# The Predictive Grey Forecasting Approach for Measuring Tax Collection
## 4 Results and Discussion  
### Accuracy of the Forecasting Model  
It is important to gauge and assess the accuracy of the forecasting model; original values of the positive and negative errors are used to calculate the estimated error value using RPE (Relative Percentage Error) ([PERSON] et al., 2010; [PERSON] et al., 2024). Previous research has shown that the forecasting results obtained by grey forecasting are more accurate than the results of the auto-regressive integrated moving average (ARIMA), which has a smaller mean absolute percentage error (MAPE) ([PERSON] et al., 2016). [PERSON] and [PERSON] (2023) recommended employing the ONGBM (1,1) and NGBM (1,1)PSO models for short-term energy consumption forecasting and combining these forecasts with those from the GM (1,1) and ARIMA(1,1,1) models. Moreover, [PERSON] and [PERSON] (2024); [PERSON] and [PERSON] (2022) indicated that grey modeling generates more accurate conclusions with limited data points than other methodologies and is better suited for forecasting under chaotic and complicated settings with discrete data points. First, we applied grey modeling to the values of indirect taxes. The calculation of indirect tax generation using the GM (1,1) is elaborated systematically below. \"\(X_{0}\)\" represents a non-negative data series for indirect tax in previous consecutive years, expressed as follows:  
\[X_{0}=(119,814,\ 237,132,\ 369,740\ldots)\]  
Then, a new \(X_{1}\) series is calculated, which is effectively a cumulative addition of the \(X_{0}\) series, which is an AGO. \[X^{1}=(238,380,\ 422,002,\ 628,914\ldots)\]Based on the following sequences, \(Z_{k}^{1}\) of \(X_{k}^{1}\) is determined as follows:  
\[Z_{k}^{1}=(1,249,118.3,\ 2,807,055.8,\ 3,306,198.3,\ 5,034,118.3)\]  
Using the least squares method, the values of \(a\) and \(b\) are obtained as follows:  
\[Y=|117318132608148608\ \ldots\ |\ B=|-238,380-422,002-628,914\ \ldots\ 1111|\]  
Then, we calculate \(\left(B^{T}.B\right)^{-1}\) for using Equation (8). \[\left(B^{T}.B\right)=|1.25042E+15-124,242,656-12,424,265,622|\] \[\left(B^{T}.B\right)^{-1}=|1.82225E-151.0291E-081.0291E-080.103572|\] \[\qquad\qquad\qquad\qquad\qquad\qquad a=-0.098,\ b=123,997.4,\ e=2.7183\] \[\qquad\qquad\qquad\qquad x_{(K+1)}^{1}=\left(1,455,398-\frac{123,997.4}{-0.098}\right)e^{-(-0.098)(k)}-123,997.4\]  
Once constants \(a\) and \(b\) are calculated, the predicted values for each year are calculated using the forecast mentioned in the above equation. Similarly, we also calculate the forecasting equation for direct taxes as shown below:  
\[x_{(K+1)}^{1}=\left(1,604,571-\frac{153,627}{-0.1}\right)e^{-(-0.1)(k)}-153,627\]  
Moreover, the results of implementing GM (1,1) show a consistent and continuous increase in the proportion of direct tax as a proportion of total tax collection (Table 1). Total revenue collection from direct tax, around INR 16.63 trillion in 2022-23, is set to become INR 30.67 trillion by 2029-30. On the other hand, indirect tax collection, which is currently around INR 13.82 trillion, is set to become INR 25.70 trillion. Total tax collection from direct and indirect taxes is set to increase from INR 30.45 trillion in 2022-23 to INR 56.37 trillion in 2029-30 (Table 1). The anticipated increase in tax income, based on the increased share of direct taxes, indicates a move towards a more progressive taxation regime ([PERSON] et al. 2024). Direct taxes becoming a bigger fraction of total revenue reflect a system in which taxation is more directly linked to income and profits, perhaps leading to lower income disparity. This can lead to a more equitable allocation of wealth, as higher earners contribute a larger portion of the government's resources; such a move has serious repercussions. Greater emphasis on direct taxation can boost fiscal stability because these taxes are more predictable than consumption-based indirect taxes. Governments can better plan and distribute resources thanks to this predictability ([PERSON] 2021).