Addition of subset and dummy variables in the Threshold Spatial Vector Autoregressive with Exogenous Variables Model to forecast inflation and money outflow
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Setiawan, Setiawan; Sohibien, Gama Putra Danu; Prastyo, Dedy Dwi; Akbar, Muhammad Sjahid; Kamil, Anton Abdulbasah Article Addition of subset and dummy variables in the Threshold Spatial Vector Autoregressive with Exogenous Variables Model to forecast inflation and money outflow Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Setiawan, Setiawan; Sohibien, Gama Putra Danu; Prastyo, Dedy Dwi; Akbar, Muhammad Sjahid; Kamil, Anton Abdulbasah (2024) : Addition of subset and dummy variables in the Threshold Spatial Vector Autoregressive with Exogenous Variables Model to forecast inflation and money outflow, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 12, Iss. 12, pp. 1-27, https://doi.org/10.3390/economies12120352 This Version is available at: https://hdl.handle.net/10419/329279 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Citation: Setiawan, Setiawan, Gama Putra Danu Sohibien, Dedy Dwi Prastyo, Muhammad Sjahid Akbar, and Anton Abdulbasah Kamil. 2024. Addition of Subset and Dummy Variables in the Threshold Spatial Vector Autoregressive with Exogenous Variables Model to Forecast Inflation and Money Outflow. Economies 12: 352. https://doi.org/ 10.3390/economies12120352 Academic Editor: Stefan Collignon Received: 18 October 2024 Revised: 29 November 2024 Accepted: 4 December 2024 Published: 19 December 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Addition of Subset and Dummy Variables in the Threshold Spatial Vector Autoregressive with Exogenous Variables Model to Forecast Inflation and Money Outflow Setiawan Setiawan 1,* , Gama Putra Danu Sohibien 2, Dedy Dwi Prastyo 1, Muhammad Sjahid Akbar 1 and Anton Abdulbasah Kamil 3 1Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya 60119, Indonesia; [email protected] (D.D.P.); [email protected] (M.S.A.) 2Department of Applied Statistics, Politeknik Statistika STIS, Jakarta 13320, Indonesia; [email protected] 3Faculty of Economics, Istanbul Gelisim University, Istanbul 34315, Turkey; [email protected] *Correspondence: [email protected] Abstract: The TSpVARX model can be used in inflation and money outflow forecasting by accommodating the reciprocal relationship among endogenous variables, the influence of exogenous variables, inter-regional linkages, and the nonlinearity of the relationship between endogenous and predetermined variables. However, the impact of some events, such as Eid al-Fitr and fuel price adjustments, still cannot be accommodated in the TSpVARX model. This condition causes inflation and money outflow forecasting using TSpVARX to be unsatisfactory. Our study is to improve the forecasting performance of the TSpVARX model by adding subset and dummy variables. We use a 12th lag subset variable to capture seasonal effects and a dummy variable to represent fuel price changes. These additions enhance the model’s accuracy in forecasting inflation and money outflow by accounting for recurring patterns and specific events, like fuel price changes. Based on the RMSE values of the training and testing data, we can conclude that forecasting inflation and money outflow using TSpVARX with the addition of subset and dummy variables is better than the regular TSpVARX. The inflation and money outflow forecasting generated after the addition of subset and dummy variables are also more fluctuating as in the movement of the actual data. Keywords: nonlinear time series; threshold spatial; SpVAR; inflation; subset variable; TSpVARX 1. Introduction Time series models used to forecast inflation and money outflow data include Autoregressive Integrated Moving Average with exogenous variables (ARIMAX), Vector Autoregressive with exogenous variables (VARX), Generalized Space-Time Autoregressive (GSTAR), Threshold Vector Autoregressive (TVAR), Spatial Vector Autoregressive (Sp- VAR), and Threshold Spatial Vector Autoregressive with exogenous variables (TSpVARX). ARIMAX is a univariate model that can accommodate the influence of predetermined variables in the form of lag endogenous variables and exogenous variables (Wei 2006; Lestari and Dini 2024). The use of the ARIMAX model in inflation modeling is performed to accommodate the influence of several variables, such as fuel increases, outliers, and interest rates. VARX is a multivariate model that can accommodate the impact of exogenous variables and the reciprocal relationship between endogenous variables (Lütkepohl 2005;Tsay 2014). Two variables that have a mutual relationship but are modeled with a single model will result in biased coefficient estimation (Stock and Watson 2020). VARX models can be used to model inflation and money outflow because inflation and money outflow have a reciprocal relationship (Adelowokan et al. 2019;Bello and Saulawa 2013). However, the VARX model does not include spatial weights that serve to distinguish the size of Economies 2024,12, 352. https://doi.org/10.3390/economies12120352 https://www.mdpi.com/journal/economies
Economies 2024,12, 352 2 of 27 the inter-regional relationship. A model that can accommodate the linkage of time series variables between regions is the GSTAR model (Sohibien 2017;Fadlurrohman 2020). Sohibien (2017) found that GSTAR with normalized cross-correlation weights is well used in forecasting the inflation of Dumai and Pekanbaru. Fadlurrohman (2020) found that GSTARX with uniform weights is better at forecasting than GSTARX with inverse distance weights. A limitation of the GSTARX model is its inability to accommodate the relationship between endogenous variables for more than one type of endogenous variable. The SpVAR model can accommodate inter-regional linkages for more than one variable (Beenstock and Felsenstein 2007;Di Giacinto 2010;Sohibien et al. 2024a). The models described above are less suitable for nonlinear patterns of relationships between endogenous and predetermined variables. The TVAR model can capture the nonlinearity of the relationship between time series variables by forming several VAR model regimes based on a predetermined threshold value (Tsagkanos et al. 2018). Although TVAR has been able to capture the nonlinearity of the relationship between endogenous and predetermined variables, the TVAR model has not been able to accommodate the interrelationship of time series variables between regions. The TSpVARX model captures complex interactions by accounting for relationships among variables, external factors, past data points, regional connections, and nonlinear patterns (Sohibien et al. 2024b). Models, such as ARIMAX, VARX, GSTARX, and SpVARX, can only be used to analyze linear relationship patterns between endogenous variables and predetermined variables. Meanwhile, the TSpVARX model can already analyze non-linear relationship patterns between endogenous variables and predetermined variables. Sohibien et al. (2024b) found that when there is a nonlinear relationship between endogenous variables and predetermined variables, the forecasting performance of the TSpVARX model is better than SpVARX. Another approach that can be used in handling nonlinear relationships is to use machine learning approaches, such as Neural Networks (NNs) and Support Vector Machine (SVM). With the flexibility of machine learning, it can approach any function so that nonlinear patterns between variables can be learned (Bharadiya 2023;Shoko and Sigauke 2023). However, the TSpVARX model has better advantages over machine learning approaches. TSpVARX provides clear rules about when the relationship between variables changes, for example when a variable crosses a certain threshold. This makes it easier for users to understand than machine learning models. In addition, the structure of the TSpVARX model is also simpler than the machine learning approach. In TSpVARX, the nonlinearity of the relationship is shown by a different model between models below the threshold and above the threshold. It causes the TSpVARX model to also be used for interpretation in understanding the relationship between variables. Meanwhile, the complexity of the machine learning approach means that it cannot be used to interpret the relationship between variables. Although the TSpVARX model has been able to accommodate many things, it still has shortcomings. The TSpVARX model of Sohibien et al. (2024b) cannot accommodate the influence of events with recurring patterns and events that include nonmetric variables. This is still a weakness if the TSpVARX model is used to forecast inflation and money outflow because the inflation and money outflow are influenced by events with recurring patterns (in this case the Eid al-Fitr event) and events that include nonmetric variables (in this case the increase and decrease in fuel prices). In addition to producing more accurate forecasts, the TSpVARX model with dummy and subset variables also has the potential to be applied in the real world, such as analyzing the impact of fuel price increases and decreases on inflation, the effect of holidays on inflation, and future economic planning based on the threshold value of the TSpVARX model. Therefore, in this study, we propose a TSpVARX with the addition of the 12th subset and dummy variables. Subset variables are lags of endogenous variables included in the model to capture recurring patterns for any given period. Dummy variables are categorical variables that are coded into numerical form (usually 0 and 1) to represent groups in the
Economies 2024,12, 352 3 of 27 data so that differences in patterns between data groups can be captured by the model. We include the 12th lag of the endogenous variable as the 12th subset variable to capture the effect of the Eid al-Fitr event on inflation and money outflow. Meanwhile, we include fuel price increases and decreases as dummy variables to accommodate the impact of fuel price adjustments on inflation and money outflow. The developed model will be applied to the forecasting of inflation and money outflow of Yogyakarta, Solo, and Semarang. 2. Methodology In this section, we will explain three things, namely the first subsection regarding the dataset, the second subsection regarding the SpVARX model, the third subsection regarding the TSpVARX model, and the fourth subsection regarding the development of the TSpVARX model with the addition of the 12th subset and dummy variables. In the first subsection, we describe the data and variables used in this study. We also explain the categories of dummy variables used in this study. In the second subsection, we explain the general form of the SpVARX model and how to estimate the coefficients of the SpVARX model. In the third subsection, we explain how the TSpVARX model is formed based on the selected threshold values and variables along with the steps to estimate the TSpVARX model coefficients. In the fourth subsection, we explain how the general form of the TSpVARX model with the addition of the 12th subset and dummy variables to accommodate the existence of a pattern that repeats every 12 months and the effect of increasing and decreasing fuel prices. We will also explain the steps to estimate the model coefficients. 2.1. Dataset The data used in this study are monthly data from January 2006 to September 2024 for the BI benchmark interest rate, the exchange rate of the rupiah against the US dollar, and Inflation of Semarang, Solo, and Yogyakarta. Besides that, we also use the money outflow data of Semarang, Solo, and Yogyakarta from January 2006 to May 2023. Inflation of Semarang, Solo, and Yogyakarta are from Statistics Indonesia. Meanwhile, the money outflow of Semarang, Solo, and Yogyakarta, the BI benchmark interest rate, and the exchange rate of the rupiah against the US dollar are from Bank Indonesia. The model used for forecasting is built using data from January 2006 to May 2023. Further forecasting will be carried out from June 2023 to December 2024. The variables used in this study are inflation of Semarang Y1 1t , inflation of Solo Y2 1t , inflation of Yogyakarta Y3 1t , money outflow of Semarang Y1 2t , money outflow of Solo Y2 2t , money outflow of Yogyakarta Y3 2t , depreciation of the rupiah against the United States dollar at t − i (X1,t−i) , BI benchmark interest rate at t − i (X2,t−i) , the 12th subset variable of Semarang inflation Y1 1,t−12 , the 12th subset variable of Solo inflation Y2 1,t−12 , the 12th subset variable of Yogyakarta inflation Y3 1,t−12 , the 12th subset variable of Semarang money outflow Y1 2,t−12 , the 12th subset variable of Solo money outflow Y2 2,t−12, and the 12th subset variable of Yogyakarta money outflow Y3 2,t−12. It is important to add dummy variables to the TSpVARX model, to capture the effect of rising and falling fuel prices on inflation. When there is an increase in fuel prices, the prices of other goods also increase. This causes an increase in inflation. Conversely, when there is a decrease in fuel prices, inflation decreases. Meanwhile, the addition of the 12th subset variable is aimed at capturing inflation patterns that recur every 12 months caused by events, such as Eid al-Fitr, Christmas, and New Year holidays. The subset variable in this study is an endogenous lag variable used in the model. This variable is different from the autoregressive variable, where if using autoregressive order 3 (AR (3)), then the endogenous lag variable used is the 1st, 2nd, and 3rd lags of the endogenous variable. In the subset variable, we only use one lag of the endogenous variable, without including the other lags of the endogenous variable. In this study, we use the 12th lag of the endogenous variable for our subset variable. The addition of the
Economies 2024,12, 352 4 of 27 12th lag of the endogenous variable aims to capture the fluctuation pattern of inflation and money outflow caused by events that recur every year (12 months), such as Eid, Christmas, and New Year. There are two types of dummy variables that we input in the modeling, namely the dummy variable of whether or not the fuel price increases and the dummy variable of whether or not the fuel price decreases. With these dummy variables, fluctuations in inflation and money outflow caused by increases and decreases in fuel prices can be accommodated by the model. The determination of dummy variable data of the fuel price adjustment is as follows: Rt=1,if the fuel price increase occurs before the 15th day of month tor after the 15th day of month t−1 0,for others St=1,if the fuel price decrease occurs before the 15th day of month tor after the 15th day of month t−1 0,for others Semarang, Solo, and Yogyakarta are the locus of our research because they are geographically close, have easy access to each other, and are included in the same Bank Indonesia region. In addition, the three cities are known as the Golden Triangle area because they are the center of economic development in Central Java both from the tourism and industrial sectors. This allows for linkages in terms of economic activity between these three cities. 2.2. The SpVARX Model The SpVARX model with spatial order one, temporal lag order p, and exogenous variable lag order qcan be written as SpVARX(1, p,q). The general form of the SpVARX model is as follows: Y1 1,t=α1 10 +γ1,(1) 11 X1 1,t−1+· · · +γ1,(q) 11 X1 1,t−q+· · · +γ1,(i) 1mX1 m,t−1+· · · +γ1,(1) 1MX1 M,t−1+· · · +γ1,(q) 1MX1 M,t−q +ϕ1,(1,0) 11 Y1 1,t−1+ϕ1,(1,1) 11 w11(1,2)Y2 1,t−1+· · · +w11(1,u)Yu 1,t−1+· · · +w11(1,N)YN 1,t−1+· · · +ϕ1,(j,0) 1rY1 r,t−j +ϕ1,(j,1) 1rw1r(1,2)Y2 r,t−j+· · · +w1r(1,u)Yu r,t−j+· · · +w1r(1,N)YN r,t−j+· · · +ϕ1,(p,0) 1KY1 K,t−p +ϕ1,(p,1) 1Kw1K(1,2)Y2 K,t−p+· · · +w1K(1,u)Yu K,t−p+· · · +w1K(1,N)YN K,t−p+ε1 1t (1a) Y2 1,t=α2 10 +γ2,(1) 11 X2 1,t−1+· · · +γ2,(q) 11 X2 1,t−q+· · · +γ2,(i) 1mX2 m,t−1+· · · +γ2,(1) 1MX2 M,t−1+· · · +γ2,(q) 1MX2 M,t−q +ϕ2,(1,0) 11 Y2 1,t−1+ϕ2,(1,1) 11 w11(2,1)Y1 1,t−1+w11(2,3)Y3 1,t−1+· · · +w11(2,u)Yu 1,t−1+· · · +w11(2,N)YN 1,t−1+· · · +ϕ2,(j,0) 1rY2 r,t−j +ϕ2,(j,1) 1rw1r(2,1)Y1 r,t−j+w1r(2,3)Y3 r,t−j+· · · +w1r(2,u)Yu r,t−j+· · · +w1r(2,N)YN r,t−j+· · · +ϕ2,(p,0) 1KY2 K,t−p +ϕ2,(p,1) 1Kw1K(2,1)Y1 K,t−p+w1K(2,3)Y3 K,t−p+· · · +w1K(2,u)Yu K,t−p+· · · +w1K(2,N)YN K,t−p+ε2 1t (1b) . . . Yn k,t=αn k0+γn,(1) k1Xn 1,t−1+· · · +γn,(q) k1Xn 1,t−q+· · · +γn,(i) km Xn m,t−i+· · · +γn,(1) kM Xn M,t−1+· · · +γn,(q) kM Xn M,t−q +ϕn,(1,0) k1Yn 1,t−1+ϕn,(1,1) k1wk1(n,1)Y1 1,t−1+· · · +wk1(n,n−1)Yn−1 1,t−1+wk1(n,n+1)Yn+1 1,t−1+· · · +wk1(n,N)YN 1,t−1+· · · +ϕn,(j,0) kr Yn r,t−j +ϕn,(j,1) kr wkr(n,1)Y1 1,t−j+· · · +wkr(n,n−1)Yn−1 r,t−1+· · · +wkr(n,n+1)Yn+1 r,t−1+· · · +wkr(n,N)YN r,t−j+· · · +ϕn,(p,0) kK Yn K,t−p +ϕn,(p,1) kK wkK(n,1)Yn K,t−p+· · · +wkK(n,n−1)Yn−1 K,t−1+wkK(n,n+1)Yn+1 K,t−1+· · · +w1K(n,N)YN K,t−p+εn kt (1c) . . . YN K,t=αN K0+γN,(1) K1XN 1,t−1+· · · +γN,(q) K1XN 1,t−q+· · · +γN,(i) Km XN m,t−i+· · · +γN,(1) KM XN M,t−1+· · · +γN,(q) KM XN M,t−q +ϕN,(1,0) K1YN 1,t−1+ϕN,(1,1) K1wK1(N,1)Y1 1,t−1+· · · +wK1(N,u)Yu 1,t−1+· · · +wK1(N,N−1)YN−1 1,t−1+· · · +ϕN,(j,0) Kr YN r,t−j +ϕN,(j,1) Kr wKr(N,1)Y1 r,t−j+· · · +wKr(N,u)Yu r,t−j+· · · +wKr(N,N−1)YN−1 r,t−j+· · · +ϕN,(p,0) KK YN K,t−p +ϕN,(p,1) KK wKK(N,1)Y1 K,t−p+· · · +wKK(N,u)Yu K,t−p+· · · +wKK(N,N−1)YN−1 K,t−p+εN Kt ,(1d) where Xn m,t−iis the m-th metric exogenous variable at the n-th location in the period t−i,
Economies 2024,12, 352 5 of 27 γn,(i) km is the coefficient of variable Xn m,t−iin equation Yn kt, Mis the number of exogenous variables, jis the lag order of the autoregressive, and iis the lag order of the metric exogenous variable from 1 to q. The SpVARX model in Equations (1a) to (1d) can be made into vector and matrix operations as follows: Y1 1,t Y2 1,t . . . Yn k,t . . . YN K,t NK×1 = x1 1,t′v1 1,t′0 0 · · · 0 0 · · · 0 0 0 0 x2 1,t′v2 1,t′· · · 0 0 · · · 0 0 . . .. . .. . .. . ..... . .. . ..... . .. . . 0 0 0 0 · · · xn k,t′vn k,t′· · · 0 0 . . .. . .. . .. . ..... . .. . ..... . .. . . 0 0 0 0 0 0 0 · · · xN K,t′vN K,t′ NK×NK(1+Mq+2Kp) γ1 1 ϕ1 1 γ2 1 ϕ2 1 . . . γn k ϕn k . . . γN K ϕN K + e1 1,t e2 1,t . . . en k,t . . . eN K,t (2) where: xn k,t′=h1Xn 1,t−1· · · Xn 1,t−q· · · Xn m,t−i· · · Xn M,t−1· · · Xn M,t−qi1×(1+Mq) vn k,t′=hYn 1,t−1Yn∗ k1,t−1· · · Yn r,t−jYn∗ kr,t−j· · · Yn K,t−pYn∗ kK,t−pi1×2Kp Yn∗ kr,t−j= N ∑ u=1 u=n wkr(n,u)Yu r,t−j γn k=hαn k0γn(1) k1· · · γn(q) k1· · · γn(i) km · · · γn(1) kM · · · γn(q) kM i′ (1+Mq)×1 ϕn k=hϕn,(1,0) k1ϕn,(1,1) k1· · · ϕn,(j,0) kr ϕn,(j,1) kr · · · ϕn,(p,0) kK ϕn,(p,1) kK i′ 2Kp×1 Let Y= Y1 1,h+1Y2 1,h+1· · · Yn k,h+1· · · YN K,h+1 Y1 1,h+2Y2 1,h+2· · · Yn k,h+2· · · YN K,h+2 . . .. . ..... . ..... . . Y1 1,TY2 1,T· · · Yn k,T· · · YN K,T with size (T − h) × NK, Zn k= xn k,h+1′vn k,h+1′ xn k,h+2′vn k,h+2′ . . .. . . xn k,T′vn k,T′ , Z= Z1 10· · · 0· · · 0 0 Z2 1· · · 0· · · 0 . . .. . ..... . ..... . . 0 0 · · · Zn k· · · 0 . . .. . ..... . ..... . . 0 0 · · · 0· · · ZN K , bn k=γn k ϕn k , β= b1 1 b2 1 . . . bn k . . . bN K , and ξ= e1 1,h+1e2 1,h+1· · · en k,h+1· · · eN K,h+1 e1 1,h+2e2 1,h+2· · · en k,h+2· · · eN K,h+2 . . .. . ..... . ..... . . e1 1,Te2 1,T· · · en k,T· · · eN K,T then if the number of
Economies 2024,12, 352 6 of 27 observations used for the model is Tand h= max(p,q), then the form of the SpVARX model in Equation (2) can be described as follows: Vec(Y)=Zβ+Vec(ξ), (3) where Vec is an operator that stacks a matrix as a column vector, Vec(Y)is a vector of size NK (T−h)×1 obtained from stacking Y, Vec(ξ)is a vector of size NK (T−h)×1 obtained from stacking ξ, Zis a matrix of size NK (T−h)×NK (1 + Mq + 2Kp), βis a vector of size NK (1 + Mq + 2Kp)×1. There are several steps to estimate the coefficient parameters of the SpVARX model with the addition of the 12th subset and dummy variables using the MLE method (Sohibien et al. 2024a,2024b). 1. Form vectors and matrices using lag orders pand qthat can produce the smallest AIC value of the SpVARX model. 2. Estimate the coefficients of the SpVARX model with OLS using the following formula: ˆ βOLS,SpVARX =Z′Z−1Z′Vec(Y) 3. Find the residuals of the SpVARX model obtained by OLS in step 2 with the following formula: Vec(ξ)OLS =Vec(Y)−Zˆ βOLS,SpVARX 4. Obtain the error covariance matrix estimator of the OLS-derived SpVARX with the following formula: ˆ ΣOLS,SpVARX = Vare1 1,tCove1 1,t,e2 1,t· · · Cove1 1,t,en k,t· · · Cove1 1,t,eN K,t Cove2 1,t,e1 1,tVare2 1,t· · · Cove2 1,t,en k,t· · · Cove2 1,t,eN K,t . . .. . ..... . ..... . . Coven 1,t,e1 k,tCoven 2,t,e1 k,t· · · Varen k,t· · · Coven 2,t,eN K,t . . .. . ..... . ..... . . Coven k,t,e1 k,tCoven k,t,e1 k,t· · · Coven k,t,e1 k,t· · · VareN K,t where en k=hen k,h+1en k,h+2· · · en k,Ti,Var(en k)=1 (T−h)(en k)′·(en k), Coven k,eN K,=1 (T−h)(en k)′·eN K. 5. Estimate the model coefficient of the SpVARX with MLE using the following formula: ˆ βMLE,SpVARX =Z′ˆ ΣOLS,SpVARX ⊗IZ−1Z′ˆ ΣOLS,SpVARX ⊗I−1Vec(Y). (4) 2.3. Threshold Spatial Vector Autoregressive with Metric Exogenous Variables (TSpVARX) The TSpVARX model is formed by dividing the observation data into several regimes based on the threshold variable Yu k,t−d and the threshold value (ς) . In each regime, the SpVARX model coefficients will be estimated so that each regime will contain SpVARX models that have different model coefficients. The TSpVARX model consisting of Gregimes with delay order d, spatial order 1, temporal lag order p, and exogenous variable lag order qcan be written as TSpVARX(G, 1, p,q,d). The general form of the TSpVARX model is as follows:
Economies 2024,12, 352 7 of 27 y= Z(1)β(1)+ε(1), whenYu k,t−d≤ς1 Z(2)β(2)+ε(2), when ς1<Yu k,t−d≤ς2 . . . Z(g)β(g)+ε(g), when ςg−1<Yu k,t−d≤ςg, . . . Z(G)β(G)+ε(G), when ςG−1<Yu k,t−d≤ςG, (5) where Yu k,t−dis the selected threshold variable, ςg−1is the selected threshold value for the lower bound of the g-th regime, and ςgis a selected threshold value for the upper bound of the g-th regime. There are several steps in generating TSpVARX coefficient estimates (Sohibien et al. 2024b). 1. Set a lag of the endogenous variable, which will be the threshold variable. 2. Determine the temporal lag order (p) based on the smallest AIC of the SpVAR model. 3. Determine the exogenous variable lag order (q) based on the smallest AIC of the SpVARX model. 4. Determine the delay limit (d) equal to the selected order pso that the threshold variable candidates are Yu k,t−1,Yu k,t−2,· · · ,Yu k,t−p. 5. For each threshold variable candidate, determine the lowest threshold value; (ςDL) is the 10th percentile of the threshold variable candidate and the highest threshold value, and (ςdU) is the 90th percentile of the threshold variable candidate so that we obtain the threshold value candidate interval as follows ςDL ≤ς≤ςUL. 6. Divide the data into two parts based on all possibilities ς and d; when Yu k,t−d>ς , the data will fall into the first regime, and when Yu k,t−d≤ς , the data will fall into the second regime. 7. Estimate the coefficients of the SpVARX model with the addition of the 12th subset and dummy variables in the first regime and the second regime for all possible data splits by using the estimation steps described in Section 2.2. 8. Calculate the ln-likelihood function values in the first regime lˆ β(1) MLE,SpVARX(d,ς) Ω(1) and the second regime lˆ β(2) MLE,SpVARX(d,ς) Ω(2) for each possible division of the data. The formula is as follows: lβ(g) MLE,SpVARX(d,ς) Ω(g)=−NKT(g) 2ln2π−1 2ln Ω(g)−1 2y(g)−Z(g)β(g)′Ω(g)−1y(g)−Z(g)β(g). 9. Calculate the total ln-likelihood with the following formula: lˆ βMLE,SpVARX(d,ς)=lˆ β(1) MLE,SpVARX(d,ς) Ω(1)+lˆ β(2) MLE,SpVARX(d,ς) Ω(2). 10. Obtain the estimated delay ˆ d and threshold values (ˆ ς) by finding the pair (d,ς) that maximizes lˆ βMLE,SpVARX(d,ς)or it can be written down: ˆ d,ˆ ς=n(d,ς),maxlˆ βMLE,TSpVARX(d,ς)o. 11. The coefficient estimator is the estimator that is used ˆ d , ˆ ς as the basis for regime division. We can write it as follows: ˆ β(g) MLE,TSpVARX with 2Regimes =ˆ β(g) MLE,SpVARXˆ d,ˆ ς
Economies 2024,12, 352 8 of 27 12. If you want to perform TSpVARX modeling up to Gregimes, there will be G − 1 threshold variables. The 1st, 2nd, . . . ,G − 2th threshold and delay estimators use the values obtained from the TSpVARX with the G−1 regime model. 13. The search for the G − 1th threshold estimator is performed by searching from the threshold value candidates in each TSpVARX model regime with G − 1 regimes so that the threshold value candidates are in the following range: ς1[G−1] L≤ς1[G−1]≤ς1[G−1] U,ς2[G−1] L≤ς2[G−1]≤ς2[G−1] U, . . . , ςg[G−1] L≤ςg[G−1]≤ςg[G−1] U, . . . , ςG−1[G−1] L≤ςG−1[G−1]≤ςG−1[G−1] U , where ςg[G−1] L is the lowest threshold value candidate derived from the 10th percentile of data in the g-th regime of the TSpVARX model with G−1 regimes, and ςg[G−1] U is the highest threshold value candidate derived from the 90th percentile of data in the g-th regime of the TSpVARX model with G−1 regimes. 14. Calculate the total ln likelihood for all possible threshold value candidates as explained in step 13 with the following formula: lˆ βMLE,SpVARX(d,ˆ ς1,ˆ ς2, . . . , ςG−1)=lˆ β(1) MLE,SpVARX(d,ˆ ς1,ˆ ς2, . . . , ςG−1) Ω(1)+lˆ β(2) MLE,SpVARX(d,ˆ ς1,ˆ ς2, . . . , ςG−1) Ω(2)+· · · +lˆ β(G) MLE,SpVARX(d,ˆ ς1,ˆ ς2, . . . , ςG−1) Ω(G). 15. The G − 1th threshold value estimator is the threshold value that maximizes the total ln likelihood. We can also write it as follows: ˆ ςG−1=nςG−1,maxlˆ βMLE,TSpVARXˆ d,ˆ ς1,ˆ ς2, . . . , ˆ ςG−1o. 16. The estimator coefficients of the TSpVARX with Gregime in the g-th regime with the addition of the 12th subset and dummy variables are the coefficients obtained by dividing the regime based on the estimation of delay ˆ d and threshold value ˆ ς1 , ˆ ς2 , . . . , ˆ ςG−1 . We can write it as follows: ˆ β(g) MLE,TSpVARXGRezim =ˆ β(g) MLE,SpVARXˆ d,ˆ ς1,ˆ ς2, . . . , ˆ ςG−1. We include a flow chart in Figure 1to improve readability and understanding of the TSpVARX model. 2.4. The Development of TSpVARX with the Addition of the 12th Subset and Dummy Variables The development of the TSpVARX model with the 12th subset and dummy variables is performed by adding the 12th lag of endogenous and dummy variables as predetermined variables in the TSpVARX model. The form of the SpVARX model with the addition of the 12th lag of endogenous and dummy variables is as follows: Y1 1,t=α110 +γ1,(1) 11 X1 1,t−1+· · · +γ1,(q) 11 X1 1,t−q+· · · +γ1,(i) 1mX1 m,t−1+· · · +γ1,(1) 1MX1 M,t−1+· · · +γ1,(q) 1MX1 M·t−q+λ1 11D1 11,t+· · · +λ1 1lD1 1l,t+· · · +λ1 1LD1 1L,t+ϕ1,(j,0) 11 Y1 1,t−1+ϕ1,(1,1) 11 w11(1,2)Y2 1,t−1+· · · +w11(1,u)Yu 1,t−1+· · · +w11(1,N)YN 1,t−1 +· · · +ϕ1,(j,0) 1rY1 r,t−j+ϕ1,(j,1) 1rw1r(1,2)Y2 r,t−j+· · · +w1r(1,u)Yu r,t−j+· · · +w1r(1,N)YN r,t−j +· · · +ϕ1,(p,0) 1KY1 K,t−p+ϕ1,(p,1) 1Kw1K(1,2)Y2 K,t−p+· · · +w1K(1,u)Yu K,t−p+· · · +w1K(1,N)YN K,t−p+ϕ1,(12,0) 11 Y1 1,t−12 +ϕ1,(12,0) 12 Y1 2,t−12 +· · · +ϕ1,(12,0) 1rY1 r,t−12 +· · · +ϕ1,(12,0) 1KY1 K,t−12 +e1 1t, (6a) Y2 1,t=α210 +γ2,(1) 11 X2 1,t−1+· · · +γ2,(q) 11 X2 1,t−q+· · · +γ2,(i) 1mX2 m,t−1+· · · +γ2,(1) 1MX2 M,t−1+· · · +γ2,(q) 1MX2 M·t−q+λ2 11D2 11,t+· · · +λ2 1lD2 1l,t+· · · +λ2 1LD2 1L,t+ϕ2,(j,0) 11 Y2 1,t−1+ϕ2,(1,1) 11 w11(2,1)Y1 1,t−1+· · · +w11(2,u)Yu 1,t−1+· · · +w11(2,N)YN 1,t−1 +· · · +ϕ2,(j,0) 1rY2 r,t−j+ϕ2,(j,1) 1rw1r2Y1 r,t−j+· · · +w1r(2,u)Yu r,t−j+· · · +w1r(2,N)YN r,t−j +· · · +ϕ2,(p,0) 1KY2 K,t−p+ϕ2,(p,1) 1Kw1K(2,1)Y1 K,t−p+· · · +w1K(2,u)Yu K,t−p+· · · +w1K(2,N)YN K,t−p+ϕ2,(12,0) 11 Y2 1,t−12 +ϕ2,(12,0) 12 Y2 2,t−12 +· · · +ϕ2,(12,0) 1rY2 r,t−12 +· · · +ϕ2,(12,0) 1KY2 K,t−12 +e2 1t, (6b)
Economies 2024,12, 352 15 of 27 Table 3. The significance test of cross-correlation between the lag of log money outflow and all endogenous variables. Endogenous Variable Log Outflow of Semarang Log Outflow of Solo Log Outflow of Yogyakarta Cross- Correlation PV Cross- Correlation PV Cross- Correlation PV Inflation of Semarang −0.136 0.050 ** −0.109 0.118 −0.064 0.355 Inflation of Solo −0.011 0.877 0.022 0.754 0.032 0.651 Inflation of Yogyakarta −0.111 0.115 −0.071 0.310 −0.027 0.702 Log outflow of Semarang 0.512 0.000 ** 0.491 0.000 ** 0.408 0.000 ** Log outflow of Solo 0.500 0.000 ** 0.568 0.000 ** 0.470 0.000 ** Log outflow of Yogyakarta 0.389 0.000 ** 0.450 0.000 ** 0.414 0.000 ** Notes: (**) significant at a 5-percent significance level. 3.2. Temporal Lag Selection (p) The selection of temporal lag (p) is performed by looking at the smallest Akaike Information Criterion (AIC) value generated from several SpVAR models formed. There are two weights used, namely uniform and normalized cross-correlation weight. The AIC of the SpVAR models can be seen in Table 4. The smallest AIC is obtained from the SpVAR (1, 1) with a constant. Table 4. Akaike Information Criterion (AIC) values based on SpVAR(1, p) model (for p= 1, 2, . . . , 4) and spatial weights. SpVAR Model AIC with Uniform Weighting AIC with Cross-Correlation Normalization Weighting Without Constant SpVAR(1, 1) −1487.846 −1486.08 SpVAR(1, 2) −1494.386 −1497.785 SpVAR(1, 3) −1422.474 −1425.865 SpVAR(1, 4) −1308.21 −1310.738 With Constant SpVAR(1, 1) −1586.181 −1587.617 SpVAR(1, 2) −1546.97 −1553.562 SpVAR(1, 3) −1453.157 −1457.315 SpVAR(1, 4) −1331.915 −1335.972 3.3. Order Selection q The next step is to select the order lag of exogenous variables (q) that will be used in SpVARX and TSpVARX modeling. The selection of order qis performed by looking at the smallest AIC generated from SpVARX models with a temporal lag order (p) of 1. We can write the general form as SpVARX (1, 1, q). The results of the AIC values formed from SpVARX (1, 1, q) models can be seen in Table 5. SpVARX models that are successfully formed using uniform weight are SpVARX models with a lag order qof one to four. Meanwhile, the SpVARX models successfully formed using the normalized cross-correlation weight are SpVARX models with a lag order qof 1, 2, and 3. The SpVARX model with a lag order qof 4 with a normalized cross-correlation weight is not successfully formed due to singularity problems in estimating the SpVARX model coefficients. A coefficient estimation that cannot be generated due to the singularity, causes us to be unable to obtain the AIC value. Based on Table 5, the smallest AIC obtained from the SpVARX model is when the order qis equal to one or both when using the uniform and normalized cross-correlation weight.
Economies 2024,12, 352 16 of 27 Table 5. Akaike Information Criterion (AIC) based on SpVAR (1, 1, q) model (for q= 1, 2, . . . , 4) and spatial weight. SpVARX Model AIC with Uniform Weight AIC with Normalized Cross-Correlation Weight With Constant SpVARX(1, 1, 1) −1584.989 −1586.072 SpVARX(1, 1, 2) −1567.85 −1569.152 SpVARX(1, 1, 3) −1545.433 −1546.449 SpVARX(1, 1, 4) −1520.337 - 3.4. Nonlinearity Test Between Endogenous and Lag of Endogenous Variables The next step is to test whether the endogenous variables are suitable to be modeled linearly with the predetermined variables. This research uses the reset test method. The hypothesis used in the linearity test with the reset test is as follows: H0.The endogenous variable fits a linearly model with predetermined variables. H1.Endogenous variables are not linearly modeled with predetermined variables. The results of the nonlinearity testing between endogenous variables and predetermined variables can be seen in Table 6. If the p-value is less than the significant level, the decision is to reject Ho, which means that the relationship between the endogenous variables and the predetermined variables is not linear. In Table 6, it can be seen that of the six endogenous variables used, three of them (inflation of Solo, inflation of Yogyakarta, and outflow of Solo) have p-values less than the five- or ten-percent significance level. The inflation of Solo and money outflow of Solo are significant at the ten-percent significance level, while the inflation of Yogyakarta is significant at the five-percent significance level. Thus, these three endogenous variables are not fit to be modeled linearly against the predetermined variables used. Table 6. Linearity test results using reset test between endogenous variables and all predetermined variables. Endogenous Variable p-Value Inflation of Semarang 0.104 Inflation of Solo 0.054 * Inflation of Yogyakarta 0.004 ** Ln outflow of Semarang 0.155 Ln outflow of Solo 0.084 * Ln outflow of Yogyakarta 0.124 Notes: (*) significant at a 10-percent significance level. (**) significant at a 5-percent significance level. The non-linear relationship between inflation and predetermined variables, such as the lag of money outflow, exchange rate depreciation, and BI benchmark interest rate is in line with the findings of several studies. Çitçi and Kaya (2023), using a sample of 149 countries, found that the exchange rate has a non-linear relationship with inflation. Gök and Bulut (2021) found that interest rates have a non-linear relationship with Turkish inflation. 3.5. Selection of Threshold Variables After we obtain the temporal lag order (p), the exogenous variable lag order (q), and the indications of non-linear relationships, the next step is to model the TSpVARX. The number of regimes (s) tried in this study are 2, 3, and 4. The TSpVARX (s, 1, 1, 1) model formed in this study uses the uniform and normalized cross-correlation weight. The value of the uniform spatial weight and the normalized cross-correlation weight can be seen in Appendix A. Because the order pis one, the threshold variables used in this study are Y1 1,t−1 , Y2 1,t−1 , Y3 1,t−1 , logY1 2,t−1 , logY2 2,t−1 , and logY3 2,t−1 . Based on Figure 4, we can see
Economies 2024,12, 352 17 of 27 that 15 of 18 TSpVARX models formed have smaller AIC values than SpVARX and 14 of 18 TSpVARX models have smaller AIC values than SpVAR. This indicates that most of the TSpVARX formed are better than SpVARX and SpVAR in terms of modeling. Based on the smallest AIC, we select the best three TSpVARX models with a uniform weight, namely the two-regime TSpVARX with the threshold variable logY2 2,t−1 , the three-regime TSpVARX with the threshold variable Y3 1,t−1 , and the four-regime TSpVARX with the threshold variable Y2 1,t−1. Economies 2024, 12, x FOR PEER REVIEW 20 of 32 formed in this study uses the uniform and normalized cross-correlation weight. The value of the uniform spatial weight and the normalized cross-correlation weight can be seen in Appendix A. Because the order p is one, the threshold variables used in this study are 1 11t Y,−, 2 11t Y,−, 3 11t Y,−, 1 21 log t Y,−, 2 21 log t Y,−, and 3 21 log t Y,−. Based on Figure 4, we can see that 15 of 18 TSpVARX models formed have smaller AIC values than SpVARX and 14 of 18 TSpVARX models have smaller AIC values than SpVAR. This indicates that most of the TSpVARX formed are better than SpVARX and SpVAR in terms of modeling. Based on the smallest AIC, we select the best three TSpVARX models with a uniform weight, namely the two-regime TSpVARX with the threshold variable 2 21 log t Y,− , the three-regime TSpVARX with the threshold variable 3 11t Y,− , and the four-regime TSpVARX with the threshold variable 2 11t Y,−. Figure 4. AIC values of SpVAR, SpVARX, and TSpVARX models of two, three, and four regimes with uniform weights. Notes: VT is the threshold variable. In Figure 5, it can be seen that 13 of 16 TSpVARX models with normalized cross-correla- tion weights have smaller AIC values than SpVARX and SpVAR. This shows that most of the TSpVARX models with normalized cross-correlation weights are better than SpVAR and SpVARX in terms of modeling. Based on the smallest AIC value, the best three TSpVARX models with normalized cross-correlation weights are the two-regime TSpVARX with the threshold variable 2 11 log t Y,−, the three-regime TSpVARX with the threshold variable 2 11 log t Y,−, and the fourregime TSpVARX with the threshold variable 2 11 log t Y,−. Figure 4. AIC values of SpVAR, SpVARX, and TSpVARX models of two, three, and four regimes with uniform weights. Notes: VT is the threshold variable. In Figure 5, it can be seen that 13 of 16 TSpVARX models with normalized crosscorrelation weights have smaller AIC values than SpVARX and SpVAR. This shows that most of the TSpVARX models with normalized cross-correlation weights are better than SpVAR and SpVARX in terms of modeling. Based on the smallest AIC value, the best three TSpVARX models with normalized cross-correlation weights are the two-regime TSp- VARX with the threshold variable logY2 1,t−1 , the three-regime TSpVARX with the threshold variable logY2 1,t−1, and the four-regime TSpVARX with the threshold variable logY2 1,t−1. Economies 2024, 12, x FOR PEER REVIEW 21 of 32 Figure 5. AIC values of SpVAR, SpVARX, and TSpVARX models of two, three, and four regimes with normalized cross-correlation weights. Notes: VT is the threshold variable. 3.6. Evaluation of TSpVARX Model Compared to SpVARX in Modeling Inflation and Money Outflow of Semarang, Solo, and Yogyakarta After we obtain the best model of TSpVARX with two, three, and four regimes, the next step is to evaluate the performance of TSpVARX and SpVARX models in forecasting the inflation and money outflow of Semarang, Solo, and Yogyakarta. An evaluation of the forecasting performance is performed by looking at the Root Mean Square Error (RMSE) for each endogenous variable. The RMSE is a value used to measure the accuracy of a model in forecasting. A lower RMSE indicates a more precise forecast result. Specifically, for the log money outflow variable, the RMSE is calculated using the residuals of the original forecasting money outflow, not in logarithm form. It is performed because the forecasting results used in reality are the original value of money outflow. According to Table 7, five of six endogenous variables obtain the smallest RMSE in the testing data when using the three-regime TSpVARX model with a uniform weight. Those five endogenous variables are Semarang Inflation () 1 11t Y,−, Solo Inflation () 2 11t Y ,− , money outflow of Semarang () 1 21 log t Y,−, money outflow of Solo () 2 21 log t Y,−, and money outflow of Yogyakarta () 3 21 log t Y,− . Meanwhile, in the training data, four endogenous variables obtain the smallest RMSE when we use a normalized cross-correlation weight. Table 7. Root Mean Square Error (RMSE) of SpVARX and TSpVARX. Data Weights Model Inflation of Semarang Inflation of Solo Inflation of Yogyakarta Money Outflow of Semarang Money Outflow of Solo Mony Outflow of Yogyakarta Testing Uniform SpVAR 1.949 0.366 0.332 3059.76 1164.92 1229.44 SpVARX 0.391 0.366 0.3 2450.76 940.89 990.31 () 1 1t Y, () 2 1t Y, () 3 1t Y, () 1 2t Y, () 2 2t Y, () 3 2t Y, Figure 5. AIC values of SpVAR, SpVARX, and TSpVARX models of two, three, and four regimes with normalized cross-correlation weights. Notes: VT is the threshold variable.
Economies 2024,12, 352 18 of 27 3.6. Evaluation of TSpVARX Model Compared to SpVARX in Modeling Inflation and Money Outflow of Semarang, Solo, and Yogyakarta After we obtain the best model of TSpVARX with two, three, and four regimes, the next step is to evaluate the performance of TSpVARX and SpVARX models in forecasting the inflation and money outflow of Semarang, Solo, and Yogyakarta. An evaluation of the forecasting performance is performed by looking at the Root Mean Square Error (RMSE) for each endogenous variable. The RMSE is a value used to measure the accuracy of a model in forecasting. A lower RMSE indicates a more precise forecast result. Specifically, for the log money outflow variable, the RMSE is calculated using the residuals of the original forecasting money outflow, not in logarithm form. It is performed because the forecasting results used in reality are the original value of money outflow. According to Table 7, five of six endogenous variables obtain the smallest RMSE in the testing data when using the three-regime TSpVARX model with a uniform weight. Those five endogenous variables are Semarang Inflation Y1 1,t−1 , Solo Inflation Y2 1,t−1 , money outflow of Semarang logY1 2,t−1 , money outflow of Solo logY2 2,t−1 , and money outflow of Yogyakarta logY3 2,t−1 . Meanwhile, in the training data, four endogenous variables obtain the smallest RMSE when we use a normalized cross-correlation weight. Table 7. Root Mean Square Error (RMSE) of SpVARX and TSpVARX. Data Weights Model Inflation of Semarang Y1 1,t Inflation of Solo Y2 1,t Inflation of Yogyakarta Y3 1,t Money Outflow of Semarang Y1 2,t Money Outflow of Solo Y2 2,t Mony Outflow of Yogyakarta Y3 2,t Testing Uniform SpVAR 1.949 0.366 0.332 3059.76 1164.92 1229.44 SpVARX 0.391 0.366 0.3 2450.76 940.89 990.31 TSpVARX with 2 Regimes 0.46 0.405 0.319 2661.00 932.80 952.01 TSpVARX with 3 Regimes 0.366 0.339 0.299 2130.05 827.40 866.93 TSpVARX with 4 Regimes 0.402 0.372 0.301 2577.12 958.13 1021.66 Cross- Correlation Normalization SpVAR 0.399 0.361 0.292 2728.032 1046.463 1098.674 SpVARX 0.389 0.359 0.297 2457.53 943.28 991.76 TSpVARX 2 with 2 Regimes 0.429 0.394 0.303 2661.60 929.26 945.37 TSpVARX 3 with 3 Regimes 0.542 0.47 0.486 2997.96 1028.87 1175.92 TSpVARX 4 with 4 Regimes 0.452 0.528 0.383 2604.83 963 892.41 Training Uniform SpVAR 0.487 0.552 0.421 1496.206 705.238 820.559 SpVARX 0.552 0.482 0.418 1442.384 687.528 806.829 TSpVARX with 2 Regimes 0.467 0.547 0.406 1412.614 677.882 797.165 TSpVARX with 3 Regimes 0.445 0.535 0.398 1426.346 698.190 795.610 TSpVARX with 4 Regimes 0.477 0.537 0.398 1426.458 695.333 795.303 Cross- Correlation Normalization SpVAR 0.488 0.553 0.421 1492.539 704.125 818.975 SpVARX 0.488 0.553 0.418 1441.095 687.130 806.098 TSpVARX with 2 Regimes 0.468 0.548 0.408 1412.025 677.884 796.828 TSpVARX with 3 Regimes 0.542 0.470 0.486 2997.964 1028.873 1175.918 TSpVARX with 4 Regimes 0.458 0.540 0.390 1405.814 675.928 790.389 In Figures 6and 7, the blue line illustrates the movement of actual data. The red line illustrates the movement of the forecasting data of the TSpVARX model, while the green line illustrates the movement of forecast data of the SpVARX model. Based on Figure 6, the up-and-down pattern of actual data can already be followed by the up-and-down pattern of forecast data generated from the three-regime TSpVARX model. However, there are
Economies 2024,12, 352 19 of 27 still some points of actual data that are still far from the forecasting inflation data. The events that cause some of the actual inflation data to be far from the forecasting data results are Christmas and New Year events (December 2006, December 2010), Eid al-Fitr events (19 August 2012, 8 August 2013), fuel price increases (24 May 2008, November 2014, 1 September 2022), and fuel price decreases (1 December 2008, 1 February 2015, 1 April 2016, 10 February 2019). In Figure 7, it can be seen that there are still actual data at several points of the training and testing periods that are far from the forecasting money outflow data when we use the TSpVARX model. The events that cause this situation are Christmas and New Year events (December 2006, December 2019) and Eid al-Fitr events (October 2007, October 2008, September 2009, September 2010, August 2011, August 2012, August 2013, July 2014, July 2015, July 2016, June 2017, June 2018, June 2019, May 2022, April 2023). The most events that cause this condition are the Eid al-Fitr events. In Table 8, we can see that the forecasting results of inflation and money outflow of Semarang, Solo, and Yogyakarta using the three-regime TSpVARX model are relatively stable after the next sixteen periods (October 2024, November 2024, and December 2024). It could be due to the use of an autoregressive lag order (p) of one that can only accommodate the data pattern of one previous period. In addition, the other influences of the Eid al-Fitr event and the adjustment of fuel prices have not been captured in the model. Economies 2024, 12, x FOR PEER REVIEW 23 of 32 the up-and-down pattern of actual data can already be followed by the up-and-down pattern of forecast data generated from the three-regime TSpVARX model. However, there are still some points of actual data that are still far from the forecasting inflation data. The events that cause some of the actual inflation data to be far from the forecasting data results are Christmas and New Year events (December 2006, December 2010), Eid al-Fitr events (19 August 2012, 8 August 2013), fuel price increases (24 May 2008, November 2014, 1 September 2022), and fuel price decreases (1 December 2008, 1 February 2015, 1 April 2016, 10 February 2019). (a) (b) (c) Figure 6. Plot of actual and forecast data using TSpVARX three regimes for (a) Semarang inflation, (b) Solo inflation, and (c) Yogyakarta inflation. In Figure 7, it can be seen that there are still actual data at several points of the training and testing periods that are far from the forecasting money outflow data when we use the TSpVARX model. The events that cause this situation are Christmas and New Year events (December 2006, December 2019) and Eid al-Fitr events (October 2007, October 2008, September 2009, September 2010, August 2011, August 2012, August 2013, July 2014, July 2015, July 2016, June 2017, June 2018, June 2019, May 2022, April 2023). The most events that cause this condition are the Eid al-Fitr events. Figure 6. Plot of actual and forecast data using TSpVARX three regimes for (a) Semarang inflation, (b) Solo inflation, and (c) Yogyakarta inflation.
Economies 2024,12, 352 20 of 27 Economies 2024, 12, x FOR PEER REVIEW 24 of 32 (a) (b) (c) Figure 7. Plot of actual and forecasting data in training and testing data periods using TSpVARX with three regimes for money outflow of (a) Semarang, (b) Solo, and (c) Yogyakarta. In Table 8, we can see that the forecasting results of inflation and money outflow of Semarang, Solo, and Yogyakarta using the three-regime TSpVARX model are relatively stable after the next sixteen periods (October 2024, November 2024, and December 2024). It could be due to the use of an autoregressive lag order (p) of one that can only accommodate the data pattern of one previous period. In addition, the other influences of the Eid al-Fitr event and the adjustment of fuel prices have not been captured in the model. Table 8. Forecasting of inflation (in percent) and money outflow (in billion rupiahs) of Semarang, Solo, and Yogyakarta using TSpVARX three regimes. Years Months Inflation of Semarang Inflation of Solo Inflation of Yogyakarta Money Outflow of Semarang Money Outflow of Solo Money Outflow of Yogyakarta 2023 Jun 0.27 0.26 0.32 1170.09 412.69 673.65 2023 Jul 0.23 0.23 0.25 1303.44 527.27 784.50 2023 Ags 0.29 0.28 0.32 1384.05 613.21 910.39 2023 Sep 0.23 0.23 0.25 1523.43 698.45 927.87 2023 Oct 0.32 0.31 0.35 1523.95 733.44 1004.58 2023 Nov 0.28 0.25 0.32 1688.13 866.91 1137.90 2023 Dec 0.36 0.34 0.39 1431.25 659.18 992.05 2024 Jan 0.31 0.29 0.34 1311.64 551.14 917.04 2024 Feb 0.21 0.21 0.24 1368.85 569.88 861.08 2024 Mar 0.36 0.35 0.39 1342.39 640.51 946.17 2024 Apr 0.36 0.33 0.40 1333.95 619.51 985.76 Figure 7. Plot of actual and forecasting data in training and testing data periods using TSpVARX with three regimes for money outflow of (a) Semarang, (b) Solo, and (c) Yogyakarta. Table 8. Forecasting of inflation (in percent) and money outflow (in billion rupiahs) of Semarang, Solo, and Yogyakarta using TSpVARX three regimes. Years Months Inflation of Semarang Inflation of Solo Inflation of Yogyakarta Money Outflow of Semarang Money Outflow of Solo Money Outflow of Yogyakarta 2023 Jun 0.27 0.26 0.32 1170.09 412.69 673.65 2023 Jul 0.23 0.23 0.25 1303.44 527.27 784.50 2023 Ags 0.29 0.28 0.32 1384.05 613.21 910.39 2023 Sep 0.23 0.23 0.25 1523.43 698.45 927.87 2023 Oct 0.32 0.31 0.35 1523.95 733.44 1004.58 2023 Nov 0.28 0.25 0.32 1688.13 866.91 1137.90 2023 Dec 0.36 0.34 0.39 1431.25 659.18 992.05 2024 Jan 0.31 0.29 0.34 1311.64 551.14 917.04 2024 Feb 0.21 0.21 0.24 1368.85 569.88 861.08 2024 Mar 0.36 0.35 0.39 1342.39 640.51 946.17 2024 Apr 0.36 0.33 0.40 1333.95 619.51 985.76 2024 Mei 0.31 0.29 0.33 1519.46 855.06 1050.96 2024 Jun 0.21 0.21 0.24 1401.90 629.53 864.87 2024 Jul 0.17 0.16 0.19 1476.63 638.73 844.42 2024 Ags 0.23 0.23 0.26 1313.11 533.11 777.51 2024 Sep 0.24 0.25 0.28 1025.16 343.08 628.31 2024 Oct 0.29 0.30 0.33 904.17 290.27 608.09 2024 Nov 0.30 0.31 0.33 805.45 241.46 565.92 2024 Dec 0.29 0.30 0.33 766.44 221.82 546.40
Economies 2024,12, 352 21 of 27 3.7. Addition of the 12th Subset and Dummy Variables in the Form of Fuel Price Adjustments Based on the previous section, it can be seen that the results of forecasting inflation and money outflow using the TSpVARX model are not so good because several forecasting values are still far from the actual data. In addition, the forecasting data of inflation and money outflow using the TSpVARX model are also still stable after the next three periods. This indicates that the influence of the increase and decrease in fuel prices and the Eid al-Fitr event have not been captured by the model. Therefore, in this section, a subset variable of the 12th lag of the endogenous variable and dummy variables of increasing and decreasing fuel prices will be added to the TSpVARX modeling. This addition is expected to help in capturing the effect of the Eid al-Fitr and fuel price adjustment event. Based on Table 9, it can be seen that there are five smallest RMSE values based on the residuals of the testing data obtained from the TSpVARX model with the addition of the 12th subset and the dummy variables. Based on the RMSE of the training data residuals, all the smallest RMSEs are obtained using the TSPVARX model with the addition of the 12th subset and dummy variables. Therefore, we can conclude that the addition of the 12th subset and the dummy variables can improve the performance of the TSpVARX model in forecasting the inflation and money outflow of Semarang, Solo, and Yogyakarta. Table 9. Root Mean Square Error (RMSE) using testing and training data based on three-regime TSpVARX and three-regime TSpVARX models with the addition of the 12th subset and the dummy variables. Data Model Inflation of Semarang Y1 1,t Inflation of Solo Y2 1,t Inflation of Yogyakarta Y3 1,t Money Outflow of Semarang Y1 2,t Money Outflow of SoloY2 2,t Money Outflow of Yogyakarta Y3 2,t Testing TSpVARX 3 Rezim 0.366 0.339 0.299 2130.05 827.40 866.93 TSpVARX 3 Rezim dengan Subset 12 dan Dummy Pt dan St 0.340 0.318 0.319 1756.249 687.680 769.774 Training TSpVARX 3 Rezim 0.445 0.535 0.398 1426.346 698.190 795.610 TSpVARX 3 Rezim dengan Subset 12 dan Dummy Pt dan St 0.335 0.419 0.315 1204.770 634.635 686.866 The addition of the 12th subset variable causes some recurring patterns of inflation and money outflow increases due to the Eid al-Fitr event to be captured in the model. Thus, the resulting forecasting in the period of the Eid al-Fitr event is better than before the subset variable is added. The addition of dummy variables in the form of increases and decreases in fuel prices causes the pattern of increase or decrease in inflation and money outflow due to fuel price adjustments to be captured by the model. Therefore, the forecasting of inflation and money outflow when fuel price adjustments occur is closer to the actual data than before the dummy variables are included. The performance comparison of TSpVARX with and without subset and dummy variables based on RMSE can be seen in Figure 8. Based on Figure 8, the RMSE value of the testing data and training data for inflation and outflow is smaller than the TSpVARX model with subset and dummy variables. This indicates that the TSpVARX model with subset and dummy variables has better performance compared to the TSpVARX model. The improvement in inflation and money outflow forecasting accuracy due to the addition of subset and dummy variables is useful for the government to take the right policy, especially in price control.
Economies 2024,12, 352 22 of 27 Economies 2024, 12, x FOR PEER REVIEW 26 of 32 and decreases in fuel prices causes the pattern of increase or decrease in inflation and money outflow due to fuel price adjustments to be captured by the model. Therefore, the forecasting of inflation and money outflow when fuel price adjustments occur is closer to the actual data than before the dummy variables are included. The performance comparison of TSpVARX with and without subset and dummy variables based on RMSE can be seen in Figure 8. Based on Figure 8, the RMSE value of the testing data and training data for inflation and outflow is smaller than the TSpVARX model with subset and dummy variables. This indicates that the TSpVARX model with subset and dummy variables has better performance compared to the TSpVARX model. The improvement in inflation and money outflow forecasting accuracy due to the addition of subset and dummy variables is useful for the government to take the right policy, especially in price control. (a) (b) (c) (d) Figure 8. Root Mean Square Error (RMSE) of three-regime TSpVARX and three-regime TSpVARX models with the addition of the 12th subset and the dummy variables of (a) inflation testing data, (b) money outflow training data, (c) inflation testing data, and (d) money outflow testing data. The comparison between the plots of actual and forecasting data can be seen in Figures 9 and 10. The blue line illustrates the movement of actual data, and the green line illustrates the movement of forecasting data from the TSpVARX model. The red line illustrates the movement of the forecast data of the TSpVARX model with the addition of the 12th subset and the dummy variables. Based on Figure 8, it can be seen that some points of increase and decrease in inflation can be captured by the TSpVARX model with the addition of the 12th subset and the dummy variables. The effects of some events on inflation captured in the model are as follows: 1. the effect of Christmas and New Year on inflation of Semarang, Solo, and Yogyakarta in December 2014; 2. the effect of Eid al-Fitr on the inflation of Semarang in August 2012; 3. the effect of Eid al-Fitr on inflation of Semarang, Solo, and Yogyakarta in July 2013; 4. the effect of a fuel price increase on inflation of Semarang, Solo, and Yogyakarta in June 2008; 5. the effect of a fuel price increase on inflation of Semarang, Solo, and Yogyakarta in November 2014; 0.366 0.339 0.299 0.34 0.318 0.319 Inflation of Semarang Inflation of Solo Inflation of Yogyakarta TSpVARX TSpVARX with Subset and Dummy Variables 2130.05 827.4 866.93 1765.25 687.68 769.77 Money Outflow of Semarang Money Outflow of Solo Money Outflow of Yogyakarta TSpVARX TSpVARX with Subset and Dummy Variables 0.445 0.535 0.398 0.335 0.419 0.315 Inflation of Semarang Inflation of Solo Inflation of Yogyakarta TSpVARX TSpVARX with Subset and Dummy Variables 1426.35 698.19 795.61 1204.77 634.34 686.87 Money Outflow of Semarang Money Outflow of Solo Money Outflow of Yogyakarta TSpVARX TSpVARX with Subset and Dummy Variables Figure 8. Root Mean Square Error (RMSE) of three-regime TSpVARX and three-regime TSpVARX models with the addition of the 12th subset and the dummy variables of (a) inflation testing data, (b) money outflow training data, (c) inflation testing data, and (d) money outflow testing data. The comparison between the plots of actual and forecasting data can be seen in Figures 9and 10. The blue line illustrates the movement of actual data, and the green line illustrates the movement of forecasting data from the TSpVARX model. The red line illustrates the movement of the forecast data of the TSpVARX model with the addition of the 12th subset and the dummy variables. Based on Figure 8, it can be seen that some points of increase and decrease in inflation can be captured by the TSpVARX model with the addition of the 12th subset and the dummy variables. The effects of some events on inflation captured in the model are as follows: 1. the effect of Christmas and New Year on inflation of Semarang, Solo, and Yogyakarta in December 2014; 2. the effect of Eid al-Fitr on the inflation of Semarang in August 2012; 3. the effect of Eid al-Fitr on inflation of Semarang, Solo, and Yogyakarta in July 2013; 4. the effect of a fuel price increase on inflation of Semarang, Solo, and Yogyakarta in June 2008; 5. the effect of a fuel price increase on inflation of Semarang, Solo, and Yogyakarta in November 2014; 6. the effect of a fuel price decrease on inflation of Yogyakarta in December 2008; 7. the effect of a fuel price decrease on inflation of Semarang, Solo, and Yogyakarta in February 2015; 8. the effect of a fuel price decrease on inflation of Semarang and Solo in February 2019.
Economies 2024,12, 352 23 of 27 Economies 2024, 12, x FOR PEER REVIEW 27 of 32 6. the effect of a fuel price decrease on inflation of Yogyakarta in December 2008; 7. the effect of a fuel price decrease on inflation of Semarang, Solo, and Yogyakarta in February 2015; 8. the effect of a fuel price decrease on inflation of Semarang and Solo in February 2019. (a) (b) (c) Figure 9. Plot of actual and forecasting data for (a) Semarang inflation, (b) Solo inflation, and (c) Yogyakarta inflation. Based on Figure 10, it can be seen that some money outflow data points, which previously could not be captured by the three-regime TSpVARX, have been captured by the three-regime TSpVARX model with the addition of the 12th subset and the dummy variables. The effects of some events on money outflow captured in the model are as follows: 1. the effect of Eid al-Fitr on the money outflow of Solo and Yogyakarta in September 2009; 2. the effect of Eid al-Fitr on money outflow of Semarang, Solo, and Yogyakarta in July 2013; 3. the effect of Eid al-Fitr on the money outflow of Semarang, Solo, and Yogyakarta in July 2018; 4. the effect of Eid al-Fitr on money outflow of Semarang, Solo, and Yogyakarta in May 2022. Figure 9. Plot of actual and forecasting data for (a) Semarang inflation, (b) Solo inflation, and (c) Yogyakarta inflation. Economies 2024, 12, x FOR PEER REVIEW 28 of 32 (a) (b) (c) Figure 10. Plot of actual and forecasting data for (a) money outflow of Semarang, (b) money outflow of Solo, and (c) money outflow of Yogyakarta. 3.8. Forecasting Results Using the Selected Model The results of forecasting using TSPVARX with the addition of the 12th subset and the dummy variables for June 2023 to May 2024 can be seen in Table 10. If we compare the forecasting result in Table 10 with Table 8, the forecasting result in Table 10 fluctuates more than in the Table 8. This indicates that the addition of the 12th subset and dummy variables can already capture the data pattern due to the Eid al-Fitr event and fuel price adjustment. By using the forecasting result generated from the TSpVARX model with the addition of subset variables and dummy variables, the government can use the right policy in controlling the inflation rate. For example, if the highest inflation forecast for the October- December 2024 period is in November 2024, then the government can take precautions, such as raising interest rates and conducting open market operations in the previous month. In addition, the government can also find out the estimated increase in inflation and money outflow during Eid al-Fitr. It can be anticipated for the government to ensure the availability of food and non-food needs for the community so that there is no scarcity. Figure 10. Plot of actual and forecasting data for (a) money outflow of Semarang, (b) money outflow of Solo, and (c) money outflow of Yogyakarta.
Economies 2024,12, 352 24 of 27 Based on Figure 10, it can be seen that some money outflow data points, which previously could not be captured by the three-regime TSpVARX, have been captured by the three-regime TSpVARX model with the addition of the 12th subset and the dummy variables. The effects of some events on money outflow captured in the model are as follows: 1. the effect of Eid al-Fitr on the money outflow of Solo and Yogyakarta in September 2009; 2. the effect of Eid al-Fitr on money outflow of Semarang, Solo, and Yogyakarta in July 2013; 3. the effect of Eid al-Fitr on the money outflow of Semarang, Solo, and Yogyakarta in July 2018; 4. the effect of Eid al-Fitr on money outflow of Semarang, Solo, and Yogyakarta in May 2022. 3.8. Forecasting Results Using the Selected Model The results of forecasting using TSPVARX with the addition of the 12th subset and the dummy variables for June 2023 to May 2024 can be seen in Table 10. If we compare the forecasting result in Table 10 with Table 8, the forecasting result in Table 10 fluctuates more than in the Table 8. This indicates that the addition of the 12th subset and dummy variables can already capture the data pattern due to the Eid al-Fitr event and fuel price adjustment. By using the forecasting result generated from the TSpVARX model with the addition of subset variables and dummy variables, the government can use the right policy in controlling the inflation rate. For example, if the highest inflation forecast for the October-December 2024 period is in November 2024, then the government can take precautions, such as raising interest rates and conducting open market operations in the previous month. In addition, the government can also find out the estimated increase in inflation and money outflow during Eid al-Fitr. It can be anticipated for the government to ensure the availability of food and non-food needs for the community so that there is no scarcity. Table 10. Forecasting result of inflation (in percent) and money outflow (in billion Rupiah) of Semarang, Solo, and Yogyakarta using TSpVARX with the addition of the 12th subset and the dummy variables. Years Months Inflation of Semarang Inflation of Solo Inflation of Yogyakarta Money Outflow of Semarang Money Outflow of Solo Money Outflow of Yogyakarta 2023 Jun 0.03 0.14 0.16 1496.36 392.09 319.76 2023 Jul 0.30 0.24 0.27 1875.41 622.51 770.40 2023 Ags 0.21 0.20 0.23 1433.36 634.06 832.20 2023 Sep 0.34 0.39 0.37 2069.90 897.37 982.46 2023 Oct 0.27 0.23 0.30 1735.58 809.57 972.46 2023 Nov 0.27 0.21 0.29 2359.80 1007.74 1253.77 2023 Dec 0.42 0.37 0.42 2293.82 797.98 1410.76 2024 Jan 0.32 0.28 0.31 1217.20 469.92 785.45 2024 Feb 0.24 0.23 0.22 1538.97 512.83 835.56 2024 Mar 0.38 0.33 0.42 1976.75 763.40 1207.94 2024 Apr 0.40 0.34 0.35 2633.24 918.13 1669.01 2024 Mei 0.29 0.23 0.33 1838.78 898.66 899.12 2024 Jun 0.24 0.17 0.19 1511.27 568.49 618.91 2024 Jul 0.21 0.16 0.18 1686.85 615.86 752.38 2024 Ags 0.26 0.19 0.19 1391.16 522.05 794.18 2024 Sep 0.37 0.31 0.27 1357.30 400.60 786.30 2024 Oct 0.37 0.31 0.33 1234.65 370.66 819.22 2024 Nov 0.43 0.38 0.35 1344.91 353.58 888.40 2024 Dec 0.39 0.34 0.35 1338.65 334.36 954.85