JSPS Grant‐in‐Aid for Scientific Research (S)Central Bank Communication Design

working paper series
No.002 (September 2018)

The Demand for Money at the Zero
Interest Rate Bound

Tsutomu Watanabe
Tomoyoshi Yabu

Research Project on Central Bank Communication
702 Faculty of Economics, The University of Tokyo,
7‐3‐1 Hongo, Bunkyo‐ku, Tokyo 113‐0033, Japan
Tel: +81‐3‐5841‐5595 E‐mail: watlab@e.u‐tokyo.ac.jp
http://www.centralbank.e.u‐tokyo.ac.jp/en/

Working Papers are a series of manuscripts in their draft form that are shared for
discussion and comment purposes only. They are not intended for circulation or
distribution, except as indicated by the author. For that reason, Working Papers may not
be reproduced or distributed without the expressed consent of the author.

The Demand for Money at the Zero Interest Rate Bound∗
Tsutomu Watanabe†

Tomoyoshi Yabu‡

September 4, 2018

Abstract
This paper estimates a money demand function using US data from 1980 onward,
including the period of near-zero interest rates following the global financial crisis. We
conduct cointegration tests to show that the substantial increase in the money-income
ratio during the period of near-zero interest rates is captured well by the money demand
function in log-log form, but not by that in semi-log form. Our result is the oppositeof the result obtained by Ireland (2009), who, using data up until 2006, found that thesemi-log specification performs better. The diﬀerence in the result from Ireland (2009)
mainly stems from the diﬀerence in the observation period employed: our observation
period contains 24 quarters with interest rates below 1 percent, while Ireland’s (2009)
observation period contains only three quarters. We also compute the welfare cost of
inflation based on the estimated money demand function to find that it is very small:the welfare cost of 2 percent inflation is only 0.04 percent of national income, which isof a similar magnitude as the estimate obtained by Ireland (2009) but much smaller
than the estimate by Lucas (2000).
JEL Classification Numbers: C22; C52; E31; E41; E43; E52
Keywords: money demand function; cointegration; zero lower bound; near-zero interest
rates; welfare cost of inflation; log-log form; semi-log form; interest elasticity of money
demand∗We would like to thank Kosuke Aoki, Yoichi Arai, Hiroshi Fujiki, Andy Levin, and Keisuke Otsu for helpful discussions. This research forms part of the project on “Central Bank Communication Design” funded
by a JSPS Grant-in-Aid for Scientific Research (No. 18H05217). Tomoyoshi Yabu gratefully acknowledges
financial support from the JSPS through the Grant-in-Aid for Scientific Research No. 17K03663.
†
Graduate School of Economics, University of Tokyo. E-mail: watanabe@e.u-tokyo.ac.jp, Website:
https://sites.google.com/site/twatanabelab/.
‡
Faculty of Business and Commerce, Keio University. E-mail: tyabu@fbc.keio.ac.jp.

1

Introduction

In regression analyses of money demand functions, there is no consensus on whether thenominal interest rate as an independent variable should be used in linear or log form. For
example, Meltzer (1963), Hoﬀman and Rasche (1991), and Lucas (2000) employ a loglog specification (i.e., regressing real money balances (or real money balances relative tonominal GDP) in log on nominal interest rates in log), while Cagan (1956), Lucas (1988),
Stock and Watson (1993), and Ball (2001) employ a semi-log specification (i.e., nominalinterest rates are not in log).
Identifying empirically the proper specification of the money demand function is important for the following reasons. First, diﬀerent specifications of the money demand function
come from diﬀerent economic models of money demand. For example, the inventory theoretic approach to money demand by Baumol (1952), Tobin (1956), and Miller and Orr(1966) implies a money demand function in log-log form. Similarly, the transactions time
approach to money demand by McCallum and Goodfriend (1989) implies a money demand
function in log-log form. On the other hand, the money in the utility function approach implies a money demand function in log-log form if the utility function is of constant relativerisk aversion (Lucas (2000)) but a money demand function in semi-log form if the utility
function is quasi-linear (Cysne (2009)). By empirically specifying the functional form of
the money demand function, we can learn which economic model is more appropriate.
Second, diﬀerent specifications of the money demand function have diﬀerent implications for the welfare cost of inflation. Lucas (2000) shows that, with a log-log money
demand function, a slight increase in the nominal interest rate from the zero lower boundgenerates a substantial decline in real money balances. Therefore, there is a significant
welfare cost of deviating just slightly from the Friedman rule (Friedman (1969)). On theother hand, with a semi-log specification, a slight increase in the nominal interest rate fromzero does not change money demand that much, so that the costs associated with it aresmaller. Therefore, with a log-log specification, most of the benefits of reducing inflation
come when moving from zero inflation to the optimal deflation rate needed to achieve theFriedman rule, but this is not the case with a semi-log specification.

2

Third, diﬀerent specifications of the money demand function have diﬀerent implications
for the conduct of monetary policy near the zero lower bound. In the case of a log-log
specification, money demand becomes infinitely large at the zero lower bound, so thatthe opportunity cost of holding money cannot go below zero. On the other hand, with asemi-log money demand function, the marginal utility of money becomes zero at a finite
level of real money balances and becomes negative beyond that level.1 Rognlie (2016), in
the context of a negative interest rate policy, shows that, with a semi-log money demand
function, the negative opportunity cost of holding money created by the negative interest
rate policy may violate the Friedman rule in the opposite of the traditional direction, sothat such a policy could deteriorate economic welfare.
Using annual data for the United States for the period 1900-1994, Lucas (2000) showsthat a log-log specification fits the data more closely than a semi-log specification, and thatthe welfare cost of inflation is substantially large. For example, when the inflation rate is
10 percent, the welfare cost reaches 1.8 percent of national income. On the other hand,extending the observation period to 2006 so that observations in the period of near-zero
interest rates are included, Ireland (2009) shows that a semi-log specification performs
better than a log-log specification, and that the welfare cost of inflation is much smaller
than estimated by Lucas (2000).
Key to identifying which specification is more appropriate – a semi-log or a log-log
specification – is, as pointed out by Ireland (2009), the inclusion of a period of nearzero interest rates when estimating the money demand function, since the diﬀerence inmoney demand between the two specifications is extremely large near the zero lower bound.
However, the period of interest rates below 1 percent in Ireland’s (2009) observation period
is very short, comprising only three quarters (2003:Q3-2004:Q1), so that his results may
1

The marginal net benefits of holding money can become negative if the storage cost of holding money(especially the storage cost of holding cash) is non-negligible. Specifically, in the context of a negativeinterest rate policy, Rognlie (2016) and Eggertsson et al. (2017) consider the case in which the storage costof holding money is positive and the marginal cost associated with holding money is positive as well. In
this case, people hold money even if financial assets other than money yield a negative interest rate, so thatthe rate of interest can be negative in equilibrium. Note that if the marginal storage cost is an increasing
function of real money balances (i.e., the storage cost is a convex function of real money balances), there
is no lower bound on the nominal interest rate. This is in sharp contrast with the case discussed by Hicks(1937), who considers a situation in which the cost of holding money is negligible, so that the rate of interest
must always be non-negative.

3

not be very robust. In this paper, we extend the observation period up until 2013, so that
we have more observations with near-zero interest rates due to monetary easing after the
global financial crisis. Specifically, we update sweep-adjusted M1 up to 2013 and repeat
the exercises conducted by Ireland (2009). We show that inclusion of the more recent
observations with near-zero interest rates changes the results substantially.2
The rest of the paper is organized as follows. In Section 2, we visually compare the loglog and semi-log specifications using annual data. In Section 3, we conduct cointegration
tests, using quarterly data, for the money-income ratio and the nominal interest rate to
examine which of the two specifications performs better. In Section 4, we check the robustness of our baseline result in three respects: (1) we check how the result changes if we use
alternative measures for the opportunity cost of holding money; (2) we allow structural
breaks in the cointegration relationships; and (3) we relax the constraint that the income
elasticity of money demand is equal to unity. In Section 5, we discuss the welfare cost of
inflation based on the estimation result of the money demand function. Section 6 concludes
the paper.

2

Data overview

In this section, we conduct a visual examination of the relationship between money demand and the nominal interest rate. We use the same annual data from 1980 onward as
Ireland (2009). Specifically, the nominal interest rate is the six-month commercial paper
rate for 1980 to 1997 and the three-month AA nonfinancial commercial paper rate from1998 onward. The nominal money balances are M1 for 1980 to 1993, and the constituent
elements of M1 are currency held by the public, non-interest-bearing demand deposits, andinterest-bearing negotiable order of withdrawal (NOW) accounts. For 1994 and later, we
use retail sweep-adjusted M1 to avoid the influence of the introduction of the retail deposit
sweep programs (see Dutkowsky and Cynamon (2003)). We extend the observation period2

A study closely related to ours is that by Mogliani and Urga (2018), who examine the stability of themoney demand function using an annual data set that includes the recent near-zero interest rate period(1900-2013). They find structural breaks in 1945 and 1976 and show that, with these two structural breaks
taken into consideration, there exists a cointegration relationship between the money-income ratio and thenominal interest rate. However, they focus only on the log-log specification and do not conduct a log-log
versus semi-log comparison.

4

up to 2013, the latest year for which retail sweep-adjusted M1 is available.
Figure 1 shows a scatter plot of the US money-income ratio (sweep-adjusted M1 relative
to nominal GDP) and the nominal interest rate. The years examined by Ireland (2009),
1980 to 2006, are represented by the circles. The newly added years, 2007-2013, are shownby x-marks. As argued by Ireland (2009), as far as the data up to 2006 are concerned,
the money-income ratio seems to increase linearly as the interest rate declines and takea finite positive value at the zero interest rate bound, so that a semi-log specification
looks like the appropriate specification. However, the more recent observations show thatwhen the interest rate is closer to the zero lower bound, the money-income ratio increases
more elastically as the interest rate approaches zero from above, suggesting that a log-log
specification provides a better fit.
To conduct a more detailed comparison between the log-log and semi-log specifications,we plot the data into the semi-log graph in Figure 2(a) and into the log-log graph in Figure2(b). Figure 2(a) shows that there exists a linear relationship between the log of the moneyincome ratio and the interest rate until 2006, but once the more recent observations areadded, this linear relationship disappears. In this sense, Ireland’s (2009) finding that thesemi-log specification fits the data well is not robust to the addition of the more recent
data. On the other hand, Figure 2(b) shows that, for the period until 2006, there appears toexist a linear relationship between the two variables – although it does not look as straightas that in Figure 2(a) for the period until 2006 – and that the linear relationship appears
to survive even when adding the more recent observations. More specifically, the nominalinterest rate is very close to zero from the start of monetary easing in 2008 onward, sothat the dots for this period line up horizontally in Figure 2(a). In contrast, in Figure2(b), the interest rate in log continues to decline even from 2008 onward, so that the linear
relationship observed until 2006 remains unchanged.

3

Cointegration tests

In this section, we conduct cointegration tests to more rigorously examine the findings in theprevious section based on casual examination of the data. Ireland (2009) conducts residualbased cointegration tests using quarterly data (1980:Q1-2006:Q4) to see which of the two5specifications is supported by the data. Specifically, if the residual from a regression of thelog of the money-income ratio on the interest rate is stationary, this means that the twovariables are cointegrated. In this case, the semi-log specification is supported by the data.
On the other hand, if the residual from a regression of the log of the money-income ratioon the log of the nominal interest rate is stationary, the log-log specification is accepted.
We follow this approach and examine in which specification a cointegration relationshipexists using the data up until 2013:Q4.
Let m denote the ratio of retail sweep-adjusted M1 divided by nominal GDP and r thethree-month US Treasury bill (TB) rate. For nominal GDP, we use the figures with baseyear 2009 rather than those with base year 2000. We start by conducting unit root tests –i.e., the augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests – for ln(m), ln(r),and r with only a constant term included to find that the null hypothesis of a unit root is
not rejected for all of the three variables. Given this result, we examine in the rest of thissection whether there exists a cointegration relationship between the variables.
We regress ln(m) on a constant and ln(r) or r to see whether the residual obtained isstationary or not.3 We employ three diﬀerent cointegration tests: the ADF test proposed
by Engle and Granger (1987); the PP test proposed by Phillips and Ouliaris (1990) withtest statistics given by Zt ; and the same PP test but with test statistics given by Zα . Somesimulation studies show that the ADF test has the least size distortions and is more reliable
than the PP tests, but that the Zα test has more power than the ADF and Zt tests (seeHaug (1996)). Based on this, Haug (1996) proposes to employ more than one cointegration
test in applied research.4 Given that the diﬀerent tests each have their relative advantages
and disadvantages, it is useful to present the results from the three tests and compare
3

Note that we assume here that the income elasticity of money demand is unity. Some previous studies
using US annual data, such as Meltzer (1963), Lucas (1988, 2000), and Stock and Watson (1993), showthat the income elasticity of money demand is close to unity. While we assume in this section that income
elasticity is unity, we will relax this assumption in the next section to see how much the result depends onit.
4
The finite sample properties of residual-based cointegration tests are similar to those of unit root tests.
Several studies show that the PP tests have more size distortions than the ADF test when MA errors havea negative autocorrelation (see Phillips and Perron (1988)) and that the ADF test has better size andpower when the errors have an AR structure (see DeJong et al. (1992)). This suggests that the PP tests areless reliable than the ADF test. On the other hand, some simulation studies suggest that the size-adjusted
power of the PP tests is higher than the power of the ADF test (see Campbell and Perron (1991))

6them. Note that Ireland (2009) reports the result of the PP test with the test statistics
given by Zt but does not discuss the other two tests.
Let us start by reproducing Ireland’s (2009) result using the same observation period(1980:Q1-2006Q4). Table 1 shows the test statistics associated with the ADF, Zt , and Zα
tests together with the static OLS estimates of the cointegrating vector, namely (α, β).5 In
the table, q represents the lag length in the case of the ADF test, and the optimal value ofq is chosen based on the Akaike information criterion (AIC). In the case of the PP tests,
q represents the truncation parameter to compute the long-run variance, and the optimalvalue of q is chosen based on Andrews’ (1991) plug-in method. Superscript a indicates the
optimal value of q in the ADF test and superscript b is the optimal value of q in the PP
tests.
The estimation results for the semi-log specification show that the null hypothesis of no
cointegration is rejected by the ADF and Zt tests but not by the Zα test at a significancelevel of 10 percent. In contrast, the results for the log-log specification show that the null
hypothesis is rejected by the ADF test at the 5 percent significance level but not by the
PP tests. These results suggest that the semi-log specification is better than the log-log
specification but the diﬀerence between the two is not that substantial, which is consistent
with what we saw in Figure 2.
As pointed out by Ireland (2009), the diﬀerence in ln(m) between the semi-log and the
log-log specification is larger when the nominal interest rate comes closer to the zero lower
bound. This suggests that it may be diﬃcult to distinguish which of the two specifications
provides a better fit unless the observation period contains a suﬃciently long period ofnear-zero interest rates. While the data used by Ireland (2009) contain a period in whichinterest rates are below 1 percent, the period is very brief, consisting of only three quarters.
This may explain why we do not obtain clear-cut results in Table 1. To see if this is thecase, we conduct the same exercise but now use an observation period with a much longer
period of near-zero interest rates.
Table 2 shows the results with the observation period extended until 2013:Q4. The
5

Note that the base year for the nominal GDP data we use in this paper is 2005, while the base year
for the nominal GDP data used by Ireland (2009) is 2000. Mainly due to this diﬀerence, our results diﬀerslightly from those presented in Table 2 in Ireland (2009).

7

results for the semi-log specification show that the null hypothesis is no longer rejectedby any of the three tests, indicating that there does not exist a cointegration relationshipbetween ln(m) and r. The test statistics are all close to zero in absolute value regardlessof the value of q. In contrast, the results for the log-log specification indicate that the test
statistics are much larger in absolute value than in the case of the semi-log specification.
In fact, the ADF test rejects the null hypothesis at a significance level of 1 percent. Taken
together, the results in Table 2 indicate that the log-log specification performs better than
the semi-log specification in the sense that it better captures the relationship between
the two variables, especially during the period of near-zero interest rates. This again isconsistent with what we saw in Figure 2.
Table 2 also presents the static OLS estimates of the cointegrating vector (α, β). For
the log-log specification, the constant is −2.0893, and the interest rate elasticity is estimated to be 0.0551. However, static OLS estimators may not be very eﬃcient, so that we
estimate cointegrating vector (α, β) based on dynamic OLS rather than static OLS. Specifically, we run a similar OLS regression but now add ∆ ln(rt−p ), . . . , ∆ ln(rt−1 ), ∆ ln(rt ),∆ ln(rt+1 ), . . . , ∆ ln(rt+p ) as explanatory variables. Note that, as long as the variables
are cointegrated, dynamic OLS estimates are asymptotically eﬃcient and conventionalt-statistics obtained from dynamic OLS estimation have conventional normal asymptotic
distributions (see Stock and Watson (1993) and Saikkonen (1991)). The estimation results
are shown in Table 3, where std(β̂) represents the HAC standard error of β̂ and is obtainedfor various values of q. The estimates of the cointegrating vector (α, β) are almost the sameas in Table 2: the constant α ranges from −2.0897 to −2.0961, and interest rate elasticity
β takes a value in the range from 0.0585 to 0.0571. The estimated value of the interest
rate elasticity is small, but the standard error associated with it is also small, so that theestimate is significantly diﬀerent from zero. Note that the interest rate elasticity estimatedhere is much smaller than 0.5, which is the estimate reported by Lucas (2000).
Finally, using a graph, let us check how well the log-log specification captures therelationship between the two variables. Figure 3 presents a scatter plot using quarterly
data. The thick and thin lines respectively represent the fitted values for the log-log andthe semi-log specification, which are calculated using the estimates of α and β in Table
8

1. It can be clearly seen that money demand increases substantially as the interest rate
approaches the zero lower bound, but the semi-log specification fails to capture this. In
contrast, the log-log specification performs well both in high and in near-zero interest rate
periods.

4

Robustness checks

In this section, we check the robustness of the results obtained in the previous section. We
check, first, how the results change when we use alternative measures for the opportunity
cost of holding money. Second, we allow structural breaks in the cointegration relationships.
Third, we relax the constraint that the income elasticity of money demand is unity.

4.1

Alternative measure for the opportunity cost of holding money

The opportunity cost of holding money is measured by the interest rates associated withfinancial assets that are outside the definition of money but are close to money. In our
exercise, we use M1 as money, so in the previous section we used the three-month TB rate
as a measure of the opportunity cost of holding money. However, there are many othercandidates that would be appropriate as a measure of the opportunity cost of holding
money. Also, it should be noted that when short-term interest rates are suﬃciently close to
the zero lower bound, people may make decisions on how much money to hold by comparing
the marginal utility of money and medium- or long-term interest rates, which are still along from the zero lower bound. If this is the case, we should use medium- or long- terminterest rates as a measure of the opportunity cost of holding money. In this subsection,
we replace the three-month TB rate with the three-year Treasury constant maturity rate
and repeat the exercise.
Figure 4 shows a scatterplot of the annual data for 1980-2013 with the money-income
ratio on the horizontal axis and the three-year rate on the vertical axis. The money-income
ratio on the horizontal axis is in log in both Figures 4(a) and 4(b), while the three-year
rate on the vertical axis is not in log in Figure 4(a) but in log in Figure 4(b). It can beclearly seen that there is no linear relationship between the two variables in Figure 4(a),indicating that the semi-log specification does not work well. On the other hand, there9appears to exist a linear relationship between the two variables in Figure 4(b), suggestingthat the log-log specification performs well.
To confirm this visual observation, we conduct cointegration tests using the quarterly
data for 1980:Q1-2013:Q4. Table 4 shows that, for the semi-log specification, the three test
statistics are all close to zero in absolute value, and the null hypothesis of no cointegration
cannot be rejected. On the other hand, for the log-log specification, the ADF and Zα
statistics are much larger in absolute value than in the case of the semi-log specification,and the null hypothesis is rejected. As for the Zt statistic, this is larger in absolute value
than that obtained in the previous section, but the null still cannot be rejected. Overall,
we conclude that the results in the previous section are robust to switching the interest
rate variable from the three-month TB rate to the three-year Treasury constant maturity
rate.

4.2

Allowing for the possibility of structural breaks

The analysis in the last section shows that, as long as we employ the observation periodused by Ireland (2009), 1980:Q1-2006:Q4, the money-income ratio and the nominal interest rate are cointegrated when the semi-log specification is used, but this cointegration
relationship disappears once we switch to the longer observation period, 1980:Q1-2013:Q4.
This raises the question why this happens. One possibility is the presence of structural
breaks associated with the cointegrating vector. Specifically, the values of α and β in the
recent period with near-zero interest rates may diﬀer from those in the earlier period.
Gregory, Nason, and Watt (1996) and Gregory and Hansen (1996) point out that if
one applies a simple cointegration test when some variables are actually cointegrated withstructural breaks, then one obtains a result biased toward accepting the null of no cointegration. To investigate this issue in more detail, we employ the tests proposed by Gregoryand Hansen (1996), which can detect a cointegrating relationship allowing for a structural
break in the intercept and the slope. Specifically, following the procedure proposed by Gregory and Hansen (1996), we (1) consider the middle 70 percent of observations as potential
candidates for the time that a structural break may have occurred, which we denote by
TB ; (2) obtain the residual from a regression of ln mt on a constant and Dt , rt (or ln rt ),10and Dt × rt (or Dt × ln rt ), where Dt is 1 for t > TB and zero otherwise; (3) conduct unit
root tests for the residual and obtain the test statistics ADF(TB ), Zt (TB ), and Zα (TB );and (4) search for the minimal values of these test statistics over all possible break points.
The test statistics obtained in this way are denoted by Inf-ADF, Inf-Zt , and Inf-Zα .
Table 5 shows the values of Inf-ADF, Inf-Zt , and Inf-Zα for the semi-log and log-log
specifications. Starting with the results for the semi-log specification, the test statistics
are much larger in absolute value than in Table 2, suggesting structural breaks may bepresent. However, we still cannot reject the null hypothesis, implying that we fail to detecta cointegration relationship even when allowing for the possibility of a structural break.6
Turning to the log-log specification, we find that the test statistics do not change verymuch from those in Table 2, implying that the possibility of a structural break is negligible
in the case of the log-log specification. The result from the ADF test indicates that thenull hypothesis of no cointegration is rejected at the 10 percent significance level.7

4.3

No restriction on income elasticity

The cointegration tests we conducted in the last section are based on the assumption that
income elasticity is unity. In this subsection, we relax this assumption and examine thecointegration relationship among three variables, i.e., real money balances, real income,and nominal interest rates.
Let M denote retail sweep-adjusted M1, Y the nominal GDP, P the GDP deflator,
and r the three-month TB rate. We conduct cointegration tests for M/P , Y /P , and r.
Specifically, we regress ln(M/P ) on a constant, ln(Y /P ), and ln(r) or r, and check whether
the residual is stationary or not. Table 6 shows the three test statistics as well as theestimates for α, β y , and β r , where α is a constant, β y is the coeﬃcient on ln(Y /P ), and β r
is the coeﬃcient on ln(r) or r.
For the semi-log specification, the income elasticity turns out to be 1.0489, which is close
to unity. All of the three test statistics indicate that the null hypothesis of no cointegration
6
The date associated with the minimal value of the test statistics is 2008:Q4, when the three month TB
rate fell below 1 percent (from 1.5% in 2008:Q3 to 0.3% in 2008:Q4).
7
Note that the significance level associated with the ADF test is lower than that presented in Table 2.
This may be due to a loss of the statistical power of cointegration tests when applying the Gregory-Hansen
test to the case in which there exists a cointegration relationship with no breaks.

11

is not rejected. For the log-log specification, the estimate of the income elasticity is 1.0666,
which is again quite close to unity. The three test statistics are all larger in absolute value
than in the case of the semi-log specification, and the test statistics associated with the
ADF test indicates that the null hypothesis is rejected at the 5 percent significance level.
Overall, the results are essentially the same as those in the last section, indicating that thefindings are robust to changes in the specification of the estimation equation.

5

Welfare cost of inflation

In this section, we calculate the welfare cost of inflation using the parameter estimatesobtained in Section 3 and compare our results with those reported in previous studies
such as Lucas (2000) and Ireland (2009). Table 7 shows the estimation results.8 In thiscalculation, we assume that the real interest rate at the steady state is 3 percent. For
example, r = 0.05 means that the inflation rate at the steady state is 2 percent, which isthe current target level set by the Fed. We start by reproducing the results by Lucas (2000)
and Ireland (2009). Lucas (2000) uses annual data for 1900-1994 to obtain α = −1.036
and β = 7 for the semi-log specification and α = −3.020 and β = 0.500 for the log-log
specification. The welfare cost of 2 percent inflation is 0.25 percent of national income in thecase of the semi-log specification and 1.09 percent in the case of the log-log specification.
These results indicate that the welfare cost of inflation is not negligible even if the inflation
rate is only 2 percent, especially in the case of the log-log specification.
Turning to the result by Ireland (2009), we use the values for α and β estimatedusing the quarterly data for the period 1980:Q1-2006:Q4, which are shown in Table 1,
to reproduce his result. As shown in the middle two rows of Table 7, the welfare costof 2 percent inflation is now reduced to 0.03 percent of national income in the case of
the semi-log specification and 0.06 percent of national income in the case of the log-log
specification.
Finally, the welfare cost calculated based on our estimates for α and β is shown in
8Lucas (2000) shows, based on Bailey (1956), that the welfare[ cost of inflation,] w(r), is given byw(r) = eα (β/(1 − β)) r1−β for the log-log specification and w(r) = eα 1 − (1 + βr)e−βr /β for the semi-log
specification. Our calculation is based on these results.

12the bottom two rows of Table 7. The results indicate that the welfare cost of 2 percent
inflation is 0.04 percent in the case of the semi-log specification and 0.04 percent in the caseof the log-log specification. Our welfare cost estimates are of almost the same size as thoseobtained for the shorter observation period used by Ireland (2009), suggesting that, as longas we use the data from 1980 onward, the estimated welfare cost of inflation is very small,irrespective of whether the recent period with near-zero interest rates is included or not.9
This is in sharp contrast with the result obtained by Lucas (2000), whose observations on
the money-interest rate relationship include the period before 1980.
Figure 5(a) compares our estimates on the welfare cost of inflation with those obtainedby Lucas (2000) and Ireland (2009). In the figure, the horizontal axis represents the nominalinterest rate, while the vertical axis shows the estimated welfare cost of inflation. Onceagain, our estimate based on the log-log specification is much smaller than the estimate byLucas (2000). Next, to compare the results we obtain based on the log-log and the semilog specification more closely, Figure 5(b) focuses on the results for interest rates below
2 percent. The figure indicates that the estimate based on the semi-log specification is aconvex function of r, while the estimate based on the log-log specification is a concave
function of r. More interestingly, when r declines from 2 to 1 percent, w(r) falls by 0.008
percentage points in the case of the log-log specification and by 0.006 percentage points inthe case of the semi-log specification, so that the changes in w(r) associated with a decline inr from 2 to 1 percent are of almost the same size in the two cases. However, when r declines
from 1 to 0 percent, w(r) in the case of the log-log specification falls by 0.009 percentage
points, which is even greater than the welfare improvement from 2 to 1 percent, but w(r)falls only by 0.002 percentage points in the case of the semi-log specification. This diﬀerence
between the log-log and the semi-log specification – that is, that the welfare improvement
associated with a decline in interest rates close to zero is larger in the case of the log-log
than the semi-log specification – has been highlighted in previous studies such as Lucas(2000) and Wolman (1997). This diﬀerence between the two specifications arises because
in the log-log specification money demand increases substantially as the interest rate falls
9This is consistent with the result obtained by Mogliani and Urga (2018), who used annual data for
1976-2013 to find that the welfare cost of inflation is very small at 0.06-0.14 percent of national income inthe case of 2 percent inflation.

13

from 1 to zero percent, but such an increase in money demand does not occur in the caseof the semi-log specification. An important implication of this diﬀerence is that it would
make more sense for the central bank to reduce inflation and thus the nominal interest
rate from 1 to zero percent, which corresponds to the optimal rate of deflation under theFriedman rule, if money demand follows a log-log functional form than a semi-log one.

6

Conclusion

Identifying the proper specification of the money demand function has important implications for the welfare cost of inflation and the conduct of monetary policy. However,in regression analyses of money demand functions, there is no consensus on whether thenominal interest rate as an independent variable should be used in linear or log form.
Specifically, Ireland (2009) showed that the semi-log specification performs better thanthe log-log specification, which stands in sharp contrast with the result obtained by Lucas(2000).
In this paper, we examined the robustness of Ireland’s (2009) results by extending
the observation period so that it includes the recent period of near-zero interest rates.
We showed through simple data plotting and formal cointegration tests that the log-log
specification performs better than the semi-log specification. Specifically, we showed thatthe semi-log specification cannot account for the substantial increase in the money-income
ratio during the period of near-zero interest rates since 2008, while the log-log specification
can.
Our result on the shape of the money demand function has important implications
for the conduct of monetary policy at the zero lower bound. A money demand functionthat takes a semi-log form implies that the marginal utility of money reaches zero at a
finite value of real money balances and becomes negative beyond that level. In this case,the opportunity cost of holding money can go below zero, and in this sense there is nolower bound on nominal interest rates, as shown by Rognlie (2016). However, our result
indicates that the marginal utility of money approaches zero as the opportunity cost ofholding money falls, but it never reaches zero. Therefore, the opportunity cost of holding
money cannot go below zero, which constrains the conduct of monetary policy.
14

We also computed the welfare cost of inflation based on our estimates of the key parameter values for the money demand function to find that the welfare cost of 2 percent
inflation is very small, suggesting that the current inflation target of 2 percent set by theFederal Reserve does not create substantial distortions in the economy.
In this paper, we were able to extend the observation period only up to 2013:Q4 due
to data limitations. However, by further extending the observation period, we may beable to learn more about the shape of the money demand function near the zero lower
bound. In addition, it would be instructive to conduct similar analyses using data fromother countries that experienced near-zero interest rates over a long period of time, such
as Japan. We leave these tasks for future research.

References[1] Andrews, Donald W. K. 1991. “Heteroskedasticity and Autocorrelation Consistent
Covariance Matrix Estimation.” Econometrica, 59(3): 817-858.
[2] Andrews, Donald W. K., and Christopher J. Monahan. 1992. “An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator.” Econometrica, 60(4): 953-966.
[3] Bailey, Martin J. 1956. “The Welfare Cost of Inflationary Finance.” Journal of Political
Economy, 64(2): 93-110.
[4] Ball, Laurence. 2001. “Another Look at Long-Run Money Demand.” Journal of Monetary Economics, 47(1): 31-44.
[5] Baumol, William J. 1952. “The Transactions Demand for Cash: An Inventory Theoretic Approach.” Quarterly Journal of Economics, 66(4): 545-556.
[6] Cagan, Phillip. 1956. “The Monetary Dynamics of Hyperinflation.” In Milton Friedman, ed., Studies in the Quantity Theory of Money, 25-117. Chicago: University of
Chicago Press.

15[7] Campbell, John Y., and Pierre Perron. 1991. “Pitfalls and Opportunities: What
Macroeconomists Should Know about Unit Roots.” NBER Macroeconomics Annual
6: 141-201.
[8] Cysne, Rubens Penha. 2009. “Bailey’s Measure of the Welfare Costs of Inflation as a
General-Equilibrium Measure.” Journal of Money, Credit and Banking, 41(2-3): 451459.
[9] DeJong, David N., John C. Nankervis, N. E. Savin, and Charles H. Whiteman. 1992.
“The Power Problems of Unit Root Test in Time Series with Autoregressive Errors.”
Journal of Econometrics, 53(1-3): 323-343.
[10] Dutkowsky, Donald H., and Barry Z. Cynamon. 2003. “Sweep Programs: The Fall of
M1 and Rebirth of the Medium of Exchange.” Journal of Money, Credit and Banking,35(2): 263-79.
[11] Eggertsson, Gauti B., Ragnar E. Juelsrud, and Ella Getz Wold. 2017. “Are Negative
Nominal Interest Rates Expansionary?” NBER Working Paper No. 24039. National
Bureau of Economic Research.
[12] Engle, Robert F., and Clive W. J. Granger. 1987. “Co-Integration and Error Correction: Representation, Estimation, and Testing.” Econometrica, 55(2): 251-76.
[13] Friedman, Milton. 1969. “The Optimum Quantity of Money.” In Milton Friedman,
The Optimum Quantity of Money and Other Essays, 1-50. Chicago: Aldine Publishing
Company.
[14] Gregory, Allan W., and Bruce E. Hansen. 1996. “Residual-Based Tests for Cointegration in Models with Regime Shifts.” Journal of Econometrics, 70(1): 99-126.
[15] Gregory Allan W., James M. Nason, and David G. Watt. 1996. “Testing for Structural
Breaks in Cointegrated Relationships.” Journal of Econometrics, 71(1-2): 321-341.
[16] Haug, Alfred A. 1996. “Tests for Cointegration: A Monte Carlo Comparison.” Journalof Econometrics, 71(12): 89-115.
16

[17] Hicks, John R. 1937. “Mr. Keynes and the “Classics”: A Suggested Interpretation.”
Econometrica, 5(2): 147-159.
[18] Hoﬀman, Dennis L., and Robert H. Rasche. 1991. “Long-Run Income and Interest
Elasticities of Money Demand in the United States.” Review of Economics and Statistics, 73(4): 665-674.
[19] Ireland, Peter N. 2009. “On the Welfare Cost of Inflation and the Recent Behavior of
Money Demand.” American Economic Review, 99(3): 1040-52.
[20] Jobst, Clemens, and Helmut Stix. 2017. “Doomed to Disappear? The Surprising Return of Cash Across Time and Across Countries,” CEPR Discussion Papers No. 12327.
Centre for Economic Policy Research.
[21] Lucas, Robert E., Jr. 1988. “Money Demand in the United States: A Quantitative
Review.” Carnegie-Rochester Conference Series on Public Policy, 29: 137-167.
[22] Lucas, Robert E., Jr. 2000. “Inflation and Welfare.” Econometrica, 68(2): 247-74.
[23] McCallum, Bennett T., and Marvin S. Goodfriend. 1989. “Demand for Money: Theoretical Studies.” In John Eatwell, Murray Milgate, and Peter Newman, eds., Money,
117-130. London: Palgrave Macmillan.
[24] Meltzer, Allan H. 1963. “The Demand for Money: The Evidence from the Time Series.”
Journal of Political Economy, 71(3): 219-46.
[25] Miller, Merton H., and Daniel Orr. 1966. “A Model of the Demand for Money byFirms.” Quarterly Journal of Economics, 80(3): 413-435.
[26] Mogliani, Matteo, and Giovanni Urga. 2018. “On the Instability of Long-run Money
Demand and the Welfare Cost of Inflation in the United States. ” Journal of Money,
Credit and Banking. doi:10.1111/jmcb.12480 (accessed September 2, 2018).
[27] Newey, Whitney K., and Kenneth D. West. 1987. “A Simple, Positive Semi-Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica,
55(3): 703-08.
17[28] Phillips, Peter C. B., and S. Ouliaris. 1990. “Asymptotic Properties of Residual Based
Tests for Cointegration.” Econometrica, 58(1): 165-93.
[29] Phillips, Peter C. B., and Pierre Perron. 1988. “Testing for a Unit Root in Time Series
Regression.” Biometrika, 75(2): 335-346.
[30] Rognlie, Matthew. 2016. “What Lower Bound? Monetary Policy with Negative Interest Rates.” http://mattrognlie.com/negative_rates.pdf (accessed September 2,2018).
[31] Saikkonen, Pentti. 1991. “Asymptotically Eﬃcient Estimation of Cointegration Regressions.” Econometric Theory, 7(1): 1-21.
[32] Stock, James H., and Mark W. Watson. 1993. “A Simple Estimator of Cointegrating
Vectors Order Integrated Systems.” Econometrica, 61(4): 783-820.
[33] Tobin, James. 1956. “The Interest-Elasticity of Transactions Demand for Cash.” Review of Economics and Statistics, 38(3): 241-247.
[34] Wolman, Alexander L. 1997. “Zero Inflation and the Friedman Rule: A Welfare Comparison.” Federal Reserve Bank of Richmond Economic Quarterly, 83(4): 1-20.

18

Table 1: Cointegration tests for 1980:Q1-2006:Q4
Semi-log function
ln(m) = α − βrα̂-1.8259β̂
1.6114

q
0
1
2
3
4
5
6
7
8

ADF
-3.032-2.521-2.305-3.801**a-3.136*-4.935***-4.623***-3.210*-3.931**Zt-3.032-3.133*-3.130*b-3.238*-3.268*-3.327*-3.410**-3.418**-3.413**Zα-14.551-15.881-15.840b-17.288*-17.700*-18.516*-19.672*-19.787*-19.716*

Log-log function
ln(m) = α − β ln(r)α̂-2.1536β̂
0.0777

q
0
1
2
3
4
5
6
7
8

ADF
-1.890-2.277-2.646-3.804**a-3.294*-3.728**-4.224***-3.356*-3.192*Zt-1.890-2.078-2.218-2.394b-2.494-2.574-2.653-2.683-2.686Zα-6.871-8.363-9.566-11.189b-12.169-12.985-13.805-14.131-14.163

Note: The test statistics Zt and Zα are computed using the Newey-West estimate of
the long-run variance (Newey and West (1987)). ***, ** and * indicate that the null
hypothesis of no cointegration can be rejected at the 1%, 5%, and 10% level, respectively.
“a” indicates the lag chosen by the AIC, while “b” indicates the closest integer to thevalue chosen by the plug-in method of Andrews (1991).

19

Table 2: Cointegration tests for 1980:Q1-2013:Q4
Semi-log function
ln(m) = α − βrα̂-1.7777β̂2.2763Log-log function
ln(m) = α − β ln(r)α̂-2.0893β̂
0.0551

q
0
1
2
3
4
5
6
7
8
q
0
1
2
3
4
5
6
7
8

ADF
-1.197-0.889-0.622-2.279-1.253-2.510-1.955-0.602-1.977aADF-1.949-2.141-2.702-2.710-3.095*-2.730-3.369*-4.195***a-4.290***Zt-1.197-1.421-1.433-1.554b-1.566-1.615-1.699-1.699-1.705Zt-1.949-2.058-2.240b-2.339-2.44-2.475-2.537-2.605-2.663Zα-4.948-6.333-6.414-7.234b-7.315-7.665-8.277-8.278-8.322Zα-9.153-10.049-11.645b-12.563-13.551-13.895-14.523-15.236-15.850

Note: The test statistics Zt and Zα are computed using the Newey-West estimate of
the long-run variance (Newey and West (1987)). ***, ** and * indicate that the null
hypothesis of no cointegration can be rejected at the 1%, 5%, and 10% level, respectively.
“a” indicates the lag chosen by the AIC, while “b” indicates the closest integer to thevalue chosen by the plug-in method of Andrews (1991).

20

Table 3: Dynamic OLS
α̂-2.0959

β̂
0.0578-2.0961

0.0583-2.09480.0585-2.0897

0.0571std(β̂)0.00350.00420.00470.00510.00340.00410.00460.00490.00330.00410.00450.00480.00380.00460.00520.0056

p
1

2

3

4

q
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8

Note: The regression equation is ln(m) = α − β ln(r). The observation period is 1980:Q1-2013:Q4. The dynamic OLS estimates for α
and β are shown for various values of the lead/lag parameter p. The
standard errors of β̂, denoted by std(β̂), are computed using the
Newey-West estimate of the long-run variance for various values of
the truncation parameter q (Newey and West (1987)).

21

Table 4: Alternative measure for the opportunity cost of holding money
Semi-log function
ln(m) = α − βrα̂-1.7555β̂2.2396

q
0
1
2
3
4
5
6
7
8

ADF
-0.657-1.093a-1.422-2.501-1.194-1.652-1.798-1.268-2.095Zt-0.657-0.992-1.160-1.354-1.394b-1.399-1.417-1.396-1.388Zα-2.507-4.199-5.166-6.389-6.657b-6.690-6.815-6.674-6.620Log-log function
ln(m) = α − β ln(r)α̂-2.1995

β̂
0.0999

q
0
1
2
3
4
5
6
7
8

ADF
-2.089-2.979-2.766-4.137***a-3.308*-3.400**-3.319*-3.491**-3.710**Zt-2.089-2.448-2.546-2.748-2.838b-2.844-2.846-2.836-2.815Zα-11.106-14.495-15.492-17.681*-18.713*b-18.778*-18.801*-18.687*-18.447*

Note: The three-year Treasury constant maturity rate is used as a measure for the opportunity cost of holding money. The observation period is 1980:Q1-2013:Q4. The test
statistics Zt and Zα are computed using the Newey-West estimate of the long-run variance (Newey and West (1987)). ***, ** and * indicate that the null hypothesis of no
cointegration can be rejected at the 1%, 5%, and 10% level, respectively. “a” indicatesthe lag chosen by the AIC, while “b” indicates the closest integer to the value chosen by
the plug-in method of Andrews (1991).

22

Table 5: Structural breaks

Semi-log specification
Log-log specification

Inf-ADF-4.521-4.875*

Inf-Zt-3.295-2.678

Inf-Zα-21.222-16.109

Note: The three test statistics (Inf-ADF, Inf-Zt , and Inf-Zα ) are computed
using the procedure proposed by Gregory and Hansen (2006). ***, ** and* indicate that the null hypothesis of no cointegration is rejected at the 1%,5%, and 10% level, respectively. For the ADF test, the lag length is chosen
based on a sequential t-test with the maximum lag length set to 8. For the
PP tests, the long-run variance is estimated using a prewhitened quadratic
spectral kernel based on Andrews (1991) and Andrews and Monahan (1992).

23

Table 6: No restriction on income elasticity

Semi-log specificationln(M/P ) = α + β y ln(Y /P ) − β r r

α̂-2.0221βˆy1.0489βˆr1.9354Log-log specificationln(M/P ) = α + β y ln(Y /P ) − β r ln(r)α̂-2.3691βˆy1.0666βˆr
0.0466

q
0
1
2
3
4
5
6
7
8
q
0
1
2
3
4
5
6
7
8

ADF
-0.919-0.740-0.684-2.287a-1.263-2.399-1.877-0.781-2.112ADF-1.785-2.038-2.839-2.980-3.219-2.964-3.623*-4.254**a-4.259**Zt-0.919-1.165-1.224-1.372b-1.409-1.471-1.556-1.569-1.585Zt-1.785-1.949-2.182b-2.324-2.446-2.502-2.576-2.641-2.684Zα-3.479-4.792-5.142-6.049b-6.296-6.699-7.279-7.378-7.489Zα-8.174-9.458-11.439b-12.759-13.940-14.514-15.275-15.964-16.456

Note: The observation period is 1980:Q1-2013:Q4. The test statistics Zt and Zα are computed
using the Newey-West estimate of the long-run variance (Newey and West (1987)). ***, ** and *indicate that the null hypothesis of no cointegration can be rejected at the 1%, 5%, and 10% level,
respectively. “a” indicates the lag chosen by the AIC, while “b” indicates the closest integer to thevalue chosen by the plug-in method of Andrews (1991).

24

Table 7: Welfare cost of inflation

Observation period
1900-1994
1980:Q1-2006:Q4
1980:Q1-2013:Q4

Functional form
semi-log
log-logsemi-log
log-logsemi-log
log-log

Parameter values
αβ-1.0367.000-3.0200.500-1.8261.611-2.1540.078-1.7782.276-2.0890.055

r = 0.13
1.17%
1.76%
0.19%
0.15%
0.27%
0.10%

Welfare costr = 0.05 r = 0.03
0.25%
0.10%
1.09%
0.85%
0.03%
0.01%
0.06%
0.04%
0.04%
0.02%
0.04%
0.03%

Note: It is assumed that the real interest rate at the steady state is 3 percent, so thatr = 0.03 corresponds to price stability, r = 0.05 to 2 percent inflation, and r = 0.13 to 10
percent inflation. The welfare costs are computed using equations (3) and (4) in Ireland(2009) for the log-log and the semi-log specification, respectively . The parameter valuesof α and β are taken from Lucas (2000) for the period 1900-1994, from Table 1 for theperiod 1980:Q1-2006:Q4, and from Table 2 for the period 1980:Q1-2013:Q4.

25

Figure 1: Money Demand, 1980-2013

0.16
US Data, 1980-2006

0.14

US Data, 2007-2013

nominal interest rate

0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.1

0.12

0.14
money/income

0.16

0.18

0.2

Figure 2: Semi-log vs. Log-log Plots(a) semi-log plot
0.16
0.14

US Data, 1980-2006
US Data, 2007-2013

nominal interest rate

0.12
0.10
0.08
0.06
0.04
0.02
0.00
-2.2

-2.1

-2.0

-1.9
ln(money/income)

-1.8

-1.7

-1.6

(b) log-log plot
-1

-2

US Data, 1980-2006

ln(nominal interest rate)

US Data, 2007-2013

-3

-4

-5

-6

-7

-8
-2.2

-2.1

-2.0

-1.9
ln(money/income)

-1.8

-1.7

-1.6

Figure 3: Estimated Money Demand Functions

0.16

0.14
semi-log

nominal interest rate

0.12
log-log
0.10

0.08

0.06

0.04

0.02

0.00
0.1

0.12

0.14
money/income

0.16

0.18

0.2

Figure 4: 3-Year Treasury Constant Maturity Rate(a) semi-log plot
0.16
0.14

U.S. Data, 1980-2006
U.S. Data, 2007-2013

nominal interest rate

0.12
0.10
0.08
0.06
0.04
0.02
0.00
-2.2

-2.1

-2.0

-1.9
ln(money/income)

-1.8

-1.7

-1.6

(b) log-log plot
-1

U.S. Data, 1980-2006

ln(nominal interest rate)-2

U.S. Data, 2007-2013

-3

-4

-5

-6
-2.2

-2.1

-2.0

-1.9
ln(money/income)

-1.8

-1.7

-1.6

Figure 5: Welfare Cost of Inflation
0.6%

(a)

This study: semi-log
This study: log-log
Lucas: log-log
Ireland: semi-log

0.5%

w(r)

0.4%

0.3%

0.2%

0.0%

0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
17%
18%
19%
20%

0.1%

r
0.020%(b)

This study: semi-log
This study: log-log

w(r)0.015%

Ireland: semi-log

0.010%

0.005%

0.000%
0.0%

0.5%

1.0%

r

1.5%

2.0%