**3. Methodology and model specification**

#### **3.1 Methodology**

This analysis aims at investigating the effect and the interrelations existing between the impact of oil price fluctuation on the monetary instrument (Exchange rate, Inflation, Interest rate). The data were sourced from the Central Bank of Nigeria (CBN), National Bureau of Statistics (NBS) and Nigeria National Petroleum Corporation (NNPC). The data cover a period of 1995–2018 and the data is monthly. All our variables are in local currency. Therefore we used oil price, the interbank exchange rate as a proxy for exchange rate data, while the prime lending rate was used as a proxy for data on the interest rate and we used consumer price index for all commodity as a proxy for inflation.

A Toda and Yamamoto model (1995) (TY-VAR) was adopted in estimating the Modified WALD Granger Non-causality test (MWALD), Forecast Error Variance Decomposition (FEVD) and Impulse Response Function (IRF).

### *3.1.1 Toda and Yamamoto model (1995) and the modified Wald test statistic (MWALD)*

According to Salisu [42], Sims [43] and Toda and Yamamoto (TY-VAR) [39], Vector auto-regressions (VARs) are one of the widely used classes of models in applied econometrics, used as tools both for prediction and for model building and evaluation. It success lied on its flexibility and ease of application when dealing with the analysis of multivariate time series.

Practitioners have recently shown that the conventional asymptotic theory does not apply to hypothesis testing in levels VAR's if the variables are integrated or co-integrated [39, 43]. And one of the deficiencies of the VAR application is the inability to ascertain the a priori expectation of the variables whether the variables are integrated, co-integrated, or (trend) stationary. This necessitates pretesting(s) for a unit root(s) and co-integration in the economic time series, asarequisite for estimating the VAR model, and also when the intentions are prioritized towards the estimation of cointegration and vector error correction model [44].

Conversely, the powers of the unit and also simulation experiments of Johansen tests for co-integrating are very sensitive to the values of the nuisance parameters in finite samples and hence not very reliable for sample sizes that are typical for economic time series [39, 45, 46].

To alleviate these problems, Toda and Yamamoto [39] as quoted by Shakya [47], Giles [48] proposes the augmented VAR modeling, that is the modified Wald test statistic (MWALD), which is more superiority to the ordinary Granger - causality tests, the method is flexible and easy to apply, since one can test linear or nonlinear restrictions on the coefficients by estimating a levels VAR and applying the Wald criterion, paying little attention or circumventing the integration and cointegration properties of the time series data [42, 44]. However, the model is not a substitute for the conventional pre-testing in time series analysis, but as a complementary to the conventional VAR [49].

In estimating the MWALD test for Granger causality, it is prerequisite to determine the maximum possible order of the integration of the basic variables (dmax). Although, the variables could be a mixture of I (0), I (1), and I (2), in such condition, dmax = 2. The determination of the optimal lag length (k) is very

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

important, to avoid overstating or understating the true value of lag, to evade biased estimates of accepting the null hypothesis when it should be rejected, vice versa. By identifying dmax and k, a level VAR model of order (k + dmax) is estimated and zero restrictions test is conducted on lagged coefficients of the regressors up to lag k. This process certifies that the Wald test statistics have an asymptotical chi-square (χ<sup>2</sup> ) distribution whose critical values can be used to draw a valid inference and conclusion [39, 44].

#### **3.2 Model specification**

The model used in this research work borrowed a leave from the Toda and Yamamoto model (1995) as iterated in the work of Saban et al. [37], their model was adopted in this paper, to finding the inter-relationship between oil price and monetary variables. While they consider Granger Non-causality and structural shift, in our model we considered Granger Non-causality test, and substitute structural shift with Impulse Response Function (IRFs) and Forecast Error Variance Decomposition (FEVD). The TY-VAR is given by:

$$y\_t = a + \beta\_1 y\_{t-1} + \dots + \beta\_{k+d} y\_{t-(k+d)} + \varepsilon\_t \tag{1}$$

Where *yt* comprises of K endogenous variables, α is a vector of intercept terms, β are coefficient matrices, and *ε<sup>t</sup>* is white-noise residuals.

#### *3.2.1 VAR modified Wald test (MWALD)*

The analysis aims at establishing the interrelationship that exist among the variables; i.e. oil price (lnoilpr), and monetary policy variable i.e. exchange rate (lnexchr), interest rates (lnintr), and inflation (lncpi). The specification considers each variable expressed as independent in the model as a function of its lag and the lag of other variables in the model. Here the exogenous error terms *ε*1*<sup>t</sup>*, *ε*2*<sup>t</sup>*, *ε*3*<sup>t</sup>*, *ε*4*<sup>t</sup>*, are independent and are interpreted as structural innovations. The realization of each structural innovation is known as capturing unexpected shocks to its dependent variable (respectively), which are uncorrelated with the other unexpected shocks (εt). Equations for the Modified Warld Test model are presented as follows;

$$\begin{split} \text{lnoilpr} &= \alpha\_{\text{l}} + \sum\_{i=1}^{\text{k}+\text{dm}} \beta\_{\text{il}} \text{lnoilpr}\_{\text{t}-1} + \sum\_{i=1}^{\text{k}+\text{dm}} \gamma\_{\text{il}} \text{lnezchr}\_{\text{t}-1} + \sum\_{i=1}^{\text{k}+\text{dm}} \delta\_{\text{il}} \text{lnoi}\_{\text{t}-1} \\ &+ \sum\_{i=1}^{\text{k}+\text{dm}} \theta\_{\text{il}} \text{lnintr}\_{\text{t}-1} + \varepsilon\_{\text{lt}} \end{split} \tag{2}$$

$$\begin{split} \text{lnexchr} &= \mathbf{e}\_{2} + \sum\_{i=1}^{\mathbf{k}+\text{dm}} \beta\_{2i} \text{lnexchr}\_{\mathbf{t}-1} + \sum\_{i=1}^{\mathbf{k}+\text{dm}} \gamma\_{2i} \text{lnoil} \text{pr}\_{\mathbf{t}-1} + \sum\_{i=1}^{\mathbf{k}+\text{dm}} \delta\_{2i} \text{lnoi} \mathbf{p}\_{\mathbf{t}-1} \\ &+ \sum\_{i=1}^{\mathbf{k}+\text{dm}} \theta\_{2i} \text{lnint} \mathbf{r}\_{\mathbf{t}-1} + \mathbf{e}\_{2t} \end{split} \tag{3}$$

$$\begin{split} \mathsf{Incpi} &= \mathsf{a}\_{3} + \sum\_{i=1}^{\mathrm{k}+\mathrm{dm}} \beta\_{3i} \mathsf{Incpi}\_{t-1} + \sum\_{i=1}^{\mathrm{k}+\mathrm{dm}} \gamma\_{3i} \mathsf{Inoilpr}\_{t-1} + \sum\_{i=1}^{\mathrm{k}+\mathrm{dm}} \delta\_{3i} \mathsf{Incarkr}\_{t-1} \\ &+ \sum\_{i=1}^{\mathrm{k}+\mathrm{dm}} \theta\_{3i} \mathsf{Inintr}\_{t-1} + \varepsilon\_{3i} \end{split} \tag{4}$$

$$\begin{split} \text{lnintr} &= \mathbf{a}\_{4} + \sum\_{i=1}^{\mathbf{k}+\mathbf{dm}} \beta\_{4i} \text{lnintr}\_{\mathbf{t}-1} + \sum\_{i=1}^{\mathbf{k}+\mathbf{dm}} \gamma\_{4i} \text{lnollpr}\_{\mathbf{t}-1} + \sum\_{i=1}^{\mathbf{k}+\mathbf{dm}} \delta\_{4i} \text{lnzakr}\_{\mathbf{t}-1} \\ &+ \sum\_{i=1}^{\mathbf{k}+\mathbf{dm}} \theta\_{4i} \text{lnçi}\_{\mathbf{t}-1} + \mathbf{e}\_{\mathbf{1t}} \end{split} \tag{5}$$

Where lnoilpr, *lnexchr*, *lncpi*, *lnintr* are the log of oil price, exchange rate, inflation rate and interest rate, while *lnoilprt*�1, *lnexchrt*�1, *lncpit*�<sup>1</sup> and *lnintrt*�<sup>1</sup> are the lag variables of oil price, exchange rate, inflation rate and interest rate in logs.
