Performance analysis of the loan repayment by poor borrowers in rural areas of saccos in Njombe, Tanzania by using time series arima modelling

Introduction

Savings and Credit Cooperatives (SACCOS) are one of the Micro Finance Institutions (MFIs) in Tanzania. They provide a wide range of savings and loan products to those individual poor borrowers in rural areas, who cannot afford to go the banks. The SACCOS are working as a bridge between these small borrowers and the formal financial institutions. These cooperative societies offer small sized loans to members to whom the formal financial institutions will not offer loans or credit in view of the various reasons most importantly the high interest rate and other formalities in processing the applications of poor people. So in other way, SACCOS are providing ways to help small and poor creditors in rural areas particularly those who, otherwise are under the poverty line. These under privileged rural persons take loans from SACCOS and return the loan payments with small interest rate. This study aims at fitting a time series model that can be used to forecast the trend of loan repayment at Kifanya SACCOS Ltd by using the Box-Jenkins procedure. ARIMA model for forecasting is recommended in this study because the model assumes that the observations of time series are statistically dependent on each another, meaning that the autocorrelation between successive values of the time series data under study is not zero (Kumar & Anand, 2015) [7] . A time series is a sequence of observations on a particular variable recorded at discrete intervals of time (Adhikari, 2013) [2] . Time series analysis involves the techniques of studying a given time series by analyzing its patterns, fitting models for those patterns and using the fits to predict the future values of that time series. Time series analysis is a very important aspect of statistics which is applicable in many fields where forecasting is highly needed. Forecasting is very important in areas of economics, business, finance, medicine, environment, industry and so on. The Box-Jenkins (or ARIMA) procedure is among the most widely used forecasting methods in time series analysis. ARIMA Models are excellent short-term forecasting linear models for a wide variety of time series. These models have shown efficiency and have outperformed complex structural models in short-term forecasting (Adebiyi, Adewumi, & Ayo, 2014) [1] . A time series model is considered to be linear when the current value of the series is a linear function of past observations (M. Zhang, 2018) [12] .

In time series analysis the interest is to obtain a suitable time series model that can be used to predict the future values of the series. The fitted model becomes useful when it is physically meaningful and observes the principle of parsimony. The principle of parsimony recommends that when we have more than one model the one with fewer parameters is preferred to the one with more parameters.

Literature Review

Time series modelling is very important in economic forecasting which plays a very crucial role in decision making process in both public and private sector. These sectors base many of their decisions on future expected economic conditions or on the predictions of specific indicators of interest such as income growth, exchange rates, inflation, interest rates, unemployment and many others. But the realization of economic outcomes is a vast, dynamic and stochastic process which makes forecasting very difficult and forecast errors unavoidable. However, forecast accuracy and reliability can be improved by using appropriate models and methods such as ARIMA and VAR models (Ghysel & Marcellino, 2018) [5] . Otooet et al. (2015) analyzed loan default at Minescho Cooperative Credit Union in Ghana by using Regression and ARIMA models. The fitted ARIMA (3,2,2) model indicated an increasing trend on the amount of loan default. Iwueze et al. (2013) [6] fitted a time Series model and showed that ARIMA (2,1,0) was the appropriate model to describe the pattern of the financial series. Ayo (2014) [1] applied ARIMA model in forecasting stock price and showed that ARIMA model is strong in short-term forecasting. Zhang (2013) [11] applied Autoregressive Integrated Moving Average (ARIMA) model, the Vector Autoregressive (VAR) model and the First-order Autoregressive (AR(1)) model in modelling and forecasting the regional GDP. The findings showed that all the three models were valid in generating short-term forecasts. Nochai & Nochai (2006) applied ARIMA Model in forecasting three types of oil palm price; farm price, wholesale price and pure oil price and showed that the proper ARIMA Models for forecasting farm price, wholesale price and pure oil price of oil palm were ARIMA (2,1,0), ARIMA (1,0,1) and ARIMA (3,0,0) respectively. In general, there is no sufficient literature on ARIMA modelling and forecasting of loan repayment particularly in savings and credit cooperative societies or credit unions. Therefore, this study adds literature on time series modelling and forecasting in the context of financial data.

Materials and Methodology 3.1 ARIMA Processes and their models

The ARIMA procedure developed by Box and Jenkins for modelling time series data is applicable when the modelling is done on a stationary time series. This procedure involves the following models.

 Autoregressive model: In autoregressive model, the forecast of a given variable is a linear combination of past values of that

variable. An autoregressive model of order p which is denoted by AR(p) is written as

Where Y t is a time series and t  is white noise.

 Moving average model: In moving average model, the forecast of a given variable is a weighted average of past forecast errors. A moving average model of order p which is denoted by MA(q) is written as

 Autoregressive moving average model: This model is a combination of an autoregressive model and a moving average model. The model is denoted by ARMA (p,q) where p and q are the orders of autoregressive component and moving average component respectively and it is written as

 Autoregressive integrated moving average (ARIMA) model: An autoregressive integrated moving average (ARIMA)

model is an ARMA model which is obtained when a time series is made stationary by differencing it say d times. This model is denoted by ARIMA(p,d,q) and is written as

Where, p, is the order of the autoregressive component d, is the number of differences needed to obtain a stationary ARMA(p, q) process q, is the order of the moving average component ε t , is white noise  Seasonal Autoregressive integrated moving average (SARIMA) model: The model is developed by including seasonal parameters in non-seasonal ARIMA models. A seasonal ARIMA model with seasonal period s is written as

are the non-seasonal AR and MA characteristic operators respectively.

are the seasonal AR and MA characteristic operators respectively.

D is the number of seasonal differences P is the order of the seasonal autoregressive process Q is the order of seasonal moving average process p, d, q and t  have already been defined in (3.4)

The Box -Jenkins Methodology

The Box-Jenkins methodology is an iterative procedure used to build an ARIMA model in three steps. The three steps involved in this procedure are;

Model identification

In this step a tentative model is suggested after studying the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of a stationary time series. Usually, the analysis starts with preliminary assessment for stationarity of the time series by plotting the data. This graph is helpful in identifying important features such as trend, seasonality and cyclic fluctuations in a given time series. Time plots are helpful in deciding whether the given time series needs to be transformed before doing any further analysis or not.

Parameter estimation

In this step methods such as the method of moments, method of maximum likelihood and the method of least squares are applied to estimate the parameters of the suggested model. Usually, this is achieved by using statistical software packages such as SPSS, STATA, Minitab, SAS and R.

Diagnosis of the model

To be useful, the tentatively identified model is examined for its adequacy. This is done by analyzing the residuals. If the fitted model is adequate then the residuals should behave like white noise and the fit can be used to generate forecasts. When the fit is not adequate, we have to go to step one where another model has to be considered.

Summary of the Box-Jenkins Methodology

The three steps involved in the Box -Jenkins procedure are summarized in the following diagram.

Data Analysis and Discussion of Results

Step 1: Model identification Before getting a tentative model, the loan repayment data are plotted as shown in the time plot displayed in figure 2. In addition, seasonal decomposition has been done to analyze the components of this series, and the results are displayed in figure 3. From these figures it is clear that the loan repayment time series is not stationary in mean as well as in variance. For that reason, the series has to be transformed. The first transformation is conducted to stabilize the changing variance of the series. This was done by applying natural logarithms to the loan repayment data, and the transformed series is displayed in figure 4. Now the transformed series appears to have stable variance. Let us denote this time series with stable variance as X t . The second transformation is conducted to remove the trend of the series. The trend is removed by differencing the series and the differenced series W t , is displayed in figure 5. By visual inspection, this series appears to be stationary but its stationarity is confirmed by conducting a formal test usually a unit root test. The unit root test that was employed here is the Augmented Dickey-Fuller (ADF) test. MacKinnon approximate p-value for Z(t) = 0.0000

The ADF test statistic is -7.809 for lag order 5 and a p-value of less than 0.0001 is recorded indicating that the series W t is stationary.

From the stationary time series on loan repayment, the ACF and PACF plots were obtained. These plots are displayed in figure 6 and they indicate that there is one significant spike at lag 1 in the ACF indicating that MA (1) might be the non-seasonal component. The PACF plot shows two significant spikes suggesting that AR(1) and AR(2) are likely to be the non-seasonal components. Therefore, the ARIMA model for this time series is likely to contain both AR and MA processes. The ACF and PACF plots of the stationary series suggested the following models; 1. ARIMA (2, 1, 0) 2. ARIMA (1, 1, 2) 3. ARIMA (1, 1, 0) 4. ARIMA (0, 1, 1)

Step 2: Model Estimation From suggested models parameter estimates were obtained by using Minitab. The best ARIMA model was selected on the basis of the following measures. In selecting the best model from the list of suggested models, we consider the values of AIC and BIC. From table 2, AIC suggests that the best model is ARIMA (1, 1, 2) whilst BIC suggests the model ARIMA (0,1,1). The use of AIC as a selection criteria for the best model is recommended here because it agrees with the smallest value of RMSE which shows that there is minimum variation with the model suggested by AIC. Therefore, the best fit for the monthly loan repayment time series is the ARIMA (1,1,2). The parameters of the fitted ARIMA (1, 1, 2) were estimated in Minitab, and they appear to be significant as shown in table 3. Step 3: Diagnosis of the model Diagnosis is done by studying the residuals to see if any pattern remains unaccounted for hence verifying that the fitted model is adequate. The fitted model is diagnosed by analysing autocorrelation in the residuals and some residual plots.

Portmanteau test:

The modified portmanteau test was performed on residuals of the fitted model and results are shown in table 3.

All p-values at different lags are greater than 0.05 indicating that the residuals are uncorrelated.

Residual plots:

The ACF and PACF plots of the residuals in figure 7 clearly show that the residuals lie within the 95% confidence limits. This indicates that, the residuals left over after fitting the models are white noise. Further analysis on residual plots is done by considering the normal probability plot, residuals versus fitted value, histogram of the residuals and the time series plot of the residuals. These plots are displayed in figure 8. The normal probability plots and histograms indicate that the residuals are approximately normally distributed. The plots of residuals against the fitted value and against observation order indicate a stable variance in the series.

In general the analysis of residuals show that the fitted ARIMA model for the loan repayment time series is adequate and therefore the fit can be used in forecasting. According to Tebbs (2013) [10] , the fitted model ARIMA (1, 1, 2) can be written in its multiplicative form suggested by Box, Jenkins and Reinsel by using equation 3.4. Now the fit takes the form

denoting the first difference of X t , B is the backshift operator which changes the time period t to time period t-1 and e t is the random error. The estimated parameters for ARIMA (1, 1, 2) which are found in the Minitab output in table 3 The fitted model (4.3) shows that the current value of the transformed loan repayment time series is a linear function of some of its past values together with its current and past shocks.

Using the Fitted Model to Forecast

Having obtained an adequate model ARIMA(1,1,2), the fit has been used to generate out-of-sample forecasts of monthly loan repayment at Kifanya SACCOS Ltd for the period starting from January 2017 up to December 2020. The forecasts along with their corresponding lower and upper 95 per cent confidence limit are displayed in figure 9.

Conclusion

The findings have shown that the fitted model for loan repayment fits well the historical data and can be used in short-term forecasting for the future values of this variable. However, in some months the forecasts may not agree strongly with the observed values as the series under study is very volatile and this increases unpredictability in its future values. But overall the loan repayment by borrowers is adequate enough and hence it is concluded that there is good governance of the SACCOS in Njombe, Tanzania.

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