# State-Dependent Model Based on Singular Spectrum Analysis Vector for Modeling Structural Breaks: Forecasting Indonesian Export

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.2. Singular Spectrum Analysis (SSA)

**Z**, then the formula is obtained as follows:

_{1}, λ

_{2}, …, λ

_{L}) are obtained in descending order (λ

_{1}≥ λ

_{2}≥ … ≥ λ

_{L}≥ 0) using the equation $\Vert S-\lambda I\Vert =0$, where

**I**is an identity matrix. The eigenvector (

**U**) is calculated with the equation $\left(S-\lambda I\right)U=0$. The eigenvector matrix consists of vectors (

**u**

_{1},

**u**

_{2}, …,

**u**

_{L}), which form an orthonormal system of the eigenvectors of matrix

**S**corresponding to the eigenvalues. The principal components $\left({\tilde{v}}_{1},{\tilde{v}}_{2},\dots ,{\tilde{v}}_{L}\right)$, are calculated using the following formula:

**G**matrix is the sum of the ${G}_{i}$ matrix, where ${G}_{i}=\sqrt{{\lambda}_{i}}{u}_{i}{\tilde{v}}_{i}^{T}$, d = L

^{*}with L

^{*}= min {L, K}. The SVD of the

**G**trajectory matrix is formulated as follows:

**G**has a rank denoted by d = max{i, where λ

_{i}> 0}, hence, when d = L, then the matrix

**G**=

**Z**. The second stage is reconstruction, which starts with grouping eigentriples according to patterns or frequency characteristics, such as trends, seasonality, cyclicality, and noise. This process partitions the ${G}_{i}$ matrix into m subsets of disjoints I

_{1}, I

_{2}, …, I

_{m}, and Equation (4) can be written as follows:

_{1}= I = {i

_{1}, …, i

_{r}} and I

_{2}= {1, …, d}\I, where 1 ≤ i

_{1}, …, i

_{r}≤ d. The next step in the reconstruction is to diagonal average each

**G**matrix into new time-series data of n denoted as follows:

_{Im}**G**matrix uses the following Formula [54]:

^{*}= min(L, K), K

^{*}= max(L, K), and $n=L+K-1$. Meanwhile ${\widehat{z}}_{i,j}={\underset{\u02dc}{z}}_{i,j}$ when L < K and ${\widehat{z}}_{i,j}={\underset{\u02dc}{z}}_{j,i}$ for others, and hence, the reconstruction method used to obtain the new time-series data is denoted by ${\tilde{Y}}^{(k)}=\left({\tilde{y}}_{1}^{(k)},\dots ,{\tilde{y}}_{n}^{(k)}\right)$. The LRF forecasting process utilizes the last (L − 1) observations from new time-series data ${\left[{\tilde{y}}_{1},\dots ,{\tilde{y}}_{n}\right]}^{T}$ multiplied by the LRF parameter coefficient $\Phi ={\left({\varphi}_{1},\dots ,{\varphi}_{L-1}\right)}^{T}$. To obtain the LRF coefficient,

**U**= [

**u**

_{1}:…:

**u**

_{r}], where

**u**

_{i}is a vector that has an element (L − 1) of vector

**u**

_{i}, then for i = 1, the first component of eigenvector

**u**

_{1}, i.e., (u

_{1,1}, u

_{2,1}, …, u

_{L}

_{-1,1}). When ${\nu}^{2}={\pi}_{1}^{2}+\dots +{\pi}_{r}^{2}$, with ${\pi}_{i}$ is the last component of vector

**u**

_{i}with $\left(i=1,\dots ,r\right)$, then the LRF parameter coefficients is formulated as follows.

#### 2.3. SSA Parameter Coefficient Mofdeling Using SDM

_{n}, and Equation (10) is formulated as follows.

#### 2.4. SSA Vector Parameter Coefficient Modeling Using SDM

**G**in Equation (7) to ensure the forecasting algorithm generates time-series data with h points in the next period as follows:

- Form a matrix $\Pi $ that is a linear operator showing orthogonal projections c ${\mathbb{R}}^{L-1}\mapsto {\underset{\_}{\zeta}}_{r}$, where ${\underset{\_}{\zeta}}_{r}=\mathrm{span}\left({\underset{\_}{u}}_{1},\dots ,{\underset{\_}{u}}_{r}\right)$, and $\widehat{\theta}$ is a modified LRF parameter coefficient based on the EKF in Equation (17).$$\Pi =\underset{\_}{U}{\underset{\_}{U}}^{T}+\left(1-{v}^{2}\right)\widehat{\theta}{\widehat{\theta}}^{T}$$
- Forming SSAV operator ${\mathcal{P}}_{VEC}:{\mathbb{R}}^{L}\mapsto {\zeta}_{r}$, which is formulated as follows:$${\mathcal{P}}_{VEC}=\left(\begin{array}{c}\Pi \\ {\widehat{\theta}}^{T}\end{array}\right)$$
- The matrix ${G}_{h}^{*}$ is a combination of matrices
**G**and ${\underset{\u02dc}{G}}_{h}$, where the matrix ${\underset{\u02dc}{G}}_{h}={\mathcal{P}}_{VEC}{\underset{\u02dc}{z}}_{i-1}$, for i = K + 1, …, K + h + L − 1. The vectors that make up the matrix ${G}_{h}^{*}$ are denoted as follows:$$|{\underset{\u02dc}{z}}_{i}=\left(\right)open="\{">\begin{array}{cc}{\underset{\u02dc}{z}}_{i}& \mathrm{for}i=1,\dots ,K\\ {\mathcal{P}}_{VEC}{\underset{\u02dc}{z}}_{i-1}& \mathrm{for}i=K+1,\dots ,K+h+L-1\end{array}$$ - Applying a diagonal averaging process to the matrix ${G}_{h}^{*}$ to obtain prediction in the 1st to n-th and the next h-th point prediction, ${\widehat{\mathbb{Y}}}_{n+h}=\left\{{\widehat{y}}_{1},{\widehat{y}}_{2},\dots ,{\widehat{y}}_{K+1},\dots ,{\widehat{y}}_{K+h+L-1}\right\}$. Therefore, the forecasting results in the next h period are denoted as follows: $\left\{{\widehat{y}}_{K+1},\dots ,{\widehat{y}}_{K+h+L-1}\right\}$.

#### 2.5. Accuracy Evaluation

## 3. Results and Discussion

#### 3.1. Modeling Indonesian Export Data Using SSAR, SSAV, SDM-SSAR, SDM-SSAV, Hybrid ARIMA-LSTM, and VARI

^{−13}, less than the 5% level. The Zivot-Andrew test was selected due to its consideration of structural changes in time-series data, consistent with the expected characteristics in Indonesian export data. The test was used to determine the presence of a unit root. Indonesian export data have considerable variability, evidenced by a substantial standard deviation of 5627.53 million US$. In addition, it had a reasonably large average of 9983 million US$. Table 1 shows a comprehensive overview of the characteristics of Indonesian export data.

^{−6}. For effectiveness, only the experimental results with the optimum parameters that yield the highest accuracy are shown in Table 2, Table 3, Table 4, Table 5 and Table 6.

#### 3.2. Simulation Study

_{i}), while the parameter π was also set differently for each scenario. However, the deliberate decision to maintain a constant variance value of 0.2 across all scenarios aims to generate data with structural breaks solely in the mean, with no consideration given to those in the variance.

_{1}= 157 and n

_{2}= 207, and large sample sizes (n ≥ 300), such as n

_{3}= 365 and n

_{4}= 405.

#### 3.3. Forecasting Indonesian Export Using SDM-SSAV

^{−5}. The actual Indonesian export, predictive, and 12-step ahead forecasting data are shown in Figure 5. The plotted forecast showed a significant decline in early 2023, followed by an improvement in export performance over the subsequent nine months, albeit with a decrease in the final three months. This decline was attributed to a seasonal factor, where exports traditionally decrease at the beginning of each year due to the holiday blues phenomenon. In addition, world geopolitical conditions have changed due to the Russian invasion of Ukraine in early 2022, which directly affected the increase in world oil prices. Uncertainty in the global geopolitical conditions has resulted in lower demand for essential export commodities from major trading partners, such as China, thereby influencing Indonesian export performance.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cao, Y.K.; Guo, H.F. The impact of global financial crisis on China’s trade in forest product and countermeasures. In Proceedings of the 2009 International Conference on Information Management, Innovation Management and Industrial Engineering, ICIII, Xi’an, China, 26–27 December 2009; Volume 4, pp. 580–583. [Google Scholar] [CrossRef]
- Afonso, A.; Blanco-Arana, M.C. Financial and economic development in the context of the global 2008-09 financial crisis. Int. Econ. J.
**2022**, 169, 30–42. [Google Scholar] [CrossRef] - Cashin, P.; Mohaddes, K.; Raissi, M. China’ s slowdown and global financial market volatility: Is world growth losing out? Emerg. Mark. Rev.
**2017**, 31, 164–175. [Google Scholar] [CrossRef] - Hu, S.; Gao, Y.; Niu, Z.; Jiang, Y.; Li, L.; Xiao, X.; Wang, M.; Fang, E.F.; Menpes-Smith, W.; Xia, J.; et al. Weakly Supervised Deep Learning for COVID-19 Infection Detection and Classification from CT Images. IEEE Access
**2020**, 8, 118869–118883. [Google Scholar] [CrossRef] - Zgurovsky, M.; Kravchenko, M.; Boiarynova, K.; Kopishynska, K.; Pyshnograiev, I. The Energy Independence of the European Countries: Consequences of the Russia’s Military Invasion of Ukraine. In Proceedings of the 2022 IEEE 3rd International Conference on System Analysis & Intelligent Computing (SAIC), Kyiv, Ukraine, 4–7 October 2022. [Google Scholar] [CrossRef]
- Priestley, M.B. State-Dependent Models: A General Approach to Non-Linear Time Series Analysis. J. Time Ser. Anal.
**1980**, 1, 47–71. [Google Scholar] [CrossRef] - Mohler, R.R. Bilinear Control Processes: With Applications to Engineering, Ecology, and Medicine; Academic Press: New York, NY, USA; London, UK, 1973; Volume 106. [Google Scholar]
- Brockett, R.W. Volterra series and geometric control theory. Automatica
**1976**, 12, 167–176. [Google Scholar] [CrossRef] - Tong, H. Non-Linear Time Series: A Dynamical System Approach; Oxford University Press: New York, NY, USA, 1990. [Google Scholar]
- Tong, H.; Lim, K.S. Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society. Series B (Methodological)
**1980**, 42, 245–292. [Google Scholar] [CrossRef] - Haggan, V.; Ozaki, T. Modelling Nonlinear Random Vibrations Using an Amplitude-Dependent Autoregressive Time Series Model. Biometrika
**1981**, 68, 189–196. [Google Scholar] [CrossRef] - Ozaki, T. Non-Linear Time Series Models for Non-Linear Random Vibrations. Journal of Applied Probability.
**1980**, 17, 84–93. [Google Scholar] [CrossRef] - Ardia, D.; Bluteau, K.; Boudt, K.; Catania, L.; Trottier, D.A. Markov-switching GARCH models in R: The MSGARCH package. J. Stat. Softw.
**2019**, 91, 1–38. [Google Scholar] [CrossRef] - Davidescu, A.A.; Apostu, S.A.; Paul, A. Comparative analysis of different univariate forecasting methods in modelling and predicting the romanian unemployment rate for the period 2021–2022. Entropy
**2021**, 23, 325. [Google Scholar] [CrossRef] - Behrendt, S.; Schweikert, K. A Note on Adaptive Group Lasso for Structural Break Time Series. Econ. Stat.
**2021**, 17, 156–172. [Google Scholar] [CrossRef] - Ito, M. Detecting Structural Breaks in Foreign Exchange Markets by using the group LASSO technique. arXiv
**2022**, arXiv:2202.02988. [Google Scholar] - Ito, M.; Noda, A.; Wada, T. An Alternative Estimation Method of a Time-Varying Parameter Model. arXiv
**2017**, arXiv:1707.06837. [Google Scholar] [CrossRef] - Gong, X.; Lin, B. Structural breaks and volatility forecasting in the copper futures market. J. Futur. Mark.
**2018**, 38, 290–339. [Google Scholar] [CrossRef] - De Gaetano, D. Forecast combinations in the presence of structural breaks: Evidence from U.S. equity markets. Mathematics
**2018**, 6, 34. [Google Scholar] [CrossRef] - Antoch, J.; Hanousek, J.; Horváth, L.; Hušková, M.; Wang, S. Structural breaks in panel data: Large number of panels and short length time series. Econom. Rev.
**2019**, 38, 828–855. [Google Scholar] [CrossRef] - Hansen, B.E. The New Econometrics of Structural Change. J. Econ. Perspect.
**2001**, 15, 117–128. [Google Scholar] [CrossRef] - Kruiniger, H. Not So Fixed Effects: Correlated Structural Breaks in Panel Data; Queen Mary University: London, UK, 2008; pp. 1–33. [Google Scholar]
- Clements, M.P.; Hendry, D.F. Intercept Corrections and Structural Change. J. Appl. Econom.
**1996**, 11, 475–494. [Google Scholar] [CrossRef] - Beaudry, P.; Galizia, D.; Portier, F. Reviving the Limit Cycle View of Macroeconomic Fluctuations; NBER Working Papers Series; National Bureau of Economic Research: Cambridge, MA, USA, 2015; pp. 1–23. [Google Scholar]
- Kantz, H.; Schreiber, T. Nonlinear Time Series Analysis; Cambridge University Press: Cambridge, UK, 2004; Volume 7. [Google Scholar]
- Broomhead, D.S.; King, G.P. Extracting qualitative dynamics from experimental data. Phys. D Nonlinear Phenom.
**1986**, 20, 217–236. [Google Scholar] [CrossRef] - Sanei, S.; Hassani, H. Singular Spectrum Analysis of Biomedical Signals; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Golyandina, N.; Nekrutkin, V.; Zhigljavsky, A. Analysis of Time Series Structure SSA and Related Techniques, 1st ed.; Chapman & Hall/CRC: London, UK, 2001. [Google Scholar]
- Hou, Z.; Wen, G.; Tang, P.; Cheng, G. Periodicity of Carbon Element Distribution Along Casting Direction in Continuous-Casting Billet by Using Singular Spectrum Analysis. Met. Mater. Trans. B
**2014**, 45, 1817–1826. [Google Scholar] [CrossRef] - Le Bail, K.; Gipson, J.M.; MacMillan, D.S. Quantifying the Correlation Between the MEI and LOD Variations by Decomposing LOD with Singular Spectrum Analysis BT. In Earth on the Edge: Science for a Sustainable Planet; Springer: Berlin/Heidelberg, Germany, 2014; pp. 473–477. [Google Scholar]
- Chang, Y.W.; Van Bang, P.; Loh, C.H. Identification of Basin Topography Characteristic Using Multivariate Singular Spectrum Analysis: Case Study of the Taipei Basin. Eng. Geol.
**2015**, 197, 240–252. [Google Scholar] [CrossRef] - Ghil, M.; Allen, M.R.; Dettinger, M.D.; Ide, K.; Kondrashov, D.; Mann, M.E.; Robertson, A.W.; Saunders, A.; Tian, Y.; Varadi, F.; et al. Advanced spectral methods for climatic time series. Rev. Geophys.
**2002**, 40, 3-1–3-41. [Google Scholar] [CrossRef] - Chao, S.H.; Loh, C.H. Application of singular spectrum analysis to structural monitoring and damage diagnosis of bridges. Struct. Infrastruct. Eng.
**2014**, 10, 708–727. [Google Scholar] [CrossRef] - Liu, K.; Law, S.S.; Xia, Y.; Zhu, X.Q. Singular spectrum analysis for enhancing the sensitivity in structural damage detection. J. Sound Vib.
**2014**, 333, 392–417. [Google Scholar] [CrossRef] - Thuraisingham, R.A. Use of SSA and MCSSA in the Analysis of Cardiac RR Time Series. J. Comput. Med.
**2013**, 2013, 231459. [Google Scholar] [CrossRef] - Bureneva, O.; Safyannikov, N.; Aleksanyan, Z. Singular Spectrum Analysis of Tremorograms for Human Neuromotor Reaction Estimation. Mathematics
**2022**, 10, 1794. [Google Scholar] [CrossRef] - Silva, E.S.; Hassani, H.; Heravi, S. Modeling European industrial production with multivariate singular spectrum analysis: A cross-industry analysis. J. Forecast.
**2018**, 37, 371–384. [Google Scholar] [CrossRef] - Silva, E.S.; Hassani, H. On the use of singular spectrum analysis for forecasting U.S. trade before, during and after the 2008 recession. Int. Econ.
**2015**, 141, 34–49. [Google Scholar] [CrossRef] - Hassani, H.; Rua, A.; Silva, E.S.; Thomakos, D. Monthly forecasting of GDP with mixed-frequency multivariate singular spectrum analysis. Int. J. Forecast.
**2019**, 35, 1263–1272. [Google Scholar] [CrossRef] - Hassani, H.; Webster, A.; Silva, E.S.; Heravi, S. Forecasting U.S. Tourist arrivals using optimal Singular Spectrum Analysis. Tour. Manag.
**2015**, 46, 322–335. [Google Scholar] [CrossRef] - Rahmani, D.; Heravi, S.; Hassani, H.; Ghodsi, M. Forecasting time series with structural breaks with Singular Spectrum Analysis, using a general form of recurrent formula. arXiv
**2016**, arXiv:1605.02188. [Google Scholar] [CrossRef] - Ghodsi, M.; Hassani, H.; Rahmani, D.; Silva, E.S. Vector and recurrent singular spectrum analysis: Which is better at forecasting? J. Appl. Stat.
**2017**, 45, 1872–1899. [Google Scholar] [CrossRef] - Rahmani, D.; Fay, D. A state-dependent linear recurrent formula with application to time series with structural breaks. J. Forecast.
**2021**, 41, 43–63. [Google Scholar] [CrossRef] - Sarkka, S. Bayesian Filtering and Smoothing; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Golyandina, N.; Zhigljavsky, A. Singular Spectrum Analysis for Time Series, 2nd ed.; Springer: Berlin, Germany, 2020; Volume 28. [Google Scholar]
- Danilov, D.L. Principal components in time series forecast. J. Comput. Graph. Stat.
**1997**, 6, 112–121. [Google Scholar] [CrossRef] - Nekrutkin, V. Approximation Spaces and Continuation of Time Series. In Statistical Models with Applications in Econometrics and Neibouring Fields; Ermakov, S.M., Kashtanov, Y., Eds.; University of St. Petersburg: St. Petersburg, Russia, 1999; pp. 3–32. [Google Scholar]
- Sasmita, Y.; Kuswanto, H.; Prastyo, D.D.; Otok, B.W. Performance evaluation of Bootstrap-Linear recurrent formula and Bootstrap-Vector singular spectrum analysis in the presence of structural break. AIP Conf. Proc.
**2023**, 2556, 050004. [Google Scholar] [CrossRef] - Ginting, A.M. An Analysis of Export Effect on the Economic Growth of Indonesia. Bul. Ilm. Litbang Perdagang.
**2017**, 11, 1–21. [Google Scholar] [CrossRef] - Xie, C.; Liu, Z.; Liu, L.; Zhang, L.; Fang, Y.; Zhao, L. Export rebate and export performance: From the respect of China’s economic growth relying on export. In Proceedings of the 2010 Third International Conference on Business Intelligence and Financial Engineering, Hong Kong, China, 13–15 August 2010; pp. 432–436. [Google Scholar] [CrossRef]
- Kalaitzi, A.S.; Chamberlain, T.W. Exports and Economic Growth: Some Evidence from the GCC. Int. Adv. Econ. Res.
**2020**, 26, 203–205. [Google Scholar] [CrossRef] - Dave, E.; Leonardo, A.; Jeanice, M.; Hanafiah, N. Forecasting Indonesia Exports Using a Hybrid Model ARIMA-LSTM. Procedia Comput. Sci.
**2021**, 179, 480–487. [Google Scholar] [CrossRef] - Djara, V.A.D.; Dewi, D.D.; Hananti, H.; Qisthi, N.; Rosmanah, R.; HM, Z.; Toharudin, T.; Ruchjana, B.N. Prediction of Export and Import in Indonesia Using Vvector Autoregressive Integrated (VARI). J. Math. Comput. Sci.
**2022**, 12, 105. [Google Scholar] [CrossRef] - Pham, M.H.; Nguyen, M.N.; Wu, Y.K. A Novel Short-Term Load Forecasting Method by Combining the Deep Learning with Singular Spectrum Analysis. IEEE Access
**2021**, 9, 73736–73746. [Google Scholar] [CrossRef] - Rahmani, D. A forecasting algorithm for Singular Spectrum Analysis based on bootstrap Linear Recurrent Formula coefficients. Int. J. Energy Stat.
**2014**, 02, 287–299. [Google Scholar] [CrossRef] - Liu, F.; Lu, Y.; Cai, M. A hybrid method with adaptive sub-series clustering and attention-based stacked residual LSTMs for multivariate time series forecasting. IEEE Access
**2020**, 8, 62423–62438. [Google Scholar] [CrossRef] - Bai, J.; Perron, P. Computation and analysis of multiple structural change models. J. Appl. Econom.
**2003**, 18, 1–22. [Google Scholar] [CrossRef] - Zivot, E.; Andrews, D.W.K. the the Great Crash, the and Unit-Root. J. Bus. Econ. Stat.
**1992**, 10, 251–270. [Google Scholar] - Royston, P. Remark AS R94: A remark on Algorithm AS 181: The W test for normality. Appl. Stat.
**1995**, 44, 547–551. [Google Scholar] [CrossRef] - Diebold, F.X.; Mariano, R.S. Comparing predictive accuracy. J. Bus. Econ. Stat.
**1995**, 13, 253–263. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) Data generated in the first scenario (n = 157); (

**b**) data generated in the second scenario (n = 207); (

**c**) data generated in the third scenario (n = 365); (

**d**) data generated in the fourth scenario (n = 405).

**Figure 4.**(

**a**). Boxplot RMSE of the SDM-SSAV and SDM-SSAR from the data-generating process n = 157 with L = 12, r = 1, horizon = 29, and smoothing factor = 5.0 × 10

^{−6}. (

**b**) Boxplot RMSE of the SDM-SSAV and SDM-SSAR from the data-generating process n = 207 with L = 12, r = 1, horizon = 24, and smoothing factor = 2.0 × 10

^{−6}. (

**c**) Boxplot RMSE of the SDM-SSAV and SDM-SSAR from the data-generating process n = 365 with L = 12, r = 1, horizon = 28, and smoothing factor = 5.0 × 10

^{−8}. (

**d**) Boxplot RMSE of the SDM-SSAV and SDM-SSAR from the data-generating process n = 405 with L = 14, r = 1, horizon = 27, and smoothing factor = 5.0 × 10

^{−8}.

**Figure 5.**The Indonesian export during January 1993–December 2022, based on predicted data (January 1993–December 2022), and the next 12-month forecasting data (January–December 2023).

No. | Statistics | |
---|---|---|

1. | Observations (month) | 360 |

2. | Mean (million US$) | 9983 |

3. | Standard deviation (million US$) | 5627.53 |

4. | Struct. breaks by Bai-Perron test | 2000 (January), 2005 (August), 2010 (February), 2018 (June) |

5. | Linearity by Teräsvirta test | p-value = 0.0342 * |

8. | Normality by Shapiro-Wilk test | p-value = 1.398 × 10^{−13} * |

9. | Stationarity by Zivot-Andrews test | Test statistics = −5.5793 *; Critical values: 0.05= −5.08 |

h | SDM-SSAV | SSAR | RRMSE | Diebold-Mariano Test (p-Value) |
---|---|---|---|---|

t | 0.1153 | 0.1354 | 0.8521 | 2.20 × 10^{−16} * |

2 | 0.1223 | 0.1434 | 0.8530 | 3.01 × 10^{−8} * |

3 | 0.1288 | 0.1512 | 0.8518 | 1.64 × 10^{−5} * |

4 | 0.1334 | 0.1588 | 0.8401 | 4.97 × 10^{−5} * |

5 | 0.1370 | 0.1662 | 0.8242 | 4.08 × 10^{−5} * |

6 | 0.1403 | 0.1734 | 0.8090 | 3.64 × 10^{−5} * |

7 | 0.1431 | 0.1804 | 0.7934 | 4.48 × 10^{−5} * |

8 | 0.1465 | 0.1872 | 0.7826 | 0.0001 * |

9 | 0.1504 | 0.1940 | 0.7754 | 0.0004 * |

10 | 0.1552 | 0.2008 | 0.7728 | 0.0012 * |

11 | 0.1605 | 0.2075 | 0.7734 | 0.0036 * |

12 | 0.1659 | 0.2143 | 0.7743 | 0.0097 * |

h | SDM-SSAV | SSAV | RRMSE | Diebold-Mariano Test (p-Value) |
---|---|---|---|---|

1 | 0.1153 | 0.1153 | 1.0003 | 0.9637 |

2 | 0.1223 | 0.1248 | 0.9802 | 0.1565 |

3 | 0.1288 | 0.1352 | 0.9525 | 0.0179 * |

4 | 0.1334 | 0.1449 | 0.9210 | 0.0024 * |

5 | 0.1370 | 0.1541 | 0.8892 | 0.0003 * |

6 | 0.1403 | 0.1630 | 0.8607 | 7.92 × 10^{−5} * |

7 | 0.1431 | 0.1710 | 0.8368 | 7.97 × 10^{−5} * |

8 | 0.1465 | 0.1791 | 0.8183 | 0.0001 * |

9 | 0.1504 | 0.1865 | 0.8064 | 0.0005 * |

10 | 0.1552 | 0.1939 | 0.8003 | 0.0016 * |

11 | 0.1605 | 0.2008 | 0.7993 | 0.0049 * |

12 | 0.1659 | 0.2076 | 0.7991 | 0.0126 * |

h | SDM-SSAV | SDM-SSAR | RRMSE | Diebold-Mariano Test (p-Value) |
---|---|---|---|---|

1 | 0.1153 | 0.1350 | 0.8543 | 2.20 × 10^{−16} * |

2 | 0.1223 | 0.1406 | 0.8698 | 5.82 × 10^{−8} * |

3 | 0.1288 | 0.1454 | 0.8857 | 8.59 × 10^{−5} * |

4 | 0.1334 | 0.1496 | 0.8916 | 0.0006 * |

5 | 0.1370 | 0.1535 | 0.8925 | 0.0013 * |

6 | 0.1403 | 0.1571 | 0.8933 | 0.0021 * |

7 | 0.1431 | 0.1603 | 0.8929 | 0.0026 * |

8 | 0.1465 | 0.1633 | 0.8976 | 0.0054 * |

9 | 0.1504 | 0.1662 | 0.9053 | 0.0161 * |

10 | 0.1552 | 0.1689 | 0.9189 | 0.0556 ** |

11 | 0.1605 | 0.1712 | 0.9376 | 0.1886 |

12 | 0.1659 | 0.1734 | 0.9569 | 0.4973 |

h | SDM-SSAV | Hybrid ARIMA-LSTM | RRMSE | Diebold-Mariano Test (p-Value) |
---|---|---|---|---|

1 | 0.1153 | 0.2987 | 0.3860 | 2.89 × 10^{−8} * |

2 | 0.1223 | 0.3030 | 0.4036 | 0.0003 * |

3 | 0.1288 | 0.3074 | 0.4190 | 0.0021 * |

4 | 0.1334 | 0.3120 | 0.4276 | 0.0062 * |

5 | 0.1370 | 0.3168 | 0.4324 | 0.0111 * |

6 | 0.1403 | 0.3218 | 0.4360 | 0.0151 * |

7 | 0.1431 | 0.3271 | 0.4375 | 0.0187 * |

8 | 0.1465 | 0.3327 | 0.4403 | 0.0216 * |

9 | 0.1504 | 0.3386 | 0.4442 | 0.0234 * |

10 | 0.1552 | 0.3448 | 0.4501 | 0.0248 * |

11 | 0.1605 | 0.3514 | 0.4567 | 0.0258 * |

12 | 0.1659 | 0.3581 | 0.4633 | 0.0255 * |

h | SDM-SSAV | VAR(2,1) | RRMSE | Diebold-Mariano Test (p-Value) |
---|---|---|---|---|

1 | 0.1153 | 0.2739 | 0.4210 | 2.2 × 10^{−16} * |

2 | 0.1223 | 0.2777 | 0.4404 | 6.29 × 10^{−10} * |

3 | 0.1288 | 0.2817 | 0.4572 | 1.13 × 10^{−6} * |

4 | 0.1334 | 0.2859 | 0.4666 | 3.25 × 10^{−5} * |

5 | 0.1370 | 0.2885 | 0.4749 | 0.0002 * |

6 | 0.1403 | 0.2867 | 0.4894 | 0.0008 * |

7 | 0.1431 | 0.2891 | 0.4950 | 0.0018 * |

8 | 0.1465 | 0.2936 | 0.4990 | 0.0036 * |

9 | 0.1504 | 0.2978 | 0.5050 | 0.0059 * |

10 | 0.1552 | 0.3029 | 0.5124 | 0.0088 * |

11 | 0.1605 | 0.3086 | 0.5201 | 0.0122 * |

12 | 0.1659 | 0.3147 | 0.5272 | 0.0161 * |

Scenario | Sample Size | π | ρ | β | η | υ | |
---|---|---|---|---|---|---|---|

1 | n = 157 | n_{1} = 50 | 10 | 2.8 | 0.009 | 0.3 | 0.4 |

n_{2} = 57 | 6 | 4.2 | −0.007 | −0.5 | −0.3 | ||

n_{3} = 50 | 12 | −1.5 | 0.002 | 0.4 | 0.3 | ||

2 | n = 207 | n_{1} = 60 | 12 | −1.5 | 0.002 | 0.2 | 0.3 |

n_{2} = 77 | 7 | 4.2 | −0.007 | −0.2 | −0.3 | ||

n_{3} = 70 | 10 | 1.5 | 0.009 | 0.3 | 0.2 | ||

3 | n = 365 | n_{1} = 120 | 8 | −1.0 | 0.003 | 0.2 | 0.2 |

n_{2} = 120 | 7 | 0.6 | −0.003 | −0.3 | −0.4 | ||

n_{3} = 125 | 9 | −1.2 | 0.002 | 0.2 | 0.2 | ||

4 | n = 405 | n_{1} = 140 | 7 | 1.3 | 0.003 | 0.2 | 0.3 |

n_{2} = 120 | 7 | 0.6 | −0.004 | −0.3 | −0.2 | ||

n_{3} = 145 | 5 | 2.1 | 0.004 | 0.2 | 0.2 |

No | Characteristics | 1st Scenario | 2nd Scenario | 3rd Scenario | 4th Scenario |
---|---|---|---|---|---|

1. | Sample size | 157 | 207 | 365 | 405 |

2. | Stationary (critical value = −5.08) | −5.57 * | −5.57 * | −5.57 * | −5.57 * |

3. | Struct. Breaks (breakpoints t-th) | 25, 50, 74, 110 | 60, 105, 137, 170 | 66, 120, 240, 306 | 60, 140, 212, 285, 345 |

4. | Linearity (p-value) | 0.01098 * | 0.00319 * | 0.00189 * | 0.00402 * |

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## Share and Cite

**MDPI and ACS Style**

Sasmita, Y.; Kuswanto, H.; Prastyo, D.D.
State-Dependent Model Based on Singular Spectrum Analysis Vector for Modeling Structural Breaks: Forecasting Indonesian Export. *Forecasting* **2024**, *6*, 152-169.
https://doi.org/10.3390/forecast6010009

**AMA Style**

Sasmita Y, Kuswanto H, Prastyo DD.
State-Dependent Model Based on Singular Spectrum Analysis Vector for Modeling Structural Breaks: Forecasting Indonesian Export. *Forecasting*. 2024; 6(1):152-169.
https://doi.org/10.3390/forecast6010009

**Chicago/Turabian Style**

Sasmita, Yoga, Heri Kuswanto, and Dedy Dwi Prastyo.
2024. "State-Dependent Model Based on Singular Spectrum Analysis Vector for Modeling Structural Breaks: Forecasting Indonesian Export" *Forecasting* 6, no. 1: 152-169.
https://doi.org/10.3390/forecast6010009