1. Introduction
Time-series forecasting is commonly considered as a technique for estimating many future domains. It has become a major interest in statistical inference and predictions for researchers and academics [
1]. Almost every aspect of life, including the economy [
2], physics [
3], agriculture [
4], and engineering [
5], has applied time-series analysis. In economics, the accurate forecasting of data from a financial time series allows investors to determine relevant strategies and techniques to reduce price risk and protect their investment from unacceptable loss. One of the most crucial applications of forecasting is in investment risk management. Thus, the scrupulous selection of an effective forecasting model is important for managing non-linearity to ensure critical input for plans and decisions. In a significant number of studies, generalized autoregressive conditional heteroskedasticity (GARCH) and exponentially weighted moving average (EWMA) models have been used for time series forecasting, especially in estimating and predicting future values given the existence of ARCH effects in return series. Individual researchers can conveniently manipulate the GARCH and EWMA models as several resources are publicly available that can be used to develop forecasts effortlessly. Bollerslev [
6] developed the GARCH model to estimate and predict future values given the existence of ARCH effects in return series in particular. The GARCH model has achieved significant success in describing in-sample forecasting characteristics [
7]. Many studies used the GARCH model, for example, a study of the stochastic representation of impulsive noise in the frequency band [
8], a study of the stock exchange market of China [
9], and a study of Bitcoin volatility [
10]. They found that the GARCH model is more accurate when compared to ordinary-least-square (OLS) and EWMA methods. From these studies, despite having publicly accessible tools to easily forecast data, the GARCH model performed better than the EWMA. Thus, our study used the GARCH model as a basis for improvement.
Extensive research is being undertaken to improve the GARCH model by combining mathematical and statistical models, thus creating new models. For instance, a Glosten–Jagannathan–Runkle–GARCH (GJR-GARCH) model was introduced to allow for the conditional variance to respond to previous negative and positive changes differently [
11], and was then applied in measuring extreme risks within a sustainable financial system [
12]. A combination of an artificial neural network (ANN) and GARCH was proposed for the volatility forecasting of daily log-returns series in a study of international stock return volatility [
13], and was then applied in a study of the Chinese energy market [
14]. More improved GARCH models were introduced for variance–covariance and correlation forecasting, including the use of orthogonal components for displaying temporal aggregated properties that are not found when working with univariate models [
15]; the use of copulas for conveniently linking univariate GARCH [
16]; and the use of realized semi-variances, semi-covariances, and semi-correlations [
17]. Most of the improved model was used for correlation components, such as covariances and volatilities of stock and bond returns [
18,
19,
20,
21]. More extensive and complex GARCH models were developed to improve the forecasting model. For example, a fuzzy GJR-GARCH model using fuzzy inference systems [
22], hybrid neural network GJR-GARCH models [
23,
24] for volatility forecasting of indexes, a mixture of integer-valued models with different distributions in GARCH model [
25], and a hidden Markov Exponential GARCH model for volatility forecasting of crude oil price [
26]. Another approach toward improving the GARCH model was introduced by using a sliding window technique to forecast future values in various fields; this was called the sliding window GARCH (SWGARCH) model [
27,
28]. These hybrid models produced more accurate results than the classic forecasting models considered in the studies.
Although these numerical models outperform the conventional forecasting models, forecasting errors are still debated regarding the issue of accuracy. Forecasting accurately facilitates researchers in decision-making; therefore, the existing models must be improved. The applications of ANN, GJR, and Markov chain methods are very popular but undesirable in this study due to their complex and lengthy computation processes. The computation is also not easy to understand and only concerned with weight adjustment [
29]. Moreover, the common neural networks are inefficient for detecting nonlinear or dynamic behaviors [
30] and are unable to independently manage data uncertainties [
31], thus decreasing the forecasting efficiency. It is important to develop a new forecasting model with a low accuracy error [
32]. In addition, the fuzzy theory in some hybrid models [
33,
34] can estimate nonlinear continuous functions uniformly with an arbitrary accuracy, thereby producing more precise results than GARCH models. However, the computations of the hybrid fuzzy models are lengthy and incomprehensible. Thus, our study considers improving the GARCH model into a more understandable and less complex model.
In previous studies, the GARCH models were sensitive to extreme values in certain seasonal months [
14,
27]. The sensitivity of the GARCH model depends on the parameter estimation using the maximum likelihood method (MLM). The likelihood method is sensitive to the selection of initial values and the data distribution. The existence of uncertainties in time-series data makes it unreliable to employ the likelihood method to manage the uncertainties because no specific distribution is known. This drawback poses a major issue in terms of the instability of time-series forecasting. The GARCH model imposes parameter restrictions that are often violated by estimated coefficients that may improperly limit the dynamics of the conditional variance process [
35]. The GARCH model is also unable to capture the influence of each variance in the observation as the model uses a long-run average variance. The calculation of the long-run average variance, which is used to determine the weight of forecast, only considers the series in its entirety; therefore, the information on different effects of the variances in each observation is disregarded. Hence, our study improves the GARCH model in terms of parameter estimation and variance calculation to search for the most effective model for forecasting time-series data.
Therefore, this study proposes a new time-series forecasting model by combining the GARCH model with two methods; the first model is the fuzzy linear regression (FLR), which is used to estimate the parameters of the model, and the second is the fuzzy sliding window (FSW) variance, which is used to determine the weight of the forecast. The FLR is used for parameter estimation to reduce the computation risk and overcome the inter-correlation problem associated with independent variables [
36]. It is capable of addressing uncertainties and handling fluctuating information and/or linguistic variables. The FLR is also more robust at handling outlier effects [
37]. When applied to finance, FLR generates better estimates than the traditional approach [
38]. This approach is very useful when facing unstable time-series data caused by uncertainties. The FSW variance is used to calculate the variance of the model. The FSW variance improves the proposed model by replacing the component of the long-run average variance in the GARCH. The FSW variance combines fuzzy theory and the sliding window technique. Applications of the sliding window technique are useful in forecasting [
39]. The technique was used to determine the adjustment weights of the forecast and a certain number of splits to create a small number of intervals [
40]. The performance of the sliding window technique enables researchers to understand how cases work and change the efficiency pattern [
41]. SWGARCH [
27] improved the GARCH model by using the sliding window technique to produce a window variance that is calculated based on principal component analysis (PCA) to determine the weight of the forecast. Although the steps are straight-forward, the determination of weight based on the PCA is unconvincing and inaccurate. Thus, the SWGARCH model cannot guarantee a highly accurate performance and further complicates the forecasting process. The new combination of the proposed model is beneficial as the FSW variance can capture the impact of each variance in the observation and retain information on various effects of each observation. This attempt is also very useful as the model can select an accurate window size, as well as determine the accurate weight of the forecast. The latter ensures the best accuracy performance. In addition, the computation of the proposed model is simple since it involves only four simple steps, allows for faster calculations, and requires a small memory space. Furthermore, the proposed model is capable of reducing calculated risk, address uncertainties in variables, and capitalizing on FLR to manage fuzzy data or linguistic variables. Therefore, this study aims to exploit the advantages of the FLR and the fuzzy window variance to address the drawbacks of the existing models. The sections below explain the algorithm of the proposed model, the implementation, the results, and the conclusion of the study.
4. Conclusions
In this study, a new model for time-series forecasting was presented. The model was presented in two phases. The first phase was fuzzy linear regression that was used to replace the maximum likelihood method in parameter estimation. The second was the fuzzy window variance used to replace the long-run average variance component. Two datasets were used to test the performance of the proposed model; the first was from economics and the second was from agriculture. The forecasted values obtained from the use of the proposed model were compared to the benchmark models. The empirical results show that the proposed FLR-FSWGARCH model produced highly accurate forecasting values and a lower mean absolute percentage error (MAPE) than the SWGARCH, GARCH, ARIMA-GARCH, and EWMA models. This demonstrates that the FLR-FSWGARCH model was capable of giving a superior forecast compared to the benchmark models. Furthermore, the results also confirmed that the proposed model was highly reliable and significantly fit for forecasting time-series data as the R2 were more than 90% for both datasets. These indicated that the combination of the FLR and FSW variance with the GARCH model was adept at addressing uncertainties and handling fluctuating information. The highlight of this work lies in the application of FLR and FSW variance, which overcomes the limitation of the GARCH. The FLR curbs the inter-correlation problem associated with independent variables, while the FSW variance captures the influence of the variance on each observation and determines the true window selection for the weight of the forecast to produce a highly accurate performance.
In view of the need for a model that is easily understood by most people, particularly forecasters, the proposed model is computationally simple and practical as it does not involve any complex computations or simulations. The equations in the proposed model can be applied in Excel to obtain the estimated parameters of models and calculate the forecasts automatically. In addition, the application of the proposed model extends beyond the fields of economics to even include use in agriculture. Therefore, the proposed forecasting model could be used as an alternative to better forecast sustainable prices and growth, thereby improving a country’s economy. The application of the FLR-FSWGARCH model will also facilitate the decision-making processes associated with forecasting practice. For further studies, more attention should be paid to the fuzzification process by imposing a bipolar fuzzy set to form the triangular fuzzy numbers. The bipolar fuzzy set is an extension of a fuzzy set whose membership degree range is (−1,1), which is different from the current study, which uses a membership degree range of (0,1). The performance of the proposed model will be more superior by using the bipolar fuzzy set as previous research has shown its advantages when handling vagueness and uncertainty over a fuzzy set.