2. Existing findings

Free riding in alliances is closely related to the concept of public goods. To the extent that security created by alliances is non-excludable and non-rival in consumption, we might expect member states to free ride on other allies' expenditures (Olson and Zeckhauser, Reference Olson and Zeckhauser1966; Becker et al., Reference Becker, Benson, Paul Dunne and Malesky2024). There are alternative theoretical accounts of examining alliance spending dynamics, such as the bargaining or US pressure perspectives (Fang and Ramsay, Reference Fang and Ramsay2010; Blankenship, Reference Blankenship2020, Reference Blankenship2021). Yet the insights based on free riding, implicitly or explicitly derived from the public goods model, remain widely shared in the public and academic discourse.

In empirical research, an intuitive strategy to operationalize free riding is treating it as an endogenous spatial lag (Yesilyurt and Elhorst, Reference Yesilyurt and Elhorst2017; see also Franzese and Hays, Reference Franzese and Hays2006). Specifically, when a member state's allies increase their military expenditures, that same country is expected to decrease its own, which makes this dynamic one of negative spatial interdependence in the outcomes. Whether this process is direct (e.g., strategic interaction) or indirect (e.g., indirect spillovers), it is inherent to intergovernmental cooperation (Flores, Reference Flores2011; Drolc et al., Reference Drolc, Gandrud and Williams2019; Cook et al., Reference Cook, Hays, Franzese, Curini and Franzese2020).

Research on NATO allies' defense spending has seen important methodological improvements. Recently, there have been a few studies that, using spatial econometric techniques, specifically address spatial interdependence among NATO allies' military expenditures (George and Sandler, Reference George and Sandler2018, Reference George and Sandler2022; Kim and Sandler, Reference Kim and Sandler2024).Footnote ¹ This research sets itself apart from existing studies on worldwide patterns of military spending, which tend to point to arms race (Goldsmith, Reference Goldsmith2007; Yesilyurt and Elhorst, Reference Yesilyurt and Elhorst2017).

The core takeaway from this research on NATO is negative and statistically significant spatial interdependence arising from the outcomes of other units, that is, defense spending of other allies. While the implied substantive interpretation is that free riding is prevalent and substantial, this interpretation suffers from two significant limitations. First, temporal dependence is omitted. Second, a substantively meaningful estimate of the free riding effect is not provided.

3. The need to directly adjust for spatiotemporal dependence

The vast majority of research based on TSCS data considers either the temporal dimension or the spatial one as a source of dependence. As noted by Cook et al. (Reference Cook, Hays and Franzese2023, 59), only a meager percentage of such articles in top political science journals have been concerned with “model[ing] both temporal and spatial dependence directly.”

Fixed effects (FE) have often been used to adjust for the underspecified dimension, that is, to correct for either within-country or within-period correlation (Cook et al., Reference Cook, Hays and Franzese2023; see also Kropko and Kubinec, Reference Kropko and Kubinec2020). However, by incorporating FE, one imposes a specific structure of correlation (time-invariant spatial clustering in the case of unit FE, or homogenous period-based shocks in the case of time FE); other solutions to deal with spatial correlation, including regional or regional-period indicators (Cook et al., Reference Cook, Hays and Franzese2023), or the method of panel-corrected standard errors (Beck and Katz, Reference Beck and Katz1995), do not resolve the distinctive challenges of two-dimensional data analysis either. More generally, the very logic of correcting standard errors prevents us from any theoretical insight into how space and time interact to define the outcome of interest.

Compared to pre-defined structures, endogenous spatial interdependence implies autoregressive patterns. This means that units influence other units in a diminishing manner, thus allowing this influence to manifest as more diffuse, without it being constrained, for instance, to a pre-defined region (Cook et al., Reference Cook, Hays and Franzese2023). Another important aspect associated with the diffuse character of spatial interdependence in the outcomes is global spillovers, as units affect other units beyond their (direct) neighbors (Yesilyurt and Elhorst, Reference Yesilyurt and Elhorst2017). Yet the potential of spatial interdependence remains underestimated if temporal dependence is overlooked.

Inadequate attention to modeling the spatial and temporal dimensions jointly leads to biased estimates of the dependence parameters (notably, overestimation of the included parameter if the other is omitted), the covariates' coefficients, and their effects (Elhorst, Reference Elhorst2001; Cook et al., Reference Cook, Hays and Franzese2023). Lately, therefore, there has been a notable methodological shift aimed at directly modeling space and time in applied research (Cook et al., Reference Cook, Hays and Franzese2023), refining some previous work (e.g., Elhorst, Reference Elhorst2001). This shift is no longer solely reliant on an empirical perspective (Yesilyurt and Elhorst, Reference Yesilyurt and Elhorst2017), but is backed by theoretical insights and analytical evidence.

Beyond the relevant methodological concerns, researchers are also interested in providing a substantive interpretation of the phenomenon of interest. Yet the lack of attention to spatiotemporal dependence undermines this endeavor, as it increases the risk that the researcher might not accurately identify the precise mechanism driving the observed data, which is both spatially and temporally correlated (Cook et al., Reference Cook, Hays and Franzese2023). Even if the researcher is only interested in accurately modeling contemporaneous spatial interdependence, neglecting the temporal dimension is likely to result in a biased coefficient of the spatial parameter, as well as inaccurate covariates' estimates, thus impairing the validity of inferences and the theoretical scope of the analysis.

There are strong reasons to expect that military spending, like other forms of budgetary expenditures, has considerable persistence over time (Cantekin et al., Reference Cantekin, Elgin and Elveren2023). While certain authors have incorporated such “habit persistence” in their analysis of worldwide defense expenditures, including within a spatial setup (Yesilyurt and Elhorst, Reference Yesilyurt and Elhorst2017; see also Goldsmith, Reference Goldsmith2007), this consideration has not systematically featured the defense economics literature (Flores, Reference Flores2011; Skogstad, Reference Skogstad2016; Kofroň and Stauber, Reference Kofroň and Stauber2024). Importantly, it has been overlooked in spatial analysis of free riding in NATO (George and Sandler, Reference George and Sandler2018, Reference George and Sandler2021; Kim and Sandler, Reference Kim and Sandler2024). An equally significant area of advancement in defense economics is the estimation of the full free-riding effect, which, to the best of our knowledge, has not been estimated in the literature so far.

4. Empirics

We directly model both the spatial and temporal dimensions in our analysis of 27 NATO allies' defense expenditures.Footnote ² We adjust for temporal dependence through the lagged dependent variable (LDV). Relative to the previous work, we update the data on defense spending measured as total spending (in millions, constant 2020 USD) up to 2021 (SIPRI, 2022). We generally follow an empirical strategy motivated by George and Sandler's (Reference George and Sandler2018, Reference George and Sandler2022) work. Our covariates thus include GDP measured in absolute value (constant 2015 USD; World Bank, 2022a) and population (World Bank, 2022b). We use the logarithmic transformations of these covariates as well as of the dependent variable and adjust our models for country FE. While adjusting for unobserved time-invariant influences, country FE also account for fixed patterns of spatial clustering in the outcome (Cook et al., Reference Cook, Hays and Franzese2023), including permanent threats.

Following George and Sandler (Reference George and Sandler2018), we look at several time periods – from 1975 to 2021, from 1991 to 2021, and from 1999 to 2021 –, to test whether there were differences in free riding across these periods. Furthermore, our models include time trends (up to the cubic polynomial), which effectively capture non-linear patterns over time (Carter and Signorino, Reference Carter and Signorino2010), including those of defense spending (Blankenship, Reference Blankenship2020).

Our empirical strategy is based on the spatial autocorrelation (SAC) model and the (spatial) maximum likelihood estimator. Recent methodological literature has identified multisource models as an improved empirical specification strategy (Cook et al., Reference Cook, Hays, Franzese, Curini and Franzese2020, Reference Cook, Hays and Franzese2023). As a reminder, SAC models mitigate the risk that the observed spatial dependence in the outcomes is actually driven by spatial effects in the residuals, a common concern in spatial analysis given the known issue that different spatial sources tend to introduce to data-generating processes similar empirical patterns (Cook et al., Reference Cook, Hays, Franzese, Curini and Franzese2020).

Furthermore, we use a non-row standardized weights matrix representing inverse distances (in hundreds of kilometers) between the capital cities. The relevance of this matrix rests on the notion that geographically closer allies may be expected to assist militarily in the case of a conventional conflict (George and Sandler, Reference George and Sandler2018). Moreover, closer allies are expected to face similar threats and develop similar defense capabilities, thus making a given country benefit more from its ally-neighbors' military spending. Besides, weight matrices derived from physical distances provide an advantage in terms of exogeneity.

We find that, compared to one-dimensional spatial models (Table 2 in the Supplementary MaterialFootnote ³), once we account for temporal autoregressive dynamics (Table 3), our models improve considerably in terms of model fit (e.g., log likelihood improves by a factor of almost 2 to 2.5 for the different time periods). The estimated spatial-lag parameter (ρ) is negative and significant, which corresponds to the results derived from the static spatial models and thus suggests free riding. Importantly, however, the spatiotemporal models (Table 3) offer a meaningfully distinct understanding relative to the exclusively spatial perspective previously established in the NATO free riding literature. They reveal that the impact of other allies' military expenditures on a given member state's defense spending extends beyond a single period and, instead, unfolds over time, ultimately proving to be substantially larger than its contemporaneous effect.

While estimated spatial parameters can offer much valuable information, they are not directly interpretable (LeSage and Pace, Reference LeSage and Pace2009; Whitten et al., Reference Whitten, Williams and Wimpy2019). Based on the operationalization of our variables, interpreting free riding makes sense if we try to answer the following question: what is the expected effect of 1 percent positive shock in all allies' military expenditures, except for a given country, on that country's defense spending for one period and all the periods? Alternatively, what is the average effect of a shock in an ally's military spending on another ally's defense expenditure, again, for one and all the periods?

In the case of a positive shock in defense spending of all allies, except for a specific country, the static models suffer from an inflation bias. More concretely, they indicate a contemporaneous spatial free riding effect, which is much greater than the short-run effect derived from the spatiotemporal models. It is greater by a factor of around three across the periods (e.g., –0.046 percent vs. –0.016 during the 1975–2021 periodFootnote ⁴). The expected effects of an increase in a single ally's military expenditure are similarly inflated in the static models.Footnote ⁵ We also observe that other covariates' coefficient-estimates turn out to be substantially smaller when we account for temporal dependence.

To provide quantitative estimates, absent the LDV, we derive the average effect of free riding from the off-diagonal elements of a full N × N effects matrix taking the form of the contemporaneous spatial multiplier, $d{ \mathbf{y}} = ( {{ \mathbf{I}}-\rho { \mathbf{W}}} ) ^{{-}1}$ (e.g., Franzese and Hays, Reference Franzese and Hays2006; Elhorst and Vega, Reference Elhorst and Vega2013). In other words, our focus is on the average indirect effect across NATO allies, supposing that the allies' defense spending could have endured a positive shock of one unit. In the LDV models, the average spatiotemporal (dynamic) effect is derived from the spatiotemporal multiplier, $d{\mathbf{Y}} = ( {{\mathbf{I}}-\varphi {\mathbf{L}}-\rho {\mathbf{W}}} ) ^{{-}1}$.Footnote ⁶ Based on this derivation strategy, Figures 1 and 2 plot the cumulative response-path estimates of interest during a specified period.Footnote ⁷ The average spatiotemporal effect of free riding (Figure 1), over time, simply accumulates the period-based averages, across NATO countries, of all the indirect effects incurred by an individual ally. For a single country's effect (Figure 2), the response-path dynamic illustrates the accumulated period-based averages, each time calculated across NATO allies, of the mean indirect effects attributable to individual countries.

Figure 1.

The average cumulative spatiotemporal (dynamic) effect of a permanent positive shock in defense spending of a member state's allies over a 15-year period.

Notes: The average effect was derived using the spatial model results obtained based on the 1999–2021 data and considering the allies that were members of NATO in 1999. Confidence intervals of 95 percent were obtained after running 1000 simulations.

Figure 2.

The average cumulative spatiotemporal (dynamic) effect of a permanent positive shock in defense spending of a single ally over a 15-year period.

Notes: The average effect was derived using the spatial model results obtained based on the 1999–2021 data and considering the allies that were members of NATO in 1999. Confidence intervals of 95 percent were obtained after running 1000 simulations.

In the spatiotemporal scenario, the effect of a permanent positive shock in allies' defense spending reverberates not only across space but also time. Specifically, Figure 1 illustrates the average cumulative spatiotemporal effect, over 15 years, for a NATO member state considering a shock of a 1 percent in defense spending of all its allies. Figure 2 illustrates the average spatiotemporal effect, assuming an increase in defense spending of a single ally.

The spatiotemporal free riding effects end up larger than suggested by the static models. The results are generally similar across the different time periods. A permanent positive shock of 1 percent in NATO members' defense spending, except for a given ally, may be expected to lead, over 15 years, to a cumulative decrease of 0.21 to 0.31 percent (depending on the time period considered) in that ally's own military expenditure.Footnote ⁸ The long-run stable-state (LRSS) responses are even more significant in magnitude.Footnote ⁹

Finally, we ran several additional models to check the robustness of our results (Table 4). To guard against the possibility that the US might be driving NATO allies' defense budgets, we tested whether the results stand once the US is dropped from the sample. Furthermore, we tested a matrix that gives special importance to US spending (the US being treated as the closest ally by assigning to the US the same distance-based weight as the one attributed to the actual closest ally). Besides, we included Russian spending, as well as the interaction term between Russian spending and geographical proximity to Russia as additional covariates to better capture the threat environment. Finally, we incorporated host-nation support (HNS) for US military presence in host countries as a proxy for US pressure.Footnote ¹⁰ None of these variations affect the robustness of our main findings.