00 (dark red).

Panel Breakdown:

* Top-Left (Graph A, Cross-Country, 1-Month): Predominantly blue, indicating positive R super 2. X G B shows the highest consistency with values between 0.71 and 0.91. R F and S V R also show strong performance. E W C is a white horizontal band at zero.

* Top-Right (Graph B, Cross-Country, 3-Month): Similar to Graph A but with slightly lighter blue shades. X G B remains the top performer with values ranging from 0.74 to 0.87.

* Bottom-Left (Graph C, Cross-Maturity, 1-Month): Shows a sharp divide. The top four models (X G B to N N) are mostly blue or light orange. The bottom four models (E W C to E Net) are dark red, with E W C showing extreme negative values like -4.05 and -3.29 in the early 2010s.

* Bottom-Right (Graph D, Cross-Maturity, 3-Month): Similar to Graph C but with more intense red across the bottom half. E W C continues to show the most significant negative performance, with values such as -4.15 and -3.25.

The results pin down where linear methods succeed, where they fail, and where nonlinear methods extend the region of positive out-of-sample performance. Against the cross-country benchmark $ {\overline{s}}_{\cdot j} $ (Graphs A and B), Elastic Net and CSR are positive in nearly every year of the 2010–2024 sample, averaging around 0.43 at the 1-month horizon and 0.34 at the 3-month horizon; PCR, restricted to five principal components, is weaker by roughly 0.20 but preserves the sign pattern. The equal-weighted combination (EWC), which strips the linear class of its coefficient structure, collapses to essentially 0 at both horizons, confirming that the cross-country signal is not an aggregation artifact. This suggests that macro fundamentals—debt-to-GDP, growth, unemployment, financing conditions—carry sufficient cross-country signal for linear specifications with regularization to recover positive predictability. Tree-based ensembles improve on this benchmark (XGBoost averages 0.81 at 1-month, Random Forest 0.76), so nonlinearity contributes in the cross-country dimension as well.

The picture changes for the cross-maturity benchmark $ {\overline{s}}_{i\cdot } $ . Linear methods fail in every cell of Graphs C and D, with negative $ {R}^2 $ in every year at both horizons. The antecedent is Doshi et al. (Reference Doshi, Ericsson, Jacobs and Turnbull2013): in a no-arbitrage CDS pricing model, slope and curvature of the term structure depend on state variables through threshold effects and regime-contingent slope changes that linear maturity interactions cannot span. Tree-based ensembles close the cross-maturity gap most completely. XGBoost is positive against $ {\overline{s}}_{i\cdot } $ at 1-month in every year of the 2010–2024 sample, averaging $ +0.49 $ . Random Forest is positive in 12 of 15 years (mean $ +0.31 $ ), with 2010, 2011, and 2022 being the three negative cells. SVR is positive in 12 of 15 years from 2013 onward, with systematically smaller magnitudes (mean $ +0.23 $ ). The feedforward network is negative in 9 of 15 years. The cleavage lies between tree-based ensembles, which capture interactions between maturity and macro state without researcher-specified structure, and the kernel and neural alternatives.

Linear cross-country predictability also weakens under regime change. Elastic Net’s 1-month $ {R}_{oos}^2 $ against $ {\overline{s}}_{\cdot j} $ drops from an average of 0.54 over 2011–2021 to 0.32 over 2022–2024. At a 3-month horizon, Elastic Net’s 2022 cell is 0.04, compared to 0.68 for XGBoost. Section C of the Supplementary Material shows that the pattern sharpens at 6-month and 12-month horizons. Nonlinear methods retain the signal (XGBoost’s 1 M mean moves from 0.85 in 2011–2021 to 0.82 in 2022–2024), so the weakening is not about fundamentals becoming uninformative but about linear coefficients failing to transport when the mapping from fundamentals to CDS is state-dependent. This suggests that the “disconnect” suggested by prior literature (e.g., Balduzzi et al. (Reference Balduzzi, Savona and Alessi2023)) is more a statement about the rigidity of linear specifications than about the information content of macro fundamentals.

Figure 3 reports the SWEAP subpanel, representing countries sharing a monetary regime and a common crisis history. Elastic Net averages 0.57 at 1-month and 0.53 at 3-month against $ {\overline{s}}_{\cdot j} $ , against 0.47 and 0.40 in Global and 0.41 and 0.30 in OECD non-EU. The same specification nonetheless shows negative cross-maturity in most years (Elastic Net mean: $ -0.52 $ at 1-month). The Doshi et al. (Reference Doshi, Ericsson, Jacobs and Turnbull2013) term-structure nonlinearity is a within-country property of the pricing mapping and is therefore not attenuated by restricting to a more homogeneous subpanel. The linear cross-maturity failure holds in the group where the linear cross-country fit is strongest.

FIGURE 3Yearly $R_{\mathit{OOS}}^2$ $ {R}_{OOS}^2 $ ROOS 2 Across Benchmarks and Horizons, SWEAP Panel

Figure 3 reports the annual average of $ {R}_{t, oos}^2 $ for the SWEAP country subgroup (Spain, Portugal, Italy, and Ireland). Rows: cross-country predictability against $ {\overline{s}}_{\cdot j} $ (Graphs A and B) and cross-maturity predictability against $ {\overline{s}}_{i\cdot } $ (Graphs C and D). Columns: 1-month (Graphs A and C) and 3-month (Graphs B and D) testing horizons. Models and formatting follow Figure 2.

FIGURE 3 Long description

A multi-panel set of four heatmaps labeled Graph A through Graph D. A shared vertical color bar on the right indicates Yearly Out-of-Sample R super 2 values, ranging from dark red at negative 1.00 to dark blue at positive 1.00, with white at 0.00. The Y-axis for all graphs lists models: X G B, R F, S V R, N N, E W C, C S R, P C R, and E Net. The X-axis spans from 2010 to 2024.

* Graph A (Top-Left): Cross-Country Predictability at a 1-month horizon. The top four models (X G B, R F, S V R, N N) show dominant blue shading with R super 2 values frequently between 0.70 and 0.94. P C R shows significant negative performance in 2021 (negative 0.96) and 2023 (negative 0.93).

* Graph B (Top-Right): Cross-Country Predictability at a 3-month horizon. Similar to Graph A but with slightly lower positive values, mostly ranging from 0.70 to 0.91 in the top four rows. P C R shows a deep red cell at negative 0.99 in 2021.

* Graph C (Bottom-Left): Cross-Maturity Predictability at a 1-month horizon. The bottom four models (E W C, C S R, P C R, E Net) are dominated by dark red, indicating strong negative R super 2 values. E W C shows extreme negative values such as negative 14.50 in 2012. The top four models show lighter blue and white cells.

* Graph D (Bottom-Right): Cross-Maturity Predictability at a 3-month horizon. Follows the trend of Graph C with intense red shading in the bottom half. E W C reaches a low of negative 15.16 in 2012. S V R shows a sharp drop to negative 3.97 in 2012.

In the OECD non-EU subpanel, the institutional setting is the opposite: no shared monetary regime, no common crisis history, and country-level fundamentals ranging from AAA-rated borrowers to investment-grade emerging markets. Figure 4 shows that nonlinear methods nevertheless deliver positive cross-maturity $ {R}^2 $ . XGBoost averages $ +0.39 $ over a 1-month horizon (14 of 15 years positive), and Random Forest $ +0.21 $ (10 of 15), while every linear specification has a negative mean. The nonlinear cross-maturity advantage, therefore, reflects the functional-form gap at the sovereign level rather than the institutional commonality visible in SWEAP.

FIGURE 4Yearly $R_{\mathit{OOS}}^2$ $ {R}_{OOS}^2 $ ROOS 2 Across Benchmarks and Horizons, OECD Non-EU Panel

Figure 4 reports the annual average of $ {R}_{t, oos}^2 $ for the OECD non-EU country subgroup. Rows: cross-country predictability against $ {\overline{s}}_{\cdot j} $ (Graphs A and B) and cross-maturity predictability against $ {\overline{s}}_{i\cdot } $ (Graphs C and D). Columns: 1-month (Graphs A and C) and 3-month (Graphs B and D) testing horizons. Models and formatting follow Figure 2.

FIGURE 4 Long description

A four-panel grid of heatmaps. Each panel shares a common X-axis representing years from 2010 to 2024 and a Y-axis listing eight models: X G B, R F, S V R, N N, E W C, C S R, P C R, and E Net.

* Graph A (Top-Left): Cross-Country Predictability at a 1-month horizon. The top four models (X G B, R F, S V R, N N) show high performance indicated by dark blue cells, with specific R-squared values like 0.92 and 0.91 visible around 2016. The bottom four models show lighter blue or white cells, indicating lower positive performance.

* Graph B (Top-Right): Cross-Country Predictability at a 3-month horizon. Similar to Graph A but with generally lighter blue shades, indicating a slight decrease in predictive power as the horizon increases.

* Graph C (Bottom-Left): Cross-Maturity Predictability at a 1-month horizon. This panel shows a stark contrast. The top models maintain some blue (positive) values, but the bottom models, particularly E W C, are dominated by dark red cells with negative values such as -3.88 and -3.42.

* Graph D (Bottom-Right): Cross-Maturity Predictability at a 3-month horizon. This panel shows the most significant negative performance, with deep red across the bottom five models. E W C shows values as low as -3.65 and -3.43.

A vertical color bar on the right side serves as the legend for all panels. It ranges from -1.00 (dark red) to 0.00 (white) to 1.00 (dark blue), labeled Yearly Out-of-Sample R super 2.

B. Four-Benchmark Aggregate Results

We now turn to full-sample averages of the yearly $ {R}^2 $ , organized by benchmark and horizon. Table 1 reports the average $ {\overline{R}}_{oos}^2 $ on the Global panel for the eight level + slope linear specifications and five nonlinear specifications, against each of the four benchmarks, at each horizon.Footnote ⁶ Table 1 is organized so that scanning a single model column from top to bottom traces the $ {R}^2 $ decomposition as the benchmark becomes more demanding; scanning a single row compares models at a fixed benchmark and horizon.

TABLE 1Four-Benchmark Decomposition of Out-of-Sample ${\overline{R}}_{\mathit{oos}}^2$ $ {\overline{R}}_{oos}^2 $ upper R bar oos 2 , Global Panel

TABLE 1 Long description

The table is organized into five forecasting horizon blocks: 1-month, 3-month, 6-month, 9-month, and 12-month. Each block contains four benchmark rows: s-bar dot dot (grand mean), s-bar dot j (cross-country), s-bar i dot (cross-maturity), and s-bar ij train (within-cell training mean).

Columns are divided into two main model classes:

1. Level plus Slope Linear Models: O L S, E W C, C S R, Lasso, Ridge, E N, P C R, and P L S.

2. Nonlinear Models: S V R, R T, R F, X G B, and N N.

Key data trends:

* For the 1-month horizon under the grand mean benchmark, X G B (Nonlinear) achieves the highest value at 0.83, while E N (Linear) leads its class at 0.52.

* As the forecasting horizon increases to 12-months, performance generally decreases. For the grand mean benchmark at 12-months, X G B remains the best nonlinear model at 0.61, and C S R is the best linear model at 0.23.

* Negative values are prevalent in the more severe benchmarks (s-bar i dot and s-bar ij train), particularly for linear models like O L S and E W C, which reach values as low as minus 11.14 and minus 10.25 respectively.

* Across almost all horizons and benchmarks, the X G B model in the nonlinear category and either the E N or C S R models in the linear category are highlighted in bold as the best-performing specifications.

The decomposition reading follows the panel’s variance structure. Moving from $ {\overline{s}}_{\cdot \cdot } $ to $ {\overline{s}}_{\cdot j} $ removes the approximately 16% of total variance attributable to between-maturity differences from the denominator, which modestly lowers every $ {R}^2 $ . Elastic Net at 1-month drops from 0.52 to 0.45, XGBoost from 0.83 to 0.81. Moving from $ {\overline{s}}_{\cdot \cdot } $ to $ {\overline{s}}_{i\cdot } $ removes approximately 73% of the variance attributable to between-country differences, which is a much larger shift: Elastic Net at a 1-month horizon falls to $ -0.76 $ , XGBoost to $ +0.44 $ . Linear specifications lose nearly all of their $ {R}^2 $ in this step; tree-based nonlinear specifications retain a substantial portion of it. The magnitude of the $ {R}^2 $ drop between the two middle benchmarks, therefore, suggests that linear specifications’ out-of-sample performance is concentrated in variance that country fixed effects would absorb in-sample, whereas tree-based ensembles generate predictions in the dimension that fixed effects cannot.

The bottom row of each horizon block reports the $ {R}^2 $ against $ {\overline{s}}_{ij} $ , the 12-month rolling country-maturity training mean. Every model is negative against this benchmark at every horizon; XGBoost is the least negative ( $ -0.64 $ at the 1-month horizon, $ -2.77 $ at the 12-month horizon), and the model ordering is preserved (XGBoost $ > $ RF $ > $ SVR $ > $ linear). This is not evidence against the macro-to-CDS link. Daily log CDS spreads are near-persistent (see Section B of the Supplementary Material), so a trailing country-maturity mean is close to a random-walk forecast, consistent with the characterization of sovereign credit spreads in Longstaff et al. (Reference Longstaff, Pan, Pedersen and Singleton2011) and Augustin (Reference Augustin2018). The difficulty of beating the $ {\overline{s}}_{ij} $ benchmark suggests that the productive macroeconomic signal lies more in the cross-country and cross-maturity dimensions than in the time-series dimension (Duffee (Reference Duffee2002)).

Reading within horizon blocks, the results are consistent with the yearly evidence in Section IV.A whereby linear methods capture cross-country but not cross-maturity predictability, and nonlinear methods extend the region of positive $ {R}^2 $ into the cross-maturity dimension and cross-country during crisis regimes. Reading across horizon blocks, the $ {R}^2 $ degrades as the horizon extends, but the relative ordering of model classes—linear versus nonlinear, and within nonlinear, XGBoost $ \approx $ RF ahead of SVR ahead of NN—is preserved. At the 12-month horizon, Elastic Net’s cross-country $ {R}^2 $ is indistinguishable from 0.

Across country groups (Section C of the Supplementary Material), the pattern is preserved without qualitative change. Linear methods capture cross-country variation but fail to capture cross-maturity variation in the OECD non-EU, EU ex-SWEAP, and SWEAP panels; nonlinear methods—XGBoost in particular—deliver positive cross-maturity $ {R}^2 $ in every subgroup at short horizons. The magnitudes differ: SWEAP countries show higher raw $ {R}^2 $ against the grand-mean benchmark at short horizons, reflecting the larger cross-country spread introduced by peripheral Eurozone sovereigns, but the within-group decomposition (the fall from $ {\overline{s}}_{\cdot j} $ to $ {\overline{s}}_{i\cdot } $ and the sign pattern it produces) is invariant.

C. Testing Predictive Accuracy

This section reports three pieces of evidence on whether the differences in out-of-sample performance documented in Table 1 and Figure 2 are statistically distinguishable. First, we report pairwise tests of predictive accuracy following Qu, Timmermann, and Zhu (Reference Qu, Timmermann and Zhu2023), a modified Diebold–Mariano procedure robust to panel dependence that delivers the raw significance of each model-versus-model comparison. Second, we report the Hansen, Lunde, and Nason (Reference Hansen, Lunde and Nason2011) Model Confidence Set at 75% and 90% confidence levels, which identifies the set of models that cannot be rejected as the best performer given the data. Third, we report Romano and Wolf (Reference Romano and Wolf2005), (Reference Romano and Wolf2016) step-down-adjusted $ p $ -values for the pairwise comparisons, controlling the family-wise error rate across the grid of tested hypotheses; this addresses the data-mining concern that the raw Diebold–Mariano $ p $ -values ignore the multiplicity of comparisons.

Panels A and B of Table 2 report the first and third tests for XGBoost against each of the 12 rivals, at the 1-month testing horizon. Technical details and the Model Confidence Set output are in Appendix A.

TABLE 2Pairwise Statistical Significance Versus XGBoost

TABLE 2 Long description

The table is divided into two main sections: Panel A for D M z-statistics and Panel B for R W adjusted p-values. Both panels evaluate four regions: Global, O E C D / E U, E U / S W E A P, and S W E A P.

Twelve rival models are listed in the first column: O L S, E W C, C S R, Lasso, Ridge, E N, P C R, P L S, S V R, R T, R F, and N N.

Key data points include:

* O L S: Panel A values range from minus 10.73 to minus 5.32. Panel B p-values range from 0.004 to 0.165.

* E W C: Panel A values range from minus 40.14 to minus 10.28. Panel B p-values range from 0.000 to 0.124.

* C S R: Panel A values range from minus 22.95 to minus 10.20. Panel B p-values range from 0.000 to 0.098.

* Lasso: Panel A values range from minus 20.15 to minus 6.85. Panel B p-values range from 0.000 to 0.124.

* Ridge: Panel A values range from minus 19.09 to minus 6.40. Panel B p-values range from 0.000 to 0.124.

* E N: Panel A values range from minus 21.19 to minus 7.95. Panel B p-values range from 0.000 to 0.124.

* P C R: Panel A values range from minus 12.76 to minus 8.69. Panel B p-values range from 0.003 to 0.050.

* P L S: Panel A values range from minus 9.53 to minus 4.86. Panel B p-values range from 0.005 to 0.167.

* S V R: Panel A values range from minus 7.54 to 3.88. Panel B p-values range from 0.005 to 0.167.

* R T: Panel A values range from 4.45 to 7.81. Panel B p-values range from 0.005 to 0.167.

* R F: Panel A values range from 4.83 to 11.52. Panel B p-values range from 0.003 to 0.167.

* N N: Panel A values range from 7.24 to 9.81. Panel B p-values range from 0.005 to 0.124.

The final row, number of rejections at 10 percent, shows 12 out of 12 for all columns except for Panel B S W E A P, which shows 2 out of 12.

Under the unadjusted test, every XGBoost-versus-rival $ z $ -statistic is large and negative, indicating lower mean squared error for XGBoost; importantly, every comparison is significant at the 1% level in every group ( $ \mid z\mid \ge 3.86 $ ). The Romano–Wolf adjustment is more demanding: in the Global, OECD non-EU, and EU ex-SWEAP panels, XGBoost’s advantage survives familywise error control against all 12 rivals at $ \alpha =0.10 $ (maximum adjusted $ p $ -value = 0.005 in Global and 0.011 in OECD non-EU, where the marginal rival is SVR in each case). In the SWEAP subgroup, only 2 of 12 comparisons survive at $ \alpha =0.10 $ (PCR at $ p=0.050 $ , CSR at $ p=0.098 $ ). The remaining models, both linear and nonlinear, are indistinguishable from XGBoost under FWER control. The SWEAP pattern reflects the narrower cross section (5 sovereigns vs. 25 in the Global panel) and the correspondingly wider posterior on the loss-differential distribution, while XGBoost retains the lowest mean squared error in every cell, as shown in Section C of the Supplementary Material.

The Model Confidence Set delivers a stronger verdict (see Table A2): at both confidence levels and across all country groups over the 1-month horizon, the MCS contains only XGBoost. That is, no rival model can be retained at the 90% confidence level as statistically indistinguishable from the best forecaster. Put differently, every non-XGBoost specification is rejected from the set of equivalent-best models.

A. Forecasting Performance Attribution

We measure predictor importance with the Shapley value (Shapley (Reference Shapley, Kuhn and Tucker1953)), which represents the unique additive and monotone allocation of a model’s forecast to its inputs. Additivity means that the predictor contributions sum to the forecast minus a model-specific baseline, so attribution is fully allocated with no residual hidden in an intercept. Monotonicity means that if predictor $ i $ raises the forecast by at least as much under one model as under another on every subset, then its Shapley value is at least as large under the first model. The two properties serve complementary roles in our setting. Additivity yields fully allocated attribution within each forecaster, and monotonicity ensures that the attributions are comparable across models fitted to the same inputs.

Formally, let $ D=\left\{1,\dots, d\right\} $ index the predictors and $ v:\mathcal{P}(D)\to \mathrm{\mathbb{R}} $ be a value function that returns a model’s forecast as a function of which predictors are “present.” The Shapley value of predictor $ i $ is

(10) $$ \overset{i^{\prime}\mathrm{s}\;\mathrm{Shapley}\ \mathrm{value}}{\overbrace{\phi_i(v)}}=\sum \limits_{S\subseteq D\backslash \left\{i\right\}}\underset{S^{\prime }s\;\mathrm{weight}}{\underbrace{\frac{\mid S\mid !\left(|D|-|S|-1\right)!}{\mid D\mid !}}}\overset{i^{\prime}\mathrm{s}\;\mathrm{marginal}\ \mathrm{contribution}}{\overbrace{\left(v\left(S\cup \left\{i\right\}\right)-v(S)\right)}}, $$

where the weight $ \mid S\mid !\left(|D|-|S|-1\right)!/\mid D\mid ! $ is the probability of forming coalition $ S $ in a random permutation of the $ d $ predictors.Footnote ⁷ Equation (10) satisfies $ {\sum}_{i\in D}{\phi}_i(v)=v(D)-v\left(0/\right) $ , so Shapley values decompose the forecast-minus-baseline into predictor-specific contributions.

A two-predictor example makes additivity concrete. For a forecaster of the term-structure slope (Term) and the debt-to-GDP ratio (Debt/GDP) with $ v\left(0/\right)=0.01 $ , $ v\left(\left\{\mathrm{Term}\right\}\right)=0.04 $ , $ v\left(\left\{\mathrm{Debt}/\mathrm{GDP}\right\}\right)=0.03 $ , and $ v\left(\left\{\mathrm{Term},\mathrm{Debt}/\mathrm{GDP}\right\}\right)=0.05 $ , equation (10) gives $ {\phi}_{\mathrm{Term}}=0.025 $ and $ {\phi}_{\mathrm{Debt}/\mathrm{GDP}}=0.015 $ .Footnote ⁸ The two sum to $ 0.04=v\left(\left\{\mathrm{Term},\mathrm{Debt}/\mathrm{GDP}\right\}\right)-v\left(0/\right) $ , and Term receives the larger attribution because its marginal forecast increase is larger whether it is added first or second.

In the empirical implementation, the value function is the model’s out-of-sample forecast with the predictors in $ S $ set at their explicand values and the predictors in $ \overline{S}=D\backslash S $ integrated over baseline observations drawn from the testing distribution:

(11) $$ v(S)=\frac{1}{\mid E\mid}\sum \limits_{x^b\in E}f\left({x}_S^e,{x}_{\overline{S}}^b\right), $$

where $ {x}^e $ is the explicand (the observation being explained), $ E $ is the set of baseline observations,Footnote ⁹ and $ f\left({x}_S^e,{x}_{\overline{S}}^b\right) $ denotes the forecast when the predictors in $ S $ take the explicand values and those in $ \overline{S} $ take the baseline values.Footnote ¹⁰

For XGBoost, equation (11) is computed with the TreeSHAP algorithm of Lundberg et al. (Reference Lundberg, Erion, Chen, DeGrave, Prutkin, Nair, Katz, Himmelfarb, Bansal and Lee2020), which exploits the tree structure to return all $ {\phi}_i $ simultaneously in time linear in the number of leaves.Footnote ¹¹ For Elastic Net, equation (11) admits the closed form $ {\phi}_i\left(f,{x}^e\right)={\beta}_i\left({x}_i^e-{\mu}_i\right) $ , where $ {\beta}_i $ is the penalized coefficient on predictor $ i $ and $ {\mu}_i={\left|E\right|}^{-1}{\sum}_{x^b\in E}{x}_i^b $ is the baseline mean.Footnote ¹² Linear attribution is therefore the deviation of the explicand from the baseline mean, scaled by a single penalized coefficient—fixed across countries, across regimes, and across maturities.

In both cases, the forecaster $ f $ at each month is the recursive 1-month-ahead estimate used to produce the out-of-sample $ {R}^2 $ reported in Section IV. The explicand $ {x}^e $ is the corresponding testing observation, and the baseline set $ E $ is the set of testing observations, so both the evaluation point and the distribution over which $ \overline{S} $ -predictors are integrated lie outside the training window.

B. Time-Varying and Cross-Sectional Attribution

Figure 5 reports the time-varying mean absolute Shapley values for XGBoost (Graph A) and the “level + slope” Elastic Net (Graph B), averaged across countries in the Global panel.Footnote ¹³ The out-of-sample window spans February 2009 through November 2024, covering the GFC aftermath, the Eurozone sovereign debt crisis, the decade of accommodative policy, the COVID-19 shock, and the 2022–2024 monetary tightening cycle.

FIGURE 5Time-Varying Importance of Macroeconomic Variables

Figure 5 displays the out-of-sample mean absolute Shapley values from XGBoost (Graph A) and the “level + slope” Elastic Net (Graph B), averaged across countries in the Global panel at each 1-month testing period. The color scale is sequential (darker indicates higher importance). Rows correspond to the predictors described in Section III.A. The out-of-sample period is February 2009 through November 2024.

FIGURE 5 Long description

A two-panel heat map display.

Panel A, titled XGBoost, shows Mean Absolute S H A P Value on a scale from 0.000 (white) to 0.200 (black). The Y-axis lists 20 macroeconomic variables including C P I, I I P, I R D, G D P, E G S, E G S forward slash G D P, G C E, G F C, I G S, I G S forward slash G D P, P C E, B O P, H U R, I N T, S P, B C I, C C I, Term Spread, F X, Debt forward slash G D P, and E P U. The X-axis spans from 2009 to 2024. In this panel, Term Spread and C C I show the highest sustained importance (darkest shading) across the entire period, particularly between 2011 and 2023. C P I shows a significant spike in importance starting around 2021.

Panel B, titled Elastic Net, uses a different scale for Mean Absolute S H A P Value from 0.0 to 4.0. It shares the same Y-axis variables and X-axis timeline. The importance distribution is more sparse compared to Panel A. The most prominent features are high-importance bands for B C I and C C I between 2010 and 2022, and a very dark, concentrated band for Debt forward slash G D P between 2011 and 2019. G F C shows a localized peak of importance around 2016.

We organize the discussion around three features that together characterize the attribution-level footprint of the cross-maturity $ {R}^2 $ failure in Section IV, namely which predictors XGBoost uses and in which regimes, which predictors Elastic Net uses and how its use differs, and how the contrast holds across the cross section of countries.Footnote ¹⁴

XGBoost uses the term-structure slope (Term Spread) persistently and inflation-related predictors conditionally. Graph A of Figure 5 shows a dominant Term Spread band across most of the 2011–2024 window; this is the only predictor that carries nontrivial attribution in every regime of the sample. The CCI is activated during the Eurozone sovereign debt crisis (2011–2014) and flares again during the COVID period (2020–2021). The year-over-year change in consumer prices (CPI) and the first difference of the short-term interest rate (INT) are essentially dormant through 2020 and then activate sharply in 2021–2024, together giving the monetary tightening cycle its distinct signature.

Fiscal ratios (Debt/GDP, EGS/GDP, IGS/GDP), the business sentiment indicator (BCI), and country-specific risk proxies (EPU, FX, equity prices) carry secondary or near-zero attribution throughout. The same three-predictor core—Term Spread, CCI, and the tightening-cycle CPI/INT pair—appears in each of the three country subgroups (Figures A3–A5 in Appendix B); the Term Spread band is sharpest in SWEAP and the tightening-cycle CPI/INT activation sharpest in OECD non-EU, but the set of active predictors and their timing are stable across groups.

Elastic Net uses debt and sentiment, not the term-structure slope or inflation. Graph B of Figure 5 is dominated by a persistent Debt/GDP band from 2011 through the mid-2020s, with CCI and BCI as secondary bands of similar structure. The three predictors that carry XGBoost’s attribution most distinctively—Term Spread, CPI, INT—are essentially white in the heatmap throughout Graph B. The tightening-cycle activation visible in Graph A is not present in Graph B; the Eurozone-crisis CCI flare is present but at a much smaller amplitude; Term Spread is unused at any point in the sample.

Expected CDS premiums from Elastic Net are therefore not a muted version of XGBoost’s but depend on a different attribution pattern entirely. The attribution asymmetry is a consequence of functional form.Footnote ¹⁵ The linear specification does not adapt its predictor use across regimes, and the predictors it does use are disjoint from those XGBoost relies on. The same ranking holds in the three country subgroups, as shown in Appendix B.

The asymmetry holds country by country. Figure 6 reports the time-averaged attribution separately for each country. For XGBoost (Graph A), Term Spread is the leading or near-leading predictor for essentially every country in the panel, with secondary concentrations on CCI (sharpest in SWEAP and the EU ex-SWEAP periphery) and on Debt/GDP (concentrated on Italy, Hungary, and Mexico). For Elastic Net (Graph B), Debt/GDP dominates, with the largest loadings falling on countries whose debt-to-GDP ratios deviate most from the panel mean—Japan, Korea, Mexico, Hungary, Italy, and the Slovak Republic. The Japan case is the cleanest illustration of the linear mechanism: sovereign CDS premiums on Japan are among the lowest in the sample, yet Elastic Net attributes the largest Debt/GDP-driven component to Japan because the closed form $ {\phi}_i={\beta}_i\left({x}_i^e-{\mu}_i\right) $ rewards deviation from the panel mean rather than spread level. The three predictors XGBoost uses most actively—Term Spread, CPI, INT—are white across every country in Graph B. The attribution asymmetry documented in Figure 5 is therefore not an artifact of cross-country averaging; it is a predictor-level property that manifests consistently across all 25 countries in the panel.

FIGURE 6Country-Specific Feature Attribution

Figure 6 displays the mean absolute Shapley values from XGBoost (Graph A) and the “level + slope” Elastic Net (Graph B), with each cell reporting a predictor’s monthly out-of-sample Shapley values averaged across the testing period for the indicated country. Columns are grouped by country group (SWEAP; EU ex-SWEAP; OECD non-EU) and ordered alphabetically within group; rows correspond to the predictors described in Section III.A. The out-of-sample period is February 2009 through November 2024.

FIGURE 6 Long description

Two side-by-side heatmaps, labeled Graph A. X G B o o s t and Graph B. Elastic Net.

Both graphs share a common Y-axis with 20 economic predictors including C P I, I I P, I R D, G D P, E G S, E G S all over G D P, G C E, G F C, I G S, I G S all over G D P, P C E, B O P, H U R, I N T, S P, B C I, C C I, Term Spread, F X, Debt all over G D P, and E P U.

The X-axis lists 24 countries grouped by region.

1. S W E A P: Ireland, Italy, Portugal, Spain.

2. E U ex-S W E A P: Austria, Belgium, Estonia, Finland, France, Germany, Netherlands, Slovak Republic, Slovenia.

3. O E C D non-E U: Canada, Chile, Denmark, Hungary, Japan, Korea, Mexico, Poland, Sweden, United Kingdom, United States.

Graph A (X G B o o s t) uses a grayscale scale from 0.00 to 0.16. It shows high feature attribution (darker cells) consistently across almost all countries for Term Spread and C C I. Specific high-impact cells include C P I for Germany and E G S all over G D P for Hungary and the United States.

Graph B (Elastic Net) uses a grayscale scale from 0.0 to 2.5. The attribution is more sparse compared to Graph A. The highest values are concentrated in the Debt all over G D P row, particularly for Italy, Slovenia, Chile, Mexico, and Poland. C C I also shows significant attribution across most countries, while Term Spread shows moderate attribution compared to its dominance in the X G B o o s t model.

C. Pricing of Macroeconomic Risks

The Shapley evidence in Section V.B shows which predictors XGBoost uses and in which regimes, but not the functional shape of the mapping. Partial dependence plots (Friedman (Reference Friedman2001)) recover that shape. We report PDPs for the six predictors with the largest XGBoost attribution in Figure 5: the term-structure slope (Spread), year-over-year log change in consumer prices (CPI), year-over-year first difference of the short-term interest rate (INT), CCI, debt-to-GDP ratio (Debt/GDP), and BCI.

The partial dependence function of a forecaster $ \hat{f} $ with respect to a predictor $ {x}_S\in \mathrm{\mathbb{R}} $ is

(12) $$ {f}_{x_S}\left({x}_S\right)=\int \hat{f}\left({x}_S,{x}_C\right)\hskip0.1em dP\left({x}_C\right), $$

where $ {x}_C $ are the remaining predictors and the expectation is over the testing-period distribution of $ {x}_C $ . Shapley values measure predictor $ i $ ’s contribution to a single forecast; PDPs measure the conditional mean forecast as a function of $ i $ alone.Footnote ¹⁶

The linear specification in Section V.B constrains each predictor’s slope to be constant across the training window. If CDS pricing is stable across regimes, the regime curves in Figure 7 should be parallel shifts; if not, they should differ in slope or shape. The results show that they differ in slope or shape, with the largest divergences occurring during the 2022–2024 tightening cycle.

FIGURE 7Partial Dependence Plots

Figure 7 Regime-averaged partial dependence functions from XGBoost for Spread (Graph A), CPI (Graph B), INT (Graph C), CCI (Graph D), Debt/GDP (Graph E), and BCI (Graph F). Regime windows: Eurozone crisis (2011-04 to 2013-03), low-rate period (2016-04 to 2018-03), COVID-19 (2019-05 to 2021-04), tightening cycle (2022-07 to 2024-06). Horizontal axes are demeaned relative to the cross-sectional mean.

FIGURE 7 Long description

A multi-panel set of six line graphs labeled Graph A through Graph F. Each graph plots Average Prediction (log C D S) on the y-axis against a specific economic variable on the x-axis. Four distinct lines represent different time regimes: Eurozone Crisis (blue dashed), Low-Rate Period (green dotted), C O V I D-19 Shock (red dash-dot), and Tightening Cycle (purple solid).

* Graph A. Term Spread: The x-axis ranges from minus 3 to 4. Most regimes show a sharp upward trend starting at 0, while the Tightening Cycle line remains relatively flat after a sharp drop between minus 1 and 0.

* Graph B. C P I (Y o Y change): The x-axis ranges from minus 4 to 6. The Tightening Cycle shows a steep, continuous positive slope, while other regimes show more moderate increases or plateaus after 1.

* Graph C. Short-Term Rate (Y o Y change): The x-axis ranges from minus 0.2 to 0.3. The Tightening Cycle shows a significant upward step-function trend between 0.1 and 0.2, while the Eurozone Crisis line trends downward.

* Graph D. C C I: The x-axis ranges from minus 20 to 50. The Eurozone Crisis line shows a deep trough between minus 10 and 0, stabilizing at a lower level than other regimes which remain relatively flat near 0.

* Graph E. Debt all over G D P: The x-axis ranges from minus 40 to 60. All regimes generally show a positive correlation, with the Eurozone Crisis and Low-Rate lines showing the steepest step-like increases as debt levels rise.

* Graph F. B C I: The x-axis ranges from minus 10 to 30. The Tightening Cycle shows a unique deep V-shaped trough centered around 5, while the C O V I D-19 Shock line shows a sharp increase after 10.

XGBoost’s response to the term spread (Graph A) is monotone increasing in the Eurozone crisis, low-rate, and COVID regimes, with ranges of approximately 1.3, 0.85, and 0.75 log CDS points, respectively. The monetary tightening-cycle curve lies near $ +0.29 $ over most of the spread range and is no longer monotonic. Similarly, CCI (Graph D) shows a monotone-decreasing Eurozone-crisis curve: the model assigns lower predicted spreads to countries with higher-than-average consumer confidence.

The economic reading of the term spread and consumer confidence is coupled with CPI and INT (Graphs B and C) and suggests that during 2022–2024, a flatter curve reflected the short-rate response to inflation rather than to deterioration in growth. Both inflation and short-term interest rates primarily affect expected CDS premiums in the tightening cycle. Thus, XGBoost treats inflation above the cross-sectional mean and the short-rate response to it as separate channels.

XGBoost’s response to Debt/GDP (Graph E) is approximately monotone increasing across three regimes, with mild nonmonotonicity around the cross-country mean, and flatter in tightening regimes. The linear component is what Elastic Net captures (see Graph B in Figure 5); XGBoost’s secondary attribution picks up the nonmonotone corrections. Graph F shows BCI as nearly flat in the pre-tightening regimes and a threshold drop from $ +0.04 $ to $ -0.11 $ log points during tightening. Elastic Net captures the pre-tightening linearity; XGBoost adds the tightening threshold.

Overall, Figure 7 describes one mechanism. XGBoost fits a different mapping in each regime, and each predictor embodies regime dependence that a linear model cannot capture. Term spread breaks in slope and level during tightening; CPI and INT switch from inert to threshold-active; CCI is monotone during the Eurozone crisis and flat elsewhere; and Debt/GDP and BCI each decompose into a linear component, which Elastic Net captures, plus regime-specific nonlinear corrections that only the tree ensemble reaches.

D. Cross-Country Extrapolation

Sections V.B and V.C establish that XGBoost learns sovereign CDS spreads via state-dependent mappings that the linear class cannot capture. They do not establish where those mappings come from. Two readings are consistent with the attribution and partial-dependence evidence. The mappings might be country-specific—that is, what XGBoost fits when it sees Italy’s macro history and Italy’s CDS history jointly—in which case the nonlinear advantage documented in Section IV reflects country-by-country overfitting that is opaque to a coefficient-based estimator. Or the mappings might be panel-wide—regularities in how macro fundamentals price sovereign risk that hold in common across the sample—in which case the nonlinear advantage reflects a structural feature of sovereign pricing that the tree ensemble learns from cross-country variation.

To distinguish the two readings, we conduct a LOO exercise. For each country $ i $ , we train every model on the remaining 24 countries and forecast country $ i $ ’s daily log CDS spreads from its macroeconomic fundamentals. The MSE is computed over country $ i $ ’s four maturities and the full out-of-sample window. Because cross-country variation accounts for roughly 73% of total panel variance (Section III), the LOO MSE measures how well each forecaster’s predictions for a held-out country preserve the cross-country ordering of CDS levels implied by the training set. Linear methods extrapolate using coefficient combinations fit to the other 24 countries; tree ensembles extrapolate using a partition of the feature space that depends on where predictor values lie in the joint distribution. The LOO exercise is thus an indirect test of which form of extrapolation transports.

Table 3 reports the results. Panel A shows that XGBoost has the lowest mean MSE across all models, at 0.45 compared to 1.11 for Elastic Net, with a median country-level ratio (calculated as $ {\mathrm{MSE}}_{\mathrm{model}}/{\mathrm{MSE}}_{\mathrm{ElasticNet}}-1 $ ) equal to $ -0.53 $ , and wins in 22 of 25 countries. The advantage is not specific to the Elastic Net benchmark. XGBoost has a lower mean MSE than every linear specification, including PLS, which has the best median MSE and the most country wins in the linear class.

TABLE 3Leave-One-Country-Out Forecasting Performance

TABLE 3 Long description

Panel A, titled Cross-Country Extrapolation by Method, lists models in two blocks.

Linear level plus slope block:

* O L S: Mean M S E 0.87, Median 0.77, Wins 13.

* E W C: Mean M S E 0.80, Median 0.80, Wins 11.

* C S R: Mean M S E 0.80, Median 0.76, Wins 11.

* Lasso: Mean M S E 1.10, Median 0.79, Wins 12.

* Ridge: Mean M S E 1.11, Median 0.77, Wins 13.

* Elastic Net: Mean M S E 1.11, Median 0.76, Wins dash.

* P C R: Mean M S E 0.84, Median 0.61, Wins 14.

* P L S: Mean M S E 0.98, Median 0.52, Wins 21.

Nonlinear block:

* S V R: Mean M S E 0.57, Median 0.37, Wins 20.

* R T: Mean M S E 0.59, Median 0.50, Wins 20.

* R F: Mean M S E 0.57, Median 0.52, Wins 20.

* XGBoost: Mean M S E 0.45, Median 0.39, Wins 22.

* N N: Mean M S E 0.64, Median 0.56, Wins 20.

Panel B, titled XGBoost Versus Elastic Net by Country Group, compares Mean M S E:

* O E C D non-E U (12 countries): XGBoost 0.58, Elastic Net 1.55.

* E U ex-S W E A P (9 countries): XGBoost 0.34, Elastic Net 0.65.

* S W E A P (4 countries): XGBoost 0.33, Elastic Net 0.83.

* All (25 countries): XGBoost 0.45, Elastic Net 1.11.

The gap between mean and median MSE for the linear methods indicates that their poor aggregate performance is driven by a handful of countries whose macro fundamentals lie in the tails of the panel distribution, where coefficient-based extrapolation fails most severely.

Among nonlinear methods, XGBoost is the strongest. Its median country-level ratio against Elastic Net ( $ -0.53 $ ) exceeds those of SVR ( $ -0.39 $ ), the regression tree ( $ -0.31 $ ), the random forest ( $ -0.25 $ ), and the NN ( $ -0.16 $ ). The cleavage between gradient-boosted trees and other nonlinear methods, identified in Section IV at the within-panel cross-maturity benchmark, reappears here at the cross-country extrapolation benchmark.

Panel B reports the XGBoost-to-Elastic Net mean MSE ratio for the three country subgroups of Section IV. The XGBoost advantage is largest for OECD non-EU ( $ -0.63 $ ), intermediate for SWEAP ( $ -0.60 $ ), and smallest for EU ex-SWEAP ( $ -0.48 $ ). The OECD non-EU group contains most of the emerging-market sovereigns in the sample, which is consistent with the extrapolation reading; the relatively smaller advantage for EU ex-SWEAP reflects these countries’ macro fundamentals being closer to the panel-typical case.

Overall, the cross-country extrapolation evidence in Table 3 supports a panel-wide reading of the mechanism. Tree-based partitioning fits state-dependent shapes in Section V.C and transports those shapes to held-out countries at a rate that orders by how close each country’s macro fundamentals are to the panel average. This is the cross-country complement to the cross-maturity result in Section IV: XGBoost’s advantage persists after removing country-specific training data because the shapes it fits are panel regularities rather than country-specific fits.