Vector autoregressive (VAR) models have become a widely used tool for analyzing multivariate time series data due to their simplicity and flexibility in capturing the linear interdependencies among multiple time series (Sims, 1980). VAR models are extensively applied across various fields, including macroeconomics, finance, public health, political science, engineering, and climatology. However, when it comes to economic data, VAR models have notable limitations. For instance, they do not adequately account for instantaneous effects or structural relationships between variables, and they fail to address the time-varying nature of volatility, where variance may fluctuate over time rather than remain constant.

To overcome the limitations of the standard VAR models and incorporate structural relationships, such as impulse responses or forecast error variance decomposition (Cooley and LeRoy, 1985; Evans and Kuttner, 1998), the structural vector autoregressive (SVAR) models were developed. The SVAR model introduces contemporaneous relationships among the variables, providing a more fluent understanding of the underlying data structure. See Bernanke (1986), Waggoner and Zha (1999). For a recent development, we refer to Breitung et al. (2004), Primiceri (2005), Fry and Pagan (2011), Lanne et al. (2017), and the papers cited therein.

In addition to the structural relationships among variables, economic and financial time series often exhibit periods of high volatility, with variance that changes over time. This has led to the development of models specifically designed to capture the volatility structure, among which the generalized autoregressive conditional heteroskedasticity (GARCH) model is one of the most prominent (Engle, 1982; Bollerslev, 1986). These models have been particularly effective in the field of finance, capturing the phenomenon of volatility clustering. Until recently, several multivariate GARCH models have been developed and widely studied, including the VECH model (Bollerslev et al., 1988), the BEKK model (Engle and Kroner, 1995), the DCC model (Tse and Tsui, 2002), and the GO-GARCH model (van der Weide, 2002). For a comprehensive review, see Bauwens et al. (2006).

Considering the unique characteristics of economic data, this paper explores the SVAR-GARCH model. Notably, by integrating the strengths of SVAR and GARCH models, the SVAR-GARCH framework serves as a powerful tool that simultaneously accounts for structural relationships among variables and time-varying volatility patterns. This combination has gained increased attention in recent research. For example, Milunovich and Yang (2013) address the identification problem in SVAR models with ARCH effects, while Lütkepohl and Netšunajev (2017) review SVAR models combined with various heteroskedastic models. In this study, we adopt the generalized orthogonal GARCH (GO-GARCH) scheme for handling SVAR-GARCH models, as it aligns particularly well with the inherent structure of the SVAR model and the independent component analysis (ICA) method, which is a conventional approach for analyzing SVAR models.

In handling the SVAR-GO-GARCH model, ICA is a natural fit since the GO-GARCH model is composed of linear combinations of independent GARCH components, which aligns with ICA’s objective of decomposing observations into independent sources. ICA is a computational technique with high speed used to separate multivariate signals into independent, non-Gaussian components. Originally developed for tasks such as blind source separation, ICA has a broad history of successful applications across various fields, such as image processing, biomedical signals (e.g., electroencephalograms (EEG) and functional magnetic resonance imaging (fMRI)), and the cocktail-party problem. For further details, refer to Stetter et al. (2000), Diamantaras and Papadimitriou (2005), and Shwartz et al. (2008). ICA has also been applied in the analysis of SVAR models by several authors, such as Hyvärinen et al. (2010) and Moneta et al. (2013), who demonstrated how the identification problem in SVAR models can be resolved through non-normality assumptions. See also Lanne et al. (2017) and Gouriéroux et al. (2017) for relevant theories and methods.

Since Page (1955), change point detection has been a core focus in time series analysis for decades, as time series data frequently experience structural or regime changes that can significantly impact both analysis and forecasting. Accurately identifying these changes is crucial, as overlooking them can lead to incorrect model specifications and misinterpretation of results. The cumulative sum (CUSUM) test has gained popularity as a conventional method for detecting change points, valued for its ease of implementation and versatility across various circumstances. Over the decades, the CUSUM test has been extensively studied and further developed across different time series models. See Inclán and Tiao (1994), Lee et al. (2003), and Lee et al. (2004). More recently, Lee (2020) introduced a location and scale-based CUSUM (LSCUSUM) test, a simplified version of the method proposed by Oh and Lee (2019), which relies solely on observations and residuals.

The LSCUSUM test has proven to be both convenient and adaptable to ARMA and GARCH-type models, with potential for hybridization with other methods, such as support vector regression (SVR) (Lee et al., 2020). In particular, to handle the change point test, Lee et al. (2023) suggested using the LSCUSUM method along with ICA specifically for SVAR models. However, their approach suffers from significant size distortion in time series with high volatility. Motivated by this limitation, we adapt the LSCUSUM test from Lee et al. (2023) to the SVAR-GO-GARCH model, offering a more robust analysis of financial time series data that accounts for both structural changes and volatility dynamics.

This paper is organized as follows. Section 2 introduces the SVAR-GO-GARCH model and proposes a new test for detecting change points in the proposed model, illustrating the change point detection algorithm using ICA. Section 3 outlines the simulation settings and results. Section 4 presents the results of the real data analysis. Finally, Section 5 provides concluding remarks.

2.1. SVAR model

Let {Y_t} be the d-dimensional VAR model of order p ≥ 1 satisfying

where Γ₁, . . ., Γ_p are d × d parameter matrices and u_t ~ (0, ∑_u) are the iid errors with Eutut′=Σu, a positive definite covariance matrix, and Eutut+h′=0 for h > 0. Notably, this model captures the lagged effects among variables but does not account for their instantaneous interactions. However, in real-world applications, variables often have immediate influences on each other. To incorporate these contemporaneous effects, a SVAR model is considered as an extension of the VAR model in (2.1), given by:

where B is a d × d matrix capturing the contemporaneous relationships between variables. Typically, the components of ε_t are assumed to be instantaneously uncorrelated, namely, Σɛ=diag(σ12,…,σd2). To ensure no influence of variables on their own, the diagonal components of B are set to 0, and the matrix B is structured to be acyclic to establish a causal ordering. This formulation allows the model to capture both the instantaneous and lagged interactions among the variables.

Remark 1. While we focus on the SVAR model with adding the term BY_t term to emphasize the relationships between variables, another approach to the SVAR model can center on the structural errors. For any nonsingular matrix P satisfying ∑_u = PP′, multiply both sides of (2.1) by P⁻¹ and then move the (I − P⁻¹)Y_t term to the right side. This yields Y_t = (I − P⁻¹)Y_t + P⁻¹A₁Y_t⁻¹ + · · · + P⁻¹A_pY_t−p + P⁻¹u_t, which corresponds to the same form as (2.2) where (I − P⁻¹) = B, P⁻¹A_i = Γ_i for i = 1, . . ., p, and P⁻¹u_t = ε_t satisfying ∑_ε= P⁻¹∑_uP′⁻¹ = I. One of the possible matrix of P in our setting (2.2) is P = (I − B)⁻¹.

In general, the SVAR model is not identifiable without making further assumptions. However, if the error term ε_t exhibits conditional heteroskedasticity, it becomes possible to identify the SVAR model without additional constraints (Lütkepohl and Netšunajev, 2017). In this paper, we assume that u_t follows the GO-GARCH model as proposed by van der Weide (2002).

2.2. SVAR-GO-GARCH model

As an extension of Model (2.2) incorporating volatilities, in line with van der Weide (2002) (see Remark 2.2 below), we introduce the SVAR-GO-GARCH model:

with ω_i > 0, α_i, β_i ≥ 0, and α_i + β_i < 1. Here, all η_i,t are iid with zero mean and unit variance. In the matrix form, Σt=diag(σ1,t2,…,σd,t2) can be expressed as:

where Ω = diag(ω₁, . . ., ω_d), α = diag(α₁, . . ., α_d), β = diag(β₁, . . ., β_d), and Et:=diag(ɛ1,t2,…,ɛd,t2).

Equation (2.3) can be rearranged to isolate the contemporaneous effects of Y_t:

yielding the VAR model:

under the assumption that I − B is nonsingular. To ensure the stationarity of Model (2.5), we assume

where is the unit disk in the complex plane. This form becomes the VAR model in (2.1), with the error terms u_t = (I − B)⁻¹ε_t consisting of a linear combination of independent univariate stationary GARCH(1,1) processes. Integrating the SVAR model with the GO-GARCH scheme allows for a comprehensive representation of economic and financial time series, capturing both contemporaneous interactions and evolving dynamic structures. This approach offers greater flexibility in modeling time-varying volatility and facilitates the identification of model parameters.

Remark 2. In van der Weide (2002), the GO-GARCH model is defined as

which restricts ε_t to have unit variances. However, in our setting, we assume that the diagonal component of the mixing matrix (I − B)⁻¹ is set to 1, rather than constraining the structural error process. This adjustment ensures that the model remains identifiable, up to permutation and sign changes.

2.3. LSCUSUM test for SVAR-GO-GARCH models

In this section, we introduce a method for detecting change points in the parameters of the SVAR-GO-GARCH model. Previous work by Lee et al. (2023) proposed a change point detection method for the SVAR model, where the proposed statistics effectively identify changes in the structural parameters. However, their method does not account for heteroskedasticity, which can lead to suboptimal detection and misinterpretation of structural breaks, especially in economic and financial contexts. To overcome this limitation, we extend the change point detection test to the SVAR-GO-GARCH model, which incorporates dynamic volatility patterns.

Given observations Y₁, . . ., Y_n, we aim to test the following hypotheses:

• ℋ₀: The parameters remain constant over time t = 1, . . ., n vs.

• ℋ₁: Not ℋ₀.

In the SVAR model, the LSCUSUM test utilizes both observations and residuals. Specifically, the test statistic is composed of either the sum or maximum of two CUSUM processes. The first process is based on the product of the conditional mean Y_t − û_t and the VAR residuals

where Γ̂_i are the least squares estimators (LSEs) of Γ_i derived from (2.5), while the second process is based on the norm of the SVAR residuals, ‖ɛ^t‖Σ^ɛ2=ɛ^tTΣ^ɛ-1ɛ^t, where ∊̂_t are obtained using ICA as described in Subsection 2.4 below and ∑̂_ε is a consistent estimator of ∑_ε based on the residuals.

To adapt this approach for our model, we modify the second term of the LSCUSUM test to account for the heteroskedasticity inherent in the model. Specifically, we replace ∑̂_ε therein with ∑̂_t, which, in view of (2.4), is obtained through the equation:

with some initial values Ê₀ and ∑̂₀, where Ω̂, α̂ and β̂ are consistent estimators of Ω, α, and β, respectively. More specifically, σ^i,t2 is obtained recursively through the equation: σ^i,t2=ω^i+α^iɛ^i,t-12+β^iσ^i,t-12, with some initial values σ^i,02 and ɛ^i,02, where ω̂_i, α̂_i and β̂_i are the consistent estimators of ω_i, α_i, and β_i. Then, ‖ɛ^t‖Σ^ɛ2 is replaced with ‖ɛ^t‖Σ^ɛ2=‖η^t‖2, where η̂_t := (η̂_1,t, . . ., η̂_d,t)^T with η̂_i,t = ∊̂_i,t/σ̂_i,t.

Consequently, we propose a new test statistic, denoted as T̂_n, instead of using the tests from Lee et al. (2023) as follows:

where

This test consists of two parts that detect changes in both the location and scale components of the model, and is therefore expressed as a function of two basic processes. Compared to estimates-based tests, which become more complex as the dimensionality of the model parameters increases, this approach provides a concise test that significantly reduces the computational burden. In particular, the test was inspired from the fact that Y_t − û_t and û_t should be nearly uncorrelated in the absence of structural changes.

Remark 3. For cases where detecting a change solely in either location or scale is of interest, although both types of changes frequently occur together in practical applications, we suggest using

to focus on location changes, or

for scale changes. See Oh and Lee (2019) and Lee (2020) for relevant references.

The newly proposed T̂_n is formulated based on the observations Y_t, the VAR residuals û_t, and the GARCH residuals η̂_t. This adjusted test allows for more accurate detection of changes in the parameters of the SVAR-GO-GARCH model, which cannot be achieved by the original test of Lee et al. (2023), as demonstrated by our simulation results. As addressed in Remark 4 below, we can obtain the null limiting distribution of T̂_n under certain conditions. Based on this, we use critical value 1.4596 for T̂_n to reject the null hypothesis ℋ₀ at the nominal level of 0.05, as in Lee et al. (2023). We can also use the values 1.3349 and 1.7120 for the levels of 0.10 and 0.01, respectively. The test’s consistency could be validated under ideal conditions. See Oh and Lee (2019), Lee (2020), and Lee and Kim (2024) for further details. Our simulation results confirm the adequacy of this approach, demonstrating the test’s stability and strong power.

Remark 4. Instead of T̂_n in (2.7), one could consider another test statistics similar to the T^nsum in Lee et al. (2023) as follows:

Lee et al. (2023) mentioned that neither of their two test statistics, T^nmax and T^nsum, for SVAR models, corresponding to T̂_n and T^n* for our models, respectively, completely outperforms the other. However, T^nmax tends to be more stable by a slight margin in most situations. As such, we recommend only using T̂_n in (2.7). An empirical study demonstrates that T^n* exhibits performance similar to that of T̂_n, although detailed results are not reported here.

Remark 5. Assuming further conditions

under ℋ₀ and using Donsker’s invariance principle (Billingsley, 1968) along with the continuous mapping theorem, it can be verified that

where W̊₁ and W̊₂ are two independent Brownian bridges. The convergence result, involving two independent Brownian bridges in the limiting null distribution, offers significant convenience in computing the critical values of the test, which enhances the practicality of the entire testing procedure in applications. The verification follows a similar approach to that in Lee et al. (2023). Refer to Lanne et al. (2017) for the asymptotic properties of the estimation. However, this property in (2.8) was only proven for SVAR models and GARCH models separately using the maximum likelihood method, see Lütkepohl and Netšunajev (2017) and Comte and Lieberman (2003), and not proven for SVAR-GO-GARCH model. Moreover, our focus is on the ICA method, not MLE, which offers the advantage of a much faster algorithm compared to the MLE. When the ICA is incorporated, verifying the asymptotic normality remains challenging and is beyond the scope of this paper.

2.4. Algorithm for computing LSCUSUM statistics via ICA

In this subsection, we summarize the steps for obtaining the components of the LSCUSUM test statistics, using ICA.

• Step 1. Computation of VAR residuals

  • Obtain the VAR estimators Γ̂₁, . . ., Γ̂ _p of the VAR model in (2.5) by the least squares method, and then, the VAR residuals û_t, as described in (2.6).

• Step 2. Computation of SVAR residuals from VAR residuals via ICA

  • Using the fastICA in Hyvärinen et al. (2010) or any other ICA algorithms, decompose û_t in Step 1 and obtain M̃ and ∊̃_t, such that û_t = M̃ ∊̃_t. Here, M̃ is a potential mixing matrix and M̃⁻¹ is a potential unmixing matrix.

  • Through the linear matching problem as described in Shimizu et al. (2006), find a row-permutation matrix P for the potential unmixing matrix, which minimizes the sum of the reciprocals of the absolute values of the diagonal elements in a row-permuted matrix, namely, ∑_i 1/|(PM⁻¹)_ii|. Set it as W̃.

  • Divide each row of W̃ by its corresponding diagonal elements to obtain W̃, and then define B̃ = I − Ŵ.

  • Let R_ij be the rank of |b̃_ij|’s where b̃_ij is (i, j)^th component of the matrix B̃. Replace b̃_ij with 0 if R_ij ≤ d(d + 1)/2 and check whether B̃ has an acyclic contemporaneous causal structure. If not so, replace b̃_ij having R_ij = d(d + 1)/2 + 1 with 0, and repeat this step until the adjusted B̃ has an acyclic contemporaneous causal structure.

  • Denoting the resulting B̃ in the above step by B̃, calculate the SVAR residuals

    ɛ^t=(I-B^) u^t,

    where ũ_t is the one in Step 1.

• Step 3. Computation of GARCH residuals and construction of the LSCUSUM test

  • For each ∊̂_i,t derived in Step 2.5, fit a GARCH(1,1) model and obtain η̂_i,t, as mentioned earlier in Section 3.

  • Calculate the LSCUSUM test statistic T̂_n in (2.7) with observations, VAR residuals, and the corresponding GARCH residuals.

Remark 6. The lag order of SVAR(p) model is that of VAR(p) model. There are some criterions such as Akaike’s information criterion (AIC), final prediction error (FPE), Schwartz criterion (SC), and Hannan-Quinn criterion (HQ). Refer to Lütkepohl (2005). Also, one can consider general order GARCH model. For GARCH model selection, see Brooks and Burke (2003). Since GO-GARCH is a combination of univariate GARCH model, one can use the criterions therein. However, it is widely accepted that GARCH(1,1) is enough in real-world applications. Refer to Hansen and Lunde (2005).

In this section, we evaluate the performance of the test T̂_n using Monte Carlo simulations. The simulations are carried out using the following SVAR(1)–GARCH(1,1) model:

where η_i,t follows a standard normal distribution. The same parameter values θ = (ω, α, β)^T are used for each i = 1, . . ., d for simplicity. We conduct the tests at the nominal level of 0.05 for n = 250, 500, 1000, and 2000. The critical value used for T̂_n is 1.4596 as mentioned earlier, refer to Lee (2020).

We determine empirical sizes and powers by calculating the rejection rate of the null hypothesis ℋ₀ over N = 1000 repetitions. To evaluate power, we set the simulations settings so that a change is occurred at the midpoint of the series, i.e., n/2 = 125, 250, 500, and 1000. The simulations consider cases where one of the parameters A, B, or θ = (ω, α, β) changes while the others remain fixed.

3.1. Empirical size

Tables 1–3 report the empirical sizes for the 3-dimensional SVAR(1)-GARCH(1,1) models. To examine performance under different levels of volatility, we fix A and B values as follows:

We then consider the three GARCH parameters settings with different volatility as follows:

with volatility increasing across these settings. Note that θ^(h), θ^(m), and θ^(l) represent highly, middle, and low volatile scenarios. For evaluating the performances of our proposed test T̂_n, we compare its size performance against the tests T^nsum and T^nmax in Lee et al. (2023).

In Table 1, T^nsum and T^nmax exhibit size distortions even with small sample sizes. These distortions become more pronounced as n increases. This outcome is expected, as larger samples are more likely to include extreme cases, particularly in high-volatility settings. In contrast, the newly proposed T̂_n does not exhibit size distortion.

Remark 7. While T̂_n’s size slightly increases with n, this can be attributed to the need for larger critical values as the sample size grows. In fact, the critical value 1.4596 is derived from simulations with n = 1000, as its limiting distributions does not have a closed form. Therefore, we recommend adjusting the critical values based on the length of the data. Empirical evidence suggests that critical values tend to increase as the data lengthens.

Tables 2 and 3 reveal even more dramatic results, that is, the size of T^nsum and T^nmax are much distorted compared to the low volatile case of θ = θ^(l). For θ^(m), the sizes of T^nsum and T^nmax are 0.200 and 0.208, respectively, even with n = 250. In the case of θ^(h), the sizes range from 0.273 to 0.482 for T^nsum and 0.290 to 0.504 for T^nmax, respectively. In contrast, T̂_n maintains its stable performance, with sizes ranging from 0.030 to 0.071 across these simulation scenarios.

Overall, T̂_n consistently provides more reliable results, with its sizes remaining closer to the nominal level of 0.05, even in highly volatile scenarios. Through the above simulations, it is evident that T^nsum and T^nmax exhibit size distortions, leading to unreliable results when a change point is present. Therefore, we focus exclusively on illustrating the performance of our proposed test T̂_n in the following subsections.

3.2. Empirical power when A changes

Next, we consider the simulation settings where the parameter A undergoes changes. Tables 4–6 present the result when B and θ remain consistent with the previous settings, that is, B = B⁽¹⁾ and θ is one of the parameters θ^(l), θ^(m), and θ^(h), while A changes. Three different settings of A are considered, denoted by A⁽¹⁾ in (3.1) and

We calculate the power for each case where the value of A changes from one to another. Table 4 presents the results for θ = θ^(l). When A changes from A⁽¹⁾ to another or vice versa, T̂_n demonstrates strong performance and approaches to 1 as n increases. This outcome is due to the simultaneous change in both the parameter values and the causal order of A. When A changes between A⁽²⁾ and A⁽³⁾, however, the power of T̂_n exhibits relatively low performance, around 0.250 for n = 250, due to the slight parameter change between A⁽²⁾ and A⁽³⁾. However, the power increases as n grows, indicating that T̂_n is effective in detecting even slight changes.

Tables 5 and 6 present results for high volatile cases with θ = θ^(m) and θ = θ^(h), respectively. Although the volatility of the data is higher than the θ = θ^(l) case, the simulation results show similar patterns with θ = θ^(l) cases. Thus, we conclude that T̂_n detects the changes in A well and remains stable under various volatility cases.

3.3. Empirical power when B changes

In this subsection, we consider the simulation settings where the parameter A undergoes changes. Tables 7–9 illustrate the result when A and θ are fixed and B changes. Here, three different settings are considered using B⁽¹⁾ in (3.1) and

Table 7 exhibit the results with A = A⁽¹⁾ and θ = θ^(l). When B changes from B⁽¹⁾ or B⁽³⁾ to B⁽²⁾, or conversely, the instantaneous causal orders are altered. As a result, the test performs well in these scenarios. However, when B changes from B⁽¹⁾ to B⁽³⁾ and in reverse, the power is notably low, especially when n = 250, where it appears below 0.08. This is because B⁽¹⁾ differs from B⁽³⁾ by only one parameter, B₂₁, which may be too small to produce a significant change. However, as n increases, the power improves, with the test achieving a power larger than 0.6 at n = 2000. Therefore, our test is capable of detecting changes effectively, even in cases of small parameter shifts. Similar patterns are observed in other cases, as reported in Tables 8 and 9, with performance improving as the sample size increases, indicating that our test is more robust even in highly volatile data, including instantaneous causal changes.

3.4. Empirical power when θ changes

In this subsection, we investigate the cases where θ changes. Specifically, we keep A = A⁽¹⁾ and B = B⁽¹⁾ fixed as defined in (3.1) and change θ from one of θ^(l), θ^(m), or θ^(h) to another, as defined in (3.2). Table 10 presents the results for cases where θ changes. The power of the test is lower for parameter changes between θ^(l) and θ^(m) compared to changes between θ^(h) and another parameter. This difference is attributed to the larger volatility difference between θ^(h) and the other parameters, compared to the difference between θ^(l) and θ^(m). Despite the smaller volatility differences, the performance of the test improves as n increases, similar to the patterns observed for changes in A and B.

Overall, it is demonstrated that T̂_n effectively detects parametric changes in SVAR(1)–GARCH(1,1) models, even in cases of small parameter changes.

In this section, we analyze a 3-dimensional time series of 100*log-returns for the exchange rates of three countries’ currencies against the US dollar: KRW (Korean Won), JPY (Japanese Yen), and SGD (Singapore Dollar). The data spans from January 1, 2021, to December 29, 2023, totaling 781 observations, and is obtainable from www.investing.com. This period covers the aftermath of the COVID-19 pandemic. Exchange rates among multiple countries are highly interconnected in the global economy. In particular, the exchange rates between Korea, Japan, and Singapore are likely to exhibit mutual influences within the global market.

Figure 1 presents the three exchange rates over time. The vertical red lines exhibits the change point detected by T̂_n, which will be discussed later. The USD/KRW and USD/JPY show a slow and gradual upward trend, while USD/SGD also trends upward but with more noticeable fluctuations. After the change point, all three exchange rates appear significantly more volatile compared to the period before the change.

Using the AIC criterion, we find that the VAR(1) model is the best fit for the time series. To assess the non-normality of the error terms in the fitted VAR models, we apply the Jarque-Bera test (Jarque and Bera, 1980; Lee et al., 2010), the Kolmogorov-Smirnov test, and the Shapiro-Wilk test to the residuals. The results indicate that all error terms deviate from normality, a finding that is further supported by the qq-plots shown in Figure 3.

Next, we obtain the fitted SVAR models using ICA and then fit the GARCH models to each variable. The resulting SVAR-GARCH model is as follows:

Based on this fitted model, we calculate the T̂_n which is 1.5627, exceeding the critical value. Therefore, we reject the null hypothesis of no changes at the 0.05 significance level and conclude that there has been a change in the log returns of the exchange rates. The detected change point is at k = 298, corresponding to February 23, 2022.

Figure 2 displays the 100*log-returns of the exchange rates. The vertical red lines indicate the detected change point, February 23, 2022. The log-returns appear to exhibit increased volatility following this change point. Notably, the log-returns for USD/KRW and USD/JPY are relatively stable before the change but become more volatile afterward. Thus, it appears that T̂_n has successfully captured this shift.

Next, we again repeat the above steps to two subseries before and after the detected change point, respectively, namely, from January 1, 2021 to February 22, 2022 and February 24, 2022 to December 28, 2023. Then, the resulting models are obtained as follows:

As the T̂_n values for the two subseries are 1.0811 and 1.2098, respectively, smaller than the critical value, no further changes are found in all cases. It can be also observed that the estimated parameter values obtained from the two subseries differ from those derived from the entire series.

Figure 4 illustrates the causal order between the variables. Blue and red arrows respectively represent positive and negative causal effects, respectively. The dashed, normal, and thick lines denote weak, moderate, and strong effects based on the absolute values of the estimated parameters, which are categorized as follows: 0.05–0.5 for weak, 0.5–1.0 for moderate, and over 1.0 for strong effects. Values smaller than 0.05 are omitted from the figure. The figure reveals that the instantaneous causal effects within the same time frame are much stronger than those observed with time lags. This suggests that fluctuations in exchange rates within the same period have a more substantial impact compared to those from the previous day. Given the interconnected nature of the global economy and its real-time responses, this result aligns with expectations.

Upon closer examination, the effects due to time differences show that appears newly, while and disappear before and after the change point. Moreover, the impact of changes to . For the instantaneous causal effects between variables, the directions of KRW_t → JPY_t and SGD_t → JPY_t are reversed, and KRW_t → SGD_t appears newly. These results exhibit that many aspects of the causal relationships between variables have changed before and after the change point.

In practice, identifying the cause of the change point is also an important issue. Notably, the detected change point date is the day before the outbreak of the Russia-Ukraine war, which began on February 24, 2022. Given the significant global economic impact of the conflict, it is reasonable to conclude that our proposed method effectively detects changes in real-world situations.

In this study, we modified the LSCUSUM test proposed by Lee et al. (2023), which detects change points in SVAR models, to apply the change point detection method to SVAR-GARCH models. For the identification of the SVAR model, we adopted the GO-GARCH scheme to address conditional heteroskedasticity. The performance of our newly proposed statistics is demonstrated to work well through several simulation settings. A real data analysis using exchange rates of Korea, Japan, and Singapore was also conducted. Notably, the proposed method identified a change point just before the outbreak of the Russia-Ukraine war, showing that our method performs well in real-world situations. Overall, our findings indicate that the LSCUSUM test is an excellent method for detecting the structural changes in SVAR-GARCH models.

We thank an AE and the anonymous referees for their valuable comments. The authors declare no conflicts of interest.