Do Currency Equivalent Monetary Aggregates Have an Edge over Their Simple Sum Counterparts?

Abstract

The currency equivalent (CE) monetary aggregates are interpreted as aggregation theoretic money stock measures by Rotemberg et al. (1995), Barnett (1991) and Kelly (2009) and are far more superior to simple sum aggregates as a policy variable. In this context, the components of four official measures of monetary constructs—M1, M2, M3 and L1—are used to construct monthly CE monetary aggregates for the period from April 1993 to June 2009. Quarterly estimates of CE aggregates are also obtained by taking quarterly averages of monthly aggregates. The empirical evidences in terms of information content, velocity behaviour and cyclical behaviour show that there is a potential gain of using CE aggregates as compared to their sum counterparts in applications of policy interest.

JEL Classification: C43 E49

Keywords

Currency equivalent monetary aggregates information content velocity behaviour India

Introduction

Theoretically, meaningful constructs of monetary aggregates should be based on economic aggregation and index number theory. In fact, the monetary aggregates that are consistent with economic theory should approximate the quantity chosen by representative economic agent that maximize his utility. Simple aggregation, used to compute the official monetary aggregates, does make little economic sense since it fails to account for the difference in the utility provided by different monetary assets. Moreover, simple sum aggregates are joint products of monetary assets that have varying degrees of ‘moneyness’. By treating all assets as perfect substitutes, simple sum aggregates do not distinguish the non-monetary services from monetary services generated by them. Thus, money stock measures using simple sum aggregation procedure compounds the non-monetary service with monetary services. Use of such variables in the policy framework is inappropriate since the error in the variables may distort the dynamic relationships which are relevant for policy analysis. These errors become wider as more interest bearing assets are being used for transaction purposes. Recent developments in the financial market such as financial innovations and improvements in the payment mechanism tend to increase the use of interest bearing assets for transaction purposes, increasing their moneyness.

In this context, the currency equivalent (CE) aggregates derived by Rotemberg et al. (1995) can be considered as an appropriate candidate for money. CE aggregates are considered as a stock measure of money that measures the share of discounted monetary services provided by an aggregate (Barnett, 1991). Barnett’s definition of economic stock of money (ESM) includes the monetary services provided by the current and future holdings of monetary assets. Further, decomposing Barnett’s economic stock money, Kelly (2009, 2011) derived the discounted stock of monetary expenditure incurred by the current portfolio of monetary assets alone which he defined as current stock of money (CSM). By definition CSM isolates the portion of each asset that functions as currency and can be considered as aggregation theoretic measure of narrowly defined money. In this context, Kelly (2011) proved that CE aggregates are unbiased estimates of CSM. CE aggregates as aggregation theoretic measures of money stock are far more superior to simple sum aggregates and use of such measures may be more appropriate in policy analysis.

Theoretical Foundations of CE Monetary Aggregates

The CE monetary aggregate was first proposed by Hutt (1963) and subsequently developed by Rotemberg et al. (1995) as an alternative measure of transaction services. The CE aggregates are constructed by assigning time varying weights to monetary assets. The weights for each monetary asset depend on the own interest rate of respective assets, relative to the return on a benchmark asset which provides no monetary services. CE aggregates can be treated as stock of currency that yields same transaction services of an aggregate consisting of different monetary assets. Accordingly, CE index assigns a weight equal to one for currency and an asset with higher rate of return receives a lower weight.

Rotemberg et al. (1995) formally derived CE aggregates from a utility framework and proved that CE aggregates approximate the aggregator function of liquidity services under certain assumption.¹ The representative consumer is assumed to derive utility from the consumption of goods and leisure. Further, the intertemporal utility function contains monetary assets and the consumption of goods and are assumed to be weakly separable. The intertemporal utility function so defined of a representative consumer is given by

\begin{matrix} V = E_{0} \sum_{t = 0}^{\infty} β^{t} u (c_{t}, T_{t}, l_{t}), & T_{t}, u_{j} > 0, & j = 1, 2, 3 \end{matrix}

(1)

where u() gives instantaneous utility and concave in all arguments, E₀ is expectation at period 0, β is an intertemporal discount factor (1>β>0), the aggregate of liquidity services (T_t) can be described as linearly homogenous function in component assets: T_t(2)fm_1tm_1tm_nt_t

where m_1tis the amount of currency, m_2t, …, m_nt_{are other monetary assets held at the period t and α_t captures the changing physical characteristics of monetary assets over time. Therefore, any meaningful measure of money should approximate the aggregator function in Equation 2.}

Assuming that the linear homogenous aggregator function is additively separable in currency and other monetary assets, Rotemberg et al. (1995) showed that the optimum quantity of monetary services chosen by an economic agent can be approximated to CE aggregate as follows:

T_{t} = \sum_{i = 1}^{n} (\frac{R_{t} - r_{i t}}{R_{t}}) m_{i t} \equiv C E_{t}

(3)

The weights assigned to each asset in the CE aggregate in equation 3 can be interpreted as marginal utility of that asset relative to that of currency as it is derived from utility function that satisfies optimality condition. Besides Rotemberg et al. (1995) argue that CE aggregates capture the changes in the monetary services provided by monetary assets due to changes in their characteristics. However, as a measure of transaction services CE aggregates make strong assumptions on aggregator function compared to divisia aggregates. Yet, as Barnett (1991) showed CE aggregates can be considered as a stock measure of money with aggregation theoretic foundations.

CE Aggregate as Stock of Money

To arrive at CE aggregate Rotemberg et al. (1995) assumed aggregator function to be additively separable in currency. Barnett (1991) observed that this assumption was far more restrictive to treat CE aggregate as a measure of flow of monetary services than what is required for divisia quantity index proposed by Barnett (1980). However, he showed that the CE aggregate can be treated as a special case of ESM which he defined as ‘sum of discounted present value of expenditure on the services of monetary assets’. Moreover, ESM so derived is consistent with aggregation theoretic foundations of divisia quantity index. He defined ESM as it enters into discounted single Fisherine wealth constraint under perfect foresight as follows:

E S M_{t} = \sum_{s = t}^{\infty} \sum_{i = 1}^{n} [\frac{p_{s}^{*}}{ρ_{s}} - \frac{p_{s}^{*} (1 - r_{i s})}{ρ_{s + 1}}] m_{i s},

(4)

where p^*_s is the true cost of living index at period s, m_is is quantity of monetary asset, r_is own interest rate of monetary asset i and the discount rate (ρ_s) for the period s is defined as

ρ_{s} = {\begin{matrix} 1 & for s = t \\ Π_{s - 1}^{u = t} (1 + R_{u}),, & for s > t \end{matrix}

(5)

Substituting Equation 5 into Equation 4 gives

E S M_{t} = \sum_{s = t}^{\infty} \sum_{i = 1}^{n} [\frac{p_{s}^{*} (R_{s} - r_{i s})}{Π_{u = t}^{s - 1} (1 + R_{u})}] m_{i s} = \sum_{s = t}^{\infty} \sum_{i = 1}^{n} (ψ_{i s} m_{i s}) / ρ_{s},

(6)

where

ψ_{i s} = p_{s}^{*} ((R_{s} - r_{i s}) / (1 + R_{s}))

Assuming that R_s, r_is and m_is follow martingale process (i.e., R_s= R_t, ris = r_it, m_is= m_it) the above equation reduces to CE aggregates

E S M_{t} = \sum_{i = 0}^{n} (\frac{R_{t} - r_{i t}}{R_{t}}) m_{i t} \equiv C E_{t}

(7)

Taking uncertainty into consideration the definition of ESM is modified by applying consumption based capital asset pricing model. Following Barnett et al. (2006, 2008) it is given as

E S M_{t} = E_{t} [\sum_{s = t}^{\infty} (Γ_{s} \sum_{i = 1}^{n} ψ_{i s} m_{i s})],

(8)

where

Γ_{s} = β^{s - t} \frac{\partial u}{\partial C_{s}} / \frac{\partial u}{\partial C_{t}},

is the subjectively discounted marginal rate of inter-temporaral substitution between consumption in the current period t and the future period s.

However, Barnett et al. (2006) showed that CE aggregates still exhibit a small downward bias since the total expenditure of monetary services assumed to follow a martingale process. In this context, Kelly (2011) showed that CE aggregate can be treated as CSM which is defined as the discounted present value of the monetary services implied by the current portfolio of monetary assets. CSM is derived from the ESM by excluding expected future monetary services. The CSM captures the portion of each monetary asset that functions as currency. Thus the CSM can be treated as an aggregation theoretic measure of narrowly defined money. Moreover, CSM is a better measure of narrowly defined traditional measures like M1 since it includes a wider range of assets. In order to derive CSM, Kelly (2009, 2011) decomposed current and future holdings of monetary assets by defining quantity of monetary asset as follows:

{\tilde{m}}_{i, t + j} = {\begin{matrix} 0 & if j = 0 \\ \sum_{r = i}^{j} Δ m_{i, t + r} & if j > 0 \end{matrix} and m_{i s} = m_{i t} + {\tilde{m}}_{i s} \forall S \geq t,

(9)

substituting Equation 9 into Equation 8 gives

\begin{matrix} E S M_{t} = E_{t} [\sum_{s = t}^{\infty} (Γ_{s} \sum_{i = 1}^{n} ψ_{i s} (m_{i t} + {\tilde{m}}_{i s}))] \\ = E_{t} [\sum_{s = t}^{\infty} (Γ_{s} \sum_{i = 1}^{n} ψ_{i s} m_{i t})] + E_{t} [\sum_{s = t}^{\infty} (Γ_{s} \sum_{i = 1}^{n} ψ_{i s} {\tilde{m}}_{i s})] \end{matrix}

(10)

The first term in the equation captures the present value of discounted monetary services of current portfolio of monetary assets; therefore, the CSM is given as follows:

C S M_{t} = E_{t} [\sum_{s = t}^{\infty} (Γ_{s} \sum_{i = 1}^{n} ψ_{i s} m_{i t})]

(11)

The CSM so defined can be equated with CE aggregates given certain assumptions.

Let us now assume the expectation of the stochastic discount factor in time period t as

E_{t} (Γ_{s}) = \prod_{u = t}^{S} {[1 + E_{t} (R_{u})]}^{- 1}

(12)

Then the equation for CSM becomes

C S M_{t} = \sum_{s = t}^{\infty} \frac{E_{t} (Σ_{i = 1}^{n} ψ_{i s} m_{i t})}{Π_{u = t}^{s} [1 + E_{t} (R_{u})]} + c o v (\sum_{i = 1}^{n} ψ_{i s} m_{i t}, Γ)

(13)

Setting $c o v (\sum_{i = 1}^{n} ψ_{i s} m_{i t}, Γ) = 0$ and assuming the return on benchmark follows a martingale process, i.e. E_t(R_s) = R_t ∀ s ≥ t, then Equation 13 can be written as

C S M_{t} = \sum_{s = t}^{\infty} \frac{(Σ_{i = 1}^{n} E_{t} (ψ_{i s}) m_{i t})}{{(1 + R_{t})}^{s - t}}

(14)

Now if the ψ_is is assumed to follow a martingale process ∀i = 1, 2, … n, the Equation 14 can be rewritten as

C S M_{t} = \sum_{s = t}^{\infty} \sum_{i = 1}^{n} [\frac{(R_{t} - r_{i t})}{{(1 + R_{t})}^{s - t + 1}}] m_{i t} = \sum_{i = 0}^{n} (\frac{R_{t} - r_{i t}}{R_{t}}) m_{i t} \equiv C E_{t}

(15)

In this context, Kelly (2011) empirically analysed and observed that CE aggregate is an unbiased estimate of CSM and it can be considered as an appropriate measure of narrowly defined money.

To consider simple sum aggregates as an appropriate measure of money stock, we need to assume the return on monetary assets (r_it) to be zero. In reality, the simple sum aggregates tend to overestimate the actual money stock and inclusion of interest bearing assets will increase this upward bias considerably. Following Barnett et al. (2006, 2008) and Kelly (2009, 2011) it can be showed that the simple sum aggregates compound both discounted present value of monetary services and the discounted present value of return yielded by the monetary assets. Thus the simple sum aggregates (SSI) can be decomposed as follows:

S S I_{t} = \sum_{i = 1}^{n} m_{i t} = \sum_{i = 0}^{n} (\frac{R_{t} - r_{i t}}{R_{t}}) m_{i t} + \sum_{i = 0}^{n} (\frac{r_{i t}}{R_{t}}) m_{i t},

(16)

where first summation term is CSM and the second term is defined as Investment Stock of Money (ISM). ISM is the present value of discounted return yielded by the monetary assets at time period t.² Compounding CSM and ISM in the simple sum aggregate makes it inappropriate as it smoothens the actual money stock. Because of the noise in the simple sum aggregates it may obscure the dynamic link between other economic variables of interest like output, interest rates, etc. In addition, the velocity behaviour of money stock based on simple sum aggregates will be erratic and may signal wrong information.

Empirical studies have documented the properties of CE aggregates and compared its performance relative to simple sum and other weighted monetary aggregates. Rotemberg et al. (1995) computed CE aggregates using US monetary data and empirically examined its performance in predicting real economic activity. They found that CE aggregates have better predictive power than their simple sum counterparts. Also Serletis and Molik (2000) using monthly data on Canadian and US simple sum, divisa and CE aggregates for a period from 1974:1 to 1999:12 observed that aggregation procedures are crucial in evaluating relationship between money and economic activity. Similarly, Serletis and Koustas (2001) found supporting evidence for long run neutrality for CE aggregates using quarterly US data over the period from 1960:1 to 1996:2.

Similarly, Barnett et al. (2006) empirically examined the properties of CE aggregates in terms of Barnett’s ESM and found that it approximates money stock with reasonable accuracy relative to simple sum aggregates. Similarly, money stock measures as defined by CE aggregates are found to contain more explanatory information on output gap than their simple sum counterparts (Kelly, 2009). Also, the errors in measurement of simple sum explain the failure of monetary policy disturbance to create negative short run correlations between nominal interest rates and money stock (Kelly et al., 2011). Recently, some studies attempted to examine the Chaotic monetary dynamics using data on CE aggregates along with divisia and simple sum aggregates. For instance, Serletis and Uritskaya (2007) investigated the dynamical structure of simple sum, divisia and CE aggregates using monthly data of United States for a period from 1959:1 to 2006:2. According to them the simple sum and divisia are more appropriate for measuring long-term tendencies while CE aggregate are related to short-term process in the economy (also see Serletis and Shintani, 2006).

So far only a few studies have done on CE aggregates using Indian data (Acharya and Gopalaswamy, 2007; Acharya and Kamaiah, 1998; Paul and Ramachandran, 2011). These studies examined the properties CE aggregates vis-a-vis simple sum aggregates in terms of their information content, stability of their demand functions, leading indicator of inflation, etc. The empirical evidence was supportive of CE aggregates in Indian context. This study attempts to add to this existing literature and documents the stylized facts of CE aggregate in Indian context.

Sample and Description of Variables

This study uses both monthly and quarterly simple sum and CE aggregates of different aggregation levels to document their properties and performance based on conventional statistical criteria. The simple sum and CE aggregates are constructed using monthly data on a number of monetary assets (m_it) that figure in the official measures of money stock in India and appropriate rate of returns (r_it) for the period from April 1993 to June 2009.³ The quarterly estimates of both simple sum and CE aggregates is constructed by taking average of monthly data in the respective quarters. The details of monetary aggregates used in this study are given in Table 1.

Table 1.

Measures of Monetary Aggregates Used in the Study

Monetary Aggregates	Monetary Components (x)
M1	Currency with the public + demand deposits with banks + other deposits with the RBI
M2	M1 + time liability proportion of the savings deposits with banks + term deposits with the contractual maturity of up to and including one year with banks
M3	M2 + term deposits with the contractual maturity of over one year with banks
L1	M3 + all deposits with the Post Office Savings Banks (excluding National Savings Certificates)

Source: Paul and Ramachandran (2011).

The components of various monetary aggregates and their corresponding interest rate proxies used in this study are given in Table 2.⁴ The data on the components of monetary aggregates, interest rates, yield on long-term government securities, Gross Domestic Product and the wholesale price index are collected from the Handbook of Statistics on Indian Economy and other publications of the Reserve Bank of India, and the interest rate on time deposits and benchmark prime lending rate of SBI, which are obtained from SBI on request.

Table 2.

Monetary Components and Corresponding Interest Rate Proxies

Monetary Components (x)	Interest Rates (r)
Currency with the public	Zero
Demand deposits (demand deposits with banks + other deposits with the RBI)	Implicit rate of demand deposits (r_DD) = r_T (1 – [BR/DD]), where r_T is 91 days treasury bill rate, BR is Bank reserve held against demand deposits and DD is demand deposits (Klein, 1974)
Term deposits with the contractual maturity of up to and including one year with banks	Interest rate on one year term deposits of SBI
Term deposits with the contractual maturity of over one year	Maximum interest rate of term deposits with the contractual maturity of over one year of SBI
All deposits with the Post Office Savings Banks (excluding National Savings Certificates)	Interest rate on postal time deposits

Source: Paul and Ramachandran (2011).

Selection of an appropriate benchmark interest rate is very crucial in the estimation of CE aggregates. Theoretically, it is a rate on bench mark asset that provides no liquidity services and is used to transfer wealth from one period to another. Thus, benchmark asset cannot be traded in the secondary market. In practice, it is either proxied by the rate of return on a least liquid asset/long maturity assets or maximum rate of return among a range of assets.⁵ Following Barnett and Spindt (1982) in this study the benchmark rate of interest (R_t) is chosen as the maximum rate among a set of market rates such as prime lending rate (PLR) of State Bank of India (SBI), yield on long-term government securities (r_gs) and the rate of return on components of the broadest aggregate (i.e., L1) and is given as

R_{t} - M a x {r_{i, t} (i = 1, 2, 3, ... n), r_{g s, t}, B P L R_{t}}

(17)

Since call money rate and rate on certificate of deposits were extremely high and volatile during a few months of the chosen sample period, call/term borrowings by financial institutions and certificate of deposits issued by commercial banks are excluded in the construction of CE aggregates.⁶

Some Stylized Facts

To begin with, we plot the CE constructs along with their simple sum counterparts. Figure 1(a–d) describes the monthly aggregates for the period April 1993 to June 2009 with solid lines indicating simple sum aggregates and dotted line their CE aggregates. Since CE aggregates measure only the discounted share of monetary services, the difference between simple sum and CE aggregates can be interpreted as the stock of investment yield contained in respective simple sum aggregate. This can be measured by the vertical gap between solid and dotted lines in the figures. The share of investment yield increases as more and more assets that provide non-monetary services are included in the aggregate. The share of investment yield in the Simple M1 as given by the vertical gap between CE M1 and M1 in Figure 1(a) is comparably smaller but constantly increases since 1999. But the vertical gap between CE aggregates and simple sum aggregates becomes wider as the level of aggregation increases (Figure 1(b–d)). Note that the divergence between CE aggregate and simple sum is more pronounced at higher level of aggregation. The size of the graph grows rapidly particularly at M3 and L1 aggregates (Figure 1(c, d)). However, the gap seems to decrease since October 2008. It is evident from the plots that the time paths of money stock measured by simple sum and CE aggregate are different even for M1. The error in the simple sum aggregates, as measured by the vertical distance between two series, may affect the short run and long run dynamics of money growth rates given the varying nature of vertical shift particularly at higher level of aggregation. A similar inference can be arrived using quarterly data which are plotted in Figure 2(a–d).

Figure 1(a).

Monthly Series of CE M1 and M1

Figure 1(b).

Monthly Series of CE M2 and M2

Figure 1(c).

Monthly Series of CE M3 and M3

Figure 1(d).

Source: Authors.

Figure 2(a).

Quarterly Series of CE M1 and M1

Figure 2(b).

Quarterly Series of CE M2 and M2

Figure 2(c).

Quarterly Series of CE M3 and M3

Figure 2(d).

Source: Authors.

To make sense of this difference in the simple sum and CE aggregates the following section documents some of the stylized facts of CE and simple sum aggregates using monthly and quarterly estimates of aggregates. A comparative analysis of CE aggregates with respect to their simple sum counterparts in terms of information content tests, velocity behaviour and cyclical behaviour.

Velocity Behaviour

Apart from growth rate of money observing trends in velocity movements may provide important insights for the policy makers. As Barnett et al. (1984) observed the trends in velocity can give useful inference about the stability of demand for money functions. Similarly, if aggregate velocity measure shows a predictable relationship with interest rates, then it can also be considered in the implementation of monetary policy (McCallum, 1989). However, by compounding the discounted non-monetary services with monetary services, estimates of velocity from simple sum aggregates tend to underestimate the true velocity measures.

The plots of velocities of various aggregates used in this study are given in Figure 3 (a–d) . Velocity of each aggregate is estimated as a ratio of annualized nominal GDP to nominal money for a period from 1997 Q1 to 2009 Q2. Prior to estimation of velocity both the annualized nominal GDP and various aggregates were transformed into logarithmic scale. Besides, each velocity estimates were normalized to unity in the first observation, i.e, 1997 Q1. Descriptive statistics of velocity estimates for CE and simple sum aggregates are given in Table 3.

The results show that simple sum aggregates underestimate the velocity at all levels of aggregation. The mean values of CE aggregates are higher than the simple sum aggregates at all levels of aggregation. Similarly, the difference between CE and simple sum aggregates becomes more pronounced as the level of aggregation increases. The dispersion of velocity around its mean as measured by standard deviation is higher with respect to simple sum aggregates at all levels of aggregation except M1. This shows that the velocity measured by CE aggregates are relatively stable than velocity measure of simple sum aggregates especially at higher level of aggregation. Similarly, the range of values of velocity of simple sum at higher level of aggregation (i.e., M3 and L1) is almost twice that of CE aggregates.

Table 3.

Descriptive Statistics of Income Velocities of Simple Sum and CE Monetary Aggregates

	CE M1	CE M2	CE M3	CE L1	M1	M2	M3	L1
Mean	0.941	0.823	0.729	0.723	0.923	0.775	0.558	0.532
Standard Deviation	0.061	0.094	0.164	0.170	0.065	0.125	0.244	0.259
Range	0.229	0.350	0.521	0.531	0.188	0.421	0.861	0.894
Minimum	0.810	0.650	0.479	0.469	0.824	0.579	0.139	0.106
Maximum	1.038	1.000	1.000	1.000	1.011	1.000	1.000	1.000

Source: Authors.

The velocity of simple sum and CE aggregates is plotted in Figure 3(a–d). The velocity of simple sum M1 and CE M1 is plotted in Figure 3(a). Even at this lower level of aggregation, the behaviour of velocity of both aggregates exhibits difference. The velocity of simple sum seems to be stable for a period from 1997 Q1 to 1999 Q1 and steadily declines afterwards. On the other hand, the velocity of CE M1 is stable for a period from 1997 Q1 to 2001 Q1. The values of velocity of CE M1 are higher compared to simple sum M1 since the first quarter of 1997. As the level of aggregation increases the divergence between simple sum and CE aggregates becomes more noticeable. The velocity of simple sum M2, M3 and L1 secularly declines throughout the sample period. On the other hand, declining trend of the velocity measure of CE M2, M3 and L1 is comparably less. Moreover, the velocity measure of these aggregates is stable since 2004q. Similarly, difference between velocity of simple sum and CE aggregates is higher for M3 and L1 levels of aggregation. This is expected as the investment yield of assets included in simple M3 and L1 causes a downward bias in these aggregates.

Figure 3(a).

Log Normalized Velocity of CE M1 and M1

Figure 3(b).

Log Normalized Velocity of CE M2 and M2

Figure 3(c).

Log Normalized Velocity of CE M3 and M3

Figure 3(d).

Source: Authors.

As Barnett et al. (1984) observed that existence of a stable demand for money function or the shifts in parameters of the function can be easily traced by the cross plots of interest rates and velocity. If velocity and nominal interest rates move in the same direction, then it can infer that the interest elasticity has correct sign. These issues are probed by plotting the velocity against interest rate variables (90 days treasury bill rate and yield on long-term government securities) and are depicted in Figure 4(a–d). Since the velocity behaviour of simple sum and CE aggregates differ considerably after second quarter of 2004 (see Figure 3(a–d)) sample period was divided into two. The first period covers data from 1997 Q1 to 2004 Q1 and second period from 2004 Q2 to 2009 Q2. The periods were differentiated using different symbols in the cross plots.

The cross plot between velocity and 90 days treasury bill rates for both simple sum and CE M1 is depicted in Figure 4(a–b). The functional shift in the velocity of both aggregates is very evident since the slope of the scatter plot between 90 days treasury bill rate and velocity of simple sum M1 turns to be negative during 2004 Q2–2009 Q2. Similarly, Figure 4(c,d) shows that same results holds even for yield on long-term government securities. Figure 5(a–d) depicts analogue plots for simple sum and CE M2 aggregates. The plot shows significant difference between simple sum and CE M2. The shift in the velocity of simple sum M2 is very much evident whether it is against 90 days treasury bill rate or yield on long-term government securities. On the other hand, the scatter plots of velocity of CE M2 against the interest rates (both 90 days treasury bill rate and yield of long-term government securities) exhibit a stable functional relation.

Figure 4(a).

CE M1 Velocity Versus 90 Days Treasury Bill Rate

Figure 4(b).

Simple Sum M1 Velocity Versus 90 Days Treasury Bill Rate

Figure 4(c).

CE M1 Velocity Versus Yield on Long-term Government Securities

Figure 4(d).

Source: Authors.

Figure 5(a).

CE M2 Velocity Versus 90 Days Treasury Bill Rate

Figure 5(b).

Simple Sum M2 Velocity Versus 90 Days Treasury Bill Rate

Figure 5(c).

CE M2 Velocity Versus Yield on Long-term Government Securities

Figure 5(d).

Source: Authors.

Similar plots for simple sum and CE M3 are given in Figure 6 (a–d). Plots of CE M3 against 90 days treasury bill rate and yield on long-term government securities continue to have a positive relation. Particularly, a stable and linear function appears to exist between the velocity of CE M3 and yield on long-term government securities. Whereas the velocity of simple sums M3 exhibits shifts and is unstable during the sample period. The velocity measures of simple sum and CE L1 shows similar patterns in Figure 7 (a–d). The results in general show that the velocity estimates derived from money stock measures using simple sum aggregates at all levels of aggregation appears to give wrong signals. Whereas the velocity measures from CE aggregates particularly at higher level of aggregation make economic sense and are relatively stable over the time period.

Figure 6(a).

CE M3 Velocity Versus 90 Days Treasury Bill Rate

Figure 6(b).

Simple Sum M3 Velocity Versus 90 Days Treasury Bill Rate

Figure 6(c).

CE M3 Velocity Versus Yield on Long-term Government Securities

Figure 6(d).

Source: Authors.

Figure 7(a).

CE L1 Velocity Versus 90 Days Treasury Bill Rate

Figure 7(b).

Simple Sum L1 Velocity Versus 90 Days Treasury Bill Rate

Figure 7(c).

CE M3 Velocity Versus Yield on Long-term Government Securities

Figure 7(d).

Source: Authors.

Information Content Test

Information content tests following Tinsley et al. (1980) and Mills (1983) are used to assess the information contained in monetary aggregates about future values of goal variables. Accordingly, the information content of a vector of goal variables (y_t) in terms a vector of indicator variable (x_t) measured in terms of reduction in expected uncertainty. In a univariate framework, the measure of information content is defined as

I (y_{t} | x_{t}) = - \frac{1}{2} \ln (1 - R^{2}),

(18)

where R² is coefficient of determination from the simple linear equation as follows: y_t(19)₀β₁x_tε_t

In a multivariate dynamic framework, this measure of information content is modified to capture the information contained in indicator variable x_t over and above the information contained in the past values of goal variable y_t. Thus, the information content measure of x_t relative to y_t in a dynamic framework is given by

I_{m} (y_{t} | x_{t}) = - 1 / 2 (\ln [\frac{1 - R_{* *}^{2}}{1 - R_{*}^{2}}]),

(20)

where $R_{*}^{2}$ and $R_{* *}^{2}$ are the multiple correlation coefficients from the following regressions: y_t(21)α(L)y_tε_t y_t(22)αLy_tβLx_tε_t

where α (L) and β (L) are finite polynomials in lag operator L. The maximum order of autoregressive process was selected based on AIC criteria. However, as Pierce (1979) observed the use of multiple co-correlation coefficient from Equation 22 can lead to ambiguous inference by compounding between-variable effect with within-variable effect. In order to overcome such problems the conventional R² is replaced by the statistic $R_{1}^{2}$ as suggested by Pierce (1979) and is defined as:

R_{1}^{2} = \frac{R S S_{1} - R S S_{2}}{R S S_{1}},

(23)

where RSS₁is the residual sum of square from Equation 21 and RSS₂ is the residual sum of square from Equation 22. $R_{1}^{2}$ measures the information contained indicator variable after accounting for the effect of past values of goal variable. The unambiguous measure of information content thus defined is

I_{*} (y_{t} | x_{t}) = - \frac{1}{2} \ln (1 - R_{1}^{2})

(24)

In this study, inflation and annual growth rate of IIP—proxy for output growth—are taken as goal variables for the monthly data. Analogously, inflation and growth rate of nominal GDP (annualized) are considered as goal variables for the quarterly series. All the growth variables were mean differenced before estimating Equations 21 and 22 and optimum lag length was selected using AIC criteria. The results of information content test for monthly and quarterly data are reported in Tables 4 and 5.

Table 4.

Results of Information Content [I_*(y_t|x_t)] of Simple Sum and CE Aggregates about Growth Rate of IIP and Inflation: Monthly Data

	Growth rate of IIP		Inflation
	SS	CE	SS	CE
M1	0.070	0.025	0.033	0.034
M2	0.027	0.019	0.022	0.026
M3	0.001	0.006	0.003	0.012
L1	0.002	0.006	0.003	0.012

Source: Authors.

Table 5.

Results of Information Content [I_*(y_t|x_t)] of Simple Sum and CE Aggregates about Growth Rate of Nominal GDP (annualized) and Inflation: Quarterly Data*

	Growth rate of NGDP		Inflation
	SS	CE	SS	CE
M1	0.064	0.118	0.041	0.075
M2	0.001	0.117	0.012	0.141
M3	0.001	0.084	0.008	0.075
L1	0.001	0.080	0.008	0.074

Note: *Sample period, Quarterly observations: 1997 Q1 to 2009 Q2.

Source: Authors.

Even though the information contained in monetary aggregates regarding respective goal variables are at best modest, CE aggregates in general performs better compared to simple sum aggregates. Monthly estimates of information content test shows that growth rates of CE aggregates particularly at higher aggregation levels (i.e., M3 and L1) contains more information regarding inflation relative to their simple sum counterparts. However, there is no considerable difference between CE aggregates and simple sum aggregates in predicting growth rate of IIP. On the other hand, quarterly estimates of information content measures show that CE aggregates performs better than simple sum aggregates at all levels of aggregation. Thus, the CE aggregates contain more information about the growth rate of GDP as well inflation and considerable reduction in prediction risk can be achieved by using CE aggregates as indicators.

Cyclical Behaviour of Monetary Aggregates

Empirical description of business cycle facts is important since it gives a summary of cyclical co-movements of economic aggregates and is useful to identify the leading, lagging and coincident indicators of economic activity. The business cycle facts as defined by Lucas (1977) refer to the statistical properties of the co-movement of cyclical components of economic aggregates with the cyclical component of real aggregate output. Similarly, co-movement of cyclical components of monetary aggregate and real aggregate output can have important implication in the selection of competing models (Serletis and Krause, 1996). This section attempts to document the pattern of cyclical behaviour of monetary aggregates and examine whether aggregation procedure makes any difference in the inference.

The procedure of Hodrick and Prescott (1980) is applied to extract cyclical components of real GDP at quarterly frequency and real IIP and monthly frequency and of various monetary aggregates at monthly and quarterly frequencies. The Hordick and Prescott filter extracts the trend component (τ_t) of a series (X_t_{) by minimizing the following equation:}

m i n τ_{t} \sum_{t = 1}^{T} (X_{t} - τ_{t})^{2 + μ} \sum_{t = 2}^{T - 1} [(τ_{t + 1} - τ_{t}) - (τ_{t} - τ_{t - 1})]^{2},

(25)

where X_t – τ_t is the filtered series, the cyclical component was computed by setting μ = 14400 for monthly series and μ = 1600 for quarterly series. All the variables are log transformed prior to the estimation.

The movement of contemporaneous and non-contemporaneous cyclical co-movements are measured by cross correlation coefficients between cyclical component of money and cyclical component of real IIP and real GDP are computed for monthly and quarterly, respectively. For monthly data up to 6-month leads and lags of H-P filtered cyclical series of real IIP and for quarterly estimates up to four-quarter leads and lags of H-P filtered cyclical series of real GDP are considered. The contemporaneous, leading and lagging natures of co-movement of various monetary aggregates with cycle is decided depending on the magnitude of correlation coefficient in absolute terms. If the correlation coefficient given by ρ(M_t, Y_t_+i) in absolute sense is the largest when i = 0, then the monetary aggregate is contemporaneously correlated with the cycle. Similarly, the series leads the cycle by i months/quarters if the absolute value of ρ(M_t, Y_t_+i) is the largest when i < 0, and lags the cycle by i months/quarters when i > 0. Similarly, the monetary aggregates are classified into acyclical when the correlation ρ(M_t, Y_t_+i) equals zero, procyclical when ρ(M_t, Y_t_+i) is significantly positive and counter cyclical when ρ(M_t, Y_t_+i) is significantly negative.

Cross correlation coefficients for cyclical components of monetary aggregates (M_t) and real IIP (Y_t+i) for monthly data are reported in Table 6. The CE aggregates at all levels of aggregation except CE M2 are acyclical, whereas the simple sum aggregates are counter cyclical. Further the simple sum M1 and M2 lags the cycle by two months and M3 and L1 lags the cycle by one month. Table 7 reports the results for quarterly data. The results show an entirely different inference on the cyclical behaviour of monetary aggregates. The CE aggregates except CE M1 counter cyclically lags by two quarters. On the other hand, the simple sum M1 is contemporaneously procyclical and simple sum M2 and M3 procyclically lags by one quarter. The simple sum M3 also procyclically lags but by two quarters.

Table 6.

Cross Correlations of H-P Filtered Simple Sum and CE Monetary Aggregates (M_t) with Real IIP (Y_t+_i): Monthly Data

M_t	t – 6	t – 5	t – 4	t – 3	t – 2	t – 1	0	t + 1	t + 2	t + 3	t + 4	t + 5	t + 6
CE M1	–0.04	–0.09	–0.14	–0.12	–0.08	–0.08	–0.10	–0.09	–0.07	–0.03	–0.08	–0.11	–0.08
CE M2	–0.09	–0.13	–0.19*	–0.19*	–0.14	–0.09	–0.10	–0.09	–0.05	–0.05	–0.06	–0.09	–0.11
CE M3	–0.09	–0.11	–0.14	–0.13	–0.06	0.00	0.02	0.01	0.03	0.02	–0.02	–0.08	–0.12
CE L4	–0.09	–0.11	–0.14	–0.12	–0.05	0.01	0.03	0.01	0.04	0.02	–0.02	–0.08	–0.12
M1	0.14	0.14	0.10	0.02	0.00	–0.01	–0.07	–0.10	–0.15*	–0.08	–0.12	–0.13	–0.08
M2	0.06	0.01	–0.07	–0.17*	–0.22*	–0.25*	–0.31*	–0.34*	–0.36*	–0.29*	–0.29*	–0.26*	–0.19*
M3	0.00	–0.05	–0.10	–0.17*	–0.19*	–0.20*	–0.22*	–0.25*	–0.24*	–0.19*	–0.19*	–0.19*	–0.13
L1	0.01	–0.04	–0.09	–0.17*	–0.18*	–0.19*	–0.22*	–0.25*	–0.24*	–0.19*	–0.19*	–0.19*	–0.13

Source: Authors.

Notes: Statistically significant correlation coefficients with highest magnitude in absolute terms are highlighted with bold entries.^*_{indicates statistical significance at 5% level.}

Table 7.

Cross Correlations of H-P Filtered Simple Sum and CE Monetary Aggregates (M_t) with Real GDP (Y_t+_i): Quarterly Data

M_t	t – 4	t – 3	t – 2	t – 1	0	t + 1	t + 2	t + 3	t + 4
CE M1	–0.15	–0.10	0.00	–0.03	–0.08	–0.07	–0.11	–0.11	–0.08
CE M2	–0.08	–0.10	–0.13	–0.27	–0.40*	–0.38*	–0.40*	–0.40*	–0.35*
CE M3	–0.25	–0.26	–0.29*	–0.43*	–0.56*	–0.57*	–0.57*	–0.53*	–0.45*
CE L4	–0.26	–0.27	–0.29*	–0.44*	–0.56*	–0.57*	–0.57*	–0.53*	–0.45*
M1	0.52*	0.60*	0.71*	0.78*	0.75*	0.61*	0.40*	0.19	0.00
M2	0.04	0.16	0.28*	0.43*	0.53*	0.56*	0.48*	0.36*	0.25
M3	–0.22	–0.11	0.01	0.17	0.30*	0.40*	0.42*	0.40*	0.39*
L1	–0.20	–0.09	0.03	0.19	0.33*	0.42*	0.41*	0.37*	0.34*

Source: Authors.

Notes: Statistically significant correlation coefficients with highest magnitude in absolute terms are highlighted with bold entries.^* indicates statistical significance at 5% level.

There are considerable differences across simple sum and CE aggregates. The CE aggregates are acyclical in monthly frequency, whereas it is found to lag counter cyclically at quarterly frequency. On the other hand, the simple sum aggregates lag counter cyclically at monthly frequencies but are procyclical at quarterly frequencies.

Conclusion

This study reviews the theoretical foundations of the CE monetary aggregates proposed by Rotemberg et al. (1995) and documents some stylized facts on the performance of CE aggregates in application of policy interest. The CE monetary aggregates are interpreted as aggregation theoretic money stock measures by Barnett (1999) and Kelly (2009). CE aggregates measures the discounted present value of monetary services of monetary assets. But the simple sum aggregates exhibit considerable error in variables as it compounds non-monetary services with monetary services. This leads to erroneous inference regarding the relationship between money and other economic variables that are relevant to policy makers.

This study constructed CE aggregates and simple sum aggregates corresponding to official definitions of money, i.e. M1, M2, M3 and L1 as recommended by Third Working Group on Money Supply for India for the sample period April 1993 to June 2009. Quarterly estimates of CE aggregates are also obtained by taking quarterly averages of monthly aggregates so constructed. The empirical evidences are found to support the theoretical superiority of CE monetary aggregates over their corresponding simple sum aggregates. The trends in velocity of simple sum and CE aggregates show considerable difference. The velocity behaviour of CE aggregates seems to be more stable. The information context test also indicates CE aggregates in general contain more information than simple sum aggregates regarding output at quarterly frequencies and inflation at monthly and quarterly models. Similarly, cyclical behaviour of CE and simple sum aggregates exhibits considerable difference, proving that inference with CE aggregates is in sharp contrast to that with simple sum aggregates.

Footnotes

Acknowledgements

We like to thank Raja Sethu Durai S. for his comments.

Notes

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