The welfare costs of inflation
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Benati, Luca; Nicolini, Juan Pablo Working Paper The welfare costs of inflation Discussion Papers, No. 21-13 Provided in Cooperation with: Department of Economics, University of Bern Suggested Citation: Benati, Luca; Nicolini, Juan Pablo (2021) : The welfare costs of inflation, Discussion Papers, No. 21-13, University of Bern, Department of Economics, Bern This Version is available at: https://hdl.handle.net/10419/242864 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Faculty of Business, Economics and Social Sciences Department of Economics The Welfare Costs of Inflation Luca Benati, Juan-Pablo Nicolini 21-13 August, 2021 Schanzeneckstrasse 1 CH-3012 Bern, Switzerland http://www.vwi.unibe.ch DISCUSSION PAPERS
The Welfare Costs of Inflation∗ Luca Benati University of Bern† Juan-Pablo Nicolini Federal Reserve Bank of Minneapolis and Universidad Di Tella‡ Abstract We revisit the estimation of the welfare costs of inflation originating from lack of liquidity satiation. We use data for the United States and several other developed countries. Our computations are heavily influenced by the recent experience of very low, even negative, short term rates observed in the countries we study. We obtain estimates that are close to those obtained by Lucas (2000), and an order of magnitude higher than those in Ireland (2009). ∗We wish to thank Peter Ireland for comments on a previous draft, and for very helpful suggestions. The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of Minneapolis, or of the Federal Reserve System. †Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001, Bern, Switzerland. Email: luca.b[email protected]e.ch ‡Federal Reserve Bank of Minneapolis, 90 Hennepin Avenue, Minneapolis, MN 55401, United States. Email: [email protected] 1
1 Introduction Weprovidenewestimatesofthewelfarecostofinflation. We follow the tradition of Bailey (1956), Friedman (1969), Lucas (2000), and Ireland (2009) in that we estimate the welfare cost using the area under the real money demand curve. Specifically, to compute the welfare cost of a given value for the interest rate, say 0we compute the integral of the real money demand curve between the lower bound for the interest rate and 0This strategy is justified by a large class of theoretical models. One such class is discussed below. There is a wide range of estimates in the literature. For a steady state interest rate of 5percent, Lucas (2000) computes the cost to be around 11percent of lifetime consumption, which is a sizeable amount. However, Ireland (2009) challenges Lucas’ interpretation of the data, and obtains an estimate of a mere 004 percent of consumption. There are two key aspects of the money demand relationship the affect the computation, as both Lucas and Ireland note. The first is the functional form adopted. The second is the values assigned to its parameters. Obviously, data is used by both authors to discipline their choices. And so will we. Our main contribution is to bring more data to the debate. We do so in two dimensions. First, we use the additional decade of data available since Ireland’s work. This is a particularly abnormal and at the same time very interesting decade, since it was characterized by several observations with very low interest rate. Thus, it helps identify the behavior of money demand in that range that, as we will discuss, is very important to identify the functional form. Second, we also use data from several other developed countries, that had similar inflation histories as the United States. Although one could certainly entertain differences across countries, this evidence is also useful to identify the functional form and the parameter values, as we show below. But, more importantly, the exploration of other countries highlights a third key feature that we bring to the analysis: the assumption regarding the true lower bound on the short term nominal interest rate. This is very relevant, since it determines the lower limit of the integral under the real moneydemandcurve.BothLucasandIreland - as most of the monetary economics literature till 2010! - assumed the lower bound to be zero. However, the negative interest rates observed in the Euro area, Sweden and Switzerland challenged that notion. Addressing this question will be at the heart of our analysis. We find that for the United States, the cost of a steady state nominal interest rate of 5percent is between 020 to 150 percent of lifetime consumption, depending on the functional form assumed and the assumption regarding the lower bound. The costs for the United Kingdom, Canada and Japan are within the same range. Estimates are larger for the Euro area, Sweden and Switzerland, where they can go as high as 2percent of lifetime consumption. Modern analysis of optimal monetary policy is typically performed with models belonging to the New Keynesian paradigm. These models consider money-less 2
economies only, so they ignore the welfare effect of lack of money satiation that we focus on. Two reasons, we believe, support this strategy. The first is the widespread belief that money is disappearing in modern economies. The second is the result in Ireland, that computes those costs to be negligible. We challenge both notions. Regarding the first reason, we present overwhelming evidence that there is no sense in which modern economies are becoming money less.1Regarding the second, Ireland bases his computations on USA data only. This is problematic, since there was already at the time substantial evidence that the standard measure of M1, that had maintained a stable relationship with interest rates and nominal output by most of the twentieth century, became unstable in the early 1980s. Fully aware of that problem, Ireland makes a very reasonable adjustment, by adding to M1 the retail sweeps that became very popular since 1994. He shows, however, that even after this adjustment, the behavior of real money demand is different from the one that prevailed between 1900 and 1980. Armed with this new monetary aggregate, and a sample that starts in 1980, Ireland argues that the welfare cost of inflation is substantially lower than the one obtained by Lucas (2000), for two different reasons. First, he shows that the log-log functional form preferred by Lucas performs much worse that the semi-log specification. Second, for the semi-log specification, he estimates a much lower semi-elasticity of the real money demand with respect to the nominal interest rate than the one calibrated by Lucas. We depart from Ireland and adopt the proposal in Lucas and Nicolini (2015), who argue that regulatory changes between 1982 and 1984 changed the availability of transactional assets in the United States. Specifically, they propose to add the Money Market Demand accounts, created in 1984, to M1.2Once these new deposits are taken into account, a remarkably stable real money demand is obtained, that behaves the same way before and after 1980. The estimates of the real money demand using the Lucas and Nicolini aggregate, that they label NewM1, imply larger estimates of the welfare cost of inflation than those obtained by Ireland. As before, the reason is two-fold. First, even using the semi-log specification chosen by Ireland, the estimated elasticity is substantially larger. Second, the evidence against the functional form used by Lucas is not as clear cut, especially if one allows for a negative lower bound on the short term interest rate. Besides finding - for obvious reasons - the argument in Lucas and Nicolini (2015) very compelling, we also report results for several other countries, for which there is no evidence of instability for the entire sample, using M1 as the monetary aggregate. Overall, the analysis for the other countries strongly support the results for the United States when using the NewM1 aggregate, in terms of both the estimated semi1A more detailed analysis with yearly data that includes more countries can be found in Benati, Lucas, Nicolini and Weber (2021). 2The Retail Sweeps that Ireland adds to M1 are a relatively low fraction of the Money Market Demand Accounts. 3
elasticity for the functional form preferred by Ireland, and the comparative weakness of the evidence against the functional specification preferred by Lucas. Our estimates of the welfare cost of lack of money satiation suggest that ignoring money in analyzing optimal monetary policy can be seriously misleading. For instance, Coibion, Gorodnichenko and Wieland (2012) make a compelling argument against increasing the inflation target in countries like the United States, in a model with frictions in the setting of prices and with recurrent, though not very frequent, episodes with the nominal interest rate at the zero lower bound. They compute the welfare effect of an interest rate of 5 percent in their preferred specification to be close to 06percent of life-time consumption. That number, that combines the cost created by price frictions and the probability to be at the zero lower bound, is well within the range of estimates we obtain for the United States. Relative to this number, the 004 estimated by Ireland does appear negligible. But 02the lowest number we estimate, is certainly not.3As it turns out, taking into account the effect that we study would reinforce the argument of their paper. On the theory side, we innovate in that we construct upper and lower bounds for the estimate of the cost. The area under the money demand curve is an almost exact measure of the welfare cost for a very general class of monetary models in the neighborhood of zero, as Alvarez, Lippi and Robatto (2019) show. We extend their results for a quite general sub-class of the models they analyze and compute exact lower and upper bounds for the estimates of the costs, using the area under the money demand curve, for any value of the interest rate. As we show, the difference between the upper and the lower bound is extremely small for the range of interest rates ever observed in the United States. We believe the formulas we derive would be useful in future work. In our analysis we follow the tradition of considering the most liquid monetary assets, that include cash and transactional deposits. We abstract from a detailed discussion of the demand for each of the components, an issue recently addressed by Kurlat (2019).4 The paper proceeds as follows. In Section 2 we discuss a family of monetary models for which we derive very tight lower and upper bounds for the welfare cost of inflation using the area under the real momey demand curve. In Section 3 we discuss the data and several Figures that, in our view, present very solid evidence in favor of stable money demand relationships for the countries we analyze. Section 4 makes formally this statement by analyzing unit root and cointegration properties of the series. It also discusses the estimation results for three different empirical specifications used in the literature, including the ones Lucas and Ireland explored. Section 5 presents our 3Coibion, Gorodnichenko and Wieland (2012) explicitly aknowledge that they do not take into account the costs derived from lack of money satiation. 4He shows that addressing these considerations in a model with imperfect competition substantially increases the estimates of the welfare cost, relative to models that ignore the creation of inside money. 4
computations for the welfare cost functions. For each interest rate level, we compute the bootstrapped distribution of the welfare costs (and therefore median estimates, and confidence intervals) expressed in percentage points of GDP. Section 6 discusses stability tests and the potential existence of non-linearities at low interest rates, as suggested by Mulligan and Sala i Martin (2000). Section 7 concludes. 2TheModel We study a labor-only economy with uncertainty in which making transactions is costly.5The economy is inhabited by a unit mass of identical agents with preferences given by 0 ∞ X =0 ()(1) where is differentiable, increasing and concave. Every period, the representative agent chooses a number of portfolio transactions that allow her to exchange interest-bearing illiquid assets for money, that is needed to buy the consumption good. The total cost of those transactions, measured in units of times, is given by a function ()where is an exogenous stochastic process. This formulation generalizes the linear function assumed by Baumol (1952) and Tobin (1956). The production technology for the consumption good is given by == where is time devoted to the production of the final consumption good and is an exogenous stochastic process. The representative agent is endowed, each period, with a unit of time that is used to produce goods and to make transactions. Thus, equilibrium in the labor market implies that 1=+() and feasibility is given by =(1 −()) It follows that the real wage is equal to . Purchases are subject to a cash in advance constraint ≤(2) where are average money balances and is the number of portfolio adjustments within each period. The variable is the only economically relevant decision to be made by the representative agent. 5The baseline model is discussed at length in Benati et. al. (2020). 5
We allow for money to pay a nominal return, that we denominate that is allowed to become negativeThis implies a point of departure from most of the literature that sets =0This is important, as we explain below, for the model to account for negative values of the short term policy rate in equilibrium, as experienced in recent years by some of the countries we analyze in the paper. At the beginning of each period, the agent starts with nominal wealth that can be allocated to money or interest bearing bonds, soarestrictiontotheoptimal problem of the agent is +≤(3) Nominal wealth at the beginning of next period, in state +1will then be given by +1 ≤(1 + )+(1 + )+(4) +[1−()] − where is the return on government bonds and is a transfer made by the monetary authority. Notice that the unconstrained efficient outcome is to allocate all the labor input to the production of the consumption good so as to set = Thus, a measure of the welfare cost of making transactions, as a fraction of consumption, is given by the value of ()in equilibrium. In the Online Appendix 1, we show that as long as the cost function ()is differentiable, an interior solution for must satisfy 2 () (1 −()) = − (5) We also show that as long as − 0the cash in advance is binding, which implies that =1 (6) so real money demand, as a proportion of output, is equal to the inverse of Note that equation (5) is independent of . Thus, secular increases in productivity do not affect the optimal solution for so the theory implies a unit income elasticity of real money demand. Note that the solution for and therefore the solution for real money demand, depends on the interest rate differential between bonds and money. As mentioned above, in most of the money demand literature, it is customary to assume that =0in which case real money demand depends on the interest rate in bonds. For further references, we let the interest rate differential between bonds and money to be ≡ − For the maximum problem of the agent to be well defined, it has to be the case that = − ≥0(7a) 6
which is the well-known lower bound on the interest rates in bonds.6The popular zero-bound restriction on policy rates is obtained from (7)plus the standard assumption in the literature that =0The analyses of both Lucas (2000) and Ireland (2009) are done under this standard assumption. The recent experience of prolonged negative short term interest rates in several countries severely challenges this notion. As condition (7)must hold in equilibrium, the challenge within the confines of the model we use can only be solved by allowing for negative values of the own return on money, namely 0at least when the short term interest rate becomes small. To allow for that possibility, we proceed as follows. As we identify our measure of money with M1 in the data, it is natural to think of the return on money as an average of the return of the two components of M1, cash and demand deposits. As for cash, a negative return can be rationalized by the risk of being lost or stolen, as Alvarez and Lippi (2013) compute using survey data.7As for deposits, we use a linear relation between their nominal return and the interest rate on bonds. Kurlat (2019) provides very strong empirical support for such a relationship. These assumptions, taken together, are consistent with the return on money satisfying =−+ ,(8) for 0and 1.8This linear relationship implies that will be negative for small enough values of In addition , together with (7)it implies that +− ≥0or ≥− (1 −) so the lower bound on the short term rate is negative. In our welfare cost computations below, we consider two combinations of values for these parameters, in addition to the standard benchmark of ==0. The functional form of the real money demand function depends on the functional form of the transactions technology (), and at this level of generality the model is consistent with many different possibilities. In what follows, and to clarify the main difference between Lucas (2000) and Ireland (2009), we consider three wellknownfunctionalformsthathavebeenusedinpreviousempiricalwork. Allofthe three functional forms exhibit a unit income elasticity, as implied by the model. The first specification is the log-log one, ln =1−ln +1 (9) that exhibits a constant interest rate elasticity equal to .Noticethatas→0real money demand goes to infinity. It is this asymptote at zero that Lucas used to argue 6Intuitively, where ()−()to be negative, the representative agent would have incentives to borrow from the government unbounded quantities and hold money. 7Alvarez and Lippi (2013) calibrate this return at -0.02 using survey data from Italy. 8Further details are provided in the Online Appendix 2. 7
motivation for the cointegration methods that we use in the rest of the paper. Figure 2 therefore shows the time series for M1 velocity and the short-term nominal rate in our sample. The data so displayed suggests that both series are I(1), and that they are cointegrated. As we now discuss, formal statistical tests strongly support this impression. 4.1 Evidence from unit root tests Table A.1 in the Appendix reports results from Elliot et al.’s (1996) unit root tests for either the levels or the logarithms of M1 velocity and the short rate. In short, the null hypothesis of a unit root cannot be rejected for nearly all countries and all series.15 In searching for a cointegration relationship between velocity and the short rate, in the next section we will therefore proceed as follows. First, taking the results from the unit root tests literally–i.e, as indication that the series contain exact unit roots–we will test for cointegration based on Johansen’s tests, which are predicated in the assumption that the series are indeed I(1). Since, however, a plausible alternative interpretation of the results in Table A.1 is that the series are local-to-unity–in which case, as shown by Elliot (1998), tests such as Johansen’s tend to perform poorly–we will search for cointegration based on Wright’s (2000) test, which is valid for both exact unit roots, and roots that are local-to-unity. All of the technical details about the implementation of the tests are identical to Benati (2020) and Benati et al. (2021),which the reader is referred to. 4.2 Cointegration properties of the data In studying a cointegration relationship between the demand for real money as a fraction of GDP and a short-term interest rate, we ought to specify a functional form, and to define a lower bound for the interest rate. In what follows we present results for the functional forms (9),(10) and (11), for two alternative tests developed in the literature. In addition, for the log-log specification, we consider three different alternatives for the lower bound on the short term interest rate.16 15For the short rate it can rejected only for Denmark (in levels) and Canada (1947Q3-2006Q4) in logarithms. For M1 velocity it can only be rejected for South Korea (in levels), whereas results for the Euro area (in levels) are ambiguous. In all of these cases we will treat rejection of the null of a unit root as a fluke. There are two reasons for this. First, if the tests were perfectly sized (which, since we are here using Cavaliere et al.’s 2014 bootstrapping procedure, should be regarded as a good approximation), with eleven countries we should expect about one rejection for any of the four tests (two series, both either in levels or in logarithms). In fact, with three rejections we obtain less than that. Second, visual inspection strongly suggests that the three series for which the null is rejected are in fact I(1). 16As explained above, these different assumptions do not affect the cointegration tests for the semi-log and the Selden-Latane, since we assumed the interest rate differential to be linear in the short-term interest rate. 12
To compute welfare costs, we consider three alternative values for the lower bound on interest rates.17 The firstcaseweexploreistheoneinwhich=0,asitis typically assumed in the literature, which implies ==0. This assumption is inconsistent with the evidence for several countries in our sample. Indeed, in the Euro area, Switzerland, and Sweden, short term interest rates have been consistently negative for the last several years (see Figure 1). In order for the theory to account for negative short-term rates, in the other two cases we assume that =−+. Kurlat (2019) very precisely estimates the slope parameter to be 015using micro-data from the US. We adopt that value. On the other hand, the constant depends on fixed costs of holding deposits, as well as the negative return in cash, related to the probability of being lost of stolen. For this parameter, we explore with two different values, that allows us to accommodate the negative interest rate experiences of the countries in our sample. Specifically, we study the cases =1 and =2which correspond to lower bounds of roughly −12percent and −24 percent, respectively. The first lower bound can account for the observations on short-term rates in Denmark, the Euro area and Sweden, but it cannot account for those for Switzerland, where the lowest value for the short term interest rate was around −18%. The second lower bound can accommodate all cases. The true lower bound on interest rates could be lower than the values we assumed: for decades it had been assumed that the lower bound on nominal interest rates was zero, and the recent experiences have shown that this is not the case. For our purposes, a natural course of action is to consider the previously mentioned range of possibilities. Table 1 reports, for any of the three money demand specifications discussed in Section 2, bootstrapped p-values for Johansen’s maximum eigenvalue test of 0 versus 1 cointegration vectors.18 For Canada we have two partially overlapping M1 series that cannot be linked, since they are slightly different. We present results based on either of them. Table 1 shows the results assuming a zero lower bound (i.e., ==0), whereas Table 1 shows the corresponding results based on the other two assumptions for the log-log case, the only one for which the results are sensitive to the lower bound assumption. Table 2 reports the 90% confidence intervals for the second element of the normalized cointegration vector based on Wright’s (2000) test. As before, Table 2 shows the results for a zero lower bound, whereas Table 2 shows the results based on the other two assumptions for the log-log. Based on Johansen’s tests (Table 1), and assuming a zero lower bound, evidence of cointegration is strong based on the Selden-Latané specification, whereas it is slightly weaker based on the semi-log, and it is materially weaker based on the log17While the cointegration tests and the parameter estimates are invariant to the assumption of the lower bound on the short rate for two of the specifications, the estimates of the welfare cost are not, since the lower bound of the integral depends on it. 18The corresponding results from the trace test are qualitatively the same, and they are available upon request. 13
log. In particular, based on Selden-Latané cointegration is not detected, at the 10% significance level, only for the first period for Canada, and almost marginally for Sweden (in this case, however, a likely explanation is that the sample period is quite short). Based on the semi-log, it is not detected for the Euro area, Sweden, Denmark, and Canada’s second sample. Table 1aBootstrapped p-valuesfor Johansen’s maximum eigenvaluetests for (log) M1 velocity and (the log of) a short-term rate for a=b=0 Money demand specification: SeldenSemiLogCountry Period Latané log log United States 1959Q1-2019Q4 0.0106 0.0302 0.4692 United Kingdom 1955Q1-2019Q4 0.0209 0.0507 0.5235 Canada 1947Q3-2006Q4 0.1758 0.0358 0.0730 1967Q1-2019Q4 0.0388 0.4241 0.0035 Australia 1969Q3-2019Q4 0.0621 0.0367 0.3081 Switzerland 1972Q1-2019Q4 0.0166 0.0547 — Sweden 1998Q1-2019Q4 0.1142 0.1137 — Euro area 1999Q1-2019Q4 0.0877 0.1242 — Denmark 1991Q1-2019Q4 0.0501 0.1449 — South Korea 1964Q1-2019Q4 0.0000 0.0831 0.0399 Japan 1960Q1-2019Q4 0.0120 0.0066 0.0117 Hong Kong 1985Q1-2019Q4 0.0189 0.0189 0.0893 Based on 10,000 bootstrap replications. Null of 0 versus 1 cointegration vectors. The last observations for the interest rate are either zero or negative. For the log-log, cointegration is not detected for the United States, the United Kingdom, and Australia, whereas the tests cannot be performed for the Euro area, Sweden, Switzerland, and Denmark, since for these countries the short rate has been either negative, or exactly equal to zero, for the most recent period. But these results are very sensitive to the assumption regarding the effective lower bound. In almost all cases, the p-values go down monotonically as the lower bound is reduced.19 In a few cases, the p-values go down substantially, most notably in the USA where the test does indeed detect cointegration at the 10% level, for the lower bound assumed to be -2.4 percent, the number consistent with the experience in Switzerland. Wright’s (2000) tests (Table 2) detect cointegration based on the Selden-Latané specification for all countries except Canada’s first sample and Denmark. Further, in 19The only exceptions are Canada for the second sample, where the p-values increase, and Hong Kong, wher they do go down relative to the benchmark, but not monotonically. In both cases, however, the p-values are below 5% for all possible assumptions regarding the effective lower bound. 14
all cases in which cointegration is detected the upper bound of the 90% confidence interval is negative. Based on the semi-log, cointegration is detected for all countries except Canada’s first sample. Finally, based on the log-log cointegration is not detected for Canada’s first sample for any value of the lower bound; for the Euro area for =−1and =−2; and for South Korea for either =0or =−1. Table 1bBootstrapped p-valuesfor Johansen’s maximum eigenvaluetests for log M1 velocity and the log of a short-term rate =0 =-1 =-2 Country Period =0 =0.15 =0.15 United States 1959Q1-2019Q4 0.4692 0.4397 0.0830 United Kingdom 1955Q1-2019Q4 0.5235 0.3566 0.2628 Canada 1947Q3-2006Q4 0.0730 0.0256 0.0230 1967Q1-2019Q4 0.0035 0.0173 0.0377 Australia 1969Q3-2019Q4 0.3081 0.1733 0.1353 Switzerland 1972Q1-2019Q4 ——0.0509 Sweden 1998Q1-2019Q4 —0.2907 0.1189 Euro area 1999Q1-2019Q4 —0.1868 0.1455 Denmark 1991Q1-2019Q4 —0.5349 0.3988 South Korea 1964Q1-2019Q4 0.0399 0.0119 0.0097 Japan 1960Q1-2019Q4 0.0117 0.0056 0.0065 Hong Kong 1985Q1-2019Q4 0.0893 0.0492 0.0321 Based on 10,000 bootstrap replications. Null of 0 versus 1 cointegration vectors. The last observations for the interest rate are either zero or negative. 4.3 Which specification do the data prefer? Overall, the results from Johansen’s and Wright’s tests in Tables 1 and 2 suggest that the data tend to prefer the Selden-Latané specification to either the semi-log or the log-log. In this sub-section we perform a more systematic model comparison exercise. Since it is not possible to nest the three money demand specifications into a single encompassing one, we proceed as follows. We start from the comparison between the semi-log and the log-log. Intuitively, the comparison between (10)and(9) boils down to whether the dynamics of log M1 balances as a fraction of GDP (i.e., minus log velocity) is better explained by the level of the short rate, or by its logarithm. For each country we therefore regress ln ()on a constant, lags of itself, and lags of either the level of the short rate or its logarithm. A natural way of interpreting these regressions is the following. Under the assumption that cointegration is indeed 15
Table 2aResults from Wright’s tests: 90% bootstrapped confidence interval for the second element of the normalized cointegration vector, based on systems for (log) M1 velocity and (the log of) a short-term ratefor a=b=0 Money demand specification: SeldenCountry Period Latané Semi-log Log-log United States 1959Q1-2019Q4 [-0.5874 -0.3432] [-0.1481 -0.0680] [-0.3473 -0.0670] United Kingdom 1955Q1-2019Q4 [-0.5323 -0.3441] [-0.1150 -0.0790] [-0.3931 -0.1969] Canada 1947Q3-2006Q4 NCD NCD NCD 1967Q1-2019Q4 [-0.4970 -0.3649] [-0.1198 -0.0358] [-0.4097 -0.2456] Australia 1969Q3-2019Q4 [-0.9216 -0.7054] [-0.1817 -0.0455] [-1.2750 -0.9747] Switzerland 1972Q1-2019Q4 [-0.4641 -0.2759] [-0.2330 -0.1289] — Sweden 1998Q1-2019Q4 [-0.3643 -0.3082] [-0.1539 -0.1219] — Euro area 1999Q1-2019Q4 [-0.6013 -0.3010] [-0.2173 -0.1653] — Denmark 1991Q1-2019Q4 NCD [-0.1393 -0.0432] — South Korea 1964Q1-2019Q4 [-0.5943 -0.5022] [-0.1485 0.0276] NCD Japan 1960Q1-2019Q4 [-1.8658 -1.0569] [-0.3175 -0.0333] [-0.6108 -0.1223] Hong Kong 1985Q1-2019Q4 [-1.0421 -0.5936] [-0.2570 -0.1009] [-0.4957 -0.0913] Based on 10,000 bootstrap replications. NCD = No cointegration detected. The last observations for the interest rate are either zero or negative.
Table 2bResults from Wright’s tests: 90% bootstrapped confidence interval for the second element of the normalized cointegration vector, based on systems for log M1 velocity and the log of a short-term rate Country Period =0, =0 =-1, =0.15 =-2, =0.15 United States 1959Q1-2019Q4 [-0.3473 -0.0670] [-0.6414 -0.1650] [-0.7745 0.0784] United Kingdom 1955Q1-2019Q4 [-0.3931 -0.1969] [-0.5916 -0.2913] [-1.6308 0.2791] Canada 1947Q3-2006Q4 NCD NCD NCD 1967Q1-2019Q4 [-0.4097 -0.2456] [-0.5516 -0.4155] [-0.7233 -0.2188] Australia 1969Q3-2019Q4 [-1.2750 -0.9747] [-1.8131 -1.1004] [-1.6781 -1.3217] Switzerland 1972Q1-2019Q4 ——[-2.0691 0.6696] Sweden 1998Q1-2019Q4 —[-0.3191 -0.2631] [-0.5678 -0.3516] Euro area 1999Q1-2019Q4 —NCD NCD Denmark 1991Q1-2019Q4 —[-0.4595 -0.1952] [-0.6246 0.3884] South Korea 1964Q1-2019Q4 NCD NCD [-0.9116 -0.8115] Japan 1960Q1-2019Q4 [-0.6108 -0.1223] [-1.1097 -0.1487] [-1.2568 0.1006] Hong Kong 1985Q1-2019Q4 [-0.4957 -0.0913] [-1.0071 -0.0942] [-1.9103 -1.0414] Based on 10,000 bootstrap replications. NCD = No cointegration detected. The last observations for the interest rate are either zero or negative.
there for all countries,20 and based on either specification, both = [ln ()]0 and = [ln ()ln()]0have a cointegrated VECM(-1) representation, which maps into a restricted VAR() representation in levels (where the restrictions originate from the cointegration relationship). The equations we are estimating can therefore be thought of as the corresponding unrestricted form of the equations for ln () in the VAR() representation in levels for either or . It is important to stress that the two specifications we are estimating are in fact nested: the easiest way of seeing this is to think of them as two polar cases–corresponding to either =1or =0–in the following representation based on the Box-Cox transformation of : ln µ ¶=+ X =1 ln µ− −¶+ X =1 Ã −−1 !+(15) We estimate (15) via maximum likelihood, stochastically mapping the likelihood surface via Random-Walk Metropolis (RWM). The only difference between the ‘standard’ RWM algorithm which is routinely used for Bayesian estimation and what we are doing here is that the jump to the new position in the Markov chain is accepted or rejected based on a rule which does not involve any Bayesian priors, as it uniquely involves the likelihood of the data.21 So one way of thinking of this is as Bayesian estimation via RWM with completely uninformative priors, so that the log-posterior collapses to the log-likelihood of the data. All of the other estimation details are identical to Benati (2008), to which the reader is referred to. Table 3reports, for either specification, and for ∈{248}themodeofthe log-likelihood. The main result in the table is that whereas the semi-log appears as the preferred functional form for the U.S. the U.K., Canada, and Hong Kong, the log-log produces a larger value of the likelihood for Australia, South Korea, and Japan, so that neither of the two specifications clearly dominates the other one.22 As we showed above, both the cointegration tests and the estimation results for the log-log are very sensitive to the assumption of the lower bound. Thus, we repeated 20If this assumption did not hold, the entire model comparison exercise would obviously be meaningless. 21So, to be clear, the proposal draw for the parameter vector ,˜ , is accepted with probability min[1, (−1,˜ |,)], and rejected otherwise, where −1is the current position in the Markov chain, and (−1˜ |)= (˜ |) (−1|) which uniquely involves the likelihood. With Bayesian priors it would be (−1˜ |)= (˜ |)(˜ ) (−1|)(−1) where (·)would encodes the priors about . 22This crucially hinges on the fact that we are here exclusively focusing on low-inflation countries. As shown by Benati et al. (2021) and Benati (2021), for high-inflation countries, and especially hyperinflationary episodes, the data’s preference for the log-log is overwhelming. 16
Table 3aModel comparison exercise, semi-log versus log-log: mode of the log-likelihood in regressions of log velocity on lags of itself and either the short rate or its logarithm p=2 p=4 p=8 SemiLog SemiLogSemiLogCountry Period log log log log log log United States 1959Q1-2019Q4 766.1394 756.6280 763.2818 751.3266 765.1439 740.3543 United Kingdom 1955Q1-2019Q4 879.6821 877.9350 898.6224 893.7504 892.1970 887.1920 Canada 1947Q3-2006Q4 820.2401 807.8379 813.8001 804.9218 802.7001 794.7403 1967Q1-2019Q4 775.0890 767.0845 775.9595 766.4531 771.9264 766.1943 Australia 1969Q3-2019Q4 650.7331 656.0624 649.9510 655.1057 642.6046 650.3903 South Korea 1964Q1-2019Q4 630.9515 633.8825 628.2222 634.6372 623.8333 628.0991 Japan 1960Q1-2019Q4 845.3632 850.7677 841.5156 848.6520 832.2577 840.2434 Hong Kong 1985Q1-2019Q4 328.0148 325.5701 326.1339 324.9236 319.8478 325.2641 For Switzerland, Sweden, Euro area, and Denmark there is no comparison because the last observations for the short rate are negative. Table 3bModel comparison exercise, semi-log versus log-log: mode of the log-likelihood in regressions of log velocity on lags of itself and either the short rate or its logarithm (p=8) Log-log Semi- =0 =-1 =-2 Country Period log =0 =0.15 =0.15 United States 1959Q1-2019Q4 765.1439 740.3543 749.5835 751.0460 United Kingdom 1955Q1-2019Q4 892.1970 887.1920 890.2113 891.0367 Canada 1947Q3-2006Q4 802.7001 794.7403 822.8528 823.7403 1967Q1-2019Q4 771.9264 766.1943 770.6245 771.9067 Australia 1969Q3-2019Q4 642.6046 650.3903 649.4391 648.9755 Switzerland 1972Q1-2019Q4 567.8522 ——559.0320 Sweden 1998Q1-2019Q4 300.9987 —300.5980 300.5340 Euro area 1999Q1-2019Q4 316.0427 —315.9802 315.6161 Denmark 1991Q1-2019Q4 402.0518 —399.5286 400.2900 South Korea 1964Q1-2019Q4 623.8333 628.0991 629.3606 628.8658 Japan 1960Q1-2019Q4 832.2577 840.2434 838.1153 836.0811 Hong Kong 1985Q1-2019Q4 319.8478 325.2641 327.9993 327.0688 The last observations for the short rate are negative.
Table 3cModel comparison exercise, Selden-Latané versus semi-log: mode of the log-likelihood in regressions of the short rate on lags of itself and either velocity or its logarithm p=2 p=4 p=8 SeldenSemiSeldenSemiSeldenSemiCountry Period Latané log Latané log Latané log United States 1959Q1-2019Q4 -22.9102 -24.0809 -5.8335 -7.3440 12.9347 10.3522 United Kingdom 1955Q1-2019Q4 -85.7350 -84.1422 -85.4391 -83.9044 -83.6446 -82.1970 Canada 1947Q3-2006Q4 -72.0532 -71.7812 -64.4770 -66.2576 -62.2194 -64.2760 1967Q1-2019Q4 -65.0057 -65.9778 -56.1253 -59.0260 -50.7916 -53.9112 Australia 1969Q3-2019Q4 -136.4591 -137.1389 -132.5116 -133.5144 -116.7487 -118.2407 Switzerland 1972Q1-2019Q4 -45.6989 -45.8396 -39.9744 -40.8984 -20.5636 -22.4888 Sweden 1998Q1-2019Q4 65.5876 65.4372 66.9821 66.9083 70.5126 68.9721 Euro area 1999Q1-2019Q4 63.8008 64.3157 64.5967 65.3372 74.7778 75.5777 Denmark 1991Q1-2019Q4 50.9544 50.7969 60.7088 60.1085 65.6600 64.4409 South Korea 1964Q1-2019Q4 -131.5950 -135.8924 -118.2770 -131.1253 -86.1317 -93.5032 Japan 1960Q1-2019Q4 -141.5147 -141.6026 -140.6631 -140.7865 -129.5219 -130.1270 Hong Kong 1985Q1-2019Q4 -65.6601 -65.7389 -60.9537 -61.4880 -50.8999 -51.6665 For Switzerland, Sweden, Euro area, and Denmark there is no comparison because the last observations for the short rate are negative.
the test for the other two assumptions about the lower bound, and also for the same three values for .Table3.presents the results for =8(results for =4and =2 are very similar, and they are reported in Tables A.1-A.1in the Online Appendix). The first feature to highlight is that for the four countries with negative rates, for which we could not do the test before, the semi-log dominates the log-log. The second is that, in line with the previous analysis, the likelihood of the log-log specification increases when the assumed lower bound is lower for most countries where the semilog is the preferred specification. This is particularly so for those countries where the cointegration tests also improve substantially for lower values of the bound, like the United States of the United Kingdom. However, with the single exception of Canada, the increase is not enough for the log-log to dominate the semi-log. Turningtothecomparisonbetweenthesemi-log and the Selden-Latané, we adopt the same logic as before, but this time we ‘flip’ the specifications for velocity on their head, by regressing the interest rate on lags of itself and of either the level or the logarithm of velocity. Once again, these two regressions can be thought of as particular cases of the nested regression =+ X =1 −+ X =1 ⎡ ⎢ ⎣³− −´ −1 ⎤ ⎥ ⎦+(16) with either =1(corresponding to Selden-Latané) or =0(corresponding to the semi-log). At first sight this approach might appear as questionable: since we are here dealing with the demand for real M1 balances for a given level of the short-term nominal interest rate, why would it make sense to regress the short rate on M1 velocity? In fact, this approach is perfectly legitimate, for the following reason. As shown by Benati (2020), M1 velocity is, to a first approximation (and up to a scale factor), the permanent component of the short-term rate,23 so that focusing (e.g.) on the SeldenLatané specification, =+ ,whereis velocity, 0are coefficients, and is the unit-root component of the short rate (), with = + ,and being the transitory component.24 This can be seen quite clearly in Figure 2 for Australia, Canada, the Euro area, Hong Kong, Sweden, Switzerland, and the U.K.. Regressing on therefore amounts to regressing the short rate on its (rescaled) stochastic trend, i.e. the dominant driver of its long-horizon variation, and it is 23This expresses in the language of time-series analysis Lucas’ (1988) point that real M1 balances are very smooth compared to the short rate. 24A simple rationalization of this fact is provided by a ‘preferred habitat’ model (see Modigliani and Sutch, 1966, and Vayanos and Vila, 2021) in which ‘long’ investors such as pension funds play an important role in money demand. The intuition is that whereas permanent shocks to the short rate shift the entire term structure of interest rates, and therefore affect the demand for M1 coming from all investors, transitory shocks only impact the short end of the yield curve, and therefore have a much smaller (and in the limit negligible) effect. 17
24 Figure 4 Informal evidence on the possible presence of non-linearities at low interest rates
these countries the short rate has consistently been positive over the entire sample period, we first consider the case in which the own return of money is zero. The results for these countries based on the Selden-Latané functional form (which, as discussed, we take as our benchmark) are reported in Figure 4..Thepoint estimates of the upper and lower bounds are depicted as continuous black lines: as we previously anticipated, in all cases the two lines are virtually indistinguishable, thus implying that the two bounds provide a very precise characterization of the welfare costs (the same holds for nearly all countries and all functional forms). The dotted and continuous red lines depict the 5th and 16th percentiles of the lower bounds, and the 84th and 95th percentiles of the upper bounds of the bootstrapped distributions. It is clear that the econometric uncertainty (captured, e.g., by the distance between the 5th and the 95th percentiles) is at least one order of magnitude larger (and likely more) than the theoretical imprecision captured by the point estimates of the lower and upper bounds. The point estimates for the welfare cost of a steady state interest rate equal to 5 percentarecloseto02percent of consumption for the United States and Japan, about 03percent for Canada while it is about 04percent of consumption for the United Kingdom. The estimate for the US is the same as the one reported by Lucas (2000) when using the semi-log specification (10) and almost one order of magnitude above the estimate of Ireland (2009) of 0037 percent. As mentioned above, the main reason for the discrepancy is the small value for the semi-elasticity obtained by Ireland, based on a different monetary aggregate. That explains most of the difference: notice from the Figure that the semi-log, used by Ireland, does imply a cost of only 015.The Selden-Latane implies an additional 005 percent. In Figure 4., we report the same point estimates using the Selden-Latané specification we report on Figure 4., together with the computations for the other two functional forms. The Figure highlights the theoretical point made by Lucas (2000), in that the log-log specification delivers substantially higher costs, about 06percent, than the 02percent of the Selden-Latane.28 However, the difference between the log-log and the other two specifications is much higher for the USA than for Japan and the UK and, to a lesser extent, than Canada. This explained by the fact that the point at which the cost implied by the log-log specification crosses the other two is quite sensitive to the estimates of particular countries. For Japan and the United Kingdom they cross about 5percent interest rate, for Canada at about 10 percent and for the United States they cross at almost 20 percent (not depicted). Although the interest rate in the countries discussed so far was always positive, the experience of some European countries suggests that the standard assumption of a zero lower bound may not be appropriate. Thus, we believe that allowing for 28Our estimate of 0.6 percent of consumption is lower than the 1.2% reported by Lucas (2000). The difference lies in that our estimate of the elasticity when the lower bound is zero, which is the one we are reporting now, is around 0.15. The elasticity used by Lucas is 0.5. The value of 0.5 is the one we obtain whan the lower bound is assumed to be -2.4 percent, a case we report below. 21
a negative lower bound cannot be completely ruled out. To explore that possibility, in Figure 4., we report the results using the three specifications for the real money demand assuming a lower bound on short term interest rate of −24percent, that corresponds to setting =015 and =2We only report the results for the United States. We compare the results with the ones in Figure 4., which correspond the the case of a zero lower bound. As it can be seen, the welfare cost of a 5 percent interest rate are twice as large as in the case of a zero lower bound: 04percent of consumption for the Selden-Latane and 03percent of consumption for the semi-log. However, for the log-log case, the increase is larger, it goes from 06to 15percent of consumption. For our second set of results, we report the welfare cost computations for Switzerland, Sweden and the Euro area. There are two differences between these countries and the ones discussed above. The first, is that they have, on average, higher money balances over output. The second, is that they experienced negative short term interest rates. As discussed above, this feature is only consistent with the notion that the own return on money is not zero, and can become negative when the short term interest rate becomes negative. In spite of that, in computing the cost of inflation, we consider three scenarios. The first, is the benchmark case of a zero own return on money. We do this in order to compare the results with the ones reported in Figure 4.:anydifference in the costs ought to be driven by the different estimated parameters only. For the other two scenarios, we follow the strategy adopted for estimation, and let =−+ where we set =015 and consider two different values for the constant, =1and =2. Figure 5 presents the results for the estimated welfare cost functions, with oneand two-standard deviations bootstrapped confidence bands. In Figure 5.,wereport the point estimates based on the Selden-Latané. If we assume the lower bound to be zero - top panel - the welfare costs of a 5 percent interest rate is about 0.5% of consumption for the Euro area and Switzerland, and a bit smaller for Sweden, more than double the ones for the United States, Canada and Japan. This is explained purely by differences in estimated parameters. In considering a lower bound that can accommodate the experiences of the Euro area and Sweden, the cost increases to 075 percent of consumption for the Euro area and Switzerland, and somewhat smaller for Sweden, as before. In the final case we consider, that is required to accommodate the experience of Switzerland, the cost can be almost 14percent of consumption for the Euro area, 12percent for Switzerland and almost 1percent for Sweden In Figure 5., we report a comparison for the point estimates for the three specifications (only two when the implied is negative) for the three alternative assumptions regarding the effective lower bound. The message from this Figure is similar to the oneinFigure4.. First, the semi-log and the Selden-Latané provide very similar results for low interest rates. Second, the log-log implies much higher welfare costs at very low rates, with the exception of Sweden for the case in which the lower bound is −12percent. However, the numbers are substantially larger than in Figure 5.,im22
23 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Switzerland (1972Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 Sweden (1998Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Percentage points of GDP Euro area (1999Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Percentage points of GDP 0 5 10 15 Short rate 0 0.5 1 1.5 2 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 3 0 5 10 15 Short rate 0 1 2 3 4 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 3 0 5 10 15 Short rate 0 1 2 3 4 Percentage points of GDP Estimate based on rm=-1+0.15*rb Standard estimate based on rm=0 Estimate based on rm=-2+0.15*rb Figure 5.a Estimated welfare cost functions based on the Selden-Latane’ specification for countries with negative interest rates (point estimates of the lower and upper bounds, 5-16 percentiles of the lower bounds, and 84-95 percentiles of the upper bounds of the bootstrapped distributions)
24 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Switzerland (1972Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 Sweden (1998Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Percentage points of GDP Euro area (1999Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Percentage points of GDP 0 5 10 15 Short rate 0 0.5 1 1.5 2 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 0 5 10 15 Short rate 0 1 2 3 4 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 3 0 5 10 15 Short rate 0 1 2 3 4 Percentage points of GDP Estimate based on rm=-1+0.15*rb Standard estimate based on rm=0 Estimate based on rm=-2+0.15*rb Log-log Semi-log Selden-Latane' Figure 5b Point estimates of the welfare costs of inflation produced by alternative specifications
plying a larger interaction of the functional form with the assumption about the true lower bound on the short-term interest rates. For example, the log-log specification implies very large welfare costs, of about 28and 26percent of consumption for the Euro area and Switzerland and about 15percent of consumption for Sweden when thelowerboundisassumedtobe−24percent. To summarize, we now describe the two most extremes scenarios. In all cases, we report the welfare cost of a 5 percent nominal interest rate. Our lowest set of estimates correspond to the first set of countries (Canada, Japan, USA and UK) for which the combined effect of the estimated parameters and the assumption of a zero bound on interest rates imply a range of estimates between 015 and 02percent of consumption. However, for the United States for example, one cannot reject the log-log specification, together with the assumption of a lower bound of −24percent. In this case, the estimated cost is 15percent of consumption. Our largest set of estimates correspond the the second group of countries (the Euro area, Sweden and Switzerland) for which the estimated cost when the lower bound is assumed to be zero are between 04and 05percent of consumption. However, these three countries experienced negative short term rates. If we assume a zero lower bound of −24and the log-log specification, the estimated costs now range between 15and 28percent of consumption. Our estimates show that under the benchmark scenario that assumes the SeldenLatané and a zero lower bound, the welfare cost of inflation evaluated at a 5% average nominal interest rate are between 0.2% and 0.5% of consumption, depending on the country, which is not a negligible number. However, they also show that they depend critically on the functional form and the underlying assumption regarding the true lower bound on the short term interest rates. The log-log specification, that cannot be clearly rejected in some countries with specificassumptionsaboutthetruelower bound, delivers much higher estimates, particularly when the true lower bound is assumed to be lower. For example, the log-log specification cannot be rejected for the USA using both Johansen’s and Wright’s tests when =−2In this case, the welfarecostat5%isestimatedtobe15percent. The same happens for Switzerland, where the cost is estimated to be 25percent. 6 Exploring Stability and Non-Linearities A main concern in working with estimated money demand curves pertains to the stability of the long-run relationship over time. As previously mentioned, even without the econometric evidence produced (e.g.) by Friedman and Kuttner (1992), the simple visual evidence had been sufficient to discredit, long ago, any notion of stability of the U.S. demand for real M1 balances. As our results make clear, the solution proposed by Lucas and Nicolini (2015) has re-established stability of the U.S. demand for M1. However, since for all of the other countries in our dataset we work with the ‘standard’ M1 aggregate, it is a legitimate question whether for (some of) these 23
countries, too, some adjustment to the standard aggregate might be required in order to obtain stability of the long-run demand for M1. 6.1 Testing for stability in the cointegration vector Table 5 reports evidence from Hansen and Johansen’s (1999) tests for stability in the cointegration vector29 for our dataset, based on any of the three money demand specifications. Only in two instances, Denmark, and Japan based on the SeldenLatané specification, the tests detect evidence of instability.30 Overall, there is very little evidence of a break in the real money demand relationship derived from the theory. This is reassuring on itself, but also in reference to the issue raised by Ireland and that has prevailed the discussion in the United States, related to a structural break in this relationship somewhere between the late 70s and the early 80s. It is the assumption of such a break that justifies focusing the analysis using only the recent data. These tests show in one hand, that once we take into account United States specific regulatory changes, there is no break in the money demand relationship. And on the other hand, they show that in other similarly developed countries that did not have the regulatory changes, the high inflation episode of the late 70s and early 80s is totally consistent with real money demand theory, using the standard M1 monetary aggregate. 6.2 Are there non-linearities in money demand at low interest rates? A conceptually related issue pertains to the possibility that, at low interest rates, money demand might exhibit sizeable non-linearities, due to the presence of fixed costs associated with the decision to participate, or not to participate, in financial markets (see e.g. Mulligan and Sala-i-Martin, 2000).31 Based on this argument, at 29On the other hand, we do not test for stability of the loading coefficients, since they pertain to the short-term adjustment dynamics of the system towards its long-run equilibrium, and they are therefore irrelevant for the purpose of computing the welfare costs of inflationinthesteady-state. Finally, we eschew Hansen and Johansen’s (1999) fluctuation tests because, as shown by Benati et al. (2021) via Monte Carlo, they exhibit, overall, a significantly inferior performance compared to the tests for stability in the cointegration vector and loading coefficients. 30This is in line with the evidence in Benati et al.’s (2021) Section 6.2. The main finding there was that evidence of breaks in either the cointegration vector or the loading coefficients vector is weak to non-existent. The estimated break dates for the cointegration vector are 2008Q1 for Denmark and 1979Q4 for Japan. The second element of the normalized cointegration vector for the first and second sub-periods is equal to -0.37 and -0.66 for Denmark, and to -0.41 and -0.74 for Japan. 31The intuition is straightforward. Suppose that the interest rate, , is initially equal to zero, and consider a household with nominal assets , which are entirely held in either cash or non-interestbearing deposits. Crucially, suppose that if the household wants to switch a fraction of its assets into bonds ,ithastopayafixed cost .Asincreases from zero to 0, unless the household will keep all of its wealth in either cash or deposits form, and only when the inequality is 24
Table 5 Bootstrapped p-valuesfor Hansen and Johansen’s (1999) tests for stability in the cointegration vector for (log) M1 velocity and (the log of) a short-term rate Money demand specification: SeldenSemiLogCountry Period Latané log log United States 1959Q1-2019Q4 0.5875 0.8030 0.9940 United Kingdom 1955Q1-2019Q4 0.5905 0.5480 0.9365 Canada 1947Q3-2006Q4 0.3535 0.6710 0.6910 1967Q1-2019Q4 0.6900 0.7945 0.6070 Australia 1969Q3-2019Q4 0.7835 0.7880 0.6950 Switzerland 1972Q1-2019Q4 0.6378 0.8102 — Sweden 1998Q1-2019Q4 0.2335 0.1690 — Euro area 1999Q1-2019Q4 0.4880 0.2915 — Denmark 1991Q1-2019Q4 0.0085 0.2605 — South Korea 1964Q1-2019Q4 0.1460 0.5835 0.4485 Japan 1960Q1-2019Q4 0.0030 0.2600 0.4030 Hong Kong 1985Q1-2019Q2 0.5280 0.4510 0.8465 Based on 10,000 bootstrap replications. Null of 0 versus 1 cointegration vectors. The last observations for the interest rate are either zero or negative. Table 6 Estimated coefficients on the short rate in SeldenLatané specifications for samples with the short rate above and below 5 per cent Based on samples with short rate: below 5 per cent above 5 per cent Estimate and 90% Median and 90% Country (55)confidence interval confidence interval Australia 0.614 0.530 [0.321; 0.763] 0.604 [0.325; 0.802] Canada, I 0.267 0.402 [0.138 0.612] 0.323 [0.248 0.399] Canada, II 0.064 0.729 [0.451; 1.110] 0.399 [0.133; 0.612] South Korea 0.584 0.351 [0.053; 0.651] 0.397 [0.305; 0.476] United States 0.072 0.573 [0.369 0.826] 0.284 [0.008 0.561] Based on 10,000 bootstrap replications. Samples with short rate below and above 5 per cent: Australia: 2009Q1-2019Q4 and 1969Q3-2008Q4; Canada, I: 1947Q3-1967Q3 and 1973Q2-1993Q2; Canada, II: 2001Q1-2019Q4 and 1973Q2-1993Q2; South Korea: 1995Q3-2019Q4 and 1964Q1-1995Q2; United States: 2001Q12019Q4 and 1972Q4-1991Q3.
sufficiently low interest rates money demand (and therefore money velocity) should be largely unresponsive to changes in interest rates, since most (or all) households simply do not participate in financial markets. The implication is that it should not be possible to reliably estimate money demand functions (and therefore the welfare costs of inflation) based on aggregate time series data, as only the use of micro data allows to meaningfully capture the non-linearities associated with the cost of participating in financial markets. Although Hansen and Johansen’s (1999) tests detect little evidence of instability in the cointegration vector, for the specific purpose of testing whether money demand curves might be flatter at low interest rates these results should be discounted for (at least) two reasons. First, as discussed by Bai and Perron (1998, 2003), when a coefficient experiences twobreaksinoppositedirections(e.g.,first an increase, and then a decrease), break tests which have not been explicitly designed to search for multiple breaks may have ahardtimedetectingthefirst break to begin with. Within the present context this could be relevant for three countries, the U.S., the U.K., and Canada. In any of these cases the short rate had been below 5% (which, based on Mulligan and Sala-iMartin, 2000, we take as the relevant threshold) at the beginning of the sample; it then significantly increased above 5% during the Great Inflation; and it has progressively decreased since the early 1980s. Under the assumption that money demand curves are comparatively flatter at low rates, this implies that the slope of the curve should have first increased, and then decreased, which is precisely the kind of circumstance in which these tests may have problems in detecting a break. Second, Hansen and Johansen’s (1999) are tests for breaks at unknown points in the sample. In principle, it should be possible to perform more powerful tests if we had strong reasons for choosing a specific threshold for the short rate, which, as mentioned, we take it to be 5%. Before delving into the econometric evidence, however, it is of interest to see what a simple visual inspection of the data suggests. Figure 6 shows informal evidence on the possible presence of nonlinearities for five countries for which both sub-samples with the short rate above, and respectively below 5% are sufficiently long. In order to provide sharper evidence, for four countries (the U.S., the U.K., Canada, and Australia) we consider long samples of annual data that we do not further analyze.32 The top row shows the raw data for M1 velocity and a short rate, whereas the bottom row shows the low-frequency components of the two series.33 The evidence in the figure speaks for itself, and it provides nearly no support to the notion that velocity– satisfied it will have an incentive to buy bonds. This implies that, under the plausible assumption that is heterogenous across the population, money demand should exhibit sizeable non-linearities (rather than a strict discountinuity) at low interest rates. 32This is because, these being annual series, for all of them at least one of the sub-samples with the short rate either above or below 5% features too few observations to produce reliable results. 33The low-frequency components have been extracted via the methodology proposed by Müller and Watson’s (2018), setting the threshold for the low frequencies to 30 years. 25
24 Figure 6 Informal evidence on the possible presence of non-linearities at low interest rates
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Online Appendix for: The Welfare Costs of Inflation Luca Benati University of Bern∗ Juan-Pablo Nicolini Federal Reserve Bank of Minneapolis and Universidad Di Tella† ∗Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001, Bern, Switzerland. Email: luca.b[email protected]e.ch †Federal Reserve Bank of Minneapolis, 90 Hennepin Avenue, Minneapolis, MN 55401, United States. Email: [email protected]
AModelSolution The problem of the agent is to maximize (1) inthemaintextbychoosing and +1 subject to (3) (4) and (2) Assume that the function ()is differentiable. If we let and be the corresponding Lagrange multipliers, the first order conditions are given by ()=+(21) ()=(22) =(1 + )+(23) =(1 + )(24) =+1 (25) The first-order conditions imply that, as long as − 0 = − and from this we obtain − ()= or − ()= (26) Note also that , as long as − 0it ought to be the case that 0which meansthatthecash-in-advanceconstraintisbinding,so = which together with feasibility =(1 −()) implies (1 −()) = Replacing on (26) above 2 () (1 −()) = − Thus, the solution for depends only on the two stochastic processes − and Note, in particular, that it does not depend on so the theory implies a unit income elasticity 33
BTheReturnonMoney As mentioned above, we assume that the monetary aggregate is the sum of cash and deposits, in fixed proportions. Thus, if we let and be the stock of deposits and cash, then =(1−)and = where ∈(01)If we let and be the nominal returns on deposits and cash, then =(1−) + As for cash, we assume that =−for some non-negative constant .Thisreflects the chances of cash being lost or stolen, which for simplicity we assume independent of the state of the economy 43 In discussing the interest rate on deposits, we refer the reader to models similar to the one we discussed above, but enlarged by modeling banks, that create deposits backed with goverenment bonds, as the one in Freeman and Kydland (2000) for example. In those models, due to costs of creating deposits, the interest rate on deposits is a function of the bond rate, so =( )with the following properties ( )≤ 1≥( )≥0(27) which implies that the spread between theinterestrateinbondsandtheinterst rate in deposits is positive, and that the interest rate on deposits is a non-decreasing function of the interest rate in bonds and that it does not change more that one to one with the interst rate in bonds.44 With these two assumptions, the diffferencebetweentheinterestrateinbonds and the interest rate on money is given by =(1−)£ −( )¤+£ −¤(28) The maximum problem is well defined for values of such that ≥0The properties specified in (27) imply that is non-decreasing on . Thus, feasibility implies that ≥minwhere min satisfies =0It immediately follows that as long as 0or (0) 0then min 0which implies that may indeed be negative. Kurlat convincingly argues that the function ( )is linear, using micro data from the United States, where the slope is very precisely estimated, while the constant is essentially zero. On the other hand, Alvarez and Lippi estimate the return on cash =−002using survey data. As a consequence, we use in the paper a liner return for money of the form =−+ Our benchmark case sets ==0which is the standard assumption in the literature. To account for the countries that experienced negative rates, we also consider 43See Alvarez and Lippi (2009) for some survey-based evidence on this cost. 44Kurlat (2019) presents ample evidence for these two properties using USA data. 34
two more cases. In both we set =015following the findings in Kurlat. We then exploretwoalternativevaluesfortheconstant∈{12}The first value implies that the lower bound on is given by − = +− ≥0 or ≥−117% which is lower than the negative rates in the Euro area and Sweden, where rates were always above −1%. However, rates in Switzerland went all the way down to -1.85%, that is why we also explore the case of =2which implies a lower bound of −235%. CTheData Here follows a detailed description of the dataset. All data are from official sources, i.e., either central banks or national statistical agencies. C.1 United States For the United States, seasonally adjusted series for nominal GDP and the standard M1 aggregate, and series for the 3-month Treasury bill rate and the 10-year government bond yield, are all from the St. Louis FED’s internet data portal, FRED II (their acronyms are GDP, M1SL, TB3MS, and GS10, respectively). The standard M1 aggregate starts in 1959Q1. Before that, the series has been linked to the series M173Q4 in the spreadsheet m1QvMd.xlsx from the Federal Reserve Bank of Philadelphia’s real-time data portal, which starts in 1947Q1. Over the period of overlapping the two M1 series are virtually identical, which justifies the linking. The series for Money Market Deposits Accounts (MMDAs), starting in 1982Q4, is from the Federal Reserve ’s mainframe. A series for currency is from the Federal Reserve’s website. C.2 United Kingdom For the United Kingdom, a seasonally adjusted series for nominal GDP (‘YBHA, Gross Domestic Product at market prices: Current price, Seasonally adjusted £m’) is from the Office for National Statistics. A seasonally adjusted and break-adjusted stock of M1 is from ‘A millennium of macroeconomic data for the UK, The Bank of England’s collection of historical macroeconomic and financial statistics, Version 3 - finalised 30 April 2017’, which is from the Bank of England’s website. Likewise, series for a 10-year bond yield and a Treasury bill rate are all from the same spreadsheet. 35
C.3 Canada For Canada, a seasonally adjusted series for nominal GDP (‘Gross domestic product (GDP) at market prices, Seasonally adjusted at annual rates, Current prices’) is from Statistics Canada. Series for the 3-month Treasury bill auction average rate and the benchmark 10-year bond yield for the government of Canada, are from Statistics Canada. M1 (‘v41552787, Table 176-0020: currency outside banks, chartered bank chequable deposits, less inter-bank chequable deposits, monthly average’) is from Statistics Canada. Data on currency are from Statistics Canada (‘Table 1760020 Currency outside banks and chartered bank deposits, monthly average, Bank of Canada, monthly’). C.4 Australia Nominal GDP (‘Gross domestic product: Current prices, $ Millions, Seasonally Adjusted, A2304418T’) is from the Australian Bureau of Statistics.Theshortrate (‘3-month BABs/NCDs, Bank Accepted Bills/Negotiable Certificates of Deposit-3 months; monthly average, Quarterly average, Per cent, ASX, 42767, FIRMMBAB90’) is from the ReserveBankofAustralia(henceforth, RBA). M1 (‘M1: Seasonally adjusted, $ Millions’) is from the Reserve Bank of Australia since 1975Q2, and from FRED II (at the St. Louis FED’s website) for the period 1972Q1-1975Q1 (over the period of overlapping, i.e. since 1975Q2, the two series are identical, which justifies their linking). 5-and 10-year government bond yields are from the RBA.Specifically, they are from the RBA’s spreadsheet ‘F2.1 Capital Market Yields — Government Bond’, which is available at the RBA’s website. A quarterly seasonally adjusted series for the ‘Unemployment rate, Unemployed persons as percentage of labour force’ has been computed by taking averages within the quarter of the corresponding monthly series from the Australian Bureau of Statistics (the series’ code is GLFSURSA). C.5 Switzerland For Switzerland, both M1 and the short rate (‘Monetary aggregate M1, Level’ and ‘Switzerland - CHF - Call money rate (Tomorrow next)’, respectively) are from the Swiss National Bank’s internet data portal. A seasonally adjusted series for nominal GDP (‘Gross domestic product, ESA 2010, Quarterly aggregates of Gross Domestic Product, expenditure approach, seasonally and calendar adjusted data, In Mio. Swiss Francs, at current prices’) is from the State Secretariat for Economic Affairs (SECO) at https://www.seco.admin.ch/seco/en/home. A series for the 10-year government bond yield is from the St. Louis FED’s internet data portal, FRED II (the acronym is IRLTLT01CHM156N). 36
C.6 Sweden For Sweden, a seasonally adjusted series for nominal GDP (‘BNPM - GDP at market prices, expenditure approach (ESA2010) by type of use, seasonally adjusted current prices, SEK million.’) is from Statistics Sweden. Series for M1 and the 3-month Treasury bill rate (‘Money supply, notes and coins held by Swedish non-bank public, M1 (SEK millions)’ and ‘Treasury Bills, SE 3M’, respectively) are from Statistics Sweden. A series for the 10-year government bond yield is from the St. Louis FED’s internet data portal, FRED II (the acronym is IRLTLT01SEM156N). C.7 Euro area For the Euro area, all of the data are from the European Central Bank. C.8 Denmark For Denmark, M1 (‘Money stock M1, end of period, Units: DKK bn.’) is from Denmark’s central bank. Nominal GDP (‘B.1GF Gross domestic product at factor cost, Seasonally adjusted, Current prices, 1-2.1.1 Production, GDP and generation of income (summary table) by seasonal adjustment, price unit, transaction and time, Units: DKK mio.’) and rwal GDP (‘B.1*g Gross domestic product, real, Seasonally adjusted, 2010-prices, real value, Units: DKK mio.’) are from Statistics Denmark. The central bank’s discounrt rate is from the central bank’s website. C.9 South Korea For South Korea, all of the data are from the central bank: nominal and real GDP (‘10.2.1.1 GDP and GNI by Economic Activities (seasonally adjusted, current prices, quarterly), Gross domestic product at market prices(GDP), Bil.Won’ and ‘10.2.2.2 Expenditures on GDP (seasonally adjusted, chained 2010 year prices, quarterly), Expenditure on GDP, Bil.Won’ respectively); M1 (‘’1.1.Money & Banking (Monetary Aggregates, Deposits, Loans & Discounts etc.), Seasonally Ajusted M1(End of), Bil.Won since 1969Q4; Before that: 1.1.Money & Banking (Monetary Aggregates, Deposits, Loans & Discounts etc.), M1(Narrow Money, End Of), Bil.Won, adjusted via ARIMA X-12); and the central bank’s discount rate. C.10 Japan A series for the discount rate is from the Bank of Japan. A seasonally adjusted series for nominal GDP is from the Economic and Social Research Institute, Cabinet Office, Government of Japan. A seasonally adjusted series for M1 has been constructed based on MA’MAM1NAM3M1MO (‘M1/Average amount outstanding/money stock’) and 37
MA’MAM1YAM3M1MO (‘M1/Percent changes from the previous year in average amounts outstanding/Money Stock’), both from the Bank of Japan. C.11 Hong Kong For Hong Kong, the HIBOR (Hong Kong Inter-Bank Offered Rate) is from the Hong Kong Monetary Authority (HKMA). M1 (‘M1, Total, (HK$ million)’) is from HKMA, and it has been seasonally adjusted via ARIMA X-12. Nominal GDP (‘GDP, HK$ million, From: Table031: GDP and its main expenditure components at current market prices, National Income Section (1)1,’) is from Hong Kong’s Census and Statistics Department. It has been seasonally adjusted via ARIMA X-12. D Why We Do Not Use Divisia Aggregates Throughout the entire paper we work with ‘simple-sum’ M1 aggregates. In this appendix we briefly discuss why we have chosen to ignore Divisia indices. A first problem is that, to the very best of our knowledge, such indices are only available for the U.S. (from the Center for Financial Stability,henceforthCFS) and for the U.K. (from the Bank of England). A second problem is that, for the U.S., the Divisia M1 series constructed by the CFS does not feature MMDAs (which are instead included in Divisia M2). This means that although the resulting index of monetary services has been constructed by optimally weighting the underlying individual assets, it suffers from the crucial shortcoming that it is not including a key component of the transaction technology. As a result, although Divisia M1 is in principle superior to the standard simple-sum M1 aggregate, it ultimately suffers from the same shortcoming of not including MMDAs. So the key question is: What is more important? Including MMDAs, or optimally weighting the underlying assets? Figure D.1 provides evidence on this, by showing the same evidence shown in Figure 2 in the main text of the paper, but this time with velocity being computed based on Divisia aggregates. The figure speaks for itself, and provides no evidence of a stable relationship between the velocity of any Divisia aggregate and its opportunity cost (computed based on the user cost series from the CFS). In particular, a comparison between the first panel of Figure D.1, and the second panel in Figure 2, clearly shows that, for the purpose of detecting a stable long-run demand for M1 in the United States, the crucial issue is including MMDAs in the definition of M1, rather than computing the aggregate by optimally weighting the underlying assets. So although, in theory, Divisia M1 possesses optimal properties, because of the specific way in which is has been constructed, within the present context such optimal properties are trumped by the fact that, exactly as its simple-sum counterpart, it does not include MMDAs. 38
25 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 Switzerland (1972Q1-2019Q4) 0 5 10 15 Short rate 0 0.5 1 1.5 Sweden (1998Q1-2019Q4) 0 5 10 15 Short rate 0 2 4 6 Percentage points of GDP Euro area (1999Q1-2019Q4) 0 5 10 15 Short rate 0 2 4 6 8 Percentage points of GDP 0 5 10 15 Short rate 0 0.5 1 1.5 2 0 5 10 15 Short rate 0 1 2 3 4 5 6 0 5 10 15 Short rate 0 0.5 1 1.5 2 2.5 3 0 5 10 15 Short rate 0 2 4 6 8 10 12 Percentage points of GDP 0 5 10 15 Short rate 0 1 2 3 4 Standard estimate based on rm=0 Estimate based on rm=-1+0.15*rb Estimate based on rm=-2+0.15*rb Log-log Selden-Latane' Semi-log Figure A.1 Estimates of the welfare costs of inflation produced by alternative specifications, based on Wright’s (2000) estimator