The traditional univariate technique for evaluating persistence

A significant attribute of many time series data is non-stationarity. The evaluation of the persistence using the traditional univariate approach of order p [AR(p)] can be written as:

Given zt symbolizes the exchange rate, ∑i=1pβi denotes the AR process, εt∼N0,σε2i.e. εt is I(0) and L is the lag operator Lzt=zt-1. Two potential situations for ∑i=1pβi exist. If ∑i=1pβi=0, subsequently zt has low persistence (or “short memory”) implying weak dependence of zt overtime. Hence, zt is I(0), i.e. zt is stationary at level. For policymaking, this scenario portends that actions taken by the central bank or monetary authority will not have a significant impact on exchange rates. Conversely, should ∑i=1pβi>0, zt is understood to be highly persistent and exhibit long memory amongst observations. In this scenario, actions taken by the central bank or monetary authority will substantially affect the exchange rate. However, the Policy impact on exchange rates can either be temporary or permanent. The effect is temporary if zt 0<∑i=1pβi<1 while it is permanent if ∑i=1pβi≥1. If the impact is temporary, it indicates that zt is mean reverting and the shocks will decay over a long horizon, However, if permanent, shocks will persist forever.

The fractionally integration model

The study also applies the fractional integration technique based on the recent findings in the literature (See ^{Canarella and Miller, 2017}; ^{Gil-Alana and Carcel, 2018}; ^{Canarella and Miller, 2016}; ^{Usman and Akadiri, 2021} and ^{Ebuh et al 2021}) that stationary economic series can be given in a fractional form and may not take integer values. The fractional integration approach indicates that the series might be fractionally integrated. Such that shocks are not permanent but die out over long horizons.

Consequently, we examine Nigeria’s nominal exchange rates Zt using the fractionally integrated model to test for persistence:

L is the lag operator LZt=Zt-1, d is any real number,  Zt is integrated of order d and symbolized by Zt≈Id.εt is a covariance stationary process with a spectral density function that is positive and is an integrated order 0(I(0)) process and finite at zero frequency. Leveraging on the binomial expansion, 1-Ld polynomial in equation (2) can be formulated in such a way that for all real d,

Hence,

Accordingly, Equation (1) is presented as follows:

Equation (5) is developed to emphasize the critical role d also plays in the calculation of the degree of persistence (^{Gil-Alana and Carcel, 2018}). Three potential outcomes exist from using d to test for persistence. Suppose, if d=0, Zt (exchange rate) exhibits low-level persistence (i.e. “short memory”) also exchange rate is termed an I(0) series. If d falls within the interval of (0,0.5) is still supposed to exhibit long memory and is covariance stationary. However, suppose d≥0.5, this implies non-stationarity. Finally, suppose, 0.5 < d < 1 accordingly Zt is considered to be mean reverting and the effect of shocks to the exchange rate will disappear over in the long run. Nevertheless, suppose d≥1, it indicates policy shock will have a permanent influence.

We employ the parametric method in analyzing fractional integration which involves the maximum likelihood estimator of ^{Sowell (1992)}. A major wrench to the fractional integration technique is the accuracy gained by leveraging on the news in the data via the parameter estimates. However, in line with ^{Sowell (1992)}, a downside of the model is the sensitivity of the parameter estimates to models considered which could be misleading due to misspecification.

Fractionally cointegrated VAR (FCVAR) for nominal exchange rates

We examine Nigeria’s Nominal exchange rates, to check if they display long-run characteristics using the fractionally cointegrated VAR (FCVAR) proposed by Johansen (2008) and later extended by Johansen and Nielsen (2010, ²⁰¹²⁾, and ^{Nielsen and Popiel (2018)}. The FCVAR allows for a fractional process of order d that cointegrates to order d-b. From a policy viewpoint, if Nominal exchange rates are cointegrated, this implies that policy targeted at one of the exchange rates will also influence other rates. Additionally, if the cointegration is fractional, the impact of policy on these rates will only disperse only after short horizons. Consider Zt as a vector of I(1) time series of the element p, the error correction form of the CVAR model is presented as follows:

The FCVAR model is a derivative of Equation (6), this is done by interchanging the Δ and L with their fractional counterparts in Equation (7):

Given Δd is the term indicating fractional difference, and Lb=1-Δb is the fractional lag operator. Hence, α and β which are the long run parameters, p×r matrices were 0≤r≤p. The rank r is the cofractional rank. The columns of β comprise the r cofractional vectors such that β'Zt are the cointegrating combinations of the variables in the system. α represents the speed of adjustment towards equilibrium for each of the variables. Γ=Γ1,…,Γk represents the short-run dynamics in the autoregressive process.(^{Nielsen and Popiel, 2018})². Given that the CVAR is a unique case of the FCVAR, the FCVAR model condenses to the CVAR variant if. The FCVAR allows for simultaneous modeling of the short-run dynamics of the system, long-run equilibrium, and the correction to divergence from the equilibrium state.

The method of estimating the FCVAR follows the following steps: (a) the optimal lag length model is estimated; (b) the cointegration rank is established; (c) the data from (a) and (b) are employed to check for fractional cointegration; and in conclusion, (d) the Likelihood Ratio (LR) test is used to compare FCVAR model with the CVAR model by placing a limiting the null. If the null hypothesis is not accepted, then the FCVAR is favored, if not, the CVAR is accepted.

Testing for structural shift in persistence

The ^{Bai and Perron (2003)} test is employed to test for structural shifts in persistence. The maximum number of structural breaks allowed in the time series is five. This procedure comprises a chronological application of test, assumed to produce an optimal result in selecting the number of breaks. The results are reported in Table 1

Preliminary analysis

Weekly data from 1999 to 2020 were used for the analyses. The nominal exchange rate pairings covered are USD-Naira, Euro-Naira, Yen-Naira, Pound-Naira, Rupee-Naira, and Yuan-Naira rates. The data for this study were sourced from the Central bank of Nigeria Statistical and Bloomberg databases. Various descriptive statistics carried out on the six nominal exchange rate variables were tabulated in Table 2. The data for the study were subdivided into two subsamples, given the results of the Bai-Perron test in Table 1 and consequently were split into Pre and post the Global financial crisis (GFC) individually. The Post-GFC captures the period of the crisis and afterward whilst the Pre-GFC is the period before September 2007. The Post-GFC stretch is so defined because of Nigeria’s peculiarity in which the effect of the GFC on macroeconomic fundamentals postdates the crisis period. Inferring from Table 2, the exchange rates for USD-Naira, Euro-Naira, Yen-Naira, Pound-Naira, Rupee-Naira, and Yuan-Naira exhibit higher volatility before the GFC than afterward.

Besides, the mean and standard deviations of the variables were higher Post-GFC than before GFC. Thus, implying the presence of more asymmetry GFC afterward on the nominal exchange rates.

The descriptive statistics and the plot of the six nominal exchange rate variables in Figure 1, clearly shows an upward trend in the nominal exchange rates after the GFC except in Yen-Naira. This upward trend in the nominal exchange rates could be attributed to the continuous depreciation of the naira. These outstanding distinctions in the behavioral pattern of the nominal exchange rates in Nigeria before and afterward of the GFC create the needed enthusiasm for empirically following the same path in the analyses.

Main results

The study examines the nominal exchange rates in Nigeria using the full, pre-GFC, and post-GFC sample periods to test for the existence and the degree of persistence of the nominal exchange rates in Nigeria. The study leveraged on the univariate and the fractional cointegrated VAR models with and without a trend. Table 3 presents the results on the traditional approach for persistence on the nominal exchange rates, which were observed to be non-stationarity as the magnitude of the persistence is close to unity for both with intercept and with intercept and trend. A further examination of the results shows that persistence is mean reverting in each sample considered, implying that the degree of persistence is temporal and not permanent. Thus, indicating that the nominal exchange rates will revert to their long-run mean.

Also, the Wald test was used to if the degree of persistence is not statistically different from one. The outcome revealed that the persistence’s were statistically insignificant in all the samples at various degrees, except the US dollar exchange rate in the pre-GFC period.

Also, we compute the half-life estimates for a shock to Nigeria’s Nominal exchange rates. The half-life fundamentally reveals the period it takes for the impact of a shock on the exchange rates to divide into two. From the results presented in Table 4, it lends credence to the findings in the traditional approach that exchange rate persistence is higher in the post-GFC period than the pre-GFC period. This finding emphasizes the need for better coordination between the fiscal and the monetary authorities, to adequately manage the exchange rates in the post-GFC period.

To complement the traditional approach, we use a fractionally integrated approach to test for persistence. From Table 5, the estimates obtained using the fractionally integrated approach were reported under two cases (with intercept and with intercept and trend) and three samples (full sample, Pre-GFC, and Post-GFC), as with the traditional approach. Several features are noticeable in Table 1. First, all the estimated values are below 0.5, showing that the six nominal exchange rates under consideration are fractionally integrated and exhibit long memory. As in the traditional approach, we also find the level of persistence to be higher for the post-GFC sample for all exchange rates than the pre-GFC sample. Leveraging on the Wald test, the values of the fractional parameter estimates were statistically less than one, contrary to the traditional approach. This result indicates that, although the Nominal exchange rates exhibit long-run memory impact of any policy shock on the exchange rate to die out, it may not be permanent.

Are nominal exchange rates fractionally cointegrated in Nigeria?

In furtherance to the objective of the study, we evaluate the likelihood of fractional cointegration amongst the various nominal exchange rates. The test for fractional cointegration is because currencies from Nigeria’s major trading partners and its reserves expose the Nigerian economy to both commodity price and trade shock. Also, Nigeria’s economy largely depends on import and trade interconnectedness, which could impact on policy actions aimed at stabilizing the exchange rate. Thus, the six nominal exchange rate variables were tested for cointegration. The study estimated the FCVAR, which is a unique variant of the CVAR with a fractional value of the order of integration. This result is achieved by first obtaining the maximum lag value, followed by the rank. These two parameters are used to obtain the order of integration. Inferring from the results in Table 6, we observe across the three sub-samples, four cointegrating relationships for the full sample and five for the Pre-GFC and Post-GFC samples, respectively. This result implies the existence of a long-run relationship amongst the nominal exchange rates of the six selected currencies with Naira. For the full sample, cointegrating relationships are better formed using the CVAR model than the FCVAR as the estimated fractional parameter is greater than 1. However, in the two sub-samples, cointegrating relationships are better formed using the FCVAR model, as the fractional parameters are both 0.010. Hence, justifying the implementation of the fractional framework. Additionally, in Table 7, we find the data to lends support to estimating the fractional parameter using FCVAR over the conventional CVAR, for the sub-samples samples, as shown by the statistically significant LR statistics. However, the CVAR model is suitable to ascertain cointegrating relationships for the full sample period.

Robustness

We employed ^{Narayan and Liu (2015)} and ^{Narayan et al. (2016)} GARCH-based unit root techniques to check for the robustness of results. The benefit of using this approach over others, is its usefulness in allowing the error term to follow a GARCH based process, as against other methods. Tule et al. (2020), ^{Salisu et al. (2020)}, ^{Salisu and Adeleke (2016)}, ^{Narayan and Liu (2015)}, and ^{Narayan and Sharma (2015)}, argue the choice of data frequency is of importance in empirical analyses. On this note, we considered, in addition to the weekly series, a monthly data frequency of nominal exchange rates to supplement the results obtained.

The results were similar to the traditional approach and the fractionally integrated approaches (Table 8 & 9) with higher degrees of persistence observed in the post-GFC period. Also, from the FCVAR estimates (Table 10 & 11), using the monthly samples, we find similar results to the weekly frequency. The results from the CVAR are superior to the FCVAR for the full sample, and the FCVAR results are superior for the Pre-GFC and Post-GFC periods, respectively. From these findings, the results are unaffected by the different data frequencies.