THE FORECASTING OF MONTHLY INFLATION IN YOGYAKARTA CITY USES AN EXPONENTIAL SMOOTHING-STATE SPACE MODEL

: The purpose of this research is to predict monthly inflation in the city of Yogyakarta with a simple and easy forecasting model that has high accuracy. The model used is the exponential smoothing-state space or known as the error, trend, and seasonal (ETS) model. This model does not have statistical assumptions, it is easy to analyze using R-package statistics which is an open-source program. This ETS model is built with a combination of trend and non-trend, seasonal and non-seasonal models as well as an additive or multiplicative errors. The monthly inflation data used in this research is secondary data obtained from the Central Bureau of Statistics (BPS) for the city of Yogyakarta from January 2015 to December 2021 with a total of 84 data. The results of this research obtained that the most suitable ETS model for predicting monthly inflation in the city of Yogyakarta is the ETS model (A, N, A). The ETS model (A, N, A) means that the error is additive (A), does not contain a trend (N) and seasonality is additive, so it is written as ETS (A, N, A). The ETS model (A, N, A) obtained in this research has an Akaike information criteria (AIC) value of 145.1996 with an RMSE forecasting accuracy value of 0.2166014 and a MAPE of 127.1662. The results of the forecasting for the next three periods show that the monthly inflation value of Yogyakarta is quite stable, there is an increase and decrease or it fluctuates slightly and is still below 10%.

Previous research on inflation prediction at the country or city level has been done by many researchers with different methods. Research on inflation prediction and consumer price index in Zambia uses the ARIMA model and Holt's double exponential smoothing (DES) model. The results of the ARIMA model research are better than the DES model, but for software tools, it is easier to use the DES model from Holt (Jere and Sianga, 2016). Another study, namely Febriyanti et al., examined the consumer price index (CPI) in the city of Yogyakarta. The consumer price index (CPI) is a component or one of the factors in calculating the value of inflation. This study uses the double exponential smoothing (DES) method from Brown with forecasting accuracy using the mean absolute percentage error (MAPE) of 0.1308443%. Brown's DES method is considered quite simple and accurate in this study (Febriyanti et al., 2021). Inflation prediction research was conducted in Kenya by Lidiema (2017) using the seasonal autoregressive integrated moving average (SARIMA) and triple exponential smoothing (TES) methods. The results of this study obtained that the best model is SARIMA by looking at the smallest forecasting accuracy, namely the mean absolute square error (MASE), the mean absolute error (MAE), and the mean absolute percentage error (MAPE). Inflation prediction research in Bandung using the seasonal autoregressive integrated moving average (SARIMA) and single exponential smoothing (SES) methods was carried out by Fajruddin and Sumitra. The results of this study show that the SARIMA model is better than the SES mode with a mean absolute deviation (MAD) forecasting accuracy of 0.117, a mean square error (MSE) of 0.023, and a Mean absolute percentage error (MAPE) of 0.72% (Fajruddin and Sumitra, 2020). Research on forecasting inflation in the city of Samarinda uses the double exponential smoothing (DES) method from Brown with the best model results for the alpha parameter of 0.3 with a mean square error (MSE) of 0.485239 (Armi, et al., 2019). Research on inflation prediction in Indonesia using the moving average method, single exponential smoothing, and double exponential smoothing results in the conclusion that the best method is single exponential smoothing with an alpha value of 1.316, MAPE of 7.76202, MAD of 0.27343, and MSD of 0,14625 (Sudibyo, et al., 2020). Monthly inflation forecasting in Malang using the ARIMA method resulted in the conclusion that the best model was ARIMA (2.0,3) with RMSE of 0.2645467, MAE of 0.2013898, and MASE of 0.6047399. The ARIMA method is not easy because statistical assumptions must be met and use special software (Farida and As'ad, 2021).
From a review of previous research, information is obtained that to predict monthly inflation, the relatively easy and simple exponential smoothing (ES) model can be used. The latest development of the exponential smoothing (ES) model, there is the development of a model by looking at the state-space known as the exponential smoothing-state space model written by Hyndman and Athanasopaulos (2018). The exponential smoothing-state space method is a decomposition model of the ES model so that it can explore data in more detail and can improve forecasting accuracy. The exponential smoothing-state space method uses research entitled "selection for the best ETS (error, trend, seasonal) model to forecast weather in the Aceh Besar District". The results of this study were to predict air temperature and sea surface temperature using the ETS (M, N, A) model, to predict the dew point, sea level pressure, and station pressure, the ETS (A, N, A) model was used to predict visibility. ETS (A, A, N) model, to predict wind speed the ETS (A, N, N) model is used (Jopifasi, et al., 2017). In this study, the exponential smoothing-state space / ETS model is used to predict monthly inflation in the city of Yogyakarta.

Research Method
This study uses secondary monthly inflation data from January 2015 to December 2021 as many as 84 data obtained from the Yogyakarta City BPS website (https://jogjakota.bps.go.id/indicator/3/1/1/inflation.html). Calculation of inflation with the consumer price index (CPI) can be calculated using the following formula (Utari et al., 2016): where, t is the month/quarter/year inflation to t CPI t is CPI month / quarter / year t CPI t-1 is CPI month/quarter/year t-1 The model used to predict monthly inflation is the exponential smoothing-state space or known as ETS (E=error, T=trand, S = seasonal). This ETS method is a decomposition of the exponential smoothing (ES) model with three models, namely: a single exponential smoothing (SES) model, a double exponential smoothing (DES) model, and a triple exponential smoothing (TES) model (Hyndman and Athanasopaulos, 2018). The SES model can be calculated as follows (Makridakis, 1998): The DES model used is Holt (two parameters). Holt's model can be calculated as follows (Makridakis, 1998): Y t is the actual data for the t period, A t is the exponential smoothing value,  is the smoothing constant no the trend, β is the smoothing constant for the trend estimate, T t is the trend estimate, Yˆthe forecast value for the future period and p is the number of periods being forecasted. The additive TES model can be calculated as follows (Hyndman and Athanasopoulos, 2018): where , , and δ are smoothing constants between 0 and 1, L = number of seasons The multiplicative TES model can be calculated as follows (Hyndman and Athanasopoulos, 2018): where , , and δ are smoothing constants between 0 and 1, L = number of seasons There are two forms of error in the ETS model, namely additive and multiplicative. There are three trends in the ETS model, namely none, additive, and additive dumped. There are three seasonal ETS models: none, additive and multiplicative. From E there are two, T there are three, and S there are three, then there are 18 possible models. This study uses one of the best models of 18 ETS models. The selected model has the smallest Akaike information criteria (AIC) value and has the highest accuracy (the smallest root mean square error / RMSE value and the mean absolute percentage error / MAPE value). The AIC value can be calculated as follows (Jopifasi, et al., 2017): This study uses the R package statistics as a software tool in the analysis. R package statistics can be downloaded for free because it is an open-source program and can be installed on Linux, macOS, and Windows operating systems. To download the R package statistics, you can do it at the following link: https://cran.r-project.org/ The following are the steps of analysis in this study are as follows: 1 H 0 :  = 0 (there is a unit root, the data is not stationary) H 1 :  ≠ 0 (no unit root, data is stationary) For the error model, there are two, namely additive (A) and multiplicative (M), while for seasonal there are three possibilities, namely None (N), additive (A), and multiplicative (M). From Figure 1, monthly inflation data shows that fluctuations from the initial period to the final period do not show an increase in fluctuations that are getting bigger, so it can be said that the seasonality is likely to be additive and there is no seasonality or none(N). From the analysis of Figure 1 The results of the monthly inflation forecast for the city of Yogyakarta from January 2022 to February 2022 there will be a decline or deflation and will then increase again in March 2022. According to the literature above, inflation is quite a under control because it is less than 10%.

Discussion
The results of this study obtained the ETS (A, N, A) model which means the model does not contain trends and contains additive seasonality, and also has additive errors. This ETS (A, N, A) model, if equated with the ARIMA model, includes a model containing seasonality so that it becomes a SARIMA model like previous studies (Lidiema, 2017;Fajruddin and Sumitra, 2020). ARIMA inflation research model that does not contain seasonality in previous studies (Jere and Sianga, 2016;Farida and As'ad, 2021). Research on monthly inflation prediction using the ARIMA or SARIMA method is quite difficult and not simple because the model must meet the statistical assumptions in the model which are sometimes difficult to fulfill. Unlike the ARIMA or SARIMA models, the ETS model which is a decomposition of the single exponential smoothing (SES), double exponential smoothing (DES) and triple exponential smoothing (TES) models do not have statistical assumptions that must be met, so the model obtained is easier and faster obtained which only chooses a model with a small AIC value and forecasting accuracy.

Conclusion
Research on monthly inflation forecasting in the city of Yogyakarta using the ETS (A, N, A) model has an AIC value of 145.1996, an RMSE of 0.2166014, and a MAPE of 127.1662. The ETS (A, N, A) model means that the data does not contain trends but contains additive seasonality with additive errors as well. Additive seasonality means that there is a stable seasonality in 12 months of the year that repeats every year. From an economic point of view, inflation is stable, sometimes a bit high in certain months, such as the long Eid holiday, New Year's Christmas, or other holidays that are rather long and will fall again if there are no holidays. Due to the long holiday, tourist visits to the city of Yogyakarta have increased and economic activity has also increased, which automatically increases the demand for goods and services with a steady supply causing the value of money to be high (inflation occurs). Inflation forecast in January 2022 rose, this may be due to the long Christmas and New Year holidays in December 2021, where the demand for goods and services increased with a fixed supply which automatically increased inflation for a moment and fell again in February 2022. Inflation forecast for March 2022 will increase slightly, it can be said to be safe or stable because the previous months did not rise continuously and were still below 10%.