As noted earlier, to examine the economic effects of defence spending in the case of the 13 Latin American countries of our sample, both linear and nonlinear causality tests will be employed. For the linear causality, we apply the Granger-causality test following the Toda and Yamamoto (1995) (hereafter TY). Following Granger’s (1969) development of the causality test, two shortcomings were identified. One relates to the specification bias and the other to the presence of spurious regression. As explained by Engel and Granger (1987), two series are regarded to be cointegrated if the linear combination of these two series is stationary, however, every variable is not. As a result, they stressed that when these two series are non-stationary and cointegrated, the Granger-causal inference will be biased. In addition, in the asymptotic distribution framework, Sims et al. (1990) have produced evidence that when applying the vector autoregressive (VAR) model, we cannot test for the causality of integrated variables in level form regardless if these variables are cointegrated. However, the TY applied in this study overcomes those problems, since it is based on augmented VAR modeling having a modified Wald test statistic (MWALD), which asymptotically as a Chi-square distribution. Moreover, since the test used here (i.e., TY) does not require any pre-tests for cointegration, it presents a better alternative over the traditional Granger-causality test. As Toda and Yamamoto (1995) explain, their test can be applied regardless of whether a series is I(0), I(1), or I(2), non-cointegrated or cointegrated of an arbitrary order producing valid estimates. This is quite suitable in our case, since the economies examined herein reveal several structural breaks^{Footnote 6} in the macroeconomic variables used. Finally, as Kuzozumi and Yamamoto (2000) suggest, the TY test should be preferred when sample sizes are small, since the distortions of the small sample properties remain in low levels, given the potential bias related to the asymptotic distribution of the applied test.

The approach developed by TY employs a modified Wald test for restriction on the parameters of the VAR (*k*), where *k* is the lag length of the model. The TY test is to artificially augment the correct order, *k*, by the maximal order of integration, say *d*
_{max}. Once this is done, a (*k* + *d*
_{max})th order of VAR is measured and the coefficients of the last lagged *d*
_{max} vectors are ignored (Menyah and Wolde-Rufael 2010).^{Footnote 7} Following this approach, the military expenditure (MGDP) and economic growth (GDP) model is given in the following VAR system:

$$\begin{aligned} \ln {\text{MGDP}}_{t} = a_{o} + \sum\limits_{i = 1}^{k} {a_{1i} \ln {\text{MGDP}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {a_{2i} \ln {\text{MGDP}}_{t - j} + } \sum\limits_{i = 1}^{k} {\beta_{1i} \ln {\text{GDP}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {\beta_{2i} \ln {\text{GDP}}_{t - j} } \hfill \\ + \sum\limits_{i = 1}^{k} {\gamma_{1i} \ln {\text{INV}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {\gamma_{2i} \ln {\text{INV}}_{t - j} } + \varepsilon_{1t} , \hfill \\ \end{aligned}$$

(1)

$$\begin{aligned} \ln {\text{GDP}}_{t} = \vartheta_{o} + \sum\limits_{i = 1}^{k} {\vartheta_{1i} \ln {\text{MGDP}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {\vartheta_{2i} \ln {\text{MGDP}}_{t - j} + } \sum\limits_{i = 1}^{k} {\lambda_{1i} \ln {\text{GDP}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {\lambda_{2i} \ln {\text{GDP}}_{t - j} } \hfill \\ + \sum\limits_{i = 1}^{k} {\nu_{1i} \ln {\text{INV}}_{t - i} + } \sum\limits_{j = k + 1}^{d\hbox{max} } {\nu_{2i} \ln {\text{INV}}_{t - j} } + \varepsilon_{2t} , \hfill \\ \end{aligned}$$

(2)

where \(\ln {\text{MGDP}}_{t}\) represents countries’ military spending as a share of GDP, \(\ln {\text{GDP}}_{t}\) countries’ annual growth rate of GDP, and \(\ln {\text{INV}}_{t}\) countries’ gross fixed capital formation as of GDP.^{Footnote 8} Support of the growth-enchasing hypothesis suggests a unidirectional Granger causality running from military expenditure (MGDP) measure to the annual growth rate of GDP \(\left( {\vartheta_{1i} \ne 0\forall_{i} } \right)\). When we have the presence of unidirectional Granger causality running from the annual growth rate of GDP to the military expenditure \(\left( {\beta_{1i} \ne 0\forall_{i} } \right)\), it is evident that the conservation hypothesis holds. On the other hand, when the feedback hypothesis is true, it suggests the existence of a bidirectional Granger causality between both variables, i.e., military expenditure and the annual growth rate \(\left( {\beta_{1i} \ne 0\forall_{i} } \right)\) and \(\left( {\vartheta_{1i} \ne 0\forall_{i} } \right)\). Finally, the absence of Granger causality between the MGDP and the growth rate \(\left( {\beta_{1i} = 0\forall_{i} } \right)\) and \(\left( {\vartheta_{1i} = 0\forall_{i} } \right)\) implies the existence of the impartiality hypothesis. Moreover, to overcome the bias of the omitted variable in growth models, a multivariate framework is applied by incorporating the measures of gross fixed capital formation in addition to military expenditure and GDP.^{Footnote 9} Before we estimate our model and proceed with the TY test, we apply several diagnostic tests (normality, serial correlation, heteroscedasticity, and the CUSUM and CUSUM of square tests) and we resort to the use of four unit roots tests to explore the degree of integration of the variables used in the empirical analysis. Namely, we apply the Augmented Dickey and Fuller (1979) (ADF) test, the Phillips and Perron (1988) (PP) test, the Kwiatkowski et al. (1992) (KPSS) test, and the Zivot and Andrews (1992) test. Finally, although tTY suggest that cointegration is not required in order for the estimates to be valid, we apply the Johansen and Juselius (1990) test as robustness check.^{Footnote 10}

As Brock (1991) illustrates, linear Granger-causality tests can have low power uncovering nonlinear causal relations, which could also exist among variables. In addition, the assumption of linear causality may act as a limiting factor when the true relationship could be nonlinear. Furthermore, since linear methods depend on testing the significance of suitable parameters only in a mean equation, causality in any higher order structure cannot be explored Diks and DeGoede (2001). For that reasons, we examine the nonlinear non-Granger causality. Baek and Brock (1992), to emphasize the limitations of the linear assumption, suggest a nonparametric statistical method for detecting nonlinear Granger causality. Whereas, Hiemstra and Jones (1994) extended their work by introducing a modified test statistic for the nonlinear causality. In addition, Diks and Panchenko (2006) (hereafter DP) develop a new nonparametric test statistic for Granger causality which enable us to avoid the problem of over-rejection observed in the frequently used test proposed by Hiemstra and Jones. As a result, in our paper, we apply the nonlinear causality test proposed by Diks and Panchenko (2006) which can be used to detect possible nonlinear causality relationship between two time series. Following the representation by Diks and Panchenko (2006, p. 1649–1657), let a strictly bivariate process \(\left\{ {\left( {X_{t} ,Y_{t} } \right)} \right\},\) and \(\left\{ {X_{t, } } \right\}\) Granger causes \(\left\{ {Y_{t,} } \right\}\) if current and past values of variable \(X\) contain information on future values of \(Y\) which is not contained in \(Y\) and \(Y_{t,}\). In addition, suppose that \(F_{X,t}\) and \(F_{Y,t}\) represent for time \(t\) the information sets of past observations of \(X_{t, }\) and \(Y_{t,}\), respectively. Then, for a stationary bivariate time series process \(\left\{ {\left( {X_{t} ,Y_{t} } \right)} \right\}, t \in {\mathbb{Z}},\text{ }\), we can represent formally that \(\left\{ {X_{t } } \right\}\) does not Granger causes \(\left\{ {Y_{t} } \right\}\) as

$$\left. {\left( {Y_{t + 1, \ldots ,} Y_{t + k} } \right)} \right|\left. {\left( {F_{X,t} ,F_{Y,t} } \right)\sim \left( {Y_{t + 1, \ldots ,} Y_{t + k} } \right)} \right|F_{Y,t} .$$

(3)

Equation (3) represents the nonlinear Granger-causality test, where ‘~’ indicates equality in distributions and \(k \ge 1\).^{Footnote 11} Then, by following Hiemstra and Jones (1994), we can define the null hypothesis of Granger non-causality as: \(H_{0} :\left\{ {X_{t, } } \right\}\) is not Granger causing \(\left\{ {Y_{t,} } \right\}\). Under the null hypothesis, \(Y_{t + 1}\) is conditionally independent of \(X_{t,} X_{t - 1} , \ldots ,\) given \(Y_{t,} Y_{t - 1} , \ldots\) For finite lags \(l_{X}\) and \(l_{Y}\), the conditional independence can be tested as

$$\left. {Y_{t + 1} } \right|\left. {\left( {X_{t}^{{l_{X} }} ,Y_{t}^{{l_{Y} }} } \right)\sim Y_{t + 1} } \right|Y_{t}^{{l_{Y} }}$$

(4)

where \(X_{t}^{{l_{X} }} = \left( {X_{{t - l_{X + 1} }} , \ldots ,X_{t} } \right)\) and \(Y_{t}^{{l_{Y} }} = \left( {Y_{{t - l_{Y + 1} }} , \ldots ,Y_{t} } \right)\). According to Diks and Panchenko (2006), Eq. (4) represents a statement about the invariant distribution of the \(l_{X} + l_{y} + 1\)-dimensional vector \(W_{t} = \left( {X_{t}^{{l_{X} }} ,Y_{t}^{{l_{Y} }} ,Z_{t} } \right)\), where \(Z_{t} = Y_{t + 1}\). By dropping the time index and assume that \(l_{X} = l_{y} = 1\), we can write \(W = \left( {X,Y,Z } \right)\) as a random vector with invariant distribution of \(\left( {X_{t}^{{l_{X} }} ,Y_{t}^{{l_{Y} }} ,Y_{t + 1} } \right)\). Under the null the conditional distribution of \(Z\) given, \(\left( {X,Y} \right) = \left( {x,y} \right)\) is the same as \(Z\) given \(Y = y\). Therefore, we can restructure (4) by taking into account the joint probability density functions \(f_{X,Y,Z} \left( {x,y,z} \right)\), where its marginals should satisfy the following relationship:

$$\frac{{f_{X,Y,Z} \left( {x,y,z} \right)}}{{f_{y} \left( y \right)}} = \frac{{f_{X,Y} \left( {x,y} \right)}}{{f_{Y} \left( y \right)}}\frac{{f_{Y,Z} \left( {y,z} \right)}}{{f_{Y} \left( y \right)}}$$

(5)

suggesting that \(X\) and \(Z\) are independent when \(Y = y\) for each fixed value of \(y\). Diks and Panchenko (2006) proved that the restated null hypothesis suggests that

$$q = E\left[ {f_{X,Y,Z} \left( {X,Y,Z} \right)f_{Y} \left( Y \right) - f_{X,Y} \left( {X,Y} \right)f_{Y,Z} \left( {Y,Z} \right)} \right] .$$

(6)

By denoting the local density estimators of a \(d_{w}\) variate random vector \(W\) as \(\hat{f}_{W} \left( {W_{i} } \right) = \frac{{\left( {2_{\varepsilon } } \right)^{{ - d_{w} }} }}{n - 1}\mathop \sum \limits_{j,j \ne 1} I_{ij}^{W} I_{ij}^{W} = I\left( {W_{i} - W_{j} < \varepsilon } \right).\),^{Footnote 12} the test statistic can be presented as

$$T_{n} \left( \varepsilon \right) = \frac{{\left( {n - 1} \right)}}{{n\left( {n - 2} \right)}}\mathop \sum \limits_{i} \left( {\hat{f}_{X,Y,Z} \left( {X_{i} ,Y_{i} ,Z_{i} } \right)\hat{f}_{Y} \left( {Y_{i} } \right) - \hat{f}_{X,Y} \left( {X_{i} ,Y_{i} } \right)\hat{f}_{Y,Z} \left( {Y_{i} ,Z_{i} } \right)} \right) .$$

(7)

For a sequence of bandwidths \(\varepsilon_{n} = Cn^{ - \beta }\) with \(C > 0\) and \(\beta \in \left( {\frac{1}{4},\frac{1}{3}} \right)\), the statistic in (7) satisfies

$$\sqrt n \frac{{\left( {T_{n} \left( {\varepsilon_{n} } \right) - q} \right)}}{{S_{n} }}\mathop \to \limits^{d} N\left( {0,1} \right)$$

(8)

where \(\mathop \to \limits^{d}\) denotes convergence in distribution and \(S_{n}\) is an estimator of the asymptotic variance of *T*
_{
n
}(·). Finally, the test statistic in Eq. (7) for nonlinear causality is asymptotically distributed as standard normal and diverges to positive infinity under the alternative hypothesis.^{Footnote 13} Before we estimate the nonlinear model, we must check if the nonlinearity assumption holds, and therefore, we utilize the BDS test (Brock et al. 1987). Under the null hypothesis, the BDS test implies that the variables under examination are identically and independently distributed (i.i.d.). However, under the alternative hypothesis, the test implies that they have linear or nonlinear dependency.