The vector autoregressive (VAR) model, used for the analysis of the effects of climatic variability on electricity energy consumption, can be specified as follows:

y t = A 1 y t − 1 + A 2 y t − 2 + ⋅ ⋅ ⋅ + A p y t − p + C D t + u t (1)

where y t = ( EEC ,   RH ,   T max ,   Wind ,   Sun ,   Rain ) is a vector of 6 observable endogenous variables namely electricity energy consumption (EEC), Relative Humidity (RH), Maximum Temperature (T_max), Wind, Sunshine (Sun) and Rainfall (Rain). D t comprises all deterministic Variables, u t is a K-dimensional unobservable zero mean white noise process, A i ; i = 1 , ⋅ ⋅ ⋅ , p and C are constraint matrices.

The general model of V A R ( p ) has many parameters, and due to complex connections and reaction between the variables in the model, they may be difficult to interpret [8]. As a consequence, numerous types of structural study are often used to summarize the complex properties of a VAR. The 3 main types of operational analysis summaries include Granger causality tests, impulse response functions and forecast error variance decomposition.

The accurateness of forecasts produced by the fitted VAR model was determined using mean absolute percentage error (MAPE) given below:

M A P E = ( 1 T ∑ t = 1 T | F t − Y t Y t | ) × 100 (2)

where T is the number of forecasts, F_t and Y_t are the predicted and observed values respectively.

Monthly climatic data on rainfall (Rain), maximum temperature (T_max), Sunshine (Sun), Wind (wind) and Relative Humidity (RH) from January 2008 to December 2018 were obtained from the Ghana Meteorological Agency while data on electricity energy consumption (EEC) for the same period was acquired from the Volta River Authority (VRA). Table 1 presents descriptive statistics for the variables considered.

The time series plots of the original data are shown in Figure 1, which revealed that both electricity energy consumption and the climatic variables in

the Yendi Municipality from 2008 to 2018 have been largely non-stationary. This is as a result of the presence of irregularities, trends, and seasonality. The mean is not constant throughout the successions.

However, stationarity in both electricity energy consumption and the climatic variables was achieved by computing the first difference of the original data as shown in Figure 2 below.

To confirm the stationarity observed in Figure 2, each of the variables in the differenced data was tested for the presence of a unit root using a test suggested by Dickey and Fuller [9]. The Augmented Dickey-Fuller (ADF) test is a unit root test for stationarity. The null hypothesis for the ADF test is that there is a unit root while the alternate hypothesis varies as per the condition applied.

Results of the ADF tests, presented in Tables 2(a)-(c), rejected the assumption of a unit root for all time series considered, implying that the relationships among the various variables analyzed below are not spurious.

A vector autoregressive model of the monthly electricity energy consumption and climatic variables was estimated with 3 lags for each variable in each equation. Each equation has 6 × 3 unrestricted coefficients plus one coefficient for a constant and one for a trend. The number of lags was chosen based on four tests: the Final Prediction Error (FPE) test [10], the Hannan Quinne (HQ) test [11], and the Information Criteria suggested by Akaike (AIC) [12] and by Schwarz (SC) [13]. These measures are defined such that the smaller the better. Two out of the four tests (HQ and FPE) indicated that as many as 3 lagged monthly values

may be sufficient. The AIC and SC suggested 12 and 1 lags respectively. Hence, based on HQ and FPE, the optimal lag length used was 3 (Table 3).

Table 4 contains parameter estimates of the fitted VAR (3) model. While some of the parameters are significant at the 0.05 level, others are not. However, for a VAR model, much more information is rather obtained from the structural analysis presented in the next sub-sections.

Granger-causality test is a basic approach used to find out whether a specific variable is influenced by developments in different variables [14]. A significant bit of leeway of this test is that they are not influenced by the ordering of the VAR framework. The results of the Granger causality test presented in Table 5 indicate that maximum temperature, rainfall, sunshine and wind all Granger causes electricity energy consumption. The only climatic variable that does not Granger causes electricity energy consumption in the Yendi municipality of Ghana is relative humidity.

Impulse response analysis was used to analyze the active relations between electricity energy consumption and the climatic variables of the VAR (3) process. The impulse responses of electricity energy consumption to climatic variables are shown in Figures 3-7. The response of electricity energy consumption to some of the climatic variables has a clear fluctuation; there is a peak positive effect

of maximum temperature on electricity energy consumption in March (3^rd month) and lowermost adverse effect of maximum temperature on electricity energy consumption in April (Figure 3). Also, the peak positive effect of sunshine on electricity energy consumption is observed in September (9^th month) while the lowest negative effect of sunshine on electricity energy consumption is observed in March (Figure 3). The response of electricity energy consumption to relative humidity, rainfall and wind did not show obvious fluctuations (Figures 4-6) have less effect almost throughout the year.

Forecast error variance decomposition (FEVD) is well-known in interpreting VAR models. Outcome of the FEVD for both electricity energy consumption and the climatic factors are shown in Figure 8 and Table 6. The results indicate that, an average of 67.15% of the pattern of the variability in the electricity energy consumption has been explained by innovations in electricity energy consumption in the past, to a large extent, about 12.13% of the pattern of the variability in the electricity energy consumption has been explained by past developments in maximum temperature, about 11.15% of the trend of the variability in

electricity energy consumption is explained by innovations in the past sunshine, 5.25% of the trend of the variability in electricity energy consumption is explained by past innovations in wind, 1.20% of the trend of the variability of electricity energy consumption is explained by past inventions of rainfall, while 1.12% of the trend of the variability in electricity energy consumption is explained by past innovations in relative humidity.

Forecasting is one of the principal motivations behind multivariate time-series studies. The VAR (3) model developed can be utilized as a prescient framework for making a forecast of electricity energy consumption. The forecasts of the differenced electricity energy consumption for the first 6 months of 2019 are shown in Figure 9 and Table 7. These forecasts indicate an increasing trend of electricity energy consumption in the Yendi Municipality in Northern Ghana.

The mean absolute percentage error (MAPE) gives an indication of the average size of forecasting error stated as a percentage of the relevant observed value irrespective of whether the forecasting error is positive or negative [15]. Using data from January 2019 to April 2019, the MAPE for the fitted model is obtained to be 4.54% (Table 8) using Equation (11), which is a sign of potentially excellent forecasts by the fitted VAR (3) model [15].