IMPLEMENTATION OF THE AUTOREGRESSIVE INTEGRATED (VARI) VECTOR MODEL FOR STRESS-TESTING ANALYSIS OF INDONESIAN BANKING

Stress testing in the banking sector is an important tool for assessing a bank's risk level and provides a basis for assisting financial institutions and regulators in making decisions by taking into account the impact of macroeconomic changes on banking risk, one of which is credit risk, which can be seen from the level of non-performing loans (NPL). bank. NPL ratio data and macroeconomic data are periodic data presented in monthly, quarterly or semi-annual periods and are categorized as time series data . At the stationarity test stage it is known that the data is not stationary, so to overcome this a differencing process is carried out so that it uses the Vector Autoregressive Integrated (VARI) model in the stress testing analysis . Based on the Granger test, there is a causal relationship between macroeconomic variables and NPL variables, so the model can be used to predict NPL values as a stress-testing analysis . The model that meets all assumptions and has the maximum lag is the VARI (5.1) model. The results of the model analysis show that the largest and most significant contribution to changes in the NPL variable is from real GDP shocks followed by shocks from Bank Indonesia's policy variables, namely the BI Rate and Inflation.


INTRODUCTION
Stress testing is test tool for measure strength and endurance system banking with do evaluation the connection between condition macro economy with risk credit from time to time. Variable risk credit used _ is ratio non-performing loans (credit problematic), because the more tall NPL ratio then the more big risk credit borne by the Bank. NPL ratio data and macroeconomic data is periodic data presented _ in period time monthly, quarterly nor semiannual and categorized as series data time. In matter this, will done analysis row time multivariate for know the connection between variable nonperforming loans (NPL) with variables macro economy (Real GDP, Inflation, and BI Rate) (Waelchli, 2016). Vector autoregressive model (VAR) which is a stochastic process model can used for see connection each other dependency from various variable with using string data time (Onwukwe & Nwafor, 2014). No all variable in study This have stationary data, so for cope with data that is not stationary in the VAR model, the differencing process is carried out. modelling row time multivariate in the form of differencing data use vector autoregressive integrated (VARI). Objective from study This is do stress testing analysis banking specifically risk commercial bank credit that is affected by the variable macroeconomy (Real GDP, Inflation , and BI Rate) using the vector autoregressive integrated (VARI) model and predicting mark ratio credit problem (NPL) of commercial and variable banks macro economy in the future come (Fauzi & Yuliati, 2021). Besides that for know impact shock variable macro economy to ratio credit problematic Commercial Banks as well as impact shock ratio credit problem to Indonesian

Optimum Lag Determination
In identifying the VARI( , ) is to determine the order of VAR( ). Determination of the optimum lag ( ) will be used in the analysis and parameter estimation for the vector autoregressive integrated (VARI) model. The length of the lag represents the degrees of freedom in the VARI model. The best model is the model that has the smallest Akaike Information Criterion, Hanna Quinn, Schwarz Information Criterion and Final Prediction Error values (William & Wei, 2006). To obtain the optimum lag, the previous step is to determine the determinant value of the residual covariance.
where is the parameter number of each equation in the VARI model.
( 1) Then proceed with using the AIC, HQ, SC and FPE values by selecting the smallest value.

Implementation of the Autoregressive Integrated (VARI) Vector Model for Stress-Testing Analysis of Indonesian Banking
= ln ( (∑ )) + 2 ∑ ln (4) where is the number of parameters in the model, is the length of the lag, (∑ )is the determinant of the variancecovariance matrix of the residuals for the model with lag , while is the number of observations. Vector Autoregressive (VAR) Sims (1980) proposed the vector autoregressive (VAR) method, which is a dynamic linear model that is widely used for forecasting economic variables in the long and short term. This model does not require a distinction between endogenous and exogenous variables because all variables in the VAR model are endogenous. In addition, the VAR model can also be used to determine cause and effect (Peña & Rodríguez, 2002). The estimation method is simple, namely the least squares method and a separate model can be made for each endogenous variable (Dijkstra, 2014). The forecasting results obtained are better than using a more complex simultaneous equation model. The VAR model with variables and orders can be written as follows.

Integrated Vector Autoregressive Model (VARI)
For example, the VAR model ( ) is not stationary, so the differencing process is carried out several times so that the data is stationary, and the VARI model ( , ) obtained by variable formulation. If = 1 and = 1, then the VARI model (1,1) can be obtained in matrix form as follows: or equal to the equation − −1 = 1 ( t−1 − t−2 ) + t (8) It is assumed that the error values follow a normal distribution, ie ~(0, 2 )and = − −1 . Then the VARI equation with = 1,2, … , is

Parameter Estimation
Parameter estimation used for the vector autoregressive integrated (VARI) model was obtained by the Ordinary Least Square method (Ordinary Least Squares Method/MKTB), namely determining the function of the sum of squared errors by changing the matrix , , into vector form. The following illustrates the VARI estimation ( , ) with the formulation of 2 variables. (10)

Implementation of the Autoregressive Integrated (VARI) Vector Model for Stress-Testing Analysis of Indonesian Banking
then equation 9 can be written as: The equation for the sum of squared errors is as follows: = − 2 + (12) Furthermore obtain derivative First function squared error against partially, and set the first derivative equation to zero. So that parameter Φestimation with the OLS method is Φ = ( ) − (13) In order to obtain the minimum squared error function, a second derivative of the squared error function is required which has a positive value so that equation 13 is derived Φas follows The estimation of the VARI model uses the OLS model in equation 14 with using time series data that has been stationary (differencing). Those models Then tested parameter significance more formerly before residual test with objective for determine the parameters that influence against VARI models. Testing is done by test that is where ̂i s the coefficient of each parameter, (̂( ) )is the standard error of each parameter. Testing the hypothesis as follows: 0 : ( ) = 0(coefficient of parameter is not significant/influence in the model) 1 : ( ) ≠ 0, = 0,1,2, . . . , (coefficient of significant/influential parameter in the model) If | ℎ | > /2,( − ) , with is the number of observations, is the number of parameters or the p-value < 0.05 then 0 it is rejected so that the parameters are significant and have an effect on the model.

Outliers
An outlier is an observation or a sub-group that is inconsistent with other observations in a data set (Barnett & Seth, 2014). In this study, the observational data was replaced with residuals, so that after obtaining the residuals from all observations, the upper and lower quartile values of the absolute residual values were determined.

Model Diagnostics Lagrange Multiplier Test
Engle developed a test to find out whether there is a problem of heteroscedasticity in time series data (ARCH effect) using the Lagrange Multiplier test (Tsay, 2005). The Lagrange Multiplier test is a test by regressing the squared residual values up to the lag , so that it will produce the regression equation as follows: a 2 = a 0 + 1 a −1 so that the test hypothesis is obtained: . If there is an ARCH effect on the residual VARI model selected, then symmetric volatility modeling using ARCH/GARCH or the GJR-GARCH model is used to deal with asymmetric volatility problems.

Autocorrelation Test
Autocorrelation is a condition where there is a correlation between the residuals in a certain period and the residuals in the previous period. Autocorrelation testing on data can use the Portmanteau Ljung-Box test which is a modification of the Portmanteau test to increase the power of the test on finite samples by replacing the residual autocorrelation coefficient so that the test hypothesis is obtained: : At least one ̃≠ 0(There is autocorrelation until the 1st lag ) Q test statistic : where is the number of residuals, ̃is the autocorrelation between residuals, is the lag. If p -value > then accept _ 0 which means that there is no significant autocorrelation until the 1st lag .

Causality
Causality test is a test that measures the strength of the relationship between two or more variables (Granger, 1969a). This starts from the absence of information regarding the influence between variables, so that the causality test can indicate whether the variable has a two-way or one-way relationship. The Granger Causality Test can be used to test the hypothesis whether a time series is useful in predicting another series (Granger, 1969b). Suppose there are two variables x and y, variable x is said to be Granger-Cause variable y if the past value of x contains information that helps predict y (Barnett & Seth, 2014). The equation for the Granger test is as follows

Implementation of the Autoregressive Integrated (VARI) Vector Model for Stress-Testing Analysis of Indonesian Banking
According to Granger (1969) to complete the causality model can be done through the F test criteria with the following formula where is the Restricted Residual Sum of Square , is the Unrestricted Residual Sum of Square , is the lag length, is the amount of observational data, is the number of parameters estimated in the unrestricted model. Suppose the variable y is the dependent variable, then the restricted model is obtained by regressing the variable y with all the lag y values without including lag x as the independent variable. The following is the form of the restricted model.
while Equation 7 is an unrestricted form of the model. There are two hypotheses used in the Granger causality test as follows 0 : X is not the cause of Granger Y 1 : X is the cause of Granger Y 0 :Y is not the cause of Granger X 1 : Y is the cause of Granger X If the calculated F value > the ( , − ) calculated value ≤ or p-value , then 0 it is rejected so that the causality conditions are met and the best VARI model can be selected.

Model Accuracy
One way to validate the accuracy of the model is to calculate the Mean Absolute Percentage Error (MAPE) value with the following formula (William & Wei, 2006).
where is the number of observations, is the actual data and ̂i s the predicted data obtained from the VARI model after the inverse differencing process .

Stress Testing Analysis
The focus of bank stress testing is on credit risk which represents the most significant banking system risk (Sorge & Virolainen, 2006). The credit risk variable used is the nonperforming loan ratio , because the higher the NPL ratio, the greater the credit risk borne by the Bank. This NPL ratio is influenced by internal factors and external factors. Macroeconomic Variables are factors that affect NPL externally. The decline in Gross Domestic Product (GDP), the increase in the Consumer Price Index (CPI) and the BI Rate (BI Rate) contributed to driving up NPLs (Indra, 2019). In calculating the Bank's capital adequacy in facing crisis situations, a stress test analysis is carried out to calculate the impact of credit quality in the face of extreme changes that are influenced by macroeconomic aspects. The stress test scenario is divided into three, namely the normal scenario, tolerable scenario, and crisis scenario (Agustin & Wiranatakusuma, 2017). The normal scenario is prepared based on forecasts for the development of the Indonesian economy, the tolerable scenario is developed to test bank resilience when economic conditions experience contraction, while the crisis scenario is developed to test bank resilience when economic conditions experience a deep contraction. Scenarios are built by adding (or subtracting) 1,2 or 3 standard deviations to the estimates produced by the VARI model [9]. The analysis component in the VAR model, namely the impulse response function (IRF), makes it possible to observe the response of this data and the future of each variable due to changes or shocks caused by other variables, while the variance decomposition provides information regarding the contribution (percentage) of changes in the variance of each variable to change in another variable.

Stability
VAR stability testing is carried out before carrying out the Impulse Response Function (IRF) and Forecast Error Variance Decomposition (FEVD) analysis. The VAR stability test is carried out by calculating the roots of the polynomial function or known as the roots of characteristic polynomial . If all the roots of the polynomial function have an absolute value less than one, then the VAR model is stable so that the resulting IRF and FEVD are considered valid. Stable VAR model, VAR can be shown if is a dimensionless coefficient matrix ( × )that represents the parameter value .

Impulse Response Function
Impulse Response Function can be used to determine the effect of one variable on the variable itself or other variables. The estimation made in this IRF is the response of a variable to a change of one standard deviation from the variable itself or from other variables contained in the model. IRF analysis uses the Cholesky Decomposition which aims to generate an impulse response that depends crucially on the ordering of the variables in the system with the following calculations (ℎ) = Γ (12) d ith Γis the parameter matrix of the VAR I model, ℎis the forecasting period , meanwhile is the Cholesky Decomposition matrix of the shock covariance variant matrix . IRF is described in form graph , with vertical axis as mark response ( default deviation ) a variable to shock and horizontal axis as time For a number of time forward after happening shock.

Forecast Error Variance Decomposition (FEVD)
Variance Decomposition aims to predict the percentage contribution of the error variance of a variable, namely the magnitude of the difference between before and after the shocks, both originating from the variable itself and from other variables. The FEVD equation can be obtained by the following illustration +1 = ϕ 0 + 1 1 (13) by value 0 and 1 used to estimate the predicted value +1 + = + + 1 2 + −2 + ⋯ + 1 −1 +1 (14) with a higher variable FEDV value explaining the higher contribution of endogenous variable variance to other endogenous variables. According to Lutkepohl (2005), the Variance Decomposition calculation is as follows: Indonesian Banking  2., 6., May 2023 ,ℎ =

Implementation of the Autoregressive Integrated (VARI) Vector Model for Stress-Testing Analysis of
(15) with = Φ , with is the lower triangular matrix of the Cholesky Decomposition matrix variance covariance , Φ = ′ with = [ 0 … 0]and is the coefficient matrix of the VARI model . FEVD analysis explains the relative importance of the diversity of each variable in the system due to shock . This analysis used For know contribution with see percentage variance every variable during a number of future time . _

RESULTS AND DISCUSSION Descriptive Research Variables
The data used in this study is time series data for the period 2003 -2021 with quarterly data positions. The results of the descriptive analysis are presented in the following table and figure.   Table 1 and Figure 1 it can be seen that the macroeconomic variables and the variable NPL ratio of commercial banks show fluctuating data. The Real GDP variable experienced a very sharp decline with the lowest value in the last 16 years of -5.32%. At the end of 2005, the inflation variable reached the highest increase of 17.10% with an average of 5.68 in the last 16 years. The BI Rate variable also achieved a very high increase as did the inflation variable, with a value of 12.75% at the end of 2005 with an average of 7.27% in the last 16 years. The variable ratio of commercial banks' NPLs reached a maximum value of 8.33% in the second

Implementation of the Autoregressive Integrated (VARI) Vector Model for Stress-Testing Analysis of Indonesian Banking
quarter of 2006, 6 months after inflation and the BI Rate reached their highest values in the last 16 years (Schechtman & Gaglianone, 2012).

Stationarity Test
Testing is done with the level of significance = %. The following table shows the results of the stationarity test. The ADF test shows that the macroeconomic variables and NPL variables are stationary differencing 1, so that the data can be used in the analysis of the vector autoregressive integrated (VARI) model. The data plots that have been stationary are presented in Figure 2 below.

Determination of Lag Parameters
Based on the AIC, HQ, SC and FPE criteria, lag 5 is obtained as the best model because it has the smallest AIC, HQ and FPE values, so the model to be used is the VARI model (5.1).

Parameter Estimation
The next process is to look at the significance level of the VARI model parameters (5.1). The significance of all parameters was partially tested using the t-test statistic so that significant parameter estimates were obtained as follows.  ( After obtaining the VARI model equation (5.1), further detection of outliers is carried out which can be seen through the residual boxplot of the model.

Model Diagnostic Test
From the LM Test it shows 0 that it is accepted ( p-value = 0.3682 > 0.05). It can be concluded that for the four equations the VARI model (5.1) fulfills the residual homogeneity requirements. The results of the Portmanteau Ljung-Box test with a significance level of 5% obtained a p-value of 0.3252, it can be concluded that the residual data has the assumption of non-autocorrelation

Causality Test
The results of the Granger causality test show that there are only 3 simultaneous causality relationships with a p-value <0.05 as follows: a. Real GDP is the cause of Granger Inflation, BI Rate and NPL, b. The BI Rate is a Granger cause of Real GDP, Inflation and NPLs, as well as c. NPL is the cause of Granger Real GDP, Inflation and the BI Rate

Stability Test
It is obtained that the value of the modulus range is less than 1 or in other words the VARI model (5.1) is stable so that it can carry out Impulse Response Function and Forecast Error Decomposition Variance (FEDV) analysis.  Figure 3 shows the response of the NPL growth variable to macroeconomic growth variable shocks (Real GDP, Inflation and the BI Rate) throughout the 60 observed periods. The real GDP variable shock of one standard deviation received a response in a negative direction in the first period and the second period of -0.118233 and -0.059429, then turned positive in the third period to the seventh period and tended to reach a balance like the condition before the shock from Real GDP in the 40th period was 0.005 until the end of the period. Figure 4 shows the Variance Decomposition value of the NPL variable, it can be seen that the dominant NPL variance is influenced by the variance of the variable itself until the 32nd period. The Real GDP variable has a fairly large percentage in influencing changes in the NPL variable compared to the Inflation and BI Rate variables.

Stress Testing Analysis
The results of backtesting using data for 2003 -2019 with the VAR model (5.1) obtained the MAPE value in accordance with the Appendix which shows the MAPE value resulting from the prediction of the GDP variable of 9.14%, which means that the model has very good accuracy, the MAPE value of the forecast variable Inflation and the NPL variable is 37.95% and 23.41% respectively, which means the model has sufficient accuracy, while the MAPE value forecasting the BI Rate variable is 11.33%, which means the model has good accuracy. Forecasting the NPL variable for the normal scenario uses the VARI model (5.1) with forecasting macroeconomic variables, so that an NPL value of 2.50% for the first quarter, 2.27% for the second quarter, 2.31% for the third quarter is obtained. and by 2.28% for the fourth quarter of 2022. Meanwhile, the tolerable scenario is constructed by adding 1 standard deviation to the estimated NPL value generated by the VARI model for 2021 as well as the crisis scenario which is constructed by adding 2 standard deviations to the estimated NPL value.

CONCLUSION
Based on the results of the analysis, modeling and forecasting of Indonesian banking stress testing using the integrated autoregressive vector model, it can be concluded that there is nonstationary in the data so that the differencing process is carried out and the ADF test results show that macroeconomic variables and NPL variables are stationary at differencing 1. AIC criteria value, HQ and FPE produce the optimal lag at lag 5, so that an integrated autoregressive vector model (VARI, (5,1)) is obtained. This model can be used for modeling the relationship between macroeconomic variables and non-performing loan variables for commercial banks as well as analysis of banking stress testing , especially for commercial bank credit risk which is influenced by macroeconomic variables. As for the results of the IRF analysis, the NPL variable for macroeconomic variable shocks of one standard deviation received a negative response to the Real GDP variable and the BI Rate variable in the first period of the shock, while the Inflation variable received a positive response in the first period of the shock. This shows that if there is a fairly high increase in inflation as well as a decrease in Real GDP and the BI Rate, it will cause the NPL variable to increase, in the sense that there is an increase in the ratio of non-performing loans to commercial banks. Meanwhile, the FEDV results state that the largest and most significant contribution to changes in the NPL variable is from real GDP shocks followed by shocks from Bank Indonesia's policy variables, namely the BI Rate and Inflation.