Introduction

Several empirical studies have investigated the determinants and stability of the income velocity of money. This is because understanding its causes and consequences can usefully inform the setting of monetary policy (see, for example, Camarero et al., 2021; Boháčik, 2022; McPhail, 1991; Caporale & Gil-Alana, 2005; Maki & Kitasaka, 2006; and Haug, 2006; Sharma & Syarifuddin, 2019; Ujunwa et al., 2022; and Oyadeyi, 2022a; Oyadeyi, 2023a). When the velocity of money is stable, it supports the monetary targeting framework. A stable income velocity of money function helps in the prediction of the influence of changes in the money supply on income and prices (Akinlo, 2012; Oyadeyi & Akinbobola, 2022; Singh, 2023; Sud, 2023). However, an unstable income velocity of money demands the use of an inflation-targeting strategy to help stabilize prices and preserve income growth.

Before the collapse of the Bretton Woods System, narrow money (M1) was considered stable in industrialized economies. However, since 1974, the conventional M1 money demand function began to overpredict the demand for money, which Goldfeld (1973) termed the case of “missing money.” The misery of the conventional money demand function increased in the 1980s as it under-predicted the velocity of money, which rose faster than expected. Implementing financial reforms in many developed countries raised doubts about using monetary aggregates to stabilize inflation rates (Kumar et al., 2010; Mohamed, 2019; Nampewo & Opolot, 2016; Oyadeyi, 2023b; Oyadeyi & Akinbobola, 2020; Singh, 2023). Suddenly, some industrial economies in the late 1980s and many developing countries afterward changed to the inflation-targeting strategy as their main instrument of monetary policy, claiming that the liberalization of their financial markets was responsible for the unstable nature of the velocity of money in their economies (Ardakani, 2023; Benati, 2020; Caporale & Gil-Alana, 2005; and Rao & Kumar, 2009; Oyadeyi, 2023c). This situation has led to several debates on the velocity of money scenario for Nigeria.

Several studies in Nigeria on the income velocity of money have empirically investigated its stability and determinants (see Nwaobi, 2002; Akinlo, 2012; Ujunwa, 2022; Oyadeyi, 2024). However, this study would extend those studies by incorporating structural breaks in the formulation of money velocity. The consensus in the literature is that incorporating structural breaks in modeling money demand is methodologically imperative (Castañeda & Cendejas, 2023; Mele & Stefanski, 2019; Omotor, 2011; Doguwa et al., 2014; Sanusi & Meyer, 2018; Oyadeyi, 2022b; Oyadeyi, 2023d). Sources of structural breaks could be due to financial reforms, innovations, or structural changes that may have taken place in the country. The reason for incorporating structural breaks is that some sudden events may have affected Nigeria’s money velocity function. Furthermore, the inflationary pressure in recent years has renewed the debate on the topic. If the velocity of money is stable, the central bank can predict the amount of money in circulation and there will be no inflationary pressure. Otherwise, it means that the central bank may not be able to accurately predict the amount of money in circulation.

As a result of the above, this paper will incorporate structural breaks within the methodological framework on the velocity of money in Nigeria. The study will also cover an extended period, using quarterly data from Q1 1981 to Q4 2023. This period extends every previous study on the Nigerian economy, spanning 172 observations. Furthermore, previous studies have limited the velocity of money function in Nigeria to include narrow (M1) and broad money (M2). This study will incorporate the narrowest version of money (M0) and broadest definitions of money (M3) into the existing definitions (M1 and M2) while accounting for its determinants and stability. Examining the determinants of four different velocity of money functions—velocity of reserve money, velocity of narrow money, velocity of broad money, and velocity of total money—offers a comprehensive understanding of the dynamics of money circulation within an economy.

For instance, the velocity of reserve money focuses on the base money, which includes currency in circulation and reserves held by banks. Understanding this helps gauge the effectiveness of central bank policies and the liquidity provided to the banking system. Furthermore, the velocity of narrow money includes currency in circulation and demand deposits. Analyzing this provides insights into the immediate spending capacity of the economy and the public’s preference for liquid assets. In addition, the velocity of broad money includes narrow money plus savings deposits, small-time deposits, and money market funds. This gives a broader view of the money supply and captures the transition from liquid assets to near-money instruments. Lastly, the velocity of total money encompasses all forms of money, including large-time deposits, institutional money market funds, and other large liquid assets. This reflects the overall liquidity available in the economy and the potential for investment and long-term financial stability.

The results of the study would hopefully be useful in contributing to effective monetary policy formulation and implementation in Nigeria. The rest of the paper is discussed as follows. The “Literature Review” section presents a review of relevant literature, while the “Methodology and Data Sources” section discusses the methodology. The “Analysis and Presentation of Results” section discusses the results, while the final section concludes the paper with relevant policy proposals.

Literature Review

Theoretical Literature Review

The income velocity of money refers to the rate at which money circulates within the economy, specifically how frequently a unit of currency is used to purchase goods and services over a year (Mishkin, 2004). This concept, also known as the velocity of circulation, indicates the level of economic activity associated with a given money supply and links money supply changes to nominal income.

Quantity Theory of Money

The income velocity of money is rooted in Fisher’s (1911) equation of exchange, forming the basis of the quantity theory of money. According to this theory, there is a direct relationship between the money supply and the general price level in an economy. Fisher’s equation of exchange posits that an increase in the money supply (M) leads to a proportional increase in the nominal income (PY), assuming the velocity (V) and the total output (T) are constant. Fisher believed that income velocity is constant over time, influenced gradually by institutions and technology.

Keynes’ Liquidity Preference Theory

Contrary to Fisher, Keynes (1936) argued that income velocity is not constant but varies with interest rates. According to Keynes, during economic booms, interest rates rise, reducing money demand and increasing the velocity of money. Conversely, during recessions, interest rates fall, increasing money demand and reducing velocity. This theory highlights the cyclical nature of interest rates and their impact on money circulation.

Milton Friedman’s Theory of Money Demand

Friedman (1959) extended the quantity theory of money by emphasizing the role of money as an asset. Friedman posited that the demand for money is influenced by factors affecting the demand for other assets, such as wealth and expected returns. Unlike Keynes, Friedman believed that the demand for money is relatively insensitive to interest rates and is stable over time. Consequently, he viewed income velocity as predictable, asserting that changes in the money supply directly affect aggregate spending.

There are two major differences between Friedman’s theory and Keynes’s. Freidman believed that changes in interest rates have little effect on the expected returns on other assets relative to money. Thus, in contrast to Keynes, he viewed the demand for money as insensitive to interest rates. In addition, he differed from Keynes in stressing that the money demand function does not undergo substantial shifts and is therefore stable. These two differences also indicate that the income velocity of money is predictable, yielding the quantity theory conclusion that money is the primary determinant of aggregate spending.

Hypotheses on Income Velocity Behavior

The behavior of income velocity has been subject to various hypotheses and empirical tests. Monetarist and Keynesian economists have produced diverse and controversial hypotheses on the behavior of the income velocity of money. Four main hypotheses are outlined as follows:

  1. 1.

    Economies of scale (Fisher, 1911): income elasticity of demand for money is less than unity, suggesting economies of scale.

  2. 2.

    Luxury good hypothesis (Friedman, 1959): elasticity of demand for money is greater than unity.

  3. 3.

    Boom cycle hypothesis (Gurley & Shaw, 1955): during economic booms, income velocity initially falls but eventually levels off as national income rises.

  4. 4.

    Income and money holding relationship: money holdings rise relative to income up to a point, then fall as income continues to rise.

Influences on Income Velocity

The movement of income velocity is influenced not just by income and interest rates but also by other factors like monetization and financial innovation. For instance, a credible electronic payment system can reduce the amount of money required for transactions, thus increasing the velocity. Conversely, an increase in cash circulation can lower the velocity of money.

In summary, the income velocity of money is a dynamic concept shaped by various theoretical perspectives and empirical findings. While foundational theories like Fisher’s and Keynes’ provide a basis, the actual behavior of income velocity is influenced by multiple factors, including technological advancements and policy changes. Understanding these nuances is crucial for effective economic management and policy formulation.

Empirical Literature Review

Empirical Literature Review on the Determinants and Stability of Velocity of Money

Adam et al. (2010) attempted to forecast the money velocity function in Tanzania from the period of 1980 to 2007. The study employed four different models, namely, a rolling trend estimator, a moving average growth estimator, a simple random walk with drift, and a reduced form vector autoregressive (VAR) model, and the results showed that the VAR model, based on the structural money demand equation, outperformed the various univariate approaches both within the sample and over a short period of out-of-sample horizon. On the contrary, Nguyen and Pfau (2010) carried out research on the determinants and stability of real money demand in Vietnam between 1999 and 2009. The paper found a stable money demand function and a long-run relationship between real money demand, foreign interest rates, income, and real stock price.

In Pakistan, Omer (2010) estimated the velocity of money functions and explored their stability in Pakistan between 1975 and 2006. The paper found that M1 velocity is determined by income, interest rates, and inflation. The study also proved that Pakistan has a stable velocity of money function. Kumar et al. (2010) conducted an empirical investigation into the level and stability of money demand (M1) in Nigeria between 1960 and 2008. The study adopted two specifications (canonical and extended forms) through augmentations of real exchange and inflation rates to capture the costs of holding money. The results suggested that the canonical specification was well-determined, the money demand relationship went through a regime shift in 1986, and the money demand was stable.

Sharma and Syarifuddin (2019), using monthly time series data, examined the determinants of the income velocity of money in Indonesia. The paper found money demand to significantly influence the velocity of money for Indonesia in the short run, while it found industrial production, revenues generated from tax, and short-term interest rates as determinants in the long run. Nunes et al. (2018), in their study of the Portuguese economy from 1891 to 1998, found the velocity of money in Portugal to be affected by both the traditional equation (per capita income and interest rate) and the institutional equation. The study also found evidence of a U-shaped velocity of money for Portugal in the long run, having its turning point in the 1960s.

Omanukwe (2010) conducted a study on the quantity theory of money in Nigeria. The paper examined the modern quantity theory of money using quarterly time series data from 1990Q1 to 2008Q4. The study used the Engle-Granger two-stage test for cointegration to examine the long-run relationship between money, prices, output, interest rate, and the ratio of demand deposits/time deposits (a proxy for financial development). The paper found convincing evidence of a long-run relationship in line with the quantity theory. Furthermore, other studies in Nigeria such as Akinlo (2006), Aiyedogbon et al. (2013), Nampewo and Opolot (2016), and Ujunwa et al. (2022) also found the money demand function to be stable for Nigeria using the cumulative sum (CUSUM) and the cumulative sum of squares (CUSUMSQ) tests. Finally, Dagher and Kovanen (2011) analyzed the stability of demand for money in Ghana using the bounds testing procedure developed by Pesaran et al. (2001) from the period 1970 to 2008. The result showed that money demand was stable, and the long-run money demand function was identified during periods of substantial changes in the financial market.

In summary, while several of these studies on the velocity of money adopted the CUSUM and CUSUMSQ tests in establishing the stability of the money demand function in their various studies, several others use different methods to establish the determinants of income velocity of money. For instance, Omanukwe (2010) used the Engle and Granger method to establish long-run relationships among the variables, while Dagher and Kovanen (2011) used the autoregressive distributed lag model (ARDL) to establish the same relationships in their study. On the other hand, Adam et al. (2010) used the VAR models in establishing the determinants of the velocity of money. One thing in common with these studies is that, despite the methods used, they found similar results in the respective economies, proving that the determinants of the income velocity of money can be established using different techniques.

Literature Review on the Impacts of Financial Development on the Velocity of Money

Ahmad (2008) studied the impact of financial deregulation on money demand in Malaysia and the implication of altered money demand patterns on Malaysian monetary policy conduct. The result displayed a stable long-run relationship between money demand in Malaysia and a greater income elasticity of money demand supported by a growing degree of monetization and substantial improvement in banking infrastructure as well as some degree of currency substitution among domestic residents. Lungu et al. (2012) empirically investigated the money demand function for Malawi and its implications for monetary policy conduct during the period 1985–2010 using monthly data. The study found that the reforms in the financial sector and the structural adjustment programs in the 1980s did not seem to affect the stability of the demand for money. Hatem and Mustafa (2013) employed a volatility model in the VAR framework for interest-free financing and the velocity of money in Sudan over the period from 1992 to 2012. The paper revealed that financial development had an influence on the narrow velocity of money (V1), while V1 was volatile and persistent in the mid-1990s but was stable after 2000.

Akinlo (2012) investigated the impact of financial development on the income velocity of money in Nigeria between the periods 1986 and 2010. The result revealed that financial development impacted the velocity of money in the long run model but not in the short run, except in the velocity of narrow money when the time deposit-currency ratio was used as a proxy for financial development. Further results revealed that the exchange rate had significant negative effects, while per capita income had positive impacts on the velocity of money. Ajibola and Nwakanma (2013) used the Polak model to carry out an empirical investigation on the money supply and income velocity of money between 1985 and 2011 in Nigeria, considering financial factors and their implications on money velocity. The findings showed convincing evidence in support of existing relationships in the long-run equilibrium of the variables. Also, the findings revealed a bi-directional causality between money velocity and variables such as exports, net foreign domestic credit, and income in Nigeria. Okafor et al. (2013) examined the determinants of the income velocity of money in Nigeria using quarterly time series from 1985:1 to 2012:4. The study revealed that financial sector development and the growth rate of stock market capitalization had a negative relationship with the velocity of money.

Al-Masaeid (2022) examined the role of financial development in Jordan’s money velocity. The results revealed that financial development, proxied by the private sector credit to GDP ratio, adversely affects the narrow velocity of money function (V1) but positively impacts the broader velocity of money function (V2). The study also found interest rates and time deposits to influence the velocity of money function in Jordan. Ng’imor and Muthoga (2015) also found financial development to adversely affect Kenya’s money velocity. It was also found that the real exchange rate negatively influences the velocity of money, while the real GDP positively influences the velocity of money function in Kenya. However, Altayee and Adam (2018) found a significantly positive relationship between the velocity of narrow money and financial development in Sudan. The study also found an unstable velocity of money in the 1990s, but this changed in the 2000s for Sudan.

In summary, while the studies under review found significant effects of financial development on the velocity of money, Akinlo (2012) was of the view that the effects of financial development on the velocity of money were only pronounced in the long run. In the short run, however, this was not the case. In addition, other studies focused more on the long-run relationships between financial development and the velocity of money. On the other hand, Akinlo (2012) considered both the short- and long-run relationships between the variables. As a result, this study will incorporate a methodology that would allow us to examine both the short-run and long-run relationships between the velocity of money and its determinants. This is to fully encapsulate how these determinants affect the velocity of money over time. However, the review comes with a few limitations, the most pertinent of which is the fact that the study could only access past articles on the topic found on the internet in repositories such as Google Scholar, Research Gate, and Academia. As a result, the literature review was only limited to these sources, excluding what was submitted to public libraries globally.

In essence, this study will extend previous works on the velocity of money by considering the four velocities of money functions as outlined earlier in the introduction section (velocity of reserve money, velocity of narrow money, velocity of broad money, and velocity of total money). Furthermore, in terms of methodology, the study will adopt the quantile ARDL method. This is because, first, the method allows us to examine both the short- and long-term relationships among the variables. Second, the quantile ARDL technique will allow us to understand the behavior of the velocity of money across the different distributions of the data set, allowing us to have a more comprehensive understanding of how these variables affect the behavior of the velocity of money in Nigeria across the different quantiles. In addition, the incorporation of structural breaks in the methodological framework of money velocity also extends previous studies on this topic, particularly in Nigeria.

Methodology and Data Sources

Methodology

The quantity theory of money posits a fundamental relationship between the money supply, the velocity of money, and economic activity. At its core, it asserts that the total nominal value of transactions in an economy (MV) is equal to the price level (P) times the real output or quantity of goods and services produced (Q). This equation highlights the role of the velocity of money (V), representing how frequently money changes hands in transactions over a given period. Traditionally, the theory assumes a stable velocity of money, suggesting that changes in the money supply (M) lead to proportional changes in the price level (P), influencing inflation rates. However, modern interpretations acknowledge that velocity can fluctuate due to shifts in consumer behavior, financial innovations, or changes in confidence levels.

Past studies have established that the factors that determine the income velocity of money are often rooted in the quantity theory of money (such as Bordo & Jonung, 1990; Ireland, 1991, 1994; Judd & Scadding, 1982; Khan, 1974; Short, 1973). This is because the theory explains details about the concept of income velocity of money and how it relates to other macroeconomic variables. Therefore, the paper rests on the quantity theory of money in examining the determinants and stability of the income velocity of money. We start by stating Fisher’s (1911) equation of exchange.

$$MV = PY$$
(1)

where M is the money stock or money supply, which is exogenously determined by the apex bank of a nation, and V is the income velocity of money, which has been defined by several studies as the average number of times each unit of currency changes hands in financing transactions during a given period. Within the framework of Fisher’s equation of exchange, the income velocity of money is assumed to be stable in the short run. Also, P is the general price level and Y is the total amount of goods and services supplied. From Eq. (1), it is evident that the money supply is positively related to the price level and output, while the income velocity of money varies directly with both output and price but inversely with the money supply. This is mathematically expressed as follows:

$$V=\frac{PY}{M}$$
(2)

Studies by Akinlo (2012), Nunes et al. (2018), and Sharma and Syarifuddin (2019) have examined the determinants of the income velocity of money. In this respect, the velocity of money (V) was considered to be a function of factors such as per capita income (y), interest rates (r), exchange rates (exc), inflation rates (πt), and financial development (fd). This could be expressed in a functional relationship as follows:

$$V_t\:=\:f\left(y_t,\;r_t,\;exc_t,\;fd_t,\pi_t\right)$$
(3)

It is also important to note that Eqs. (2) and (3) express factors that determine the size and magnitude of the income velocity of money in an economy. Meanwhile, based on the focus of the study, a dummy variable (dvt) is introduced into the functional Eq. (3) to capture structural breaks, where the variables take the value of zero during periods where structural breaks were not found and one for periods where structural breaks were found. Therefore, Eq. (3) is re-specified as follows:

$$V_t\:=\:f\left(y_t,\;r_t,\;rexc_t,\;fid_t,\;\pi_t,\;dv_t\right)$$
(4)

Hence, re-specifying Eq. (4) from its implicit form to its statistical form gives

$${v}_{t}={\delta }_{1}+{\delta }_{2}{y}_{t}+{\delta }_{3}{r}_{t}+{\delta }_{4}rex{c}_{t}+{\delta }_{5}fi{d}_{t}+{\delta }_{6}{\pi }_{t}+{\delta }_{7}d{v}_{t}$$
(5)

The statistical form of Eq. (5) is expressed in econometrics as follows:

$${v}_{t}={\delta }_{1}+{\delta }_{2}{y}_{t}+{\delta }_{3}{r}_{t}+{\delta }_{4}rex{c}_{t}+{\delta }_{5}fi{d}_{t}+{\delta }_{6}{\pi }_{t}+{\delta }_{7}d{v}_{t}+{\varepsilon }_{t}$$
(6)

where Vt is the income velocity of money (V); this is measured as the ratio of gross domestic product (GDP) to the money supply (M), and m is proxied by reserve money (M0), narrow money (M1), broad money (M2), and extended broad money (M3). These different categories of money stock were used to account for four different velocities of money equations to gain an understanding of the behavior of the different velocities of money functions. yt is the per capita income (pci), and it is measured by dividing nominal GDP by the population; rt is the interest rate; rexct is the real exchange rate; fdt is the financial development indicator proxied by the ratio of credit to private sector (cps) to GDP; πt is the inflation rate, and \({\epsilon }_{t}\) is the error term.

In line with Eq. (6), the quantile ARDL model proposed by Cho et al. (2015) is specified for examining the determinants of income velocity of money in Nigeria. The reason for adopting this methodology is that the study evaluates the stability of relationships across the different quantiles, which provides a more adaptable econometric framework for examining the relationship between the velocity of money and its primary determinants. In contrast to linear ARDL models, the quantile ARDL model introduces potential asymmetries in the way the velocity of money responds to changes in its determinants across different quantiles, thereby offering distinct advantages. Furthermore, the quantile ARDL model, according to Shahbaz et al. (2018) and Olasehinde‑Williams et al. (2024), outperforms linear approaches in numerous respects methodologically.

First, it is capable of accommodating locational asymmetry, which is characterized by the possibility of parameter changes based on the position of the dependent variable within its conditional distribution. Second, the quantile ARDL model simultaneously depicts the short-term dynamics across various quantiles of the velocity of money’s conditional distribution as well as the long-term relationship between the velocity of money and its determinants. Third, the model determines quantile-varying cointegration coefficients in the near term, a nuance that is frequently disregarded by conventional techniques (Xiao, 2009; Olasehinde‑Williams et al., 2024). Fourth, the quantile ARDL model accounts for fluctuations in the cointegrating coefficient over the innovation quantile that are caused by disturbances. Finally, quantile ARDL utilizes a data-driven methodology, in contrast to the nonlinear autoregressive distributed lag (NARDL) model, which enforces nonlinearity through a predetermined threshold (Shin et al., 2014). Therefore, the quantile ARDL method is a preferable choice for accurately modeling the determinants of income velocity of money in Nigeria due to these attributes.

Thus, to specify the quantile ARDL, we begin by specifying the traditional ARDL model.

(7)

The ect is the error correction term, and \(\omega\) is a parameter that indicates the speed of adjustment to the equilibrium level after a shock. Following Cho et al. (2015), Eq. (7) is re-specified to accommodate the ARDL framework across different quantiles.

$$\begin{gathered} {Q_{V_t}} = \alpha (\lambda ) + \sum\limits_{i = 1}^{qi - 1} {\partial v(\lambda )\Delta {v_{t - 1}}} + {\tau_v}(\lambda ){v_t} + \sum\limits_{i = 1}^{qi - 1} {\ell y(\lambda )\Delta {y_{t - 1}}} + {\tau_y}(\lambda ){y_t} \hfill \\ + \sum\limits_{i = 1}^{qi - 1} {\hbar r(\lambda )\Delta {r_{t - 1}}} + {\tau_r}(\lambda ){r_t} + \sum\limits_{i = 1}^{qi - 1} {\vartheta exc(\lambda )\Delta ex{c_{t - 1}}} + {\tau_{exc}}(\lambda )ex{c_t} \hfill \\ + \sum\limits_{i = 1}^{qi - 1} {\psi fid(\lambda )\Delta fi{d_{t - 1}}} + {\tau_{fid}}(\lambda )fi{d_t} + \sum\limits_{i = 1}^{qi - 1} {\omega \pi (\lambda )\Delta {\pi_{t - 1}}} + {\tau_\pi }(\lambda ){\pi_t} \hfill \\ + \sum\limits_{i = 1}^{qi - 1} {\delta dv(\lambda )\Delta d{v_{t - 1}}} + {\tau_{dv}}(\lambda )d{v_t} + {\varepsilon_t}(\lambda ) \hfill \\ \end{gathered}$$
(8)

The short-run effects of the relationship between the velocity of money and its determinants are captured by the parameters in Eq. (8). We can obtain the long-term effects by re-specifying Eq. (8).

$${Q_{V_t}} = \eta (\lambda ) + {K_t}\iota \phi (\lambda ) + {N_t}(\lambda )$$
(9)

where K is the vector of regressors, which serves as the determinant of the income velocity of money. To avoid any effect of serial correlation in the error term, Eq. (9) above shall be re-specified as

$$\begin{gathered} {Q_{\Delta {V_t}}} = a + {b_1}{v_{t - l}} + {b_y}{y_{t - 1}} + {b_{exc}}ex{c_{t - 1}} + {b_r}{r_{t - 1}} + {b_{fid}}fi{d_{t - 1}} + {b_\pi }{\pi_{t - 1}} + {b_{dv}}d{v_{t - 1}} + \sum\limits_{i = 1}^\rho {{\partial_i}\Delta {v_{t - 1}}} + \hfill \\ \sum\limits_{i = 1}^{qi - 1} {{\ell_i}\Delta {y_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\hbar_i}\Delta {r_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\vartheta_i}\Delta ex{c_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\psi_i}\Delta fi{d_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\omega_i}\Delta {\pi_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\delta_i}\Delta d{v_{t - 1}}} + {\mu_t}(\lambda ) \hfill \\ \end{gathered}$$
(10)

Finally, the quantile ECM model that would be estimated is specified below:

$$\begin{gathered} {Q_{\Delta {V_t}}} = a(\lambda ) + {b_1}(\lambda ){v_{t - l}} - {b_y}(\lambda ){y_{t - 1}} - {b_{exc}}(\lambda )ex{c_{t - 1}} - {b_r}(\lambda ){r_{t - 1}} - {b_{fid}}(\lambda )fi{d_{t - 1}} - {b_\pi }(\lambda ){\pi_{t - 1}} \hfill \\ - {b_{dv}}(\lambda )d{v_{t - 1}} + \sum\limits_{i = 1}^{\rho - 1} {{\partial_i}(\lambda )\Delta {v_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\ell_i}(\lambda )\Delta {y_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\hbar_i}(\lambda )\Delta {r_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\vartheta_i}(\lambda )\Delta ex{c_{t - 1}}} + \hfill \\ \sum\limits_{i = 1}^{qi - 1} {{\psi_i}(\lambda )\Delta fi{d_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\omega_i}(\lambda )\Delta {\pi_{t - 1}}} + \sum\limits_{i = 1}^{qi - 1} {{\delta_i}(\lambda )\Delta d{v_{t - 1}}} + {\mu_t}(\lambda ) \hfill \\ \end{gathered}$$
(11)

Data Sources and A Priori Expectations

To achieve the objectives of the study, we employed quarterly data from 1981 to 2023 from the Central Bank of Nigeria (CBN) statistical bulletin 2023 edition. The study adopts four velocities of money functions, which are the velocity of reserve money (VMO), velocity of narrow money (VM1), velocity of broad money (VM2), and velocity of total money (VM3). The reason why we considered these four velocities of money functions is because using VM0 or VM1 alone as a proxy for the velocity of money supply has notable limitations that can impede its effectiveness in fully capturing money circulation within an economy. Furthermore, VM0 and VM1 exclude components of VM2 (broad money), such as demand deposits, savings accounts, and time deposits, which are significant parts of the money supply. Finally, it also omits money market instruments and other financial products that contribute to liquidity in the economy. These components are critical for understanding the full extent of money available for spending and investment.

Per capita income represents income and can have either a positive or negative effect on the velocity of money. Friedman (1959) postulated that income may have a negative relationship with the velocity of money since an increase in the savings-to-income ratio would raise the money-to-income ratio when the economy is developing. Also, Short (1973) noted that the relationship may be negative if the elasticity of money demand is greater than one. However, Fry (1988) noted that the relationship may be positive for developed countries. Moreover, the life cycle hypothesis helps explain the direct relationship between the velocity of money and income by illustrating how individuals’ consumption and saving behaviors change over their lifetimes due to a change in income.

For inflation, the a priori expectation is a positive sign between inflation and the income velocity of money, implying an inverse relationship with holding domestic money and, hence, a positive relationship with the velocity of money. Thus, a rise in the inflation rate of a domestic country would lead to a fall in the demand for that domestic currency, and the income velocity of money would increase since people would want to hold less of that currency or more of other currencies. Furthermore, anticipated and unanticipated changes in inflation may help explain this relationship. For instance, anticipated inflation might lead to increased velocity as people try to spend money quicker to avoid erosion of purchasing power.

For the interest rate, the a priori expectation should be positive since the interest rate is incorporated as a measure of the opportunity cost of holding money since a substitution can occur between money and alternative financial assets. Hence, a rise in the rate of interest leads to a higher cost of holding money so the velocity of money should increase. It is also important to note that the velocity of money may also influence interest rates, leading to endogeneity concerns. Regardless, the expected relationship between both variables is expected to be positive. The exchange rate can have either a positive or negative sign, but it is mostly expected to have a positive relationship with money velocity due to increased international trade led by economic reforms. If the domestic currency is expected to depreciate, the domestic portfolio holders would prefer to hold foreign assets. As a result, depreciation would lead to a higher cost of holding local currency so velocity would increase. On the other hand, the degree of financial integration and capital mobility may also determine the relationship between the exchange rate and the velocity of money in the economy. For instance, on rare occasions, when funds are moved out of the domestic economy for investments or savings abroad, the domestic money supply used for transactions decreases, reducing the velocity of money.

The theoretical expectation between financial development and the income velocity of money is expected to be positive since financial innovations would enhance the reduction in cash holdings with the use of money substitutes, thereby increasing money velocity. In measuring financial development, the study uses the ratio of private sector credit to GDP. This is because previous studies on the topic are of the view that this measure is a better measure of financial development in the Nigerian economy compared to other measures such as the ratio of money supply to GDP or the ratio of stock market capitalization to GDP (Oyadeyi, 2023c, 2023d). The description and measurements of variables and their sources are presented in Table 8, which is displayed in the Appendix.

Analysis and Presentation of Results

Pre-Analyses

We start our analyses by examining the trends of the different velocity of money functions in Nigeria. From Fig. 1, the income velocity of money, measured as the ratio of nominal GDP to the money supply, shows that the income velocity of money has undergone significant changes since 1981. Even though there were periods between 1981 and 1989 when the velocity of money for VM0, VM1, and VM2 slightly fell, it rose more often. They began to fall again in 1991. Between 1996 and 2009, VM1 and VM2 fell, while VM0 fluctuated. Despite a sharp rise in the velocity of money functions in 2010, we can say that they continued to fall between 2011 and 2021, albeit VM0 rose between 2015 and 2017, and VM1 rose between 2012 and 2015. Furthermore, the four velocities of money functions rose slightly in 2022 before falling again in 2023. The growth in financial transactions, financial innovations, financial deepening, advancements in financial liberalization, and service automation may have led to changes in the income velocity of money in Nigeria over the years.

Fig. 1
figure 1

Plots of income velocity of base money (VM0), the velocity of narrow money (VM1), the velocity of broad money (VM2), and the velocity of total money (VM3)

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The descriptive statistic results in Table 1 showed that the mean and median values of all the variables lie within their maximum and minimum values showing a good level of consistency, while the income velocity of extended broad money (VM3) displayed the least variability. The skewness statistics revealed that all the variables were positively and negatively skewed, while the kurtosis statistic exceeded three for all the variables except for the log of per capita income (lnPCI) and log of exchange rate (lnexc), meaning that the other series follows a leptokurtic distribution, while lnPCI and lnEXC follow a platykurtic distribution. The correlation matrix results in Table 2 showed that the variables are not strongly correlated and are fit for running the regression analysis. Furthermore, in the case of Nigeria, the data suggests that per capita income has a negative correlation with the four different velocities of money functions, which is not in line with the quantity theory of money. Furthermore, financial development and exchange rates have negative correlations with the four velocities of money functions. On the other hand, interest rates and inflation rates have a negative correlation with VM0 and VM3, which is not in line with the quantity theory, while they have a positive correlation with VM1 and VM2, in line with the quantity theory of money.

Table 1 Descriptive characteristics of the variables
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Table 2 Correlation matrix
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The unit root test results in Table 3 were examined using augmented Dickey and Fuller (1979) tests (ADF), Phillips and Perron (1988) tests (PP), Perron and Vogelsang (1993) tests (PV), and Zivot and Andrews (1992) tests (ZA). We included the PV and ZA unit root tests due to the incorporation of structural breaks in our model. The significance of the ZA and PV unit root test is that it identifies the period of structural break endogenously and identifies a break date associated with the shift. Since the ZA and PV unit root tests are superior to the ADF and PP tests due to the importance of structural breaks in our model, we, therefore, accept the ZA and PV unit root test results. The ZA and PV unit root test results show that the variables are stationary in their differenced form. Therefore, the ARDL model would be adopted to examine the determinants of income velocity of money in Nigeria.

Table 3 Unit root tests
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To examine the possibility of structural breaks in our model, we employ the Bai and Perron (2003) multiple breakpoint tests for each model (VM0, VM1, VM2, and VM3). The Bai-Perron technique suggests different dates for the four respective models. The results are presented in Table 4. Furthermore, we examined the lag length criteria among each of the estimated four velocities of money models and selected the appropriate model based on the Schwarz Bayesian Information Criterion (SBC). The reason the SBC was selected is because it is generally preferred over the Akaike Information Criterion (AIC) for selecting the lag length of a model, particularly when dealing with large sample sizes, as it is consistent, more parsimonious, and tends to provide better predictive performance by avoiding overfitting. In contrast, the AIC is not as consistent and tends to overfit (select models with too many parameters) as the sample size grows larger. The SBC test revealed a lag length of one for all four velocities of money models. These results are presented in Table 5. Since the presence of regime shifts must be confirmed in each model, we employ the ARDL technique to confirm the significance of the break dates in the velocity of money function for Nigeria.

Table 4 Proposed break dates using the Bai-Perron multiple breakpoint tests
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Table 5 Lag length selection criteria
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Main Results Using Quantile ARDL Technique

To confirm the main results, the study adopted the quantile ARDL estimation method to verify the short-run and long-run effects of the determinants of income velocity of money across the different quantiles (25%, 50%, and 75%). Tables 6a and b present the results using the quantile ARDL method. To begin with, from Table 6a, the quantile estimates for the velocity of reserve money show that per capita income had a positive effect on VMO across the three different quantiles, both in the short run and long run. This implies that as incomes rise, households and businesses are likely to increase their spending and investment. This heightened activity means that money changes hands more frequently, boosting the velocity of reserve money and leading to greater consumption and production. The exchange rate on the other hand had no significant effect on VMO in the lower quantile, but this changes in the median quantile in the long run. While the effect in the upper quartile was negative and significant both in the short run and long run, since central banks rely on the velocity of money to gauge the effectiveness of their monetary policies, a decline in the velocity of reserve money due to adverse exchange rate movements can diminish the impact of monetary policy measures such as interest rate adjustments on the economy.

Table 6 (a) The results using the quantile ARDL technique for the velocity of the base money and narrow money. (b) The results using the quantile ARDL technique for the velocity of the broad and total money
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On the other hand, interest and inflation rates have no effect on VMO in the short and long run. However, financial development has both short- and long-run negative and significant impacts on the velocity of reserve money. This suggests that as the financial sector grows and becomes more sophisticated, the rate at which reserve money circulates in the economy decreases, thereby slowing the circulation of money within the economy. Finally, the error correction term (ECT) starts with a higher speed of 32% at the lower quantile, reducing to 15% at the middle quantile and 6.8% at the upper quartile. This implies that the speed of adjustment varies across the different quartiles, coming with the appropriate negative sign and being significant at 1%. This indicates that there is a convergence to the long-run steady state after a short-term disequilibrium, confirming the existence of cointegration within the model. For the dummy variables, the break dates of Q3 1991 (the removal of capital account restrictions), Q1 2010 (the recovery from the global financial crisis), and Q4 2017 (the recovery from the recession) did not affect the velocity of reserve money. However, this changed across the different quartiles as events that occurred during these periods had significantly negative effects on the velocity of base money in the long run.

For the velocity of narrow money, as displayed in Table 6a, the significantly positive effects of income on VM1 across the different quartiles were in line with the results of VM0. These effects were increasing as we moved from the lower quartile to the higher quartile, indicating that income has a stronger impact on the velocity of narrow money over the long term. This means that income has positive effects on the velocity of narrow money in Nigeria. Consequently, increased income will foster spending and investments by households and businesses, boosting the velocity of narrow money. On the other hand, exchange rates do not impact the velocity of narrow money in the short term. In the long term, however, it has a significantly negative effect on VM1 across the three quartiles. These effects were also stronger as we moved from the lower to the upper quartile, indicating that the negative effects of the exchange rate on the velocity of narrow money are persistent and become higher over the long term. This result suggests that a decline in the velocity of reserve money due to adverse exchange rate movements can diminish the impact of monetary policy measures on the economy.

Furthermore, the interest rates had insignificant short-run effects on the velocity of narrow money across the different quartiles. However, this changes in the long term as the effect of interest rates becomes pronounced, particularly in the lower and middle quartiles, while the significant effect of interest rates in the upper quartile dissipates in the long term. This means that interest rates only affect the velocity of narrow money in the middle and lower quartiles. When interest rates negatively impact the velocity of narrow money, it suggests that higher interest rates lead to a slower circulation of narrow money within the economy. The results of inflation on VM1 were in line with the results of interest rates on VM1. This implies that higher inflation rates lead to a slower circulation of the most liquid forms of money within the economy.

Financial development has both short- and long-run negative and significant impacts on the velocity of narrow money. This suggests that as the financial sector grows and becomes more sophisticated, the rate at which the most liquid forms of money circulating in the economy declines, thereby slowing the circulation of money within the economy. Finally, the ECT starts with a higher speed of 12.7% at the lower quantile, reducing to 5.7% at the middle quantile and 1% at the upper quartile. This implies that the speed of adjustment varies across the different quartiles, coming with the appropriate negative sign and being significant at 1%. This indicates that there is convergence to the long-run steady state after a short-term disequilibrium even though it is very slow, particularly at the 75% quartile.

The structural break date of Q3 1994, during the beginning of the re-regulation era, had significant short-term effects at the 25th quartile, while the structural break date of Q2 1987, when the economy was beginning to witness the effects of SAP, had negative effects on the income velocity of narrow money in the short run. However, at the 75th quartile, the structural break dates of Q2 1987 (negative), Q3 1994 (positive), and Q1 2010 (positive), representing the periods when SAP was implemented, the re-regulation era, and during the recovery of the global financial crisis, all had significant effects on the velocity of narrow money. In the long run, they all had positive effects on the velocity of narrow money across the three different quartiles, while the liberalization of economic activities in the year 2000 had negative effects on the velocity of narrow money.

For the velocity of broad money (VM2), Table 6b showed that per capita income had a positive effect on the velocity of broad money across the three different quantiles, both in the short run and long run. This implies that as incomes rise, households and businesses are likely to increase their spending and investment. This heightened activity means that money changes hands more frequently, boosting the velocity of broad money and leading to greater consumption and production. Furthermore, exchange rates had significant and negative effects on the velocity of broad money both in the short run and long run and across the three quartiles. This suggests that central banks decline in the velocity of broad  money due to adverse exchange rate movements can diminish the impact of monetary policy measures on the economy.

On the other hand, interest rates had significantly positive effects on the velocity of broad money in the lower 25th quartile in the short run. This significant effect swung across the 50th and 75th quartiles in the short run. In the long run, however, interest rates had positive effects on VM2 across the 25th, 50th, and 75th quartiles with roughly the same magnitude. This suggests that higher interest rates lead to a higher rate of circulation of broad money within the economy. The inflation rate on the other hand had insignificant effects on the velocity of broad money across the different quartiles in the short run. In the long run, however, inflation had no effect on VM2 in the lower 25th quartile. However, this changed significantly in the median and 75th quartiles as inflation had significantly positive effects on VM2, with increasing effects along the higher quartiles. This suggests that inflation rates lead to a higher circulation of the broad forms of money within the economy.

Financial development had no significant impact on VM2 in the short run across the different quartiles. In the long term, however, the effects of financial development were negative on the velocity of broad money in Nigeria, increasing from 59 in the 25th quartile to 60% in the middle quartile before retreating to 57% in the 75th quartile. The implication suggests that as the financial sector grows and becomes more sophisticated, the rate at which broad money circulates in the economy decreases, thereby slowing the circulation of money within the economy.

Finally, the error correction term (ECT) starts with a higher speed of 30% at the lower quantile, reducing to 14.3% at the middle quantile and 12% at the upper quartile. This implies that the speed of adjustment varies across the different quartiles, coming with the appropriate negative sign and being significant at 1%. This indicates that there is a convergence to the long-run steady state after a short-term disequilibrium, confirming the existence of cointegration within the model. For the dummy variables, the break dates of Q2 1987 (implementation of SAP) had negative effects on the velocity of broad money in the 25th and 50th quartiles, while the effect became insignificant for the 75th quartile. On the other hand, Q4 1994 (a return to the re-regulation era) and Q1 2001 (the liberalization of economic activities in Nigeria) did not affect the velocity of broad money in Nigeria in the short term. In the long term, however, the implementation of SAP (Q2 1987) and the re-regulation of economic activities (Q4 1994) had positive effects on the velocity of broad money, while the liberalization of economic activities (Q1 2001) had significantly negative effects on the velocity of broad money in Nigeria.

The final velocity of money function to consider is the velocity of total money (VM3). This is the broadest definition of the income velocity of money. The findings from Table 6b suggest that income had positive effects on the broadest definition of the velocity of money across the three quartiles in the short and long term. Consequently, increased income will foster spending and investments by households and businesses, boosting VM3. On the other hand, the exchange rate, interest rates, and inflation rates do not significantly affect VM3 in the short term across the different quantiles. But this changed over the long term as their effects became significant and negatively impacted VM3 across the lower, middle, and upper quartiles. For exchange rates, this suggests that a decline in the broadest definition of income velocity of money due to adverse exchange rate movements can diminish the impact of monetary policy measures on the economy. For interest rates and the inflation rate, this suggests that higher interest rates and inflation rates lead to a slower circulation of the broadest definition of money velocity in Nigeria.

Financial development on the other hand had significantly negative effects on the velocity of the broadest measure of money across the different quartiles and in the short and long run. In the short run, while its effect increased from the lower 25th quartile to the 50th quartile, it slightly fell in the 75th upper quartile. In the long run, however, the negative effects of financial development diminished as we moved up the quartiles. This suggests that as the financial sector grows and becomes more sophisticated, the rate at which the broadest definition of the velocity of money circulates in the economy declines, thereby slowing the circulation of money within the economy. Finally, the ECT started with a higher speed of 75% at the lower quantile, reducing to 21.1% at the middle quantile and 15% at the upper 75th quartile. This implies that the speed of adjustment varies across the different quartiles, coming with the appropriate negative sign and being significant at 1%. This indicates that there is convergence to the long-run steady state after a short-term disequilibrium even though it is very slow, particularly at the 75% quartile.

The structural break dates of Q1 2015, during the general elections in Nigeria, and Q1 2020, during the COVID-19 pandemic, did not significantly affect VM3 in the short run across the quantiles. But this changed over the long term, showing that the general elections and the COVID-19 pandemic had significantly long term negative effects on the velocity of total money in Nigeria.

Finally, the study conducted the diagnostic test using the Ramsey RESET tests. The findings, as displayed in Table 7, showed that since the probability values are above 0.05, there is no strong evidence to reject the null hypothesis that the model is correctly specified. A p-value greater than 0.05 suggests that there is no significant evidence of omitted variables or incorrect functional form in the model. This means the model is correctly specified.

Table 7 Diagnostic testusing the Ramsey RESET technique
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The Quantile Process Coefficients

The quantile process coefficients show the change in the coefficients as a result of the change in the quantiles. It shows the marginal effects of a change in the model as a result of a change in the distribution over the long run. The graphs in Fig. 2 show that income had a positive effect on VM0, but this fell in the 20th quartile before rising again in the 30th quartile, up until the 70th before slightly falling. The exchange rate had negative effects, just like the main result showed. It was initially rising before falling up until the 70th quartile and rising afterwards. Interest rates on the other hand had both positive and negative effects but were overall negative within the 10th quantile to the 40th quantile before swinging positively from the 50th quantile. Inflation generally had a positive effect up until the 40th quantile before swinging back to positive in the 90th quantile. Financial development however had negative effects on all the periods. Therefore, these coefficients have been able to show us how these variables affect the velocity of money across the different quantiles.

Fig. 2
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Quantile process coefficients for the velocity of reserve money. Source: author’s computation

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The results in Fig. 3 were in line with Fig. 2 for per capita income, showing that it had positive effects on the velocity of narrow money across the distribution. Furthermore, the exchange rate had negative effects all through the distribution on VM1, peaking in the 10th and 90th quantiles. Interest rates on the other hand were positive across the distribution on VM1, while inflation was negative in the 10th and 90th quantiles. Financial development (CPS) on the other hand was negative throughout the distribution. For Fig. 4, income had positive effects on VM2, and this was rising as the quantile rose. The exchange rate had negative effects, and its effect was falling up until the 80th quantile on the velocity of broad money. Furthermore, the effects of interest rates were positive and falling up until the 40th quantile. It rose slightly in the 50th quantile before falling again until the 80th quantile. Inflation on the other hand rose until the 80th quantile before falling. Finally, financial development was negative throughout the distribution and fell in the 30th quantile before rising until the 60th quantile and falling until the 80th quantile before rising afterwards.

Fig. 3
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Quantile process coefficients for the velocity of narrow money. Source: author’s computation

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Fig. 4
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Quantile process coefficients for the velocity of broad money. Source: author’s computation

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Finally, Fig. 5 shows that income had a positive effect on VM3, although this effect fell along the distribution except for the 30th quantile. On the other hand, the exchange rate had a negative effect on VM3 except for the 90th quantile when it swung to positive changes on VM3. Interest rates on the other hand had negative effects on VM3 falling up until the 90th quantile. Inflation, however, had positive effects until the 40th quantile. But this effect was falling as the quantiles rose. The effect of inflation on VM3 was negative from the 50th quantile. Finally, financial development had negative effects on VM3 across the entire distribution, rising in the 20th quantile, falling to the 40th quantile, before rising again up until the 80th quantile. Overall, these findings from Figs. 2, 3, 4, and 5 show that the effects of income, interest rates, inflation rates, exchange rates, and financial development vary across the different quantiles, ensuring that the results are solid in understanding the effects of those variables that serve as determinants of income velocity of money in Nigeria.

Fig. 5
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Quantile process coefficients for the velocity of total money. Source: author’s computation

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In addition, the four ARDL models were fitted against their historical velocity of money data (VM0, VM1, VM2, and VM3), and they performed very well as they were able to track the cyclical nature of income velocity of money in the four equations for Nigeria (see Fig. 6a–d). To confirm the stability of our models, we adopted the cumulative sum (CUSUM) and cumulative sum of square (CUSUMSQ) tests proposed by Brown et al. (1975). It is based on the cumulative sum of the recursive residuals which plots the cumulative sum together within the 5% critical lines. The CUSUM results in Fig. 7a, b showed that they were not within the 5% critical lines. On the other hand, while Fig. 8a was within the 5% critical line, and Fig. 8b was not. Furthermore, Fig. 9a is not within the two critical lines, and Fig. 9b is within the two critical lines. Finally, while Fig. 10a is within the two critical lines, and Fig. 10b is not. For those models that were not within the two critical lines, this indicates parameter instability as some parts of the line drifted out of the two critical bounds. This implies that the income velocity of money in Nigeria is unstable. The instability in the model implies that the inflation-targeting strategy is the appropriate monetary policy instrument to be adopted by the CBN in its monetary policy formulations. However, if the CBN decides to adopt monetary targeting policies, then this may not guarantee the use of effective monetary policy for the country.

These findings are in line with earlier studies such as Nwaobi (2002) and Kumar et al. (2013), who also found the income velocity of money not to be stable in Nigeria. This result contrasts with previous studies such as Akinlo (2006), Owoye and Onafowora (2007), and Aiyedogbon et al. (2013). A key reason for this may be that these studies did not incorporate structural breaks within their methodological framework, while they did not consider the use of quantile regressions for their estimations of Nigeria.

When the velocity of money is unstable, it becomes difficult for policymakers to predict the impact of changes in the money supply on economic variables such as inflation, output, and employment. For instance, an increase in the money supply might not lead to the expected increase in economic activity if the velocity of money decreases simultaneously. Central banks often use targets for money supply growth to control inflation. However, if the velocity of money is unstable, the relationship between the money supply and inflation becomes uncertain, making it challenging to set effective targets.

The implication of the instability in the velocity of money means that inflation rates can become more volatile and less predictable, which Nigeria currently witnesses today. Policymakers might find it harder to control inflation because the expected relationship between money supply changes and price levels is disrupted. Furthermore, interest rates are a primary tool for influencing economic activity and money velocity. However, if the velocity is unstable, the impact of interest rate changes on the economy becomes less predictable. This can lead to either an underestimation or an overestimation of the necessary rate adjustments to achieve desired economic outcomes. Therefore, their strategy should be focused on taming inflation by targeting inflation so as to taper inflation to sustainable levels. This will require consistent review and analysis until the velocity of money becomes stable and predictable for Nigeria.

More importantly, policymakers must rely on a broader set of economic indicators, beyond just money supply and interest rates, to gauge the health of the economy and the effectiveness of monetary policy. This includes real-time data on consumer spending, investment, and financial market conditions. An unstable velocity of money necessitates a more adaptive and responsive policy framework. Central banks may need to adjust their strategies more frequently based on incoming economic data and emerging trends.

In essence, the instability of the velocity of money in Nigeria poses significant challenges to effective monetary policy. Policymakers need to adopt a more nuanced and flexible approach, closely monitoring a wide range of economic indicators and being prepared to adjust their strategies as needed. Addressing underlying structural issues in the economy and enhancing financial sector development are also crucial steps toward achieving greater stability in the velocity of money, thereby improving the effectiveness of monetary policy in managing economic growth, inflation, and overall financial stability.

Fig. 6
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a Plot of the actual and fitted velocity of reserve money (VM0). b Plot of the actual and fitted velocity of narrow money (VM1). c Plot of the actual and fitted velocity of broad money (VM2). d Plot of the actual and fitted velocity of total money (VM3)

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Fig. 7
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a CUSUM test result of the velocity of reserve money. b CUSUMSQ test result of the velocity of reserve money

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Fig. 8
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a CUSUM test result of the velocity of narrow money. b CUSUMSQ test result of the velocity of narrow money

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Fig. 9
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a CUSUM test result of the velocity of broad money. b CUSUMSQ result of the velocity of broad money

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Fig. 10
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a CUSUM test result of the velocity of total money. b CUSUMSQ test result of the velocity of broad money

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Conclusion

This paper investigated the determinants and stability of the income velocity of money using quantile ARDL regressions and the cummulative sum techniques, after accounting for structural breaks in its modeling approach. The study adopted the Bai-Perron multiple structural break approach to identify the break dates, the PV and ZA unit root test with a structural break to confirm the stationarity properties, the quantile ARDL framework to examine the determinants of money velocity, and the CUSUM and CUSUMSQ tests to examine the stability of velocity of money in Nigeria from the period of Q1 1981 to Q4 2023. The quantile ARDL technique allowed us to understand the marginal effects of a change in the model as a result of a change in the distribution across the different quantiles and over the long run.

The results suggest that per capita income, exchange rates, interest rates, inflation, and financial development are key determinants of the velocity of money in the four equations. The structural break dates of Q3 1991 (removal of capital account restrictions), Q1 2010 (the recovery from the global financial crisis), and Q4 2017 (the recovery from the economic recession) were found to have significant effects on the velocity of reserve money (VM0) in the long term. Furthermore, the break dates of Q2 1987 (the adoption of SAP), Q3 1994 (the return to the re-regulation era), Q4 2000 (the liberalization of the economy), and Q1 2010 (economic recovery from the global financial crisis) significantly influenced the velocity of narrow money in the long term. In addition, the structural break dates of Q2 1987 (the adoption of SAP), Q4 1994 (the return to the re-regulation era), and Q1 2001 (the liberalization of economic activities) were significant influences on the velocity of broad money. Finally, the break dates of Q1 2015 (general elections) and 2020Q1 (COVID-19 pandemic outbreak) affected the velocity of total money in the long run in Nigeria. Therefore, the study provides the following implications on the determinants of the velocity of money in Nigeria.

For per capita income, policymakers can invigorate economic activity by implementing measures that boost per capita income through education, job creation, and equitable economic policies. By investing in education, they can enhance workforce skills and productivity, leading to higher wages and improved living standards. Job creation initiatives, such as infrastructure projects and support for small businesses, can reduce unemployment and increase disposable income. On the impact of the exchange rate on the velocity of money, ensuring a stable exchange rate environment fosters economic confidence and stability, as businesses can plan and invest without the fear of sudden currency fluctuations affecting their costs and revenues.

To manage inflation and interest rates, the central banks must play a crucial role in managing economic activity by adjusting interest rates to control inflation and stimulate growth. Since interest rate was found to influence the velocity of money in Nigeria, therefore, interest rate reforms should be a component of the broad package and should be targeted at aiming for and facilitating financial intermediation and monetary management as well as enhancing economic growth. Lowering interest rates can encourage borrowing and spending, which boosts economic activity, while raising rates can help cool down an overheated economy and control inflation. This directly impacts the velocity of money, as higher spending and investment increase the frequency at which money circulates through the economy. Maintaining moderate and stable inflation preserves purchasing power, ensuring that consumers and businesses feel confident about spending and investing, which further influences money velocity.

On the impact of the financial sector on the velocity of money, strengthening the financial sector through regulatory reforms, promoting financial inclusion, and embracing technological advancements enhances market efficiency and the velocity of money. Regulatory reforms can ensure a stable and transparent financial system that protects consumers and promotes fair competition. Promoting financial inclusion by expanding access to banking services for underserved populations can stimulate economic activity, as more people can save, invest, and borrow. Embracing technological advancements, such as digital payment systems and fintech innovations, can streamline transactions, reduce costs, and increase the speed at which money circulates in the economy.

Finally, since the velocity of money is unstable and unpredictable for Nigeria, this means that inflation rates can become more volatile and less predictable, and policymakers might find it harder to control inflation because the expected relationship between money supply changes and price levels is disrupted. Therefore, targeting interest rates may either lead to an underestimation or overestimation of the necessary rate adjustments to achieve desired economic outcomes. Thus, implementing the inflation targeting strategy will help policymakers tame inflation to levels that would not be growth-harming. Finally, adopting a more nuanced and flexible approach, closely monitoring a wide range of economic indicators, and being prepared to adjust strategies as needed are important components of adopting the inflation targeting strategy. Therefore, addressing underlying structural issues in the economy and enhancing financial sector development are also crucial steps toward achieving greater stability in the velocity of money, thereby improving the effectiveness of monetary policy in managing economic growth, inflation, and overall financial stability.

In summary, by understanding and focusing on these key determinants—per capita income, exchange rates, interest rates, inflation, and financial development—policymakers can craft informed strategies to manage economic growth and stability. This holistic approach ensures that the benefits of economic policies are felt broadly across the economy, ultimately bolstering overall financial health and enhancing the velocity of money, which is crucial for sustained economic progress. By implication, the instability of income velocity of money calls for more monitoring and sound monetary policies to ensure stable and predictable velocity of money in Nigeria.

The study’s limitations stem from the notion that while quarterly data provides a more granular view than annual data, it may still miss short-term fluctuations and seasonal variations that can impact the velocity of money on a monthly basis since most macroeconomic data are gathered on a monthly basis (except for GDP in the case of Nigeria which is gathered quarterly, and this was the reason why the study adopted the quarterly framework). In the case of future research on this area, authors can target using mixed methods to help accommodate monthly and quarterly data within the same estimation framework to provide more insights on the determinants and stability of money demand in Nigeria.