REMODELLING MACROECONOMIC TRILEMMA AND CENTRAL BANK BEHAVIOUR IN NIGERIA: A MARKOV-SWITCHING APPROACH

This study investigates macroeconomic trilemma and Central Bank behavior in Nigeria. The period of investigation spans the quarterly period of 1981 – 2017. Upon the data stability condition of Zivot-Andrew unit-root test with structural breaks, the Markov Switching Dynamic Regression was employed as the technique of analysis. With a validated trilemma hypothesis, the study found that the trilemma constraints hold for the Nigerian economy but at the expense of the autonomy of the monetary authority. Being the policy variable of the Central Bank of Nigeria, the exchange rate was found to follow two regimes of fixed and managed-float regimes. The results also showed that political risk was found insensitive to the regimes of exchange rate while the foreign sector was considered as a moderating factor for the behavior of the monetary authority; irrespective of the exchange rate regime.


Introduction
The idea of the classical (macroeconomic) trilemma is well rooted in theoretical exposition. It came with the realization that there are sets of macroeconomic objectives that requires trade-off for proper economic management. As a result, the interest rate has been identified as the single most important policy variable for the attainment of internal balance in the economy (see Fisher, 1930;McKinnon, 1973;Shaw, 1973;Phelps, 1967;Muth, 1961;Wicksell, 1936). However, the interest rate parity condition suggests that how independently the central bank would control for the domestic interest rate to achieve this internal balance is restricted by the behavior of the foreign interest rate in the base economy (see Masih, 2005). This has further implications on the economy's balance of payment position and the value of her domestic currency. Consequently, there would be exchange rate instability if the monetary authority independently determines the domestic interest rate and allows free flow of financial capital (Dooley, Folkert-Landau and Garber, 2013).
In order to stabilize the domestic currency and independently determine the domestic interest rate, the monetary authority would, therefore, have to embark on capital repression policy (see Gan, 2014;Kim and You, 2016). On the other hand, an economy that seeks to pursue financial openness policy alongside the stability of the exchange rate should be prepared to trade off the autonomy of its monetary policy. The monetary policy, then, becomes endogenously determined by the movement in the exchange rate (see Goh and McNown, 2015;Georgiadis and Mehl, 2015). It is this difficulty to simultaneously achieve these three policies of monetary autonomy, exchange rate stability and financial openness; as propounded by the duo of Mundell (1963) and Fleming (1962) that is known as the macroeconomic trilemma.
In Nigeria, available data show that the Nigerian economy has been characterized by increasing interest rate regimes for the period 1970 -2016. In 1970, the prime lending rate was 7.0 percent and this rate was consistently maintained till 1974. Up till 1981, a single digit interest rate of 7.75 was maintained but since 1982, the economy has been on double digit interest rate hovering around 20.0 to 30.0 percent. At times where the domestic interest rate has exceeded the foreign interest rate, the outcomes have been increased net financial flows. This happened in the periods 1970 -1974 and 2011 -2015 with corresponding average net capital flows of N66.96Million and N11,946.175Million respectively. To complete the insights for macroeconomic trilemma, the domestic currency depreciated to N2.026 in 1986 and exchanged for N193.28 by the end of 2015. It is expected that the accumulation of external reserves should afford the economy to maintain a middle-ground position of the trilemma constraint (Aizenman and Ito, 2012). A period of foreign reserve accumulation occurred in Nigeria only between 2003 and 2008. The foreign reserve increased correspondingly from US$10Million to US$50Million.
Also, it is believed that the central bank faces both ex-ante and ex-post constraints to its independence. The former, which is the ex-ante constraint, is such that the central bank board is usually appointed by the political authorities. Being appointed by the political officeholders suggests that the independence of the monetary authority can be compromised. In fact, Smaghi (2007) posited that legal provisions are only necessary but not sufficient conditions to ensuring the independence of central bank. This position is of greater concern for developing economies where it is said that poor institution still largely exists. As such, the autonomy of its monetary policy strategies would be less circumspect. It implies that the functional independence of the central bank cannot be ascertained if the personal independence cannot be guaranteed; especially when monetary policies are used to achieve fiscal policy goals (Golomejic, 2011). The latter; which is the ex-post constraint, is such that the central bank also plays the role of government's bank. With this, the central bank can be influenced to make some unpopular decisions to protect the interest of the government of the day, particularly to remain in office and even possibly embezzle public funds when providing interventions in certain capital projects or some special needs with the claim of fulfilling its avowed role of economic management. Also, the monetary authority can be compelled to embark on lowering the interest rate or even charge negative interest rate so as to obtain cheap or costless capital purely for political motives in the economy. In this vein, monetary policy becomes instrument of obtaining fiscal objectives either directly through imposition of inflation tax or indirectly, through a discriminate but favourable interest rate to government borrowings (Dooley and Isard, 1980;Bartolini and Drazen, 1997). It is this latter position that is known as the self-serving government view of financial repression enunciated in the writings of Alesina and Roubini (1992) and Nordhaus (1975).
Beckmann, Ademmer, Belke and Schweickert (2017) put it succinctly when they classified these ex-ante and ex-post constraints as the opportunistic business cycle and partisan business cycle theories, respectively. The authors, however, attributed developed economies, being the focus of their investigation, with partisan preferences. This was conditioned on the fact that these economies have strong institutions to dissuade opportunistic business cycles (Park, Buntaine and Buch, 2017). For a developing economy such as Nigeria, however, both the partisan preferences and the opportunistic business cycle views matter for the autonomy of the monetary authority.
These two forms of political business cycles, which are constituted into issues of institution, can delimit the potency of its monetary policy strategies. Even though the Central Bank of Nigeria Act of 2007 seeks to set the monetary authority in Nigeria to be truly autonomous, the fact that the central bank governor and its board are being appointed by the executive indicates that it cannot be truly independent. It is the collection of these issues that makes it imperative that this study investigate the role of institution in the behaviour exhibited by the Central Bank of Nigeria when confronted with macroeconomic trilemma. In addition to this introductory section, Section 2.0 reviews extant literature while Section 3.0 sets forth the methodological framework for investigating macroeconomic trilemma and the behaviour of Central Bank of Nigeria. In Section 4.0, estimates for the specified models were obtained and findings were discussed while Section 5.0, being the last, provides conclusion and proffer policy suggestions.

Literature Review
The theoretical linkage among financial openness, exchange rate and monetary policies is anchored on the Mundell-Fleming framework and the Compensation thesis. The compensation thesis is predicated on an accounting framework of double entry principle. It illustrates sterilization effects by an automatic mechanism within a stock-flow accounting analysis. As a post-keynesian theory, the compensation thesis is anchored on the assumption that the supply of money is endogenous, demand-led and has a horizontal curve. There is a possibility of multiple natural rates of interest since the short-term interest rates are exogenous and set by the central bank. Within the compensation thesis, causality moves from credit to money aggregates to reserves, Inflation and output growth. The theory suggests that all sectors of the economy need buffers as adjustment factors.
On the contrary, the standard Keynesian view, which is enshrined in the Mundell-Fleming framework, suggests that the supply of money is exogenous and can be viewed as a vertical curve. Rates of interest are endogenous. In this case, there exists a natural rate of interest and causality runs from reserves or high-powered money to money and then to credit and excess money growth causes price inflation. The compensation thesis is an extension of the reflux principle evidenced in the earlier writings of Nurkse (1947), Kaldor (1980)  As a way of general overview, the empirical literature falls under three major strands. The first strand relates to those studies that tests for the validity of the trilemma hypothesis, the second strand examines its impacts on macroeconomic fundamentals while the third strand debated the adjustments in the methods and methodologies. The first and second categories relate to those studies that test for the trilemma hypothesis and further investigate for its impacts on macroeconomic fundamentals either as country-specific studies or as cross-country studies.
Examples of empirical works in this strand include the studies of Cheng and Qian (2009) The major results of these studies are that, with the accumulation of international reserves, emerging market economies are adopting policy choices that are converging to a 'middleground' position of the trilemma constraint since the past two decades. For industrialized developed economies; however, there has been more divergence of the triad policies with a combination of high exchange rate stability and financial openness being most prominent. This usually occurs at the expense of monetary independence; largely due to regional integrations among countries within the European Union. The common conclusion reached is that the trilemma hypothesis is still binding. More so, the convergence of the triad policies to a 'middleground' position was found to be a pre-condition for financial stability, but not for output stabilization, in developing as well as emerging economies. But generally, to stabilize the real sector, convergence policy must be complemented with high international reserve holdings. A condition not necessary for developed industrialized economies. More so, the interest rate parity condition between countries has been employed in various studies 4 . For the extent of financial integration, the popular Feldstein and Horioka (1980) remains largely prominent while later writers have strengthened the literature with additional measures 5 .
Also, the measure of monetary independence has not been without its criticisms 6 . It is the critique of these foundational measures that informed that other new measures as well as gauges were suggested in the empirical literature for testing the validity or otherwise of the trilemma hypothesis. Schularik and Ward (2015) suggested a wavelet analysis where oscillations and cyclical properties of financial variables were accounted for. The result obtained was found consistent with other alternative measures; including the trilemma index. Popper, Mandilaras and Bird (2013) and Mandilaras (2015) introduced new gauge of stability which was bounded and correspondingly non-Gaussian. The study found that the combination of fixed exchange rates and financial market openness is the most suitable arrangements for either low or high-income countries. This is a possibility for emerging economies, only under a regional financial arrangement as obtained by Aizenman and Ito (2012)

Theoretical Framework and Models Specifications
A major contribution of this study is to modify and extend the basic Mundell-Fleming framework in such a way that both macroeconomic trilemma and financial trilemma would be nested together; both being the double trilemma facing an economy, and concomitantly, trace how it alters the behaviour of the central bank. Practically, the modification is that, within the framework of the classical (macroeconomic) trilemma, the assumption of perfect asset substitutions must be relaxed. Consequently, the condition of uncovered interest parity, which presupposes risk-neutral investors, is modified to include a risk premium. These modifications become imperative since there are imperfect asset substitutions in developing economy such as Nigeria. In fact, investors are usually risk-averse in developing economies due to innumerable uncertainties. More so, the uncovered interest parity (UIP) condition cannot be satisfied without the inclusion of a risk premium since empirical studies have not established strong evidence for the validity of the UIP condition; even in the long-run situation (Chinn and Meredith, 2004). An extension of the Mundell-Fleming framework is to account for the political economy of central bank behaviour in a developing economy such as Nigeria where the role of institution cannot be under-estimated.
The basic Mundell-Fleming framework is anchored on five (5) major assumptions of risk-free arbitrage condition; imperfect competitive firms with constant domestic price level and elastic supply of domestic output; imperfect capital mobility; perfect asset substitutions and small country assumption that is not capable of influencing the global dynamics of prices and income.
As presumed, an evaluation of these assumptions validates that there is imperfect capital mobility in Nigeria. Capital is imperfectly mobile in a developing economy such as Nigeria since cross border capital transactions are usually sequenced and movement of financial capital is often gradual (Bankole and Ayinde, 2014). Importantly, the Mundell-Fleming model contributes two main theoretical elements to the autarky framework of IS-LM. Firstly, there exists an extra output demand element to capture economic interaction with other economies of the world. This is the net export component of the real sector equilibrium. Secondly, there is uncovered interest parity (UIP) condition that is predicated on risk-neutral behaviour of investors. These two conditions are weighty.
The first contribution indicates that the balance of payment (BoP) equation would be included in obtaining the simultaneous equations that comprise set of endogenous and exogenous variables, thus, the name IS-LM-BP framework. The second condition suggests there is risk-free arbitrage where the ratio of the forward to the spot exchange rate will equate the interest rate differential between the two assets; measured in domestic currency. This is given as; i is the corresponding return on the foreign assets. Equation (1) is the risk-free arbitrage condition. To the extent in which the investors are riskaverse, the forward rate differs from the spot rate at a risk premium given as  . This is to compensate for the perceived riskiness of holding the domestic assets in terms of the foreign assets. The risk premium is defined as; ,, Substituting equation (2) into (1) gives; From equation (4), the uncovered interest parity (UIP) condition holds when the risk premium This implies that the expected change in the exchange rate from period t to period t+k equals the current interest rate differential. As such, the investors become risk-neutral between holding domestic assets or foreign assets. Including a rational expectation hypothesis as the market value of expected exchange rate is not observable. Then, we have; Therefore, substituting equation (5) into (4) gives; Under the risk-neutral condition of the investors, the last two terms are assumed orthogonal to the interest rate differential.
Relaxing the assumption of risk-neutral investors, however, we have an empirical model for uncovered interest parity condition with risk premium given as; Where; a non-zero o  captures the risk premium component of the UIP condition.
This study is, therefore, anchored on the augmented Mundell-Fleming model. This model has four building blocks of goods market equilibrium, money market equilibrium, balance of payment equilibrium and the uncovered interest parity (UIP) condition. Equation (7) is the UIP condition with risk premium component and the goods market equilibrium is made up of: Where; C is the consumption, I is the level of investment, G is the government expenditure; which is exogenously determined, NX is the net export. The Mundell-Fleming framework presupposes that consumption is determined by the disposable income () YT  , investment is determined by the interest rate ( r ) and the net export is a function of income () Y and the real exchange rate () e . The () e measures the level of competitiveness given as Where; E is the nominal exchange rate and * P P is the ratio of foreign price level to the domestic price level.
The third equilibrium is the money market equilibrium given as; (10) Equation (10) suggests that the demand for money is determined by both the interest rate and aggregate income in the economy.

BP
CA KA ………………………………………………………………………………… (13) Where; CA is the current account component while KA is the capital account component. Given that the current account captures the trade balance () NX and that the capital account being largely determined by a shift parameter ( k ) and the interest rate differential of the returns on domestic asset and foreign asset; equation (13) becomes; (14) Equations (8), (12), (14) and (7) are four systems of equations that include the real sector (goods market) equilibrium, money market equilibrium, external sector equilibrium and the UIP condition respectively. These give a modified Mundell-Fleming model thus;

……………………………………………………….(15)
A solution to the simultaneous equations above would partition the variables into a set of exogenous variables Alternatively, equation (15) is re-arranged such that the domestic interest rate becomes the subject of the formular for a single regression equation. This yields a behavioural model for this study, given thus; Equation (16) is the economic model for this study and it suggests that the domestic interest rate is a function of income, real money balances, net export and some control factors such as level of financial development.
Recall from equation (12) With perfect foresight assumed, the expected change in exchange rate is taken as the actual change. Therefore, equation (20) becomes; ……………………………………………………………………… (21) Equation (21) is the functional model that relates to how the central bank behaves in the face of macroeconomic trilemma and the behavioural variable for the monetary authority is the changes in the exchange rate. It is important to note that money supply would no longer be exogenous but rather becomes endogenous in nature; especially for a developing economy such as Nigeria where the monetary authority seek to stabilize the exchange rate. As a result, the central bank sells domestic currency to buy foreign currency in order to avoid an appreciation and buys domestic currency to sell foreign currency in order to avoid depreciation.

Framework for Markov Switching Dynamic Models
Considering an unobserved states or regimes which is said to follow a Markov chain process (Quandt, 1972;Goldfeld and Quandt, 1973;Mills and Wang, 2006;Guidolin, 2011a;2011b), the evolution of exchange rate can be modeled as a state-dependent intercept term for k states;  - (Hamilton, 1989;1990). The probability distribution of the Markov Switching model is expected to follow a logistic distribution (Appendix 1; Hamilton, 1994;Chen and Shen, 2007 for modeling frameworks) With a reflection of the dynamic interactions among the variables, the lags of exchange rates are also introduced as explanatory variables in equation (32)

Descriptive Statistics and Data Stability Conditions
In order to obtain the statistical properties for the variables included for models specifications, the descriptive statistics among these variables were found appropriate. As detailed in Also, the standard deviation value of 3.90 for political risk factor in Nigeria shows that the political risk in the country does not deviate substantially from the expected value. Both the kurtosis value of 2.69 and Jarque-bera statistics of 10.21 strongly lend credence to the normal distribution of political risk in the country for the period under review.
On the other hand, the growth of real gross domestic product (denoted as rgdp ) is averagely 1.21 but with -1.31 skewness value. This shows that economic growth in Nigeria has averagely been on a downswing and has skewed negatively from the expected value by 1.31. This is an indication that economic growth in Nigeria has been grossly non-performing for the period under consideration. In addition, growths of money supply ( 2 gm ) also has similar behaviour with 5.71 mean values. This indicates that price in Nigeria have been appreciably controlled as the average price index; caused by monetary growth, has not reached double digit

Source: Authors with Data Obtained from the Central Bank of Nigeria (CBN) Statistical Bulletin (2018)
In terms of data stability conditions, the test statistics for the unit-root and stationarity tests detailed in Table 2 suggests that some of the variables are unit-root as well as non-stationary at levels while some other variables are non-unit-root as well as stationary at levels. Particularly, the conditioning variable or the variable of interestthe change in the exchange rate ( e  )is non-unit-root and stationary at levels. The use of the Augmented Dickey Fuller (ADF) and the Phillip-Perron (PP) confirm the non-unit-root of the change in the exchange rate while the Kwiatkwoski-Phillips-Schmidt-Shin (KPSS) confirm its stationarity. This is so as the variable ( e  ) has test statistics of -10.923 and -10.928 that are greater than their values at the critical level; given as -3.478 for both, at the 1 percent level. As such the null hypothesis of unit-root is rejected; even at the 1 percent level of significance.
Other variables that are also stationary at the 1 percent level are growth of money supply ( 2 gm ) and the rate of inflation ( inf r

Macroeconomic Trilemma and Central Bank Behaviour in Nigeria
The behaviour of the central bank in the face of macroeconomic trilemma can be examined on the bases of how the monetary authority employs the use of monetary policy instruments to achieve an overall health of the economy with trade off of macroeconomic objectives; including economic growth (through increased investment) and balance of payment equilibrium (through interest rate differential and exchange rate stability). Also, the attainments of these macroeconomic fundamentals are to be examined in order to determine how they will alter the stability of monetary policy variables such as the interest rate and the exchange rate. The ex-ante behaviour relates to how the central bank uses the monetary policy instruments to achieve macroeconomic objectives while the ex-post behaviour indicates how the extent of attaining these macroeconomic objectives affect the stability of interest rate and exchange rate.
Specifically, the ex-post behaviour is usually considered as the reaction function of the central bank (Siri, 2012). It is the combination of both the ex-ante behaviour and ex-post behaviour that summed up the behaviour of central bank in an economy.
Usually, the changes in the rate of interest in the economy are affected by the level of money supply. Basically, the level of money supply provides the direction for the rate of interest. Figure   1 implies that money supply has increased steadily since 1996:Q1 with negligible decline in the first quarters of 2009, 2014 and 2017. Expectedly, if these increases in money supply have been absorbed into the economy, the level of interest rate should decline and further stimulate investment opportunities. However, the rate of interest appeared not particularly affected by the changes in the supply of money (see Figure 2). During these periods, the reserve requirements of the deposit money banks in Nigeria remain stable until 2011Q1. While the increase in the reserve requirements is considered a contractionary monetary policy, the money supply still increased steadily. The implication is that other counteracting expansionary policy would have neutralized that effect (see Figure 1). A further implication of the foregoing is that even though the money supply is expected to be an endogenous variable in an economy such as Nigeria which operates a managed-float foreign exchange rate regime and sometimes fixed peg regime, money supply has been largely exogenously determined by the monetary authority. More so, the interest rate in Nigeria did not always moves with that of the foreign interest, indicated by the US interest rate (see Figure 2). By the simple interest rate parity condition and given a capital mobility condition, the domestic interest rate is expected to be at par with that of a referenced foreign interest rate.
For the case of Nigeria, however, it is evident that this is not always the case. This further lends credence to the independence of monetary policy in Nigeria.
Nonetheless, the exogenous determination of money supply by the central bank of Nigeria has been carried out at discretion rather than sticking to rules. This is obvious from the graphical trend of both the targeted and actual money supply depicted in Figure 3. Except for the periods 1996 -1997, 2004, 2009, 2011 and 2018; the actual money supplied into the economy deviated from the targeted amount by the monetary authority throughout the periods 1985 -2018. For the period 1985 -1990, there was no target earmarked by the central bank of Nigeria for its monetary aggregates. While the targeted money supply was relatively stable between 1996 and 2005, that actual money supplied is cyclical throughout. In whole, this suggests that the behaviour of the central bank towards money supply in Nigeria is discretional rather than being at a rule (see Figure 3). More importantly, a variable that can affect the level of broad money supply when endogenously determined is the movement in the level of exchange rate. For the periods 1981 -1985 and 1993 -1998, the rate at which the domestic currency (the Naira in this case) exchanges for a unit of foreign currency (the US$ in this case) were relatively stable; this occurs for same the reason of fixed exchange rate regime but at different dimensions. Nigeria's exportable oil in the international foreign market which snowballed into depreciation of her domestic currency (see Figure 5). By 2017, the foreign reserve has started building up again. It was because the free fall of the naira has to be prevented that the monetary authority subscribed to fixed exchange rate regime as against subjecting the currency to the dictates of the market forces.

Policy Variables of Central Bank Behaviour in Nigeria
Consequent upon the non-parity condition of the domestic interest rate in Nigeria to its foreign counterpart (see Figure 1), it becomes expedient to acknowledge that the exchange rate, therefore, becomes the policy variable of the monetary authority in Nigeria. This is because the exchange rate alters the direction of money supply in the economy and the supply of money, then, impacts on the domestic interest rate. This underscores the reason why the interest rate cannot be the policy variable. The interest rate is endogenously determined by the exchange rate.
Again, the fact that the assumption of perfect capital mobility does not hold for a developing import-dependent economy such as Nigeria suggests that interest rate would be endogenously determined rather than being the variable for strategic monetary policy in the country. This is importantly so in that the monetary policy must review its interest rate from time to time to align the economy to the dynamics in the global marketplace.

Political Risks and Central Bank Behaviour in Nigeria.
In order to capture the political economy of central bank behaviour in Nigeria, this study considers the political risk factor as the most appropriate measure of institutional factor in the country. This is because it is believed that the central bank; being the government's bank, can do the biddings of public office holders and can be influenced to behave differently from what the economic situation demands; mainly for political reasonslike winning elections, perpetuating themselves in office, stepping down unpopular decisions due to ethnic, religious and cultural pressures and, even, pressures from opposition parties. The exegeses of these political factors tend to alter the behaviour of the central bank in a developing economy such as Nigeria. It is from the foregoing that the behaviour of the central bank in Nigeria is likely to be affected by political risks; even in the face of the macroeconomic trilemma.
Stemming from the foregoing, a cursory overview of political risks confronting Nigeria for the period (1981Q1 -2015Q4) under review is highly imperative in this sub-section of the study.
More so, how these political risks have influenced the direction of monetary policies in the country will be highlighted. As a decision rule, it is expected that the higher the sum total of the weight, the lower the risk and vice versa. For assessment purposes, a weighted sum less than 50 percent is considered very high risk; between 50 -60 percent is high risk; 60 -70 percent is moderate risk; 70 -80 percent is low risk while 80 -100 percent is very low risk. The trend in Figure 25 borders on political risks for the country and the behaviour of monetary policies during this period. These monetary policies are growth of money supply ( 2 gm ), the rate of interest ( int r ) and the growth ratio of credit to the private sector to the GDP ( __ cps gdp grwt )see between 50 -60 percent benchmark (see Figure 6). This implies that there were even more periods of very high political risks in the country. Curiously, the interest rate is highest during the period 1992Q2 -1994Q4. This implies that investment will be stifled, and growth will be impeded. The implication for macroeconomic trilemma is that there would be high influx of capital transactions into the country due to portfolio adjustment of the global investors. If this influx of capital transaction exceeds the productive capacity of the economy at that point in time, there would be financial instability.
Consequently, the growth ratio of credit to the private sector to GDP is highest during this period. This suggests that credit expansion is highest during this period. It is an indication that there is high financial instability in this period in Nigeria than any other period under study.
Another period of relatively high interest rate in the country is 1991Q1 -2002Q2. It is observed that there is no financial instability during this period as the growth ratio of credit to the private sector to the GDP is almost lowest during this period. This suggests that the productive capacity of the economy has been able to absorb the influx of financial flows that was necessitated by the increase in interest rate. The growth of money supply is highest between 2007Q3 -2008Q3. This suggests that the monetary authority embarked on expansionary monetary policy during the period of very high political risks. This is also evident during the earlier period of very high political risk (1981Q1 -1983Q2).

Data Stability Condition
Data stability condition is to ascertain the stationarity or otherwise of the variables included in the model. It also provides an insight into the estimation procedure employed for analyses. In order to avoid misleading results, the data stability test conducted for this study is the Zivot-Andrew (2002) unit-root with structural breaks. This is essential to examine the presence of break-points in the variables as a series considered to be non-unit-root but with structural breaks would lead to wrong choice of method of analyses. As detailed in Table 1, it is evident that the variables for estimations of macroeconomic trilemma and central bank behaviour in Nigeria have different break periods and the levels of stationarity also differ. Only the change in the exchange rate, growth of the economy and output gap that are non-unit-root at levels, other variables such as the political risk, interest rate, inflation rate, net export and growth of money supply that are all unit-root at levels. This suggests that the use of deterministic models would not be appropriate for this study. This further justifies the use of the Markov Switching models which can capture different break-point period within a structural analysis in a modeling framework such as this.

The Trilemma Constraint and Policy Trade-Off in Nigeria
As earlier posited, the starting point for a study on investigating the trilemma hypothesis is to ascertain if the trilemma constraint holds and identify the binary choice variables of the triad policy that is optimal for the economy. One decision rule to validate the existence of the trilemma hypothesis and that the trilemma constraint holds is that the estimates as well as the coefficients for these triad policies; when regressed against a constant, should all be positive and must have high goodness-of-fit. Also, the trade-off to determine the binary policy choice, when regressed against one another, must be negatively signed. As detailed in Table 2, it is evident that all the coefficients for the triad policies are positively related to the constant and highly statistically significant; at least, at the 5 percent level.
Also, the regression estimates to determine the appropriate binary combinations show that, on the one hand, trade-off significantly exists between monetary independence and financial openness (Panel D,  Sengupta (2013, 2014). This implies that monetary policy becomes an endogenous variable in Nigeria as the central bank has to ensure it sells domestic currency to buy foreign currency in order to avoid the pressure for currency appreciation during the period under consideration. On the other hand, the monetary authority buys domestic currency in order to sell foreign currency so as to avoid the pressure for currency depreciation. This is to ensure exchange rate stability in the country.
The statistically significant trade-off between monetary independence and financial openness in Nigeria is highly instructive. The insight from this is that for the monetary authority in Nigeria to independently determine interest rate in the economy, it must significantly control the flow of capital transactions in and out of the economy. The implication is that sudden withdrawal of short-term capital; known as hot-money, must be avoided and that surge in the quantum of longterm capital must be in tandem with the developmental pace of the economy. In fact, liberalization on capital account transactions must be gradual and sequenced. Otherwise, the injection of additional and withdrawal of excess liquidity by the monetary authority will becomes less effective in driving the growth of the economy. there is a dichotomy to both the inclusion of political risks factor and without. It is evident that under managed-float regime, all explanatory variables for macroeconomic trilemma do not differ in terms of size and significant. The only exception is the growth process of the Nigerian economy (indicated as rgdp) which, with political risk factor, is significantly negatively related to exchange rate behaviour at the 5 percent level with -0.170 coefficient. Specifically, the political risk is inversely related to the exchange rate movement in Nigeria with -0.159 coefficients with significantly T-statistics value at the 5 percent level with 0.017 probability values. The implication is that an increase in the value of political risk (which simply denotes a reduced political risk) would results in a fall in exchange rate (that is, an appreciation).

Estimation of Regime Switching Model with Macroeconomic Trilemma
Therefore, the results show that a lower political risk will engender an exchange rate appreciation in the Nigerian economy.
This is also the case under a fixed exchange rate regime with -4.182 coefficient and 0.000 probability value. Interestingly, though, when the economy is faced with macroeconomic trilemma and the exchange rate regime is fixed, then, the growth of the economy have significant impacts on the exchange rate movement with -127.964. This suggests that economic growth enhances exchange rate appreciation. Also, the growth of money supply has -0.967 coefficients; albeit insignificantly at the 5 percent level. A possible intuition for macroeconomic trilemma here is that with fixed exchange rate regime, increase in monetary growth could create a pressure for currency appreciation. Both the foreign interest rate and net export are positively signed with 9.875 and 0.0003 coefficients respectively. These estimates are significant at the 5 percent level too. The implication is that an increase in the foreign interest rate, which translates to reduced financial openness; consequent upon portfolio adjustment, would lead to pressure for foreign exchange rate devaluation. The positive coefficient of 0.0003 for the net export would significantly create a pressure for exchange rate devaluation under a fixed exchange rate regime; either with or without political risks. This is unlike the case of managed-float regime where the increase in the net export and foreign rate of interest creates pressure for exchange rate appreciation; albeit insignificantly. This indicates that there is a 96.1 percent that the exchange rate regime will be of managed-float in the next quarter if it is currently operating at the managed-float in the current quarter. In fact, the duration of persistence is approximately 26 quarters under the managed-float regime and 1.44 quarters during the fixed exchange rate regime. This is equivalent to four and a half (4 1 /2) years and one (1) quarter of persistence respectively. The cross probabilities of transition suggest that it is much easier to transit from a fixed exchange rate regime to a managed-float regime by 69.4 percent while it is near impossible to transit from managed-float to fixed exchange rate regime by a paltry 3.9 percent chance (Panel B, Table 4). However, when political risk is considered, the probabilities of persistence dropped to 93.9 percent and 20.8 percent respectively while the cross probabilities of persistence increased to 6.1 percent from a managed-float to a fixed exchange rate regime and 79.2 percent from a fixed exchange rate regime to a managed-float regime. The conclusion here is that political risk speeds up the transition of regime switching but reduced the degree of persistence of the exchange rate regimes.

Conclusion and Recommendation
The study showed that the trilemma constraints hold for the Nigerian economy as the trilemma hypothesis was validated. The appropriate binary choice variables to counteract the effect of the trilemma hypothesis are the combination of both exchange rate stability and financial openness.
This implies that the monetary authority in Nigeria cannot crave for autonomy in the face of macroeconomic trilemma. With this, it implies that money supply becomes an endogenous monetary policy in Nigeria as it would be used as the variable of adjustment to stabilize the exchange rate in the economy. The implication from this is that to stabilize exchange rate in Nigeria, the monetary authority has to continually intervene through the use of money supply. .
There are two regimes of exchange rate in Nigeria and these are identified as the fixed exchange rate and managed-float regimes. With the exchange rate being the policy variable of the monetary authority, political risk was found insensitive to the regimes of exchange rate.
In conclusion, political risk faced by the economy is less important in determining the behaviour of the monetary authority in the face of macroeconomic trilemma than even the binary choices of the triad policies made. However, foreign interest rate and net export were sensitive to the type of exchange rate regime. It leads to exchange rate appreciation, and significantly so, under a managed-float regime but results in exchange rate depreciation given a fixed exchange rate regime. Stemming from this conclusion, the followings are recommended. First, the endogenous determination of the money supply indicates that there would be economic prosperity when there is effective exchange rate management and barriers on cross-border financial capital flows are sequenced. Also, political institutions should be put in check such that impulses from these institutions would not affect the dynamics of macroeconomic trilemma and, consequently, the behavior of the Central Bank of Nigeria. In order to counteract the negative effects of foreign interest rate and net exports, government should create enabling environment to reduce the risk premium on investments and stimulate aggregate demand in the economy.

*Availability of data and material
The data for this study are available on request.

*Competing interests
It is instructive to note that there is no competing interesting for this study. The two authors with corresponding email addresses provided here are the contributed to the completion of this manuscript.

*Funding
The African Economic Research Consortium (AERC) is greatly appreciated for creating the