**4. Empirical results and analysis**

#### **4.1 Stationarity tests**

Although, the Todo-Yamamoto model, the MWALD test was introduced for ease of estimation by circumventing the presence of unit roots pre-testing problem, nevertheless, there is the need to determine the maximum order of integration of the variables, which is necessary for estimation of The MWALD test for Granger causality by Toda and Yamamoto [39]. Therefore, we ran the test for the Augmented Dickey-Fuller (ADF) test, Phillips – Perron (PP) test and Kwiatkowski– Phillips–Schmidt–Shin (KPSS) unit root test, to ascertain the stationarity of the variables [45, 50–54].

From **Tables 1** and **2**, the unit-roots tests confirmed all our process to be considered integrated at the first difference and 1% level of significance using Augmented Dickey-Fuller (ADF) test and Phillips – Perron (PP).

While Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) in **Table 3** is in contrast to ADF and PP which indicated that the variables are at levels. This corroborates with the work of Yakubu and Abdul Jalil in their test of stationarity. A quick check on the line graphs in **Figure 1** indicated that all the variables are at first difference I (1). Therefore, we stick to ADF and PP, and agree that dmax = 1.

#### **4.2 Modified Wald (MWALD) test for Granger causality**


The Modified Wald (MWALD) Test for Granger Causality requires the determination of optimal lag which is presented in **Table 4**. By default, we use LR: sequentially modified LR test statistic, FPE: Final prediction error, AIC; Akaike

*Note: \*\*\*, \*\* and \* denote significance at 1%, 5% and 10% respectively. ADF test the null hypothesis of 'not stationary' against the alternative of 'stationary'. Source: E-views Version 9 software was used in the estimation.*

*Impact of Oil Price Fluctuation on the Economy of Nigeria, the Core Analysis for Energy… DOI: http://dx.doi.org/10.5772/intechopen.94055*


*Note: Just like the ADF, the PP unit root test has the null hypothesis of 'not stationary' against the alternative, which is 'stationary'. \*, \*\* and \*\*\* indicate the level of significance at 10%, 5% and 1% respectively. Source: E-views Version 9 software was used in the estimation.*

#### **Table 2.**

*PP stationarity tests.*


*Note: In contrast to ADF and PP, KPSS unit root test has the null hypothesis of 'stationarity' against the alternative, 'not stationary'. \*\*\*, \*\* and \* represent 1%, 5% and 10% level of significance respectively. Source: E-views Version 9 software was used in the estimation.*

#### **Table 3.**

*KPSS stationarity tests.*

#### **Figure 1.**

*Graphical representation of original series at I(1) for oil price (doilpr), exchange rate (dexcri), CPI (dcpi) and interest rate (dintr).*

information criterion, SBC: Schwarz information criterion and Hannan-Quinn information criterion to determine the optimal lag for the estimation of VAR system. The SC and HQ minimize its value at lag 2 while LR and FPE minimizes at lag 3. According to Liew [55], Asghar and Abid [56] Estimating the lag length of the autoregressive process for a time series is imperative in econometrics. The selection is done to minimize the chance of underestimation while at the same time maximizing the chance of recovering the true lag length. Another important aspect of the lag selection criteria is to overcome the structural break. Though, studies indicated that HQC is found to surpass the rest by correctly identifying the true lag length. In contrast, AIC and FPE are better choices for a smaller sample. In **Table 4** out of the two criteria, we propose three lags (lag 3) as the optimal lag.

#### **4.3 Correlation matrix for TY-VAR**

The orthogonal impulse response are based on recursive causal ordering, if the ordering is reversed different sets of structural shocks will be identified, and this gives a different impulse response function (IRF) and forecast error variance decomposition (FEVD), except if the error terms contemporaneous correlations are low [57]. According to Lutkepohl [58] given a sample size of T, the determinant of the reordering of the variables is given by �1*:*<sup>96</sup> ffiffiffi *T* <sup>p</sup> .

The ordering of variables suggested by Sims (1981, 1980) as iterated in the work of Yakubu and Abdul Jalil [44], Duasa [46], is to start with the most exogenous variables in the system and ended by the most endogenous variable. **Table 5** shows the residual correlation matrix result, the result shows that there is no instantaneous correlation between the variables because the variables are not significantly different from zero (at a 5% level of significance) [59]. This is based on the sample size in this analysis, we need at least a correlation of 31% that is above 5% level of significance to satisfy the call for reordering of the variables. Since there is no strong correlation among the variable we assumed the arrangement of our variables are in order.
