Testing the Martingale Difference Hypothesis (MDH) with Structural Breaks: Evidence from Foreign Exchanges of Nigeria and South Africa

This study tests for MDH in two prominent foreign exchange (FX) markets in Africa, Nigeria and South Africa using three benchmark currencies (euro, dollar and pound sterling). Data utilized cover time series closing rate data set of five-day weekly frequency spanning December 14, 2001 to September 26, 2014. The study considers both the linear and nonlinear measures for MDH with better size and power properties. We also capture structural break endogenously from the data stream using Perron (2006) unit root test with structural break. Three striking findings are discernible from our analyses. First, on average, the South African FX market appears to be more efficient than the Nigerian FX market. Thus, the latter may be more susceptible to speculations than the former. Second, ignoring significant structural breaks may render statistical inferences invalid. Third, the choice of methodology does matter when testing for MDH of foreign exchanges in Africa.

Keywords: Martingale Difference Hypothesis (MDH); structural breaks; Nigeria; South Africa; FX market

In the literature, several studies have examined the market efficiency hypothesis (MEH) based on whether the series follows a random walk hypothesis (RWH), the martingale hypothesis (MH) or martingale difference hypothesis (MDH). Most of these studies have evaluated MEH for different FX markets both in developed and developing countries including emerging and transition economies. A cursory review of some of the recent existing related studies is provided in Table 1 (see Charles & Darné, [9] for a review of earlier papers). Dominant in the empirical literature of FX markets are those that studied the euro and US dollar markets.[1] For example, some studies essentially focus on the euro exchange rates (see Al-Khazali & Koumanakos, [2]; Belaire-Franch & Opong, [5]; Charles & Darné, [9]; Chen, [12]; Cheung, Su, & Choo, [14]; Yang, Su, & Kolari, [43]); some on the US dollar exchange rates (see Azad, [4]; Chang, [8]; Lee, Pan, & Liu, [27]; Lima & Tabak, [29]; Tweneboah, Amanfo, & Kumah, [40]), and a few others combine different FX markets including euro and US dollar markets (see Al-Khazali, Pyun, & Kim, [3]). The empirical findings although differ slightly across countries or regions and FX markets, the weak-form efficiency, however, seems prominent.

Table 1. A Cursory Review of the Literature.

6 Source: Authors' compilation Note: AVR = Automatic Variance Ratio test; GS = Generalized Spectral test; AQ = Automatic (data-driven) Box-Peirce test; DD = Durlauf-Deo test; GAP = Generalized Andrews and Ploberger test.

Some of the early techniques used to test for MEH particularly MH or MDH include but not limited to the single variance ratio (VR) test of Lo and MacKinlay ([32]); the multiple VR test of Chow and Denning ([17]); and the Wright ([42]) sign and rank VR tests. Notable recent developments in this regard include the wild bootstrap Chowtrap Chowor martingale d06); power-transformed test of Chen and Deo ([13]); the generalized spectral (GS) test of Escanciano and Velasco ([19]); the wild bootstrap automatic variance ratio (AVR) test of Kim ([23]); the automatic portmanteau (AQ) test of Escanciano and Lobato ([20]); and the joint sign test of Kim and Shamsuddin ([24]).[2] In a more explicit representation as demonstrated in the papers of Charles and colleagues (Charles, Darné, & Fouilloux [11]; Charles, Darné, & Kim, [10]), these tests have been broadly classified into linear and nonlinear measures of dependence. The linear measures are the VR tests and their various modifications while the spectral and generalized spectral tests are usually used as the nonlinear measures (see Table 2 for clarity of these classifications). In essence, some studies have employed a particular category of the techniques (for example, Al-Khazali & Koumanakos, [2]; Chen, [12]; Charles & Darné, [9]; Chis, [15]; Kumar & Kamaiah, [25], consider the linear approach) while a few others employ the two for robustness purpose (see for example, Charles et al., [11]; and Al-Khazali et al., [3]).

Table 2. Classification of Tests for MDH.

7 Source: Compiled by the authors.

The main objective of this study is to investigate whether Foreign Exchange (FX) markets (for euro, pound sterling and US dollar) in Nigeria and South Africa exhibit martingale difference hypothesis (MDH) and to also verify whether accounting for structural breaks matters. The latter may be important when testing for market efficiency since the two countries, at one time or another, considered policy options; chief among which is the liberalization of the financial system with attendant consequences on the behavior of their exchange rates. In addition, a number of studies have documented the presence of structural shifts in exchange rates of both countries. For example, May ([34]) reveals that copious structural breaks are apparent in the exchange rates of South African Rand data. The paper argues that these several shifts are attributable to the financial market liberalization that has been implemented in South Africa since 1995. Similar evidence was also obtained by Akinboade and Makina ([1]) for South Africa. In Nigeria, the naira exchange rates have responded to both external and internal shocks including policy adjustments. For example, Salisu and Mobolaji ([37]) find robust structural breaks that coincide with the period of global financial crisis for the Nigerian naira/USD exchange rate. In addition, Salisu ([36]) provides some stylized facts about exchange rate management in Nigeria and also finds that the behavior of the exchange rate in Nigeria tends to change over short periods of time.

Drawing from the foregoing, therefore, it will not be surprising if MDH is biased in the presence of significant structural breaks. Accounting for structural breaks in hypothesis testing is not new and has in fact become a standard practice in the literature particularly when dealing with models with time series component. Examples include unit root test, cointegration test, and parameter stability test, among others.

The efficiency of the FX markets of these two countries within the framework of MDH has not been well documented in the literature. In the light of the growing dominance of portfolio investments of Nigeria and South Africa in Africa, analyzing the efficiency of financial markets including FX market is important for profit maximizing investors. Therefore, this study essentially focuses on the Nigerian and South African foreign exchanges as the two most prospective markets in Africa. This is attested to by their membership of the World Federation of Exchanges (WFE). Consequent upon this recognition by WFE, these two markets stand a chance of gaining better credibility and visibility. In addition, these markets have been beneficiaries from the WFE's unrivaled promotion of market standards, reliable statistics and inestimable advocacy for fairness, transparency and efficiency. All together, these lead to a stronger regulatory environment and engendered global best practices in both markets, and now reduce the level of complicity that usually characterizes foreign exchange markets in developing economies which may subject empirical investigations to cumbersome considerations. From the empirical standpoint, it indicates that the number of unknown factors would have been extricated and only left with the fundamentals that can be largely explained within the specified models. This further reduces the sum of error residuals and both the predicted and estimated values would be tightly close.

Recently, Charles et al. ([10]) conducted a Monte Carlo experiment to compare power properties of alternative tests for the MDH and they find that the wild bootstrap AVR test shows the highest power against linear dependence, while the GS test performs most desirably under nonlinear dependence. Thus, we adopt both the wild bootstrap AVR and wild bootstrap GS tests for the MDH in order to capture both linear and nonlinear dependence of foreign exchange returns. This study also accounts for structural breaks using the Perron ([35]) generalized structure for analyzing structural breaks with unit roots since the non-rejection of the null hypothesis of unit root or linear dependence may be due to the presence of structural breaks. Therefore, ignoring these breaks when they are significant may bias the results. Some preliminary analyses are provided in section 2. The econometric methodology of the study is described in section 3. Section 4 renders the discussion of results and section 5 concludes the paper.

In this section, we render some preliminary analyses of the FX returns for both Nigeria and South Africa (as the two representative foreign exchanges in Africa). We consider the Nigerian naira and South African rand in relation to some international reference currencies such as the pound sterling, euro and US dollar. The preliminary analyses are partitioned into three. First, we provide some descriptive statistics such as the Mean, Standard Deviation, Skewness and Kurtosis including formal tests such as the Normality [Jarque-Bera] test, Serial Correlation [Ljung-Box] test and Heteroscedasticity [ARCH LM] test. The unit-root test with structural breaks is also carried out in order to identify any possible structural break(s) in the FX returns. This affords us the opportunity to further investigate if any marked difference(s) exists in the MDH results between the pre- and post-breaks as well as whether the results differ from the full sample case. The second phase involves graphical illustration of the selected FX returns in order to depict the behavioral patterns of these series for the period under consideration.

The FX return is computed on a continuously compounded basis for a particular exchange rate i at time t as given below:

(1)

Graph

Where is the exchange rate returns of a given country at time ; is the exchange rate of that country at time , while represents one period lag in the exchange rate. For the purpose of clarity, an increase in the FX return here denotes depreciation in the domestic currencies (naira in the case of Nigeria and rand in the case of South Africa) relative to the referenced currencies while a decrease implies an appreciation. Importantly, we examine these exchanges for the weak form efficiency test with time series closing rate data set of five-day weekly frequency spanning December 14, 2001 to September 26, 2014. Thus, we utilize 667 observations for the return series. The main sources of data are the Central Bank of Nigeria and South African Reserve Bank databases. We denote the naira to pound sterling, euro and US dollar with NP, NE and NU respectively while RP, RE and RU are the respective proxies for the rand to pound sterling, euro and US dollar.

We carry out preliminary analyses using FX return series since the use of returns is required when testing for martingale difference hypothesis (MDH). The return series is preferred to level series in order to circumvent the problem of nonstationarity usually encountered with the level series (see Escanciano & Lobato, [20]). Thus, we test the weak form efficiency of the foreign exchange market of the two prominent African countries by examining whether the exchange rate returns of these countries are Martingale Difference Sequence (MDS) or not.

Table 3 presents the statistical properties of the return series. Starting with the mean returns, both the naira and rand recorded positive mean returns across the reference currencies. Also, for both countries, the US dollar recorded the lowest positive mean returns while the euro recorded the highest. There are at least four implications of this finding. First, both the naira and rand seem to have witnessed more depreciations than appreciations against the reference currencies over the period under consideration. Second, the rate of depreciation of the naira is slightly higher than the rand. Third, the rate of depreciation of both the naira and rand against US dollar is lower than the pound sterling and euro, on average. Fourth, in terms of returns to a profit maximizing investor, the US dollar gives the lowest returns while the euro gives the highest returns, on average, in both cases.

Table 3. Statistical Properties for Nigerian and South African Exchange Rates and their Returns.

8 Source: Authors' Computation. Note: NP, NE and NU are the exchange rates of the naira to the pound, euro and US$ respectively. Also, RP, RE and RU are the exchange rates of the rand to the pound, euro and US$ respectively. The suffix _R represents returns. ***, ** and * denote statistical significance at 1%, 5% and 10% respectively. Ljung-Box Q-statistic and ARCH LM tests are performed on the residual of AR(1). Also, nR2 statistic is reported for the ARCH LM test.

In terms of the risks associated with the returns, judging by the standard deviation values, we find that the US dollar has the highest risk in both countries while the euro has the lowest risk for the naira and the rand is less risky against pound sterling. Thus, in both countries, higher risks suggest higher potential losses. Also, the standard deviation can be used to measure the extent of volatility in time series. Extending this application to the return series, we observe that the US dollar is more volatile than the other reference currencies in both scenarios.

In relation to the distribution of the return series, the positive skewness of the returns suggests higher possibilities of depreciation of both the naira and rand relative to the reference currencies. Equally, the kurtosis values show that the return series are leptokurtic in nature; that is, the tail is heavier than normal. The Jarque-Bera statistics including the skewness and kurtosis indicate that exchange rates of the naira and rand are non-normally distributed.

Furthermore, the Ljung Box Q-statistic for squared residuals shows that higher order serial correlation is significant, while the results of ARCH LM test at both lags 5 and 10 reveal that the null hypothesis of no ARCH effects is strongly rejected. The graphical illustration in Figure 1 is also a clear demonstration of the presence of volatility clustering in the behavior of the selected FX returns. The joint statistical properties of non normality, higher order serial correlation and conditional heteroscedasticity in financial time series such as exchange rate suggest the presence of conditional mean dependence which could be validated under the assumption of martingale difference hypothesis.

Graph: Figure 1. Graph for Level and Return Series.

In testing for this hypothesis, as earlier mentioned, we employ the wild bootstrapping procedure for both Automatic Variance Ratio (AVR) and Generalized Spectral (GS) tests. Wild bootstrapping procedure is a recent development in addressing the issue of serially uncorrelated non Martingale processes especially in the presence of significant conditional heteroscedasticity. It has high size and power advantage in determining linear and nonlinear dependence in conditional mean of the financial returns.

In addition, studies have shown that the predictability of financial returns is inconsistent over time. In other words, any structural change in financial markets may have significant effect on the weak form efficiency hypothesis of the markets (see for example, Charles et al., [10]; Lazăr, Todea, & Filip, [26]). Thus, we employ unit root test with structural break by Perron ([35]) to determine the break points/dates as well as the stationarity or otherwise of return series. The result is presented in Table 4.

Table 4. Unit Root Tests with Structural Breaks.

9 Source: Authors' Computations. Note: We extract appropriate Critical values from Perron (1997) which are –5.28 and –4.62 at 1 and 5% levels of significance respectively.

The unit root estimates show that all the return series are non-unit-root in the presence of structural breaks. In effect, this suggests that the justification for the consideration of structural break in our analysis cannot be discounted. Drawing from the results of the structural breaks, we find that all the rand exchange rates maintain the same structural break dates. Similarly, with the exception of the naira to the euro, virtually all the exchange rates for the naira also maintain the same structural break dates. Essentially, the case of the South African rand is predicated on the aftermath effect of the global financial cum economic crisis of 2007 to 2009. It seems that the South African currency appears immune to the vagaries of the global crisis but the attendant crisis effect became apparent when it was already tapering off in other economies of the world (see Tella, Yinusa, & Ayinde, [39]). For the case of the Nigerian currency, it was the attendant effects of financial market reform that began around 2004 and 2005 that accounted for the noticeable structural changes in its foreign exchange market (see Soludo, [38]). Consequent upon the foregoing, the empirical results for the MDH, in this study, are presented in three scenarios – namely: the full sample period, the period before structural break, and the period after. In the section that follows, we describe the methodology for the MDH tests both the linear and non-linear dependence.

Conventionally, the MDH is a non-mean-reverting process often referred to as conditional mean independence series (see Goldberger, [21]). Basically, the usual procedure is to evoke the serial non-correlation assumption despite the fact that uncorrelated sequence is not necessarily martingale sequence even though a martingale differenced system (MDS) is an uncorrelated sequence. This basic truism about an MDH series gives marked distinctions between time domain tests (such as the VR tests) and the frequency domain tests (such as the GS tests). The former tests only consider serial correlation and do not directly test for MDS while the latter tests capture the MDS non-linear dependence relations. In effect, uncorrelated process is only a necessary but not a sufficient condition for MDH (see Domingruez & Lobato, 2003; Escanciano & Lobato, [20]; Lim & Luo, [28]).

The automatic variance ratio (AVR) test is one of the notable techniques employed in evaluating the efficiency of financial market. The workhorse measure of the MDH, which was anchored on the assumption of serial correlation, is the VR test. Basically, the basic measure was the asymptotic test proposed by Lo and MacKinlay ([31]) but due to the identified shortcomings (see Charles & Darne, [9]), various extensions of the individual Wald tests by Chen and Deo ([13]) coupled with the joint Wald statistics by Chen and Deo ([13]) and Kim and Shamsuddin ([24]) have been suggested. Also, the VR tests with single rank and sign by Wright ([42]) and the VR test with multiple ranks and signs by Belaire-Franch and Contreas ([6]) have been proposed. However, the attraction of the AVR test proposed by Choi ([16]) over the conventional VR tests reviewed above is evident in its data dependent nature (see Charles et al., [11]). Further modifications to Choi's ([16]) original AVR test were made by Kim ([23]) with the inclusion of the wild boostrapping features for small sample cases – basically to improve the asymptotic properties of size and power – which was identified as the prime defect of the conventional VR test of Lo and Mackinlay ([31]). Following a bootstrap procedure (see Veka, [41]) and under the condition of serial non-correlation, Choi ([16]) proposed an AVR test of the equation (2):

(2)

Graph

However, Kim ([23]) recommends wild bootstrap of Mammen ([33]) to strengthen AVR against small sample properties especially when the series of interest (say) is subject to conditional heteroscedasticity. Kimmmen (1993)i (1999) propoAVR test employed in this study is an iterative procedure which contains three basic stages:

• Form a boostrap sample of size T as for with being a random sequence with zero mean and unit variance
• Calculate with
• Repeat item (1) and (2) times to form a bootstrap distribution

The two-tailed p-value of the test can be obtained by dividing the number of absolute values of greater than the absolute value of by the total number of bootstrap samples . As such, a two-tailed probability of the equation below results from the bootstrap procedure:

(3)

Graph

Equation (3) is a ratio of the excess absolute AVR value by the total number of bootstrap samples B.

A non-linear asymptotic compliant test of the linear Automatic Variance Ratio (AVR) type was proposed by Escanciano and Velasco ([19]). Unlike the linear AVR test, which was of the generalized Kolmogorov Smirnov form, the non-linear test of generalized spectral test (GST) uses the Cramer-von Mises norm to obtain the statistic of the form:

(4)

Graph

With the condition that the null hypothesis for MDS is equivalent to; which implies that, where is a real number. Escanciano and Velasco ([19]) considered a pairwise approach that makes use of available data in the sample and at the same time avoids high dimensional integration. Thus, they present a null hypothesis for testing MDH as:

(5)

Graph

wherea.s. are the pairwise regression functions. This implies that no matter what value a previous realization of has taken, the expected future value of remains the same. Compactly, the null hypothesis gives:

(6)

Graph

Where; represents an auto-covariance measure in a non-linear framework; represents any real numbers.

The alternative hypothesis is the negation of the null defined as:

(7)

Graph

Thus, the conditional mean dependence measures, can be viewed as a generalization of the usual autocovariances to measure the conditional mean dependence in a nonlinear time series framework.

On the basis of Theorem 1 in Bierens ([7]), the following characterization is used to represent the null hypothesis:

(8)

Graph

It follows that under the null hypothesis, and the process for testing the null hypothesis is proposed by the authors as

(9)

Graph

In order to evaluate the distance between and zero for all possible values of and , a norm has to be chosen. Escanciano and Velasco ([19]) suggest the appropriateness of Cramer von Mises norm:

(10)

Graph

where is a weighting function. Furthermore, they suggest the use of either the normal or exponential cumulative distribution function (CDF) as the weighting function. In this study, we employ normal CDF and the test statistic for computing is thus

(11)

Graph

The null hypothesis of MDH is rejected when becomes sufficiently large. As the distribution of under the null depends on the data-generating process, wild bootstrapping procedure is employed to generate p-values as suggested by Escanciano and Velasco ([19]).

The results depicted in Table 6 are the probability values of the AVR test and the GS test over the full sample and sub-samples. As previously noted, the sub-samples are based on the break dates obtained from the structural break test. Before we proceed to interpreting the results, it is important to define the underlying null hypotheses for the tests used in evaluating the efficiency of foreign exchange markets in Nigeria and South Africa. The AVR test has a null hypothesis of absence of linear dependence in foreign exchange returns while for the GS test, the null hypothesis is that there is absence of non-linear dependence. A non-rejection of these respective hypotheses suggests the presence of MDH; either in linear or non-linear form while a rejection connotes the absence of MDH. Also, the presence of MDH is an indication that the market under consideration is efficient while it is inefficient if there is absence of MDH. If a market is efficient (i.e. it follows MDH), then, the past and current values do not feed into the future values of foreign exchanges and therefore, it becomes difficult to 'circumvent' the market for arbitrage activities. If a market is inefficient (i.e. it does not follow MDH), then, investors in the market can make abnormal returns through speculations. Table 5 further describes the parameters used for the interpretation of MDH test.

Table 5. Null Hypothesis of MDH and Significance Tests.

Table 6. Probabilities of Rejection of Null Hypothesis.

10 Source: Authors' computations. Note: NP, NE and NU are the exchange rates of the naira to the pound, euro and US$ respectively. Also, RP, RE and RU are the exchange rates of the rand to the pound, euro and US$ respectively. Also, B represents the number of bootstraps

We begin with the AVR test results which capture the linear dependence formation of the foreign exchange series under consideration. The results are presented in Table 6 and the data utilized here span the (full sample) period of December 14, 2001 to September 26, 2014 with bootstrapping procedure of 300 and 500. With the exception of the rand–euro exchanges (proxied by RE), it is evident that the MDH cannot be rejected for the South African rand in relation to the international referenced currencies. The conclusion is based on the obtained probability values which are greater than the 0.10 significance level. However, the contemporaneous exchanges of the naira to these referenced currencies (proxied as NU, NE and NP respectively) reject the MDH since the probability values for the full sample case lie within the critical region of a 10% level of significance. This suggests that the rand is, at least, weak form efficient implying that investors cannot make abnormal gains on returns through arbitraging activities and speculative exercises. This also implies that the current values of exchange rates in South Africa for these referenced currencies are respectively independent and do not provide any viable information for the future trends in these exchanges that could warrant arbitrage activities. The full sample of exchange rates for Nigeria are found to be inefficient and of non-random walk process in that it becomes possible for investors and traders alike, to generate excess returns over time through speculations. However, this interpretation should be considered with some level of caution, especially under a linear dependence hypothesis, as sluggishness of the Nigerian foreign exchange series to market information might blur the random walk process. In essence, the naira may be efficient if closely examined non-linearly (see Veka, [41]). Therefore, it is important to consider both the linear and non-linear dependence tests for MDH.

Interestingly, for the rand case, the non-linear martingale investigation of the GS test confirms the non-rejection of MDH for all the three exchange rates. This finding is similar to the linear counterpart involving the AVR test with the exception of RE (see Table 6). These imply that the South African foreign exchange market is efficient both linearly and non-linearly. However, for the naira, we find that naira-euro (NE) and Naira-US Dollar (NU) are martingale difference series implying that FX speculators cannot generate abnormal returns through speculation for these naira exchange rates. More specifically, the naira exchange rates for euro and pound sterling are only efficient non-linearly. In other words, the rand is both linearly and non-linearly independent while the naira could only be said to be largely non-linearly independent.

Importantly, the linear and non-linear independence of returns on rand exchange rates is an indication of nervous market conditions and the indifference nature of market participants as optimal trading positions may have been sustained for some periods. This may not be applicable to the naira exchange rates which are only non-linearly independent.

There are two sub-samples defined as Pre-Break and Post-Break and the results are also presented in Table 6. The findings are quite instructive. In relation to the Pre-Break, except for the naira–dollar exchange (proxied as NU), the probability values obtained indicate that both the naira and rand exchange rates are consistent with the MDH at the 10% level. In addition, both the GS test and AVR test do not conflict as regards this evidence. On the contrary, when post structural break was considered, the results indicate that virtually all the naira and rand exchange rates are non-martingale difference series based on the AVR test and therefore are linearly inefficient during this period. On the basis of GS test however, the exchange rates are efficient as virtually all the probability values exceed the chosen level of statistical significance.

Intuitively, one could infer that the global financial crisis does not significantly impact on the behavior of the exchanges between the naira and the rand. However, the non-linear efficiency for all the scenarios of full sample and sub-samples (before and after structural breaks) aligns with the rational expectation of the participants in the foreign exchange markets. These findings contradict the adaptive market efficiency hypothesis of Lo ([30]) as the efficiency of the foreign exchange markets for these two African countries are not influenced by changing market conditions after the global financial crisis. However, the expectation here is that downward swings in the markets consequent upon the global financial crisis will alter the degree of efficiency in the foreign exchange market. A close to zero p-values, as obtained for the linear dependence case after break, indicate a significant departure of foreign exchange market from efficiency (see Lazăr et al., [26]). Considering the linear dependence for all the periods, however, it is found that these foreign exchanges are reflective of available information as market efficiency responds to changing market conditions after the global financial crisis. These imply that these exchanges are linearly dynamic but non-linearly static in nature; given the structural changes (see Veka, [41]). In essence, accounting for structural breaks when they are significant is inevitable in order to render valid and meaningful inferences.

Despite the mixed results obtained, given the linear and non-linear dependent models of the AVR and GST respectively, it is still evident that structural breaks significantly alter the behavior of foreign exchanges in African economies. This suggests that macroeconomic episodes, global financial and economic positions as well as imbalances impacted on the South African and Nigerian foreign exchange markets. Thus, deliberate actions need to be pursued by relevant authorities in Nigeria and South Africa to deal with shocks that may cause significant shifts in the behavior of their foreign exchange markets. A more recent example is the current global oil price shock that appears to have altered the path of the Nigerian exchange rate owing to the structure of the economic base of the country which is predominantly oil. This is a strong attestation to the vulnerability of foreign exchange markets (particularly in Nigeria) to structural breaks.

Specifically, the bail-out measures and other forms of government interventionist programs towards correcting macroeconomic fundamentals during the global crisis should be taken with caution so as not to provide the wrong signals to investors capable of promoting speculations. Usually, the government is always concerned with current account imbalances and could consequently sterilize the economy through monetary policy impulses in order to check the outflow of hot money and by implication reduce capital flight. More so, round-trippers and speculators might want to take advantage of this to scoop the foreign exchange markets off foreign currencies so as to create artificial scarcity and earn rents on exchange rate differentials. In addition to the use of sterilization control earlier suggested, the government should put in place strong institutional framework to curtail, if not totally eliminate, the excesses of round-trippers.

Besides, the non-linear independence of the Nigerian foreign exchange market is an indicator of nervous market conditions and the discrimination of investors as well as market participants to different market conditions. It indicates that investors would behave differently to different market circumstances. Intuitively, a period of boom would create a pseudo immunity posture that would suggest that the foreign exchange market is completely immune from the vagaries of global exchange rate behavior. However, a downswing in market situation may result to panic withdrawal of 'hot money' and would expose the potential danger facing the Nigerian foreign exchange market. This portends grave consequences for policy direction. While the behavior of the investors appears rational, the government should ensure that the market is free of distortions that are capable of undermining its level of efficiency. Policy choice in this direction includes reduction in the exchange rate differentials between the parallel market and the interbank rates. This can be achieved through disincentivizing activities that promote arbitraging and speculative tendencies.

This study tests for MDH in two prominent FX markets in Africa namely Nigeria and South Africa. The return series of the Nigerian naira and the South African rand relative to the US dollar, euro and pound sterling are used to test for the efficiency of the two FX markets. As a preliminary procedure, relevant statistical properties of the return series are provided and discussed with economic and statistical implications. The study also determines structural breaks endogenously from the data stream using Perron's ([35]) unit root test with structural break. Furthermore, it employs recent techniques of the variance ratio and spectrum based tests which involve wild bootstrapping procedure. Thus, wild bootstrap automatic variance ratio (AVR) test by Kim ([23]) and Generalized Spectral (GS) test by Escanciano and Velasco ([19]) are employed, particularly as both are confirmed to be consistent against autoregressive conditional heteroscedasticity (ARCH) in the financial returns. The tests have also been verified to have high power and size advantage in determining linear and nonlinear dependence in the conditional mean of financial returns respectively.

Empirical results from both the GS and AVR tests show that the two FX markets prominently follow the MDH before the structural break. However, after the structural break, the results differ markedly between the two tests. While the GS test indicates that the African foreign exchanges are martingale difference, the estimates of the AVR test support the non-rejection of MDH. Meanwhile, the full sample case substantially validates the MDH but not without lack of consensus between the two tests. This is evident in the naira–pound and naira–dollar exchanges with no martingale process for the AVR test. In addition, the rand is both linearly and non-linearly independent while the naira seems more non-linearly independent.

Overall, three striking findings are discernible from our analyses. First, on average, the South African FX market appears to be more efficient than the Nigerian FX market. Thus, the latter may be more susceptible to speculations than the former. The implication of this finding is that the South African rand does quickly incorporate available information from the movements in these internationally referenced currencies of dollar, pound sterling and euro, both directly and indirectly, while the same cannot be said of the Nigerian naira which only directly incorporates available information from the behavior in these referenced currencies but not otherwise. This further lends credence to the submission that the rand is quite efficient than the naira. This suggests that international portfolio investors/FX traders/FX speculators would prefer the Nigerian economy to that of the South African as a safe haven for their stakes since abnormal returns can be realized from arbitrage activities and speculations. However, investors in foreign direct investment would prefer the South African economy to that of Nigeria as the economic activities of the former are less likely to be susceptible to distortions compared to the latter since the market is found to efficiently react to existing market information. This finding portends a grave danger for the Nigerian economy as its investment portfolio will be characterized by 'hot money' which can be easily and quickly withdrawn in the face of market and economic uncertainties. Again, the Nigerian economy will also always experience the case of capital flight; especially in this globalized world where barriers to capital account liberalization are to be removed in order to promote trade and finance among economies of the world. Nonetheless, effective capital controls geared towards discouraging capital flight should be pursued by the Nigerian government.

Second, ignoring significant structural breaks may render inferences invalid. This indicates that without accounting for structural breaks in MDH, institutional investors and the investing public may be wrongly advised as to the efficiency or otherwise of foreign exchange markets in Africa. Third, the choice of methodology does matter when testing for MDH. This suggests that researchers in the areas of international finance would have to address the peculiarity of each data with the right methodology. In essence, a more systematic procedure beyond the basic modelling framework should be considered to understand the peculiar features of the financial market being analyzed.