Revisiting the guns-butter tradeoff: a wavelet analysis of the US and Britain

ABSTRACT This study applies continuous wavelet analysis to examine defense-education and defense-healthcare relations in the US and Britain. It discovers four empirical patterns that have not been shown in the existing literature. First, the defense-welfare tradeoff rarely occurs at cycles less than 6 years and hence is not a short-run phenomenon. Second, very noticeable bilateral tradeoffs between education and defense can be detected. The effect, however, is more pronounced in the direction from education to defense. Third, the defense-welfare tradeoff is much less likely to occur in defense-healthcare relations than defense-education relations. Fourth, a structural change in the defense-healthcare relationship occurred during the 1960s, after which the defense-healthcare connection became primarily complementary. Together, the established patterns question the assumption that the defense sector has a dominant power in budget allocation. They also raise new theoretical and empirical questions demanding future research efforts.

the defense budget to welfare programmes that are more growth-prone. Despite the academic and public enthusiasm for the topic, the empirical literature on the defense-welfare relations has yielded highly mixed results.
I argue that the ambiguity in the present literature reflects researchers' diversified responses to four methodological challenges on the way to a solid understanding of the defense-welfare relationship. The first is about which welfare programmes to be included in an analysis. The second concerns the issue of nonlinearity in the time domain. The third is about the varied defense-welfare relationship in the frequency domain. The fourth is the choice between a time-series and a panel design. This study revisits the topic of the guns-butter tradeoff by paying due attention to these methodological challenges. In specific, it applies a continuous wavelet analysis to examine the defense-education and defense-healthcare relations in the US and Britain with the assistance of century-long time-series observations. Several empirical patterns have been discovered. First, the crowd-out effect rarely occurs at cycles that are less than 6 years. Thus, the defense-welfare tradeoff is not a short-term phenomenon. Second, a highly noticeable tradeoff between education and defense can be detected. The effect, however, is more pronounced in the direction going from education to defense. Third, a tradeoff is much less likely to be seen in the defense-healthcare relation than in the defense-education relation. Fourth, there is a structural change in the defense-healthcare relation during the 1960s, after which the relation becomes primarily complementary. Overall, the empirical patterns established in this research tend to question the assumption that the defense sector has a dominant power in budget allocation.
The rest of this study is organized as follows. Section 2 reviews the empirical literature on the guns-butter tradeoff. Section 3 develops a research design by reflecting on the present literature. Section 4 introduces data and methods in detail. Section 5 reports and summarizes findings. Section 6 concludes by discussing valuable directions for future research.
The empirical literature on the defense-welfare tradeoff Empirical research on the defense-welfare tradeoff starts with a few developed economies. The pioneering work by Russett (1969), for instance, reports that military spending in the US can crowd out education and healthcare expenditures. According to the same research, such a defense-dominating tradeoff can also be found in Britain and France, but not in Canada. Focusing on several subcategories of healthcare spending, both Peroff (1976) and Peroff and Podolak-Warren (1979) confirm the existence of defensehealthcare tradeoff in the US Follow-up research, however, generate contradictory results. Caputo (1975), for instance, reports a complementary relationship in which defense and welfare spending increase or decrease simultaneously. According to the research, US defense spending is shown to be positively correlated with healthcare spending but not related to education spending. In Australia and Sweden, defense spending always correlates positively with education and healthcare expenditures. Only in Britain, defense spending is shown to squeeze out resources from education but have no significant impact on healthcare programmes. It is interesting to note that even Russet himself later argues that US defense spending does not affect its education and health expenditures if the question is assessed in a marginal sense (Russett 1982).
Focusing on the overall welfare spending, Domke, Eichenberg, and Kelleher (1983) report no crowd-out effect from defense to welfare in the US, Britain, France, and Germany. In the same vein, Heo and Bohte (2012) show no evidence for a tradeoff between defense and welfare in the US More recently, research on G7 (Zhang et al. 2017) and OECD (Lin, Ali, and Lu 2015) countries provide evidence of a complementary relationship between defense and welfare expenditures using panel data. In sum, findings from developed economies are mixed and inconclusive.
Results from developing and newly industrialized economies are also conflicting. Take Latin America, for instance. Ames and Goff (1975) first report defense and education expenditures to be complementary in the region. Verner (1983) claims that the complementary relationship between defense and education applies only to some regional countries. Apostolakis (1992) argues, however, that there is overwhelming evidence in support of a crow-out effect going from defense to social programmes. Asia is another region that has attracted attention from explorers of the guns-butter tradeoff. Rather than treating regional countries as a whole, studies of the area often focus on a single country (for a noticeable exception, see Coutts et al. 2019). Turkey is one such country that has received particular attention. Ozsoy (2002), for instance, shows that an increase in defense spending reduces both education and healthcare expenditures. Follow-up research by Yildirim and Sezgin (2002) confirms the defense-healthcare tradeoff. It, however, shows the defense-education relation in Turkey to be complementary. The rise of China as a global power makes the country another hotspot for the tradeoff explorers. Wang (2014) reports a crowd-out effect going from defense to the combined spending on education, science, culture, and healthcare. Xu et al. (2018), nevertheless, claims to detect a crowd-out effect going from education to defense.
Finally, several studies intend to draw a general conclusion on the defense-welfare tradeoff by pooling many countries into single research (Dabelko and McCormick 1977;Deger 1985;Hess and Mullan 1988;Fan, Liu, and Coyte 2018). Again, no general agreement seems to emerge from this strand of research by now. For instance, Deger (1985) shows that defense spending can crowd out education spending based on a sample of 50 developing countries. The result, however, is questioned by later research. After studying another sample of 77 developing countries, Hess and Mullan (1988) conclude that defense and education expenditures correlate positively with each other.

Reflection and research design
I argue that the contradictory results of the present literature are the outcome of researchers' diversified responses to four major methodological issues involved in the inquiry into the potential guns-butter tradeoff. A quick inspection of the existing research reveals that some of the methodological issues have not been well received in the literature.
First, the tradeoff between defense and welfare expenditures might be conditioned upon which welfare programmes to be included for an investigation. For instance, military spending may squeeze out budgetary recourses from education but may not do the same to healthcare. The varied choices across different research on which welfare programmes or their combinations to be considered thus contribute to the ambiguous results shown in the existing literature.
Second, the tradeoff between guns and butter, if it exists, might be nonlinear due to structural changes over time. For instance, a tradeoff between defense spending and a welfare programme may only exist in some periods of history but not others. If some researchers focus on a period that coincides with the occurrence of a tradeoff, a crowd-out effect is then detected and reported. However, other researchers will report no evidence for such an effect if they choose to study another period that witnesses no tradeoff. Partially due to the lack of long time-series data, nonlinearity has not yet been well treated in the present empirical literature.
Third, the relationship between defense and welfare expenditures might vary in the frequency domain. For instance, some scholars have argued that governments can, in the short run, avoid any guns-butter tradeoff by either levying additional taxes or issuing bonds to cover the budgetary deficit. Nevertheless, such practices will have far-reaching implications for the entire economy and influence the allocation of budgetary resources between defense and non-defense items in the long run. Therefore, the suggested tradeoff between defense and welfare is more likely to be seen in the long run than in the short run. This concern over the diversified defense-welfare relations across cycles has not been satisfactorily modelled in the previous research. The reason is that traditional time-domain approaches to time series analysis, the method powerhouse of the present literature, cannot differentiate dynamic patterns at different cycles.
Finally, early research on the tradeoff often applies time-series methods to a small number of countries. Such a research design requires long series for each country and limits the number of candidate countries that can be studied. Due to data availability, later research begins to use panel date that primarily utilizes cross-national variance to develop generalized statements on the relationship between defense and welfare budgets. As Mintz and Stevenson (1995) noted, however, early findings based on this panel strategy depend critically on the homogeneity of the included countries. Although development in panel data analysis has provided new techniques intended to deal with both time-invariant and time-variant heterogeneity, efforts to include both cross-sectional and time-series variations in an analysis inevitably encounter practical and theoretical complications, many of which are not readily solvable. Practically, the attempt to include more countries in an analysis will run into the obstacle of minimal data availability in the time-series dimension. As a result, researchers are forced to narrow their research horizon for over-time dynamics and hence are unable to deal with the issue of nonlinearity as discussed above. In theory, the additional complication in the panel context can be best illustrated in the case of unit root test. Unlike in the time-series context, unit root tests in the panel context face a difficult choice between assuming a series for all units to be stationary and assuming it is stationary only for some of the units. Although the former is more reasonable in theory, it is unlikely to be the case in practice. Thus, the latter is more often used in applied work, but its implications are elusive. As a result, it is not hard to imagine that some series that are shown to be nonstationary in a time-series design might appear to be stationary in the panel context or vice versa. Since the results of unit root tests fundamentally decide the later operations of analysis in both the panel context and the context of standard time-series analysis, the choice between time-series and panel designs might have further contributed to the contradictory results of the present literature.
It is not hard to see that a set of solutions to the above-mentioned methodological challenges is needed for serious research on the guns-butter tradeoff. Paying due attention to the challenges, solutions adopted in the current study are specified as follows. To the concern of which welfare programmes to be included, the answer is relatively simple. Education and healthcare are the two most frequently examined welfare spendings because they are closely related to the quality of labour input, and therefore, have important implications for the well-being of an economy. This study thus examines the defense-welfare relationship by focusing on potential defense-education and defensehealthcare tradeoffs. Some previous research combines education and healthcare expenditures with other welfare expenditures such as public housing and unemployment benefits to develop a mega category and then examines the correlation between defense spending and the overall spending of the mega type. The current research does not follow such a strategy for two reasons. First, that strategy ignores the varied ability across welfare programmes in the competition with the defense sector for more budgetary resources. Second, a mega category of welfare spending error on the side of an artificially created tradeoff between defense and non-defense budgets.
This study adopts the method of continuous wavelet analysis to deal with the second and third methodological challenges. The availability of long time series provides a prerequisite to better deal with the second challengenonlinearity in the time domain. To solve the problem from a parametric perspective, however, requires methods such as threshold or regime-switching models (Tong 1983;Hamilton 1994;Tsay and Chen 2018). Studies in such directions are rarely seen in the empirical literature on the defense-welfare tradeoff. Regarding the third challenge, although discussions on the difference between short-run and long-run defense-welfare relations can be found in previous research, adequate modelling of the issue with frequency-based methods, such as spectral analysis, has not been done in the existing literature. It is important to note that purely time-based and purely frequency-based approaches cannot provide useful information in the other domain. Hence, an approach that can process information from both dimensions is needed. Continuous wavelet analysis is a time-frequency approach that maps information contained in a time-series signal onto the time and frequency domains simultaneously. Due to the merit, it has gained popularity in fields such as physics, engineering, neuron science, and finance. The technical details of the method are introduced in the next section.
The final methodological issue is about the choice between time-series and panel designs. This study examines defense-education and defense-healthcare relations in the US and Britain with a time-series approach. Two concerns drive the decision to use on a time-series design. First, pooling too many countries will significantly decrease the over-time availability of data and limit our ability to search for any crowd-out effect from a dynamic perspective. Second, panel methods that can simultaneously deal with time-domain nonlinearity and frequency-domain cyclical information are underdeveloped. This research focuses on the US and Britain for three reasons. First, the empirical literature on defense-welfare tradeoff starts with the two countries and have examined them extensively. Unfortunately, most of the research about them was done several decades ago and didn't benefit from the methodological advancement in recent years. As a result, findings of those studies might suffer significantly from methodological issues that were not even known when the studies were done. Thus, a reexamination of the cases with up-to-date technology is needed. Second, as the consecutive global hegemony in modern history, Britain and the US provide a category of their own. Empirical patterns shared by the countries might offer us new clues for valuable directions of research. Third, data collection efforts on the US and Britain make long series available to researchers, which have produced fruitful results (Barro 1987;Ward, Davis, and Lofdahl 1995;Smith 2020). Such a luxury in terms of data availability does not usually apply to other countries. This study applies continuous wavelet analysis to the US and British data mentioned above. Morlet and his colleagues (Morlet et al. 1982a(Morlet et al. , 1982b first developed wavelet analysis based on the research of Gabor (1946) to provide time-localized frequency analysis of signals. Although the method originates in geophysics research, mathematicians have long been the principal contributors and promoters of the approach (Woodward, Gray, and Elliott 2017). Today, the method has gained its momentum in many scientific disciplines such as physics, engineering, environmental science, and life science (Crowley 2007;Walker 2008;Addison 2016). Among social scientists, economists led the use of wavelet-based methods in the 1990s (Ramsey, Usikov, and Zaslavsky 1995;Ramsey and Lampart 1998). Since then, it has become a popular instrument for the analysis of financial and economic series where nonlinearity and nonstationarity are normally encountered. Recently, Aguiar-Conraria, Magalhães, and Soares introduced the method to scholars of political science and public policy in a series of research articles (2012,2013). The remainder of this section reviews critical concepts and measures of continuous wavelet analysis involved in this study. It is important to note that comprehensive treatment of the method can be found in Vidakovic (1999), Percival and Walden (2000) as well as Heil and Walnut (2006). Hubbard (1998) provides a non-technical introduction to wavelet analysis.

Data and method
By translating with the time shift parameter t and dilating with the scale factor parameter s of a mother wavelet c t,s , researchers can generate a series of daughter wavelets.
For a discrete series x t , its continuous wavelet transform (CWT) with respect to a chosen mother wavelet c t,s is defined as where * is the conjugate form. If the mother wavelet is a complex one, the CWT can equivalently be expressed in the following polar form where i is the imaginary unit and f x (t, s) is the phase angle on a complex plane. For two time series x t and y t , their cross-wavelet transform is defined as The strength of the connectivity between the two series can be gauged by their wavelet coherency R xy (t, s) = |l{W xy (t, s)}| l{WPS x (t, s)}l{WPS y (t, s)} where l{ · } denotes a smoothing operator. WPS x (t, s) and WPS y (t, s) are the wavelet power spectrum of series x t and series y t , respectively. Take series x t , for instance. Its wavelet power spectrum is defined as The direction of influence between the two series can then be inferred from the crosswavelet phase angle f xy (t, s) = tan −1 I(W xy (t, s)) R(W xy (t, s)) where I stands for the imaginary part and R stands for the real part. If f xy (t, s) [ 0, p 2 , the two series move in phase and series x leads series y. If f xy (t, s) [ − p 2 , 0 , the two series are also in phase, but series y leads series x. The two series move out-of-phase with series y leading series , the two series also move out-of-phase, but series x leads series y. This research adopts the Morlet wavelet for CWT. The wavelet is defined as where v 0 is dimensionless frequency. Following the existing literature, it is set 6 to strike an ideal balance between time and frequency localization. The next section applies continuous Morlet wavelet analysis to the data introduced at the beginning of this section. The analysis is operated with WaveletComp, an R package developed by Roesch and Schmidbauer (2018). Figure 1 presents the US federal expenditures on defense, education, and healthcare as a percentage of nominal GDP. Characterizing features of the series are summarized as follows. Regarding defense spending, there are two noticeable spikes. The first is related to World War I, in which the federal defense budget reaches 17% of the GDP. The second spike is associated with World War II, and the federal defense budget reaches 35% of the GDP in 1945. The Korea War pushes the defense budget to 11% of the GDP, the third height for the entire period of 1902-2019. However, a vibrated declining trend of defense spending can be observed after that. Among the three budget programmes in consideration, eduction receives the least support from the federal government. Except for 1948-1949, 1972, and 1975-1981, federal spending on education is normally below 1% of the GDP. Federal spending on healthcare has experienced two distinct periods. From 1902 to 1966, it lingers at a deficient level and never reaches even half percent of the GDP. From 1967 onward, it shows an upward linear trend. Since its pass of 1% of the GDP in 1969, it has passed 2%, 3%, 4%, and 5% in 1981, 1992, 2003, and 2009, accordingly. In recent years, the linear increase in healthcare spending begins to slow down and stabilizes at a level barely below 6% of the GDP.

Results
Before proceeding to the wavelet analysis of the guns-butter tradeoff in the US context, it might be interesting to see what conventional time-series methods say about the data series. The upper half of Table 1 presents unit root tests for defense, education, and healthcare budgets. It is important to note that, in opposition to the ADF test (Dickey and Fuller 1979), the KPSS test (Kwiatkowski et al. 1992) has the null hypothesis that a series is stationary. The results confirm that defense spending is stationary in level and education spending is stationary after taking the first difference. Although we can be sure that healthcare spending is nonstationary in level, the results on its first difference are inconclusive, indicating that further difference might be needed. Because defense spending is integrated of order 0 while education spending and healthcare budgets are integrated of order 1 or higher, we cannot use cointegration methods to study the gunsbutter tradeoff as we specified (Hamilton 1994;Pfaff 2008). It is still mathematically possible to run regression-based models between defense spending in level and other differenced spending series. Such a practice is substantively and theoretically problematic, however. First, empirical results with such regression-based models cannot answer the theoretical questions we ask. Second, the theoretical foundation of such empirical analysis is elusive. In our case, why do policymakers in the US intend to trade-off between defense budget, the growth rate of the education budget, and the acceleration of healthcare spending? These inconvenient facts with conventional time-series methods provide good motivation to analyze our research topic from a wavelet perspective. Unlike traditional time-series approaches, wavelet analysis can be directly applied to nonstationary time series (Aguiar-Conraria and Soares 2014). Figures 2 and 3 depict the wavelet coherency of the defense-education nexus and the defense-healthcare nexus in the US, respectively. As defined by Equation (5), wavelet coherency ranges from 0 to 1, and a higher value indicates greater strength of the connectivity. To test the statistical significance of the defense-welfare connections, wavelet coherency is assessed against surrogate data generated by 5000-time Monte Carlo simulations of the red-noise process (Torrence and Compo 1998). The contour lines delineate the 5-percent level of significance. The arrow sign indicates the cross-wavelet phase angle  as defined by Equation 7, which is used to infer the existence and direction of tradeoff effect. The translucent area is the cone of influence (COI). It is introduced to deal with the edge effect accompanied by CWT. It is important to note that results shown in the COI are artifacts and should be ignored. For the convenience of interpretation, cycles in years are reported for the frequency dimension. A couple of dynamic structures regarding the defense-education nexus can be detected according to Figure 2. First, there is no evidence for a tradeoff between federal defense and education expenditures for short cycles. For the band of cycles that are less than 6 years, the nexus between defense and education is either significantly complementary or not statistically significant. Second, observations of tradeoff gradually emerge as moving to longer periods. For 6-10 year cycles, complementary relation between defense and education only occurs from the late 1990s to the early 2010s. For the same band of cycles, a crowd-out effect going from defense to education can be seen in the 1940s, 1950s, and 1980s. Third, a very pronounced tradeoff between defense and education can be detected at cycles longer than 10 years. For 12-24 year cycles, a crowd out effect going from education to defense can be seen from the 1920s to the early 1950s and again from the second half of the 1960s to the early 1980s. Note that there seems to have perfect synchronization between defense and education expenditures at 3-5 year cycles in the early1920s. However, we should refrain from getting into it because the region overlaps heavily with the COI. This principle applies to all of the upcoming analysis.
The connections between federal defense and healthcare expenditures in the time and frequency domains are represented in Figure 3. A quick comparison with Figure 2 immediately reveals the sparse distribution of statistically significant zones here. Thus, connectivity, positive or negative, is much less an issue for the defense-healthcare relations. In fact, we see little evidence for either a crowd-out effect or a complementary effect at all cycles before the mid of 1960s. From the mid of 1960s to the early 2000s, complementary connections between defense and welfare expenditures can be detected at 3-10 year cycles. From the 2000s onward, however, it seems that healthcare spending begins to crowd out the spending on defense at 3-4 year and 16-year cycles. Figure 4 shows British defense, education, and healthcare expenditures over time. Defense spending at the beginning of the nineteenth century was characterized by a plateau due to the Napoleonic Wars. For that period, the defense budget accounted for about 10 percent of the nominal GDP. That plateau, however, is dwarfed by the two spikes associated with the two world wars. Defense spending reaches 45% of the GDP during World War I and 52% during World War II. After the Korea War, it has experienced a linear decline and recently stabilizes at about 1.75% of the GDP. A general upward trend can be detected in both education and healthcare expenditures. The education spending first passed half percent of GDP in 1878. Though bumped during the two world wars, it continues to increase until 1975. After reaching the historical apex of 6 percent in 1975, it vibrates within the bound between 3.5% and 6% of the GDP. A comparison with the federal education budget in the US shows that the British central authority has allocated more resources on education in the modern age. Such a difference and its implications, however, should not be overinterpreted for two reasons. First, the US has a federalist system, in which local governments take more responsibility for public education. Second, private schools in the US have a more significant role to play than their European counterparts. British healthcare budget first took off in 1921 by doubling itself from 0.4% to 0.8% of the GDP. Since then, it enjoys a nearly linear growth until reaching the all-time height of 7.5% in 2010. After that, it has stabilized within the 7-7.5 percent bound.
As in the US case, we first look at what conventional time-series methods would say about the data. The lower half of Table 1 presents the unit root tests for the defense, education, and healthcare budgets in the British case. The results confirm that defense spending is integrated of order 0 while education and healthcare budgets are integrated of order 1. As a result, we cannot use cointegration methods to examine the tradeoff between guns and budget. It is still mathematically reasonable to run regressions between defense spending in level and other difference-transformed budgets. However, similar to the US case, such a practice is not substantively and theoretically sensible. Again, conventional time-series methods have a stroke of bad luck with our data, and hence we proceed to wavelet analysis.
The wavelet coherency between defense and education expenditures in Britain is shown in Figure 5. For cycles less than 6 years, there is no evidence for a crowd-out effect between the two series. In fact, the evidence points to either no relation or a complementary relation for such short-run cycles. For cycles longer than 6 years, however, pronounced crowd-out effect can be observed, where the direction of influence can be either going from defense to education or going from education to defense. On the one hand, defense spending crowds out education spending in three time-frequency zones. The first is between 1910 and 1930 at the 7-10 year cycles. The second is between 1880 and 1920 at the 28-40 year cycles. The third is between 1850 and 1870 at 60-64 year cycles. On the other hand, spending education also crowds out the defense budget. From the late 1890s to the early 1960s, education spending squeezes out defense spending at 8-28 year cycles. Overall, the bilateral crowd-out effect is more pronounced in the direction going from education to defense. Figure 6 provides the wavelet coherency between defense and healthcare expenditures in Britain. A quick comparison with Figure 5 shows the relative scarcity of statistically significant zones here. The relationship between the two series is seen to be primarily complementary for cycles of less than 6 years. Two exceptions in modicum do occur in the 1870s and the 1980s, where defense spending crowds out the healthcare budget. For cycles longer than 5 years, a structural change seems to happen in the 1960s. From the 1910s to the 1960s, tradeoffs between defense and healthcare can be detected. In most cases, it is the defense spending that crowds out the healthcare budget. The only exception is a crowd-out effect going from healthcare to defense occurs in the 1920s at the 14-year cycle. From the 1960s onward, however, no tradeoff between the defense and healthcare expenditures can be detected. Instead, they are only shown to be complementary to each other at the 5-10 year cycles. To sum, the above analysis of continuous wavelet coherency has disclosed several empirical patterns in the American and British regarding the defense-welfare relationship. First, the crow-out effect between defense and welfare programmes rarely occurs at cycles of less than 6 years. It implies that guns-butter tradeoff is unlikely a short-term phenomenon. This result echoes the call of Mintz and Stevenson (1995) that research on the defense-welfare nexus should pay more attention to their long-term linkages. Second, although noticeable bilateral tradeoff between defense and education budgets can be detected, education spending shows a more pronounced ability to crowd out defense expenditure. Third, the tradeoff between budget programmes is more likely to occur in the defense-education relation than in the defense-healthcare relation. Fourth, there is a structural change in the relationship between defense and healthcare spendings during the 1960s, after which the defense-healthcare nexus becomes primarily complementary. Overall, the patterns mentioned above tend to question the assumption that the defense sector has a dominant power in the allocation of budgetary resources.

Conclusions
The contribution of this research is twofold. First, it establishes several empirical patterns concerning the defense-welfare relationship in the US and Britain that have not been discovered in the existing literature. Second, it demonstrates continuous wavelet analysis as a useful instrument in studying the budgetary consequences of defense spending. Theses encouraging results raise new questions and highlight valuable directions of research.
The empirical patterns revealed on the time-frequency plane give rise to four theoretical questions demanding further research efforts. First, the crowd out between defense and welfare expenditures is not a short-term phenomenon. The postpone of budgetary tradeoffs is possible because the government might choose to levy additional taxes or issue bonds to cover the budgetary deficit. As Mintz and Stevenson (1995) point out in a review article, such practices to postpone the decision will have far-reaching implications for the entire economy and influence the allocation of budgetary resources between defense and non-defense items in the long run. They thus call for more theoretical research to explore the causal mechanisms behind long-term guns-butter tradeoff, which might be indirect and conditioned upon intermediate variables. An inspection of the past 25-year research, however, reveals how little has been done on this front. This research raises the question again and demands more theoretical attention to the long-term nexus between defense and welfare expenditures. Second, this study also shows that defensewelfare tradeoff is more likely to occur between defense and education spendings than between defense and healthcare spendings. This finding asks for some theoretical research on why that is the case. Third, this study shows a structural change in the defense-healthcare nexus in the 1960s in both the US and Britain, after which the relationship becomes primarily complementary. It is interesting to know what social, political, or economic developments at the time help explain the epochal change. Finally, from the perspective of collective action, interest groups of the education sector are thought to be weaker than pro-defense groups that are well organized and financed. The finding of a more prominent crowd-out effect going from education to defense is thus counterintuitive. Together with other results, this finding tends to challenge the common assumption that the defense sector enjoys a dominant influence over budgetary decisions. Explanations of the unexpected weakness of the defense sector, presumably borrowing strength from the recent development of the public choice literature, is an area that deserves more attention.
Empirically, there are two directions that worth our efforts. First, it is not clear if findings based on the US and Britain are unique to their club of global hegemony or generalizable to other great powers. Therefore, applying the current research design to examine the defense-welfare relationship in other great powers might be an important direction to pursue. Second, continuous wavelet analysis has proved itself as a powerful instrument for time-frequency analysis. Since defense spending has significant policy implications for an array of macroeconomic variables, such as inflation and growth, to which the issue of cycles is an integral part, it is interesting to see more wavelet analysis of other policy consequences of the defense budget.

Disclosure statement
No potential conflict of interest was reported by the author.

Funding
Xi'an Jiaotong-Liverpool University provides the financial support for the open-access publication of this article.

Notes on contributor
Yu Wang received his Ph.D. in politics from New York University in 2009. Since then, he has worked at the Chinese University of Hong Kong (2009Kong ( -2015, the University of Nottingham Ningbo China (2015China ( -2017, and the University of Iowa (2018-2020). He is currently an assistant professor of international studies at Xi'an Jiaotong-Liverpool University. He has published 15 refereed articles.