Forecasting GDP of Nepal using Autoregressive Integrated Moving Average (ARIMA) Model

Background : Globally many research are working on modeling and forecasting of gross domestic product (GDP). The trend and pattern will help the planner and policy maker to make future monetary policy. The aim of this research is to find the ARIMA model and forecasting. Methods : Box-Jenkins methodology was use for the modeling and forecasting of annual GDP series of Nepal from 1990/91 to 2019/20. Eviews 10 software was use for data analysis. Results : Using the Box-Jenkins methodology this research examine the number of ARIMA family model that describe the annual GDP series and the appropriate model is ARIMA(1,1,1). Conclusions : This research concluded that ARIMA(1,1,1) is the model which capture the GDP series of Nepal for this period.


INTRODUCTION
GDP represents the total value of all goods and services produced within a country's borders over a specified period, usually a year. 1 It includes everything from the products manufactured by businesses to the services provided by individuals, such as healthcare, education, and transportation. 2 GDP is considered an important indicator of a country's economic health and growth, as it reflects the total income generated within a country's economy. It is used to measure changes in the standard of living, economic growth, and productivity of a country. 3 However, it should be noted that GDP alone may not necessarily indicate the well-being of the people or the distribution of wealth in a country, as it does not account for factors such as income inequality or non-monetary measures of well-being, such as access to healthcare, education, and social services. 4 Nominal GDP is the total monetary value of the final product in terms of current market prices produced from all productive sectors within a country during a year. If GDP rises from one year to the next, one of two things must be true: (1) the economy is producing a larger output of goods and services, or (2) goods and services are being sold at higher prices. 5 Economists want to separate these two effects while they study the changes in the economy over time. In particular, they want a measure of the total quantity of goods and services the economy is producing that is not affected by changes in the prices of those goods and services.
They use real GDP to measure this fact. Real GDP is the total monetary value of final goods and services produced from all productive sectors in terms of constant prices (or base year prices) within a country during a year. [6][7] There are three ways from which the GDP of any country can be measured. In the production approach GDP includes the sum of gross value added of the various institutional sectors or the various industries plus taxes and less subsidies on products. In the expenditure approach GDP includes the sum of final uses of goods and services by resident institutional units (actual final consumption and gross capital formation), plus exports and minus imports of goods and services. 8 While in the income approach GDP is the sum of uses in the total economy generation of income account (compensation of employees, taxes on production and imports less subsidies, gross operating surplus and mixed income of the total economy). 9 Now Nepal is moving toward achieving the sustainable development goals and graduating from the status of least developed country. The support from international community in democratic transition of Nepal, to some extent, is expected to facilitate the country for making progress toward achievement of the Millennium Development Goals. In an attempt toward this direction, growth in GDP remains as the main target variable for Government of Nepal for setting up effective and efficient strategies and policies for economic development. In this backdrop, it is necessary to provide a precise forecast of GDP in order to develop meaningful vision of the future trend of Nepalese economy. Framing appropriate strategies and policies for economic development with proper allocation of funds towards priority sectors requires a good estimate of GDP for some period ahead. It is only possible by using appropriate time series model for forecasting (Rana, 2019). Nowadays globally many academician and policy maker are working on the modeling and forecasting the behavior of gross domestic product (GDP). The evolution of academic and policy interest in this area has been basically geared by the fact that GDP is considered as an important index of national economic development. Besides, GDP also helps judging the operating status of macro economy as a whole. 10 From the last few years many theoretical and empirical attempts have witnessed the growing academic interest on GDP growth and its determinants. Lucas (1988) explains that increasing concern toward GDP growth is being accelerated by the need to achieve higher rate of economic growth in both developed and developing nations. [7][8][9][10][11] Many researches are carried out in GDP to find the trend and various factors responsible for such change. Some of the factors are living standard, quality of life, lower investment on productivity. 12 Based upon the forecasted value of GDP Rastra Bank in making the various planning and policy. 13 Gross domestic product (GDP) is one of the most important indicators of national economic activities for countries. 14 For the time series modeling and forecasting of GDP series we used Box and Jenkins (1976) methodology which is also known as ARI-MA (Auto-Regressive-Integrated-MovingAverage) methodology. 2

METHODS
A cross sectional study was conducted by taking 30 years' time series GDP data of Nepal from 1990/91 to 2019/20 for the modeling and forecasting using Box-Jenkins methodology. For the data analysis Eviews 10 software was used. 2 Autoregressive integrated moving average (ARIMA) model was used in this research. An ARIMA model has three component functions: AR (p), the number of lag observations or autoregressive terms in the model; I (d), the difference in the nonseasonal observations; and MA (q), the size of the moving average window. An ARIMA model order is depicted as (p,d,q) with values for the order or number of times the function occurs in running the model. Values of zero are acceptable. In order to model a time series data with this approach, at first the series must be stationary. Statistically, the series is confirmed to be stationary if the n values seem to fluctuate with constant variation around a constant mean. If time series data set is non-stationary, differencing process is used to make it stationary. If the first order differences of the original time series values are also non-stationary, then second order differences are used to produce stationary time series values. Since the essence of engaging an ARIMA model is to forecast a series, the B-J methodology uses four steps: identification, estimation, diagnostics and forecasting. 2

Identification for stationary
Upadhyay et., Forecasting GDP of Nepal using Autoregressive Integrated Moving Average (ARIMA) Model (ACF) starts with a high value and declines slowly, indicating that the series is non-stationary. Also the Q-statistic of Ljung-Box which was smaller than 0.05, so we cannot reject the null hypothesis that the GDP series is non-stationary. For the further conformation ADF unit root was carried out.   From the above figure 5 showed correlogram of first difference of Ln(GDP). The coefficient of autocorrelation (ACF) starts with a high value and declines slowly, indicating that the series is nonstationary. Also the Q-statistic of Ljung- Box (1978) at the 24th lag (only 12 lags values were reported) has a probability value of 0.000 which is smaller than 0.05, so we cannot reject the null hypothesis that the GDP series is non-stationary in first difference.  showed that GDP series is non stationary (p-value<0.05). This indicates that there is no enough evidence to reject the null hypothesis of unit root. The p-value of Augmented Dickey Fuller test is less than 0.05 so we have sufficient evidence to say that the first difference of Ln(GDP) is stationary. This indicates that Ln(GDP) data is stationary in first differences. Therefore for our model ARIMA (p,d,q), So, the order of differencing is 1 i.e d=1. Above correlogram was used to determine the model ARMA (p,q), i.e. the values of parameters p and q. Which is already mentioned in the above figure, an AR(p) model has a PACF that truncates at lag p and an MA(q)) has an ACF that truncates at lag q. The orders of autoregressive and moving average process has been determine by observing the value of ACF and PACF of the second difference of GDP. The ACF of first difference of Ln(GDP) series is significantly decrease after at lag order 1, 2. This implies that the autocorrelation of the successive pair of observations in time period 1, 2 So, the tentative order of moving average process can be 1, 2 (that is q=1, 2). The PACF of first difference of Ln(GDP) series is significantly decrease after lag order 1,2,6. So, the tentative order of autoregressive process can be 1 (that is p=1,2,6). So, the tentative models are ARIMA(1,1,1), ARIMA (1,1,2), ARIMA(2,1,1), ARIMA(2,1,2), ARIMA (6,1,1), ARIMA(6,1,2)

Estimatmation Process
After identifying the tentative ARIMA model, in the next steep need to find the best ARIMA model. In to select the best ARIMA model Adj. R 2 , SER, AIC and SIC of all tentative model was carried out. Appropriate model should have most significant coefficients, lowest volatility, highest R 2 , and lowest AIC and SBIC.

Table 3. Result of ARIMA (p,d,q) Model fitting
Above table 3 showed the best model fitting among all the models. The best fitted model is ARIMA (1,1,1) which have more significant coefficient, lowest volatility, high adj R 2 , and lowers AIC as well as SIC. Diagnostics Figure 6. Correlogram of residuals for ARIMA (1,2,1) From the above table the appropriate model is (1,1 ,1). A flat correlogram is ideal. If a lag is significant re-estimated the model. The correlogram of the residuals is flat (all the values are lies in between standard error bound -95% CI) which indicate that all information has been capture. So the forecast will be based on this model.

Figure 7. Ljung-Box test for squared residuals (autocorrelation test)
Ljung-Box test for squared residuals (autocorrelation test). Above figure 7 showed the result of Ljung-Box test for squared residuals.   Table 3

Forecasting
Essence of fitting an ARIMA model is to forecast future values of the series by using the past value of the series itself. The forecast is based on the final selected model. In order to verify whether the forecast is correct or not need to plot the forecast graph. The forecast is based on the ARIMA model for second differenced GDP.

DISCUSSION
By sing Box and Jenkins (1976) methodology the best ARIMA model is (1, 1, 1) while the research of Rana forecast annual time series of GDP in Nepal from mid July, 1960 to mid-July, 2018 he found the best ARIMA model as (0,1,2). 14 Wei and al. (2010) use data from Shaanxi GDP for 1952-2007 to forecast country's GDP for the following 6 years. Applying the ARIMA (1,2,1) model they find that GDP of Shaanxi present an impressive increasing trend. 10 Maity and Chatterjee (2012) examine the forecasting of GDP growth rate for India using ARIMA (1,2,2) model and a time period of 60 years. The results of their study showed that predicted values follow an increasing trend for the following years. 8 Hisham use Sudan GDP during the period 1960-2018 by using the Box-Jenkins methodology the appropriate model is ARIMA (1,1,1). 5