A new approach to estimating the natural rate of interest
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Benati, Luca Working Paper A new approach to estimating the natural rate of interest Discussion Papers, No. 21-08 Provided in Cooperation with: Department of Economics, University of Bern Suggested Citation: Benati, Luca (2021) : A new approach to estimating the natural rate of interest, Discussion Papers, No. 21-08, University of Bern, Department of Economics, Bern This Version is available at: https://hdl.handle.net/10419/242859 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 A New Approach to Estimating the Natural Rate of Interest Luca Benati 21-08 May, 2021 Schanzeneckstrasse 1 CH-3012 Bern, Switzerland http://www.vwi.unibe.ch DISCUSSION PAPERS
A New Approach to Estimating the Natural Rate of Interest∗ Luca Benati University of Bern† Abstract Building upon the insight that M1 velocity is the permanent component of nominal interest rates–see Benati (2020)–I propose a novel, and straightforward approach to estimating the natural rate of interest, which is conceptually related to Cochrane’s (1994) proposal to estimate the permanent component of GNP by exploiting the informational content of consumption. Under monetary regimes (such as inflation-targeting) making inflation I(0), the easiest way to implement the proposed approach is to () project the monetary policy rate onto M1 velocity–thus obtaining an estimate of the nominal natural rate–and then () subtract from this inflation’s sample average (or target), thus obtaining the real natural rate. More complex implementations based on structural VARs produce very similar estimates. Compared to existing approaches, the one proposed herein presents two key advantages: (1) under regimes making inflation I(0), M1 velocity is equal, up to a linear transformation, to the real natural rate, so that the natural rate is, in fact, observed; and (2) based on a high-frequency estimate of nominal GDP, the natural rate can be computed at the monthy or even weekly frequency. In the U.S., Euro area, and Canada the natural rate dropped sharply in the months following the collapse of Lehman Brothers. Likewise, the 1929 stock market crash was followed in the U.S. by a dramatic decrease in the natural rate. Keywords: Natural rate of interest; money velocity; structural VARs; unit roots; cointegration. ∗I wish to thank Giuseppe Cavaliere, James Hamilton and Peter Ireland for useful suggestions, and Juan-Pablo Nicolini for kindly providing data on Money Market Deposit Accounts for the United States. Usual disclaimers apply. †Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: luca.b[email protected]e.ch 1
1 Introduction Since the outbreak of the financial crisis, the natural rate of interest has been one of the most intensely discussed issues in both policymaking circles and academia. Currently, there are two approaches to estimating the natural rate. In the first, which was originally proposed by Laubach and Williams (2003), the natural rate is modelled as an I(1) process, usually a pure unit root;1it is embedded within a semi-structural framework also featuring a Phillips curve; and it is extracted from the data via the Kalman filter. In the second approach the natural rate is instead estimated based on a fully-specified DSGE model.2 In this paper I illustrate a novel, and straightforward method to estimate the natural rate of interest, which in line with the recent non-DSGE literature I define as a pure unit root process, specifically as the permanent component of the ex post real short-term (monetary policy) rate. The approach is conceptually related to Cochrane’s (1994) proposal to estimate the permanent component of GNP by exploiting the informational content of consumption, and builds upon the insight that M1 velocity3is, to a close approximation, the permanent component of nominal interest rates (see Benati, 2020). This suggests that in the same way as, as argued by Cochrane (1994), consumption can be treated as a good estimate of permanent GNP, M1 velocity can be regarded as a reliable estimate of the permanent component of the nominal short-term rate, , i.e. of the nominal natural rate. Further, basic economic logic suggests that is driven by ()permanentinflation shocks (via the Fisher effect), and () permanent shocks to the real natural rate of interest, i.e., = + ,where is the permanent component of inflation, and is the real natural rate. This implies that under monetary regimes, such as inflation-targeting, that cause inflation to be I(0)4–so that =0–permanent shifts in M1 velocity, ,uniquely reflect permanent fluctuations in the natural rate of interest, so that, e.g., =+ +,whereis a ‘small’5I(0) component, and the rest of the notation is obvious. Under these regimes, the easiest way to implement the proposed approach is to (1) project the monetary policy rate onto M1 velocity–thus obtaining an estimate of the nominal natural rate–and then (2) subtract from this inflation’s sample average (or target), thus obtaining the real natural rate. 1See e.g. Holston, Laubach and Williams (2017), and Fiorentini, Galesi, Pérez-Quirós and Sentana (2018). To be precise, in these papers the natural rate is modelled as the sum of two pure random walks (see Holston et al.’s equations 6, 8 and 9, and Fiorentini et al.’s equations 3, 5 and 6), one being trend GDP growth, and the other an additional ‘catch-all’ factor. 2See e.g. Del Negro, Giannone, Giannoni, and Tambalotti (2017). 3Defined as the ratio between nominal GDP and nominal M1, i.e. as the inverse of M1 balances expressed as a fraction of GDP. 4See Benati (2008). 5In the sense of explaining close to nil of fluctuations in velocity. 2
More complex implementations based on structural VARs produce very similar estimates. If, on the other hand, over the sample period inflation had been I(1), so that 6=0, in order to compute the real natural rate it is necessary to purge the nominal natural rate of permanent inflations shocks. This can be accomplished, e.g., based on a cointegrated SVAR for M1 velocity, the short rate and inflation (and possible other series). Compared to existing approaches, the one proposed herein presents two advantages. First, since under regimes making inflationI(0)M1velocityisequal,uptoa linear transformation, to the real natural rate, this implies that, under such regimes, the real natural rate of interest is observed. This is of obvious interest to policymakers, and (e.g.) it implies that a consistent decrease in M1 velocity under such a regime–such as the progressive fall that has been going on in several inflationtargeting countries since the early 1990s–provides direct evidence of a fall in the natural rate. Second, since M1 is observed (at least) at the weekly frequency, and interest rates are observed on a continuous basis, based on a high-frequency estimate of nominal GDP the natural rate can in principle be computed at the monthy, or even weekly frequency. In an application based on monthly data I show that in the U.S., Euro area, and Canada the natural rate fell sharply in the months following the collapse of Lehman Brothers. More generally, my evidence suggests that in all of the countries I analyze herein the real natural rate has been declinining at least since the early 1990s, and that at the end of the sample, in 2019, it had highly likely been negative in several of them. The paper is organized as follows. The next section discusses the data, whereas Section 3 discusses the close conceptual similarity between the present work and Cochrane (1994). Section 4 estimates the nominal natural rate of interest, whereas Section 5 explores the integration properties of inflation by monetary regime. Section 6 estimates the real natural rate, whereas Section 7 discusses the advantages of the proposed approach compared to existing alternatives, and computes monthly natural rate estimates for the U.S., the Euro area, the U.K., and Canada. Section 8 discusses two applications of the proposed methodology, pertaining to the evolution of the natural rate during the Great Depression, and to the impact of the COVID pandemic. Section 9 concludes. 2TheData Online Appendix A describes the data and their sources in detail. In brief, nearly all of data are from the datasets assembled by Benati (2020) and Benati, Lucas, Nicolini and Weber (2021), which for the post-WWII period I have updated to 2019Q4.6 All of the series are standard, with the single exception that, following Lucas and 6With the exception of the exercise in Section 8.2 I exclude the year 2020 from all samples, in order to avoid that my results could be distorted by the impact of the COVID pandemic. 3
22 Figure 1a M1 velocity and a nominal short-term interest rate over the post-WWII period
22 Figure 1b M1 velocity and a nominal short-term interest rate for West Germany and selected pre-World War II samples
Nicolini (2015), Benati (2020), and Benati et al. (2021), for the United States I consider, instead of the standard M1 aggregate produced by the Federal Reserve, one of the modifications that had originally been suggested by Goldfeld and Sichel (1990, pp. 314-315) in order to restore the stability of the long-run demand for M1. This alternative M1 series–which Lucas and Nicolini (2015) label as ‘New M1’–is obtained by adding to the standard M1 aggregate Money Market Deposits Accounts (MMDAs). The rationale for doing so is that MMDAs perform an economic function very similar to that of the ‘checkable deposits’ included in the standard M1 series (on this see the discussion in Lucas and Nicolini, 2015). Figure 1shows, for the eight countries analyzed herein, M1 velocity and a shortterm nominal interest rate over the post-WWII period. Visual impression suggests the following three facts, which as shown by Benati (2020) are indeed confirmed by a proper econometric analysis: () M1 velocity and the short rate are both I(1); () the two series are cointegrated; and, crucially, () up to a linear transformation, M1 velocity is, essentially, the stochastic trend of the short rate. Figure 1shows the same type of evidence for selected pre-World War II samples and for West Germany. The evidence for Portugal during the interwar period, with the hump-shaped fluctuation in the short rate being mirrored by a corresponding fluctuationinvelocity,is qualitatively in line with the post-WWII evidence in Figure 1. This is also the case, although to a lesser extent, for interwar Japan, with both series exhibiting an overall downward trend. For all other countries, however, the lack of any discernible trend in the short rate is mirrored by the broad flatness of M1 velocity. This is especially the case for Finland, Portugal (1892-1913), the United States, and to a lesser extent for Argentina and West Germany. This evidence naturally suggests that the large fluctuations in M1 velocity that have characterized the post-WWII period have been caused, under a stable demand for M1 balances as a fraction of GDP, by permanent fluctuations in both inflation and the real natural rate of interest injecting a unit root in nominal short-term interest rates. On the other hand, as the evidence in Figure 1shows, when nominal interest rates do not exhibit any trend, M1 velocity is likewise essentially flat. In turn, this suggests that, to the extent that ()inflation will remain under the control of the monetary authority, and therefore I(0), and () the decline in the real natural rate of interest will ultimately stop, the fall in velocity that has been going on since the early 1980s (see Figure 1) will also cease. I now turn to discussing the conceptual similarity between the present work and Cochrane’s (1994) analysis for consumption and output. 3 Conceptual Similarity With Cochrane (1994) The best way to illustrate the approach I am advocating herein is to highlight its close conceptual similarity with Cochrane’s (1994) proposal to estimate the permanent component of GNP by exploiting the informational content of consumption. 4
24 Figure 2 United States: fractions of forecast error variance explained by the permanent shock, based on cointegrated VARs featuring either (i) consumption and GDP or (ii) M1 velocity and the Federal Funds rate (with 1and 2-standard deviations bootstrapped confidence bands)
25 Figure 3 Estimates of the nominal natural rate computed by projecting the short rate on M1 velocity (with 1and 2-standard deviations bootstrapped confidence bands)
mator (as described in Hamilton, 1994) imposing one cointegration vector. I set the number of bootstrap replications to 10,000. Based on each bootstrapped, artificial sample ,with= 1, 2, ..., 10,000, I then perform exactly the same operations I previously performed based on the actual data. When estimating the nominal natural rate by projecting the short rate onto M1 velocity, I therefore estimate (7) based on the bootstrapped short rate and bootstrapped velocity, i.e. I run the OLS regression =+ + ,where and are the bootstrapped short rate and velocity for replication . This produces an estimate of the nominal natural rate for bootstrap replication , i.e. ˆ =ˆ +ˆ , and of the associated transitory component, ˆ =ˆ −ˆ −ˆ . When working with cointegrated VARs identified via long-run restrictions, on the other hand, I compute ˆ as in Blanchard and Quah (1989), i.e. by re-running history only conditional on transitory shocks. In this way, based on either approach I build up the bootstrapped distribution of the transitory component of the short (or shadow) rate, which I then use in order to compute confidence bands for the transitory component, and therefore, as a result, also for the permanent component. The following main results emerge from the two sets of figures: () as already mentioned, the simple, projection-based methodology produces results that are qualitatively the same, and quantitatively close to those produced by the alternative approach based on cointegrated SVARs. For reasons of simplicity and especially robustness,23 in what follows I will uniquely focus on the results produced by the simpler approach. () Whereas for the U.S. using shadow rates does not produce materially different estimates, for the U.K. and especially the Euro area this is not the case (see Figures A.5 and A.9 in the Online Appendix). This reflects the fact that, for the latter two countries, the difference between the shadow rate and the official monetary policy rate has been significantly greater than for the U.S.. For these three countries, in what follows I will exclusively focus on the results based on the official monetary policy rate, but the full sets of results based on the shadow rates are available upon request. () As one would expect, for all countries the estimated nominal natural rate behaves as a very low-frequency component of the short-term rate. () As shown in Figure 4, the nominal rate gap–defined as the difference between the short rate and the nominal natural rate, i.e. as −ˆ in (8)–exhibits a strong negative contemporaneous correlation with the detrended unemployment rate. This 23If M1 velocity were exactly equal to the nominal natural rate, the projection-based approach would exactly capture the latter. Under these conditions the SVAR-based approach could not improve upon this estimate, because the projection-based approach would rely on an observed linear transformation of the natural rate. To the extent that, as documented by Benati (2020), in fact we are close to such ideal situation, this argument approximately holds. On the other hand, a permanent-transitory decomposition based on a cointegrated SVAR is significantly more complex than a simple OLS regression, and as such the results it produces are likely more sensitive to issues such as lag order selection, and initial conditions (i.e., when the sample starts). 11
26 Figure 4 The nominal rate gap and the detrended unemployment rate
is in line, e.g., with the evidence in King and Watson (1996, pp. 38-39 and Figure 2) that ‘[the band-pass filtered cyclical components of] nominal interest rates and output are positively correlated’, and it has a straightforward interpretation in terms of counter-cyclical monetary policy.24 () Before the collapse of Lehman Brothers, the probability that the nominal natural rate had been negative had consistently been (close to) nil. Since then, however, it has materially increased in Canada, the Euro area, Sweden, and the U.S., whereas it has exhibited little variation in the remaining countries. In particular, at the end of the sample the probability was equal to 95 per cent in Sweden, 50 per cent in Canada, and around 60 per cent in both the Euro area and the U.S.. I now turn to discussing the integration properties of inflation. 5 Monetary Regimes and the Stochastic Properties of Inflation Table C.1 in Online Appendix C reports results from tests for multiple breaks at unknown points in the sample in the mean of inflation based on the methodology proposed by Bai and Perron (1998, 2003).25 For Australia, Canada, New Zealand, Norway, Sweden, and the United Kingdom I focus on the sample period since the introduction of inflation targeting;26 for the Euro area I consider the period since the start of European Monetary Union, in January 1999; and for the United States theperiodfollowingthebreakinthemeanofinflation identified by Levin and Piger (2004), in 1992Q2. The null hypothesis of no breaks in the mean of inflation cannot be rejected for any country. 24Interestingly, for the U.S. the relationship between the nominal rate gap and the detrended unemployment rate had been put temporarily offkilter by the introduction of Money Market Deposits Accounts (MMDAs) in 1982Q4. After a brief period of adjustment, however, the relationship strongly reasserted itself since the second half of the 1980s. This confirms the meaningfulness of working with Lucas and Nicolini’s (2015) ‘New M1’ aggregate: what it suggests is that in fact New M1 is the equivalent, for the period since the 1980s, of the standard M1 aggregate for the previous period. Benati (2021) presents additional evidence on this based on a comparison between the evolution of M1 velocity and of long-term interest rates. 25In performing the tests I exactly follow the recommendations of Bai and Perron (2003), with the only difference that, instead of relying on the asymptotic critical values tabulated in Bai and Perron (1998), I bootstrap the -values via the procedure proposed by Diebold and Chen (1996), setting the number of bootstrap replications to 10,000. 26In Canada, New Zealand, Norway, Sweden, and the United Kingdom inflation targeting was was introduced, respectively, in February 1991, February 1990, March 2001, January 1993, and October 1992. As for Australia, which never formally announced an inflation target, I consider the period since mid-1994 (specifically, since 1994Q3), when the ReserveBankofAustraliastarted to target inflation de facto. 12
Table 1 Exploring inflation persistence by monetary regime Bootstrapped p-values for Hansen MUB Country Period =2 =4 =6 =8 estimate of I: Regimes with clearly-defined nominal anchors Australia 1994Q3-2019Q4 0.0012 0.0148 0.0908 0.3618 0.28 [0.05 0.51] Canada 1991Q1-2019Q4 0.0000 0.0002 0.0002 0.0009 -0.08[-0.350.19] Euro area 1999Q1-2019Q4 0.1015 0.2230 0.1919 0.3032 0.66 [0.46 0.88] New Zealand 1990Q1-2019Q4 0.0000 0.0001 0.0060 0.0029 -0.21[-0.510.10] Norway 2001Q2-2019Q4 0.0016 0.0047 0.0616 0.1582 0.22 [-0.09 0.52] Sweden 1998Q1-2019Q4 0.0000 0.0210 0.0640 0.1115 0.30 [-0.15 0.78] United Kingdom 1992Q4-2019Q4 0.0000 0.0001 0.0030 0.0060 -0.40 [-0.66 -0.16] United States 1992Q2-2019Q4 0.0081 0.0501 0.0704 0.0090 0.49 [0.35 0.64] II: Previous periods Australia 1972Q2-1994Q2 0.1627 0.7399 0.5881 0.2010 1.01 [0.78 1.07] Canada 1967Q2-1990Q4 0.2220 0.4366 0.3495 0.2957 0.90 [0.73 1.03] Euro area 1970Q2-1998Q4 0.4598 0.4598 0.7095 0.8493 1.01 [0.92 1.04] Norway 1978Q2-2001Q1 0.0014 0.0272 0.1735 0.1527 0.49 [0.19 0.81] United Kingdom 1955Q2-1992Q3 0.0410 0.1942 0.1752 0.2245 0.87 [0.74 1.02] United States 1959Q2-1992Q1 0.3491 0.3266 0.3062 0.3316 0.92 [0.85 0.99] Based on 10,000 bootstrap replications. With 90% bootstrapped confidence interval. Table 1 reports bootstrapped p-values27 for Elliot, Rothenberg, and Stock (1996) unit root tests for inflation, together with Hansen (1999) ‘grid bootstrap’ medianunbiased (MUB) estimates of the sum of the autoregressive coefficients ()inAR() representations for inflation.28 In both cases I set the number of bootstrap replications to 10,000. As for the sample periods, I consider both the previously mentioned monetary regimes featuring clearly-defined nominal anchors,29 and, depending on data availability, the previous periods. The evidence in Table 1 confirms the findings in Benati (2008). In particular, for regimes with clearly-defined nominal anchors, () the point estimates of produced by Hansen’s procedure range between -0.40 and 0.66, and the upper limits of their bootstrapped 90%-coverage confidence interval range between -0.16 and 0.88: based on Hansen’s procedure there is no evidence that, under these regimes, inflation may have been I(1). () By the same token, based on Elliot et al.’s tests a unit root in inflation is 27-values have been computed by bootstrapping 10,000 times estimated ARIMA(,1,0) processes. 28For Hansen’s (1999) procedure, I select the lag order as the maximum between the lag orders selected by the Schwartz and Hannan-Quinn criteria, and I set the ‘step’ in the grid of possible values for to 0.01. 29Strictly speaking, the U.S. Federal Reserve introduced an inflation target only in January 2012. In what follows I consider the entire period since 1992Q2 because, even before the introduction of a formal target, the Fed’s monetary policy had been characterized since the end of the Volcker disinflation by a strong, although generic committment to price stability. 13
strongly rejected for Canada, New Zealand, the United Kingdom, and the United States, and for Australia, Norway, and Sweden it is rejected, at the 10 per cent level, for all lag orders except =8. Only for the Euro area a unit root cannot be rejected for any lag order. For the previous periods, which had been largely dominated by the Great Inflation episode, the opposite is true. Starting from Hansen’s MUB estimates of ,thepoint estimate is borderline explosive for Australia and the Euro area, and for four countries (Australia, Canada, Euro area, and United Kingdom) the 90 per cent confidence interval includes 1, whereas for the United States, with an upper bound equal to 0.99, this is almost the case. Likewise, based on Elliot et al.’s tests the null of a unit root cannot be rejected for any lag order for Australia, Canada, the Euro area, and the United States, whereas evidence is mixed for Norway, and for the United Kingdom it can be rejected only for =2. These results confirm Benati’s (2008) main finding that whereas for sample periods dominated by the Great Inflationexperienceitistypicallynotpossibletorejectthe null hypothesis of a unit root in inflation, under monetary regimes, such as inflation targeting, featuring a clearly-defined nominal anchor (or, in the case of the United States before the introduction of an inflation target, a generic, but strong and credible committment to keeping inflation low and stable), inflation has consistently been I(0). Under this respect, the results from Elliot et al.’s tests for the Euro area should be quite heavily discounted for two reasons. First, the visual evidence in Figure A.1 in the Online Appendix clearly suggests that the collapse of Lehman Brothers, which unleashed the most violent phase of the Great Recession, was associated with a dramatic, highly persistent, but ultimately transitory fall in Euro area inflation, from an average of 2.01 per cent over the period 1999Q1-2008Q3,30 to 1.05 per cent over the period 2008Q4-2017Q3. Over the subsequent period inflation has progressively converged towards 2 per cent. A possible, and (I would argue) plausible interpretation of the lack of rejection of a unit root for the period 1999Q1-2019Q4 is therefore that it is the figment of a very large negative transitory shock, which in a small sample can easily be confused for a permanent one. Second, in spite of such persistent downard shift in inflation, inflation expectations (as measured by the ECB’s Survey of Professional Forecasters) have remained well-anchored,31 thus suggesting that agents have correctly interpreted the shift as temporary. In what follows I will therefore work under the assumption that, for the sample periodsreportedinTable1as‘regimeswithclearly-defined nominal anchors’, inflation has consistently been I(0). I now turn to discussing the estimates of the real natural 30In fact, in line with Benati (2008), for the period 1999Q1-2008Q3 the bootstrapped -values for Elliot et al.’s tests are equal to 0.1077, 0.0416, 0.0249, and 0.0134, and Hansen’s (1999) MUB estimate of is 0.41 [0.02 0.82]. 31See in particular Figure A.3 in the Online Appendix of Benati (2020). The figure shows the inflation forecasts from the ECB’s Survey of Professional Forecasters at three alternative horizons, 1-, 2-, and 5-years ahead. Over the entire period since 1999Q1, the 5-years ahead forecast has fluctuated between 1.8 and 2.0 per cent. 14
27 Figure 5 Estimates of the real natural rate for monetary regimes making inflation I(0) (with 1and 2-standard deviations bootstrapped confidence bands) computed by projecting the short rate on M1 velocity
rate for such monetary regimes. 6 Estimating the Real Natural Rate Figure 5 shows, for monetary regimes with clearly-defined nominal anchors,32 the ex post short-term real rate, computed as the difference between the short-term nominal rate and inflation, together with the estimated real natural rate, which has been computed by subtracting inflation’s sample average from the nominal natural rate estimates shown in Figure 3.33 Figure A.10 in the Online Appendix shows the corresponding estimates based on cointegrated SVARs identified long-run restrictions. For all countries, the estimated real natural rate behaves, as expected, as a very lowfrequency component of the ex post short-term real rate. The following main results emerge from the two figures: () consistent with both conventional wisdom, and previous evidence–see in particular Holston et al. (2017), and Fiorentini et al. (2018)–in all countries natural rate estimates have been consistently trending downwards over the entire sample period. The decrease has been especially marked for Australia, New Zealand, the U.K., and Canada: since the first half of the 1990s, the point estimate of the natural rate has fallen by about 6 per cent for the first three countries, and by about 8 per cent for the fourth. By the same token, in the U.S. it has fallen by about 6 percentage points since the peak of 4.1 per cent reached in the second half of the 1990s around the time of the ‘New Economy’, whereas in both the Euro area and Sweden the decrease since the start of the new millennium has been equal to about 4 percentage points. ()Different from the corresponding results for the nominal natural rates discussed in point () of Section 4, in several countries the probability that the real natural rate had been negative had already been increasing before the collapse of Lehman Brothers. This is the case especially for the Euro area, Norway, and the U.K.. Following Lehman’s collapse the probability has markedly increased in all countries except Norway. In particular, at the end of the sample the probability was equal to 100 per cent in both the Euro area and Sweden, whereas in Canada and the U.S. it was slightly greater than 90 per cent. This evidence provides support to the conjecture, firstadvancedbySummers(1991),thatfollowinglargenegativeshocksthe natural rate might fall below zero. () Further, in several countries the estimates are quite sobering, especially towards the end of the sample. In the U.S., for example, the point estimate has been equal to about -2 per cent since 2014, whereas in Canada, the Euro area and Sweden it has reached, in 2019, -1.9, -1.7, and -2.4 per cent, respectively. The only two countries for which in 2019 the point estimate was still (barely) positive were New 32The sample periods are the same reported in Table 1. 33Once again, in order to eliminate some low-frequency variation, all series have been smoothed a 4-quarter centered moving average. 15
Zealand and Norway. These estimates are very similar to those produced by Fiorentini et al. (2018) basedonamodified (and, they argue, superior) version of the methodology originally proposed by Laubach and Williams (2003), and more recently used, e.g., by Laubach and Williams (2016) and Holston et al. (2017). For example, in Fiorentini et al.’s (2018) Figure 10, the U.S. natural rate decreased from 2.5-3 per cent in the second half of the 1990s to about -2 per cent in 2016, whereas that for the Euro area fell from 2 per cent in 2000 to about -1 per cent in 2016. These figures are very close to those in Figure 5 in the present work. On the other hand, my estimates are lower than those found in Holston et al. (2017), but based on Fiorentini et al.’s (2018) arguments those estimates should be regarded as less reliable. EstimatesoftherealnaturalrateaslowasthoseinFigure5,aswellasinFiorentini et al. (2018), raise an obvious question: Are they plausible? Could the real natural rate truly sink that low? This question is best addressed by focusing on ()therelationship between the natural rate and the ex post real rate, and () the behavior of GDP and inflation over the sample period. Let us consider for example Sweden, with an estimated natural rate of -2.5 per cent at the end of 2019. At first sight, this number might appear to some researchers as manifestly absurd. It becomes however much less absurd, and much more plausible, when one considers that since Lehman’s collapse (and in fact since the beginning of the millennium) the natural rate has closely tracked the dramatic decrease in the ex post real rate: this suggests that on average the Riksbank ’s monetary policy has been broadly neutral, and that the fall in the ex post real rate it has engineered by decreasing the monetary policy rate was simply a reaction to the progressive fall in the natural rate. The evolution of prices and output is consistent with this: since the 2008-2009 recession annual inflation and GDP growth have both been broadly stable, the former slowly increasing from about 1 per cent in early 2010 to 2.5 per cent at the end of 2019, and the latter fluctuating around an average of about 2 per cent. A very similar argument can be made for the remaining countries. This suggests that central banks have been broadly tracking the natural rate, and that the progressive decreases in ex post real rates across the board have simply reflected the underlying fall in the natural rates. In turn, this suggests that the estimates in Figure 5 are likely plausible.34 Finally, it is worth highlighting how, in line with Taylor (2008, 2009), for the U.S. a comparison between the ex post real rate and the estimated natural rate suggests that monetary policy had been highly expansionary during the years immediately preceding the outbreak of the financial crisis. In particular, in 2004 the ex post real rate had been below the natural real rate, on average, by about 300 basis points. I now turn to discussing the advantages of the methodology I am advocating compared to existing approaches. 34It is also worth recalling that DSGE-based estimates are often much more volatile. For example, in Barsky et al.’s (2014) Figure 1 the U.S. natural rate has fluctuated, since the early 1990s, between about -7 and about 12 per cent, i.e. over a range of nearly 20 percentage points. 16
7 Advantages of the Proposed Approach Compared to existing approaches to the estimation of the natural rate, the one proposed herein features two advantages, which I discuss in turn. 7.1 Under monetary regimes making inflation I(0) the real natural rate is observed As discussed in Section 3.3, under monetary regimes making inflation I(0)–so that, in (9), =0–permanent shifts in M1 velocity uniquely reflect, to a first approximation, permanent fluctuations in the real natural rate of interest. In fact, as long as in expression (4) is ‘small’, '+ ,sothat '(−): in plain English, under such regimes the real natural rate of interest is, up to a linear transformation, observed. An immediate implication is that a consistent decrease in M1 velocity under a monetary regime causing inflation to be I(0)–such as the protracted fall in velocity that has been going on in several inflation-targeting countries since the early 1990s–provides direct evidence of a fall in the real natural rate of interest. The fact that the approach I am advocating herein relies on a series that, under monetary regimes making inflation I(0), is essentially a linear transformation of the real natural rate highlights a stark difference with existing approaches (either DSGEor non-DSGE based), none of which exploits a series with such a strong informational content for the real natural rate. 7.2 Computing high-frequency estimates of the natural rate Since interest rates are observed on a continuous basis, and M1 is observed (at least) at the weekly frequency, all a researcher needs in order to compute high-frequency estimates of the nominal and real natural rates of interest is a corresponding highfrequency estimate of nominal GDP. Interpolating quarterly GDP to the monthly frequency35 is routinely done in the literature (for the United States, see e.g. Bernanke, Gertler, and Watson, 1997, and Stock and Watson, 2012). The recent work of (e.g.) Lewis, Mertens, and Stock (2020) about tracking the economic impact of the COVID pandemic has shown how to perform a similar interpolation at the weekly frequency.36 Based on a weekly estimate of nominal GDP, and weekly observations for M1 and nominal interest rates, a central bank could therefore, in principle, produce weekly estimates of nominal and real natural rates. Figure 6 presents estimates of nominal and real natural rates at the monthly frequency for Canada, the Euro area, the United Kingdom and the United States, 35To the very best of my knowledge, Canada and the U.K. are the only countries producing official monthly estimates of real GDP. U.K. estimates start however in 1997, so that in the present work I have relied on the unofficial estimates from NIESR (for details, see Online Appendix A). 36The be precise, Lewis et al. (2020) focus on real GDP, but their methodology can obviously also be applied to nominal GDP. 17
caveats they are subject to. At the same time, taken at face value they suggest that the two crises had a very similar impact on the natural rate. 9Conclusions Since the early 1980s it has been conventional wisdom among macroeconomists and policymakers that monetary aggregates contain little useful information for monetary policy. In this paper I have shown that, in fact, a specific transformation of a monetary aggregate, the velocity of M1, contains crucial information about the evolution of the real natural rate of interest. Building upon the the insight that M1 velocity is the permanent component of nominal interest rates (see Benati, 2020), I have proposed a new and straightforward approach to estimating the natural rate of interest, which is conceptually related to Cochrane’s (1994) proposal to estimate the permanent component of GDP by exploiting the informational content of consumption. Under monetary regimes (such as inflation-targeting) making inflation I(0), the easiest way to implement the proposed approach is to ()projectthemonetarypolicy rate onto M1 velocity–thus obtaining an estimate of the nominal natural rate–and then ()subtractfromthisinflation’s sample average (or target), thus obtaining the real natural rate. More complex implementations based on structural VARs produce very similar estimates. Compared to existing approaches, the one proposed herein presents two key advantages: (1) under regimes making inflation I(0), M1 velocity is equal, up to a linear transformation, to the real natural rate, so that the natural rate is, in fact, observed; and (2) based on a high-frequency estimate of nominal GDP, the natural rate can be computed at the monthy or even weekly frequency. In the U.S., Euro area, and Canada the natural rate dropped sharply in the months following the collapse of Lehman Brothers. Likewise, the 1929 stock market crash was followed in the U.S. by a dramatic decrease in the natural rate. 21
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A Results from unit root tests Table A.1 reports bootstrapped -values for Elliot, Rothenberg, and Stock (1996) unit root tests (with an intercept, but no time trend) for M1 velocity and a short-term nominal interest rate. The -values have been computed by bootstrapping estimated ARIMA(,1,0) processes via the procedure proposed by Diebold and Chen (1996), setting the number of bootstrap replications to 10,000. Table A.1 Bootstrapped p-values for Elliot, Rothenberg, and Stock unit root testsfor velocity and a short rate Country Period =2 =4 =6 =8 M1 velocity Australia 1972Q1-2019Q4 0.9557 0.9573 0.9508 0.9841 Canada 1982Q3-2019Q4 0.2122 0.3114 0.3555 0.1592 Euro area 1970Q1-2019Q4 0.0733 0.3247 0.2133 0.1701 New Zealand 1990Q1-2019Q4 0.8679 0.8850 0.8748 0.7521 Norway 1978Q1-2019Q4 0.3307 0.3386 0.2024 0.2365 Sweden 1998Q1-2019Q4 0.7846 0.7036 0.7687 0.6471 United Kingdom 1963Q1-2019Q4 0.9884 0.9647 0.9436 0.9268 United States 1959Q1-2019Q4 0.8686 0.8662 0.8576 0.8340 Short rate Australia 1972Q1-2019Q4 0.4980 0.5122 0.5269 0.4098 Canada 1982Q3-2019Q4 0.0792 0.4167 0.4657 0.5730 Euro area 1970Q1-2019Q4 0.5228 0.6106 0.6629 0.5678 New Zealand 1990Q1-2019Q4 0.1123 0.0809 0.0258 0.0972 Norway 1978Q1-2019Q4 0.4458 0.5890 0.6803 0.5231 Sweden 1998Q1-2019Q4 0.3808 0.3850 0.4061 0.5515 United Kingdom 1963Q1-2019Q4 0.5847 0.4865 0.5016 0.5134 United States 1959Q1-2019Q4 0.3162 0.2854 0.2914 0.2031 Based on 10,000 bootstrap replications. In nearly all cases, evidence of a unit root for either series is very strong. The only exception is the short rate for New Zealand, for which the null of unit root is instead near-uniformly rejected. In what follows I will proceed under the assumption that all nominal interest rates have been I(1) over the sample periods analized herein,39 and that the rejection of the null of a unit root for New Zealand is a statistical fluke, possibly due to small-sample issues. There are two reasons for doing so. First, even a perfectly sized test, by definition, incorrectly rejects the null hypothesis per 39This is an important qualification. Under metallic standards–for which inflation had been uniformly I(0), and in fact most of the time statistically indistinguishable from white noise (see Benati, 2008)–it is often possible to reject the null hypothesis of a unit root in short-term interest rates, as one would logically expect if the natural real rate featured comparatively little variation (this evidence is available upon request). 25 Appendix
centofthetimeattheper cent level. When performing many statistical tests, such as in the present case, a certain fraction of ‘fluke rejections’ of the null should therefore be logically expected. Sure enough, the data I am using herein have not beenrandomlygeneratedaspartofaMonteCarloexperiment, 40 but the basic logic of this argument still holds. For example, taking the argument literally–i.e., as if we were here dealing with a Monte Carlo experiment featuring independent random draws–the five rejections (at the 10 per cent level) of the null of a unit root reported in Table A.1 represent 7.8 per cent of the overall number of tests reported in the table, i.e. smaller than the 10 per cent of ‘fluke rejections’ we would expect from a perfectly sized test with independent Monte Carlo artificial samples. Second, visual evidence (see Figure 1) strongly suggests that all nominal short rates have been non-stationary over the sample periods analized herein. B Evidence on Cointegration Between M1 Velocity and the Short Rate Table B.1 reports, for bivariate systems featuring M1 velocity and a short-term rate, () bootstrapped -values for Johansen’s maximum eigenvalue tests of the null hypothesis of 0 versus 1 cointegration vectors, and () 90%-coverage bootstrapped confidence intervals for the second element of the normalized cointegration vector based on Wright’s (2000) methodology. As for Johansen’s tests, following Benati (2020) and Benati et al. (2021), I bootstrap them via the procedure proposed by Cavaliere et al. (2012, henceforth CRT).41 I select the VAR lag order as the maximum42 between the lagorderschosenbytheSchwartzandtheHannan-Quinncriteria 43 for the VAR in levels, and I estimate the VECM based on Johansen’s estimator as detailed in Hamilton (1994). As for Wright’s (2000) test, since it has been designed to be equally valid for data-generation processes (DGPs) featuring either exact or near unit roots, following Benati (2020) and Benati et al. (2021) I consider two alternative bootstrapping procedures, corresponding to either of the two possible cases. The first procedure involves bootstrapping as in CRT the cointegrated VECM estimated by imposing one cointegration vector. This procedure is the correct one if the data feature exact 40In particular, the data for individual countries are not independent random draws, since all countries experienced common events such as the Great Inflation of the 1970s, the disinflation of the early 1980s, the spread of globalization, and the 2008-2009 financial crisis. 41For details see Online Appendix B.2, which also discusses Monte Carlo evidence on the performance of CRT’s procedure. 42I consider the maximum between the lag orders chosen by the SIC and HQ criteria because the risk associated with selecting a lag order smaller than the true one (model mis-specification) is more serious than the one resulting from choosing a lag order greater than the true one (over-fitting). 43On the other hand, I do not consider the Akaike Information Criterion since, as discussed (e.g.) by Luetkepohl (1991), for systems featuring I(1) series the AIC is an inconsistent lag selection criterion, in the sense of not choosing the correct lag order asymptotically. 26
unit roots. For the alternative possibility in which the two series are near unit root processes, I proceed as follows. Based on the just-mentioned cointegrated VECM estimated by imposing one cointegration vector, I compute the implied VAR in levels, which by construction features one, and only one, eigenvalue equal to 1.44 Ithenturn suchexactunitrootVARintoitscorresponding near unit root VAR, by shrinking the single unitary eigenvalue to =1—0.5×(1/), where isthesamplelength. 45 The bootstrapping procedure I implement for the second possible case, in which the two series are near unit root processes, is based on bootstrapping such near unit root VAR. In practice the two procedures produce near-identical results, and in Table B.1 I therefore uniquely report results based on bootstrapping the VECM estimated by imposing one cointegration vector. Table B.1 Bootstrapped p-values for Johansen’s maximum eigenvalue tests for M1 velocity and a short-term rate, and 90% bootstrapped confidence intervals for the second element of the normalized cointegration vector based on Wright’s (2000) tests p-values for maximum Results from Country Period eigenvalue testsWright’s test Australia 1969Q3-2019Q4 0.0661 [-0.9255 -0.7013] Canada 1967Q1-2019Q4 0.0279 [-1.1642 -0.1032] Euro area 1999Q1-2019Q4 0.0896 [-0.6013 -0.2970] New Zealand 1990Q1-2019Q4 0.1584 [-0.1643 -0.0642] Norway 1978Q1-2019Q4 0.0992 [-0.1268 -0.0868] Sweden 1998Q1-2019Q4 0.1136 [-0.3642 -0.3081] United Kingdom 1955Q1-2019Q4 0.0201 [-0.5323 -0.3441] United States 1959Q1-2019Q4 0.0985 [-0.5634 -0.3672] Based on 10,000 bootstrap replications. Null of 0 versus 1 cointegration vectors. Based on Wright’s (2000) tests, the null hypothesis of cointegration is never rejected. Likewise, at the 10 per cent level Johansen’s tests reject the null of 0 cointegration vectors for all countries except Sweden (marginally), and New Zealand (with a-value of 0.1584). As in Benati (2020), in what follows I will therefore proceed under the assumption that M1 velocity and the short rate are cointegrated in all samples. Online Appendix B.4 reports results from Hansen and Johansen’s (1999) Nyblom-type tests for stability in either the cointegration vector, or the vector of loading coefficients, in the estimated VECMs, and discusses Monte Carlo evidence on the performance of the tests. In short, evidence of breaks in either the cointegration vector or the loading coefficients is nearly non-existent. In particular, based on either 44Bootstrapping this VAR would be equivalent to bootstrapping the underlying cointegrated VECM, that is, it would be correct if the data featured exact unit roots. 45Once again, for details see Online Appendix B.2. 27
the Selden-Latané specification–which, as discussed, appears to be the one preferred by the data for low-inflation (and therefore low-interest rates) countries–or the loglog, not a single break in either the cointegration vector or the loading coefficients is identified. As for the semi-log, no break in the cointegration vector is identified for any country, whereas only for Norway a break in the loading coefficients is identified, although the -value, at 0.0935, is essentially borderline. C Computing Permanent and Transitory GDP by Projecting GDP on Consumption The left-hand side panel of Figure C.1 in this appendix shows the transitory component of U.S. GDP obtained by projecting log real GDP onto log real consumption, i.e. the residual from the cointegrating regression ln =+ln +,(C.1) where and are real GDP and real consumption, respectively (the two series are described in Online Appendix A.2.9., and their unit root and cointegration properties are discussed in footnote 7 in the main text). I estimate (C.1) asimpleOLS regression, but near-identical results are produced by Stock and Watson’s (1993) dynamic OLS estimator. Two main findings emerge from the figure. First, the estimated transitory component of GDP captures remarkably well the peaks and troughs of the post-WWII U.S. business cycle as established by the NBER BusinessCycle Dating Committee (i.e., the vertical blue and red bars in the figure). Second, the transitory component interprets a sizeable portion of the fall in output associated with the Great Recession as permanent: this is clearly highlighted, e.g., by the fact that whereas the troughs of annual real GDP growth associated with the Volcker recession and the Great Recession had been equal to -2.6 and -3.9 per cent, respectively, the troughs of the corresponding transitory components of GDP obtained by projecting log real GDP onto log real consumption had been equal to -4.0 and -2.8 per cent, respectively. As a matter of logic, the only possible interpretation of this is that, when viewed though the lenses of consumption, the latter recession had been characterized by a significantly greater decrease in permanent GDP than the former. The right hand-side panel of Figure C.1 provides simple, but powerful corroborating evidence that this may in fact had been the case. The figure shows log real GDP and rescaled log real consumption,46 together with () up to 2004Q4, the HP-filtered trend of log real GDP,47 and () starting from 2005Q1, the forecast of the HP-filtered trend, which I computed recursively by exploiting the fact that, in the state-space 46I rescaled log real consumption as in Cochrane’s (1994) Figure III, i.e. by adding to it the mean log ratio between GDP and consumption. 47I set the smoothing parameter to the standard value of 1600 for quarterly data, but qualitatively similar results are produced by alternative plausible values of the parameter. In order to compute 28
31 Figure C.1 Computing transitory GDP by projecting GDP on consumption
representation of the Hodrick-Prescott filter, the second difference of the HP trend is white noise (see e.g. Harvey and Jaeger, 1993, and King and Rebelo, 1993), so that the trend, , evolves according to =2−1−−2+,withbeingashock. Following the collapse of Lehman Brothers, both GDP and consumption have significantly fallen short compared to the forecast of the HP trend. In particular, at the end of 2019 the shortfall had been for both series equal to about 12 per cent. Crucially, the fact that this has equally held for both GDP and consumption logically suggests that the shortfalls are permanent: otherwise, by the Permanent Income Hypothesis, consumption would be close to the HP trend. D TheMonthlySeriesUsedinSection7.2 As discussed more extensively in Online Appendix A.3, for Canada and the U.K. monthly seasonally adjusted real GDP estimates are available respectively from Statistics Canada and from the U.K.’s National Institute for Economic and Social Research (NIESR). As for the Euro area I have interpolated seasonally adjusted quarterly real GDP based on Stock and Watson’s (2012) methodology, using monthly seasonally adjusted industrial production as the interpolator series. In order to compute nominal GDP, for any of the three countries I have then interpolated to the monthly frequency the quarterly seasonally adjusted GDP deflator based on Stock and Watson’s (2012) methodology, using the monthly seasonally adjusted core CPI as the interpolator series. For the United States, seasonally adjusted monthly series for real and nominal GDP are from Stock and Watson (2012) until 2010, and from IHS Markit, a consultancy, after that. Finally, a crucial component of Lucas and Nicolini’s (2015) ‘New M1’ aggregate that is used herein (see the discussion in Section 2), i.e. Money Market Deposit Accounts (MMDAs), is available only at the quarterly frequency. As discussed in Online Appendix A.3.4, I have therefore interpolated MMDAs to the monthly frequency as in Stock and Watson (2012), using as monthly interpolator the seasonally adjusted series for ‘Total Checkable Deposits’ from the Federal Reserve Board. The rationale for using this interpolator series is exactly the same originally advanced by Goldfeld and Sichel (1990, pp. 314-315), and then reiterated by Lucas and Nicolini (2015), for including MMDAs within an expanded, and economically more sensible definition of M1 (see Section 2): MMDAs perform an economic function which is very similar to that of the checkable deposits included in the standard M1 series. Therefore, on the one hand it makes sense to include them within an economically sensible definition of M1; on the other hand, it makes sense to use total checkable deposits as the monthly interpolator for quarterly MMDAs. the HP trend I only use data up to 2004Q4 because it could possibly be argued that during the years immediately preceding the financial crisis U.S. GDP had been significantly above trend. Using data up to 2008Q3, however, produces near-identical results. 29
Post-WWII period A quarterly seasonally adjusted series for nominal GDP (Gross Domestic Product, GDP Billions of Dollars, Quarterly, Seasonally Adjusted, Annual Rate)isfromFREDII(acronymisGDP).MonthlyseriesfortheFederalFundsrate and for 3-, 5-, and 10-year government bond yields are also from FRED II (acronyms are FEDFUNDS, GS3, GS5, and GS10). All interest rate series have been converted to the quarterly frequency by taking averages within the quarter. As mentioned in the main text, M1 is constructed as the sum of the standard aggregate produced by the Federal Reserve and of Money Market Deposit Accounts (MMDAs). The former series is from the St. Louis FED (acronym is M1SL, ‘M1 Money Stock, Billions of Dollars, Quarterly, Seasonally Adjusted’), whereas MMDAs data are from the Federal Reserve’s mainframe, and they have been kindly provided by Juan-Pablo Nicolini. A quarterly seasonally adjusted series for real GDP (Real Gross Domestic Product, Billions of Chained 2009 Dollars, Quarterly, Seasonally Adjusted Annual Rate) is from FRED II (acronym is GDPC1). A quarterly seasonally adjusted series for real chainweighted consumption of non-durables and services has been computed based on the data in Tables 1.1.6, 1.1.6B, 1.1.6C, and 1.1.6D of the National Income and Product Accounts produced by the U.S. Department of Commerce’s Bureau of Economic Analysis. A quarterly seasonally adjusted series for the ‘Civilian unemployment rate, persons 16 years of age and older’, has been computed by taking averages within the quarter of the corresponding monthly series from the U.S. Bureau of Labor Statistics (the FRED II code is UNRATE). A.3 Monthly data A.3.1 Canada A monthly seasonally unadjusted series for M1 (‘v41552787, Table 176-0020: M1B (gross) (currency outside banks, chartered bank chequable deposits, less inter-bank chequable deposits) (x 1,000,000)’) is from Statistics Canada. The series has been seasonally adjusted via ARIMA X-12 as implemented in Eviews. A monthly series for the Bank rate (i.e., the Bank of Canada’s monetary policy rate) is from the Bank of Canada. A monthly seasonally adjusted series for real GDP (‘Real GDP, Total economy, 1986 constant prices’) is from Statistics Canada. I interpolated to the monthly frequency a quarterly seasonally adjusted series for the GDP deflator from Statistics Canada (‘GDP deflator, Seasonally adjusted, 2016A000011124’, acronym is v62307282) as in Stock and Watson (2012), using as interpolator series a monthly seasonally adjusted core CPI series from Statistics Canada (‘Consumer Price Index (CPI), all-items excluding eight of the most volatile components as defined by the Bank of Canada and excluding the effect of changes in indirect taxes, seasonally adjusted’, acronym is v112593706). Finally, I computed a monthly seasonally adjusted series for nominal GDP as the product of the interpolated real GDP and GDP deflator series. 7
A.3.2 Euro area All of the data are from the European Central Bank (ECB): a monthly seasonally adjusted series for M1 (‘Euro area (changing composition), Outstanding amounts at the end of the period (stocks), MFIs, central government and post office giro institutions reporting sector - Monetary aggregate M1, All currencies combined - Euro area (changing composition) counterpart, Non-MFIs excluding central government sector, denominated in Euro, data Working day and seasonally adjusted’; ECB code is BSI.M.U2.Y.V.M10.X.1.U2.2300.Z01.E); a monthly seasonally unadjusted series for the 3-month Euribor rate (‘Euro area (changing composition) - Money Market - Euribor 3-month - Historical close, average of observations through period - Euro, provided by Reuters, average of observations through period (A)’; ECB code is FM.M.U2.EUR.RT.MM.EURIBOR3MD_.HSTA); a monthly seasonally adjusted series for the consumer price index (‘Euro area (changing composition) - HICP - Overall index, Monthly Index, European Central Bank, Working day and seasonally adjusted’; ECB code is ICP.M.U2.Y.000000.3.INX); and a monthly seasonally adjusted series for industrial production (‘Euro area 19 (fixed composition) - Industrial Production Index, Total Industry - NACE Rev2 Eurostat; working day and seasonally adjusted’; ECB code is STS.M.I8.Y.PROD.NS0010.4.000). Then, I interpolated to the monthly frequency quarterly seasonally adjusted series for real GDP and the GDP deflator from the ECB as in Stock and Watson (2012), using as interpolators the previously mentioned monthly seasonally adjusted series for industrial production and the consumer price index, respectively. Finally, I computed a monthly seasonally adjusted series for nominal GDP as the product of the interpolated real GDP and GDP deflator series. A.3.3 United Kingdom A monthly seasonally unadjusted series for the core CPI (‘CPIH Index: Excluding Energy, food, alcoholic beverages & tobacco 2015=100’, acronym is L5KB) is from the Office for National Statistics (henceforth, ONS), and it has been seasonally adjusted via ARIMA X-12 as implemented in Eviews. A monthly series for the Bank rate (i.e., the Bank of England’s monetary policy rate) available since 1694 is from from the Bank of England’s website. A monthly seasonally adjusted M1 series (LPMVWYT, ‘Monthly amounts outstanding of monetary financial institutions’ sterling and all foreign currency M1 (UK estimate of EMU aggregate) liabilities to private and public sectors (in sterling millions) seasonally adjusted’) is from the Bank of England’s website. A monthly seasonally adjusted series for real GDP is from the National Institute for Economic and Social Research (NIESR), and it has been kindly provided by Garry Young. I interpolated to the monthly frequency a quarterly seasonally adjusted series for the GDP deflator from the ONS (‘Implied GDP deflator at market prices: SA Index’, acronym is L8GG) as in Stock and Watson (2012), using the previously mentioned monthly seasonally adjusted core CPI series as interpolator 8
series. Finally, I computed a monthly seasonally adjusted series for nominal GDP as the product of the interpolated real GDP and GDP deflator series. A.3.4 United States A monthly seasonally adjusted series for the the core PCE deflator (‘Personal Consumption Expenditures Excluding Food and Energy (Chain-Type Price Index)’) is from the U.S Bureau of Economic Analysis. Seasonally adjusted monthly series for real and nominal GDP are from Stock and Watson (2012) until 2010, and from IHS Markit, a consultancy, after that2(originally, the series used to be produced by another consultancy, Macroeconomic Advisors). IHS Markit’s production notes for its monthly real GDP series states: ‘Note: IHS Markit’s index of Monthly GDP (MGDP) is a monthly indicator of real aggregate output that is conceptually consistent with real Gross Domestic Product (GDP) in the NIPA’s. The consistency is derived from two sources. First, MGDP is calculated using much of the same underlying monthly source data that is used in the calculation of GDP. Second, the method of aggregation to arrive at MGDP is similar to that for official GDP. Growth of MGDP at the monthly frequency is determined primarily by movements in the underlying monthly source data, and growth of MGDP at the quarterly frequency is nearly identical to growth of real GDP.’ A monthly series for the Federal Funds rate is from FRED II (acronym is FEDFUNDS). Finally, I interpolated to the monthly frequency a seasonally adjusted quarterly series for Money Market Deposit Accounts (MMDAS) as in Stock and Watson (2012), by using, as monthly interpolator, the seasonally adjusted series for ‘Total Checkable Deposits’ from the FRB H.6 release from the Federal Reserve Board.The rationale for using this interpolator series is exactly the same originally advanced by Goldfeld and Sichel (1990, pp. 314-315), and then reiterated by Lucas and Nicolini (2015), for including MMDAs within an expanded, and economically more sensible definition of M1 (see the discussion in Appendix A in the main text of the present work): MMDAs perform an economic function which is very similar to that of the checkable deposits included in the standard M1 series. Therefore, on the one hand it makes sense to include them within an economically sensible definition of M1; on the other hand, it makes sense to use total checkable deposits as the monthly interpolator for quarterly MMDAs. 2See at: https://ihsmarkit.com/products/us-monthly-gdp-index.html 9
B UnitRootandCointegrationPropertiesofthe Data B.1 Unit root tests Tables B.1-B.1report bootstrapped p-values3for Elliot, Rothenberg, and Stock (1996) unit root tests for M1 velocity, a short-term nominal interest rate, up to three long-term nominal interest rates (depending on data availability for each individual country), and the corresponding long-short spreads. All tests are with an intercept, but no time trend. In nearly all cases, evidence of a unit root for M1 velocity and nominal interest rates is very strong, whereas the null of a unit root is near-uniformly rejected for the long-short spreads. The only exceptions to this pattern are ()the long-short spreads for the Euro area, for which the null of a unit root is near-uniformly not rejected, and ()theshortratefortheEuroareaandnominalinterestrates for New Zealand, for which the null of unit root is instead near-uniformly rejected. As for (), as discussed in the main text, I proceed under the assumption that all of the long-short spreads are in fact I(0), and that the results from Elliot et al.’s (1996) unit root tests for the Euro area are a statistical fluke, possibly due to smallsample issues. There are two reasons for doing so. First, as discussed in the main text, basic economic theory suggests that any permanent shock to nominal interest rates–originating from either permanent inflation shocks, or permanent shocks to the Wicksellian (i.e., natural) real rate of interest–has an identical long-run impact on all nominal interest rates at all maturities. The implication is that, whatever the origin of permanent shock to nominal interest rates, the spreads will ultimately remain unaffected, and will therefore be I(0). Second, with very few exceptions, this is in fact what the data suggest: since the end of the Napoleonic wars–i.e., since when high-quality macroeconomic data start being consistently available–the null of unit root in long-short interest rates’ spreads can near-uniformly be rejected. Taken together, the logical/theoretical argument, and the empirical evidence since the end of the Napoleonic wars, naturally suggest that lack of a rejection of the null of a unit root in a long-short spread is likely a statistical fluke. As for (), as discussed again in the main text, I proceed under the assumption that all nominal interest rates have been I(1) over the sample periods analized herein, and that the rejection of the null of a unit root is, once again, a statistical fluke possibly due to small-sample issues. There are two reasons for doing so. First, it is important to remember that even a perfectly sized test, by definition, incorrectly rejects the null hypothesis per cent of thetimeattheper cent level. When performing many statistical tests, such as in 3-values have been computed by bootstrapping 10,000 times estimated ARIMA(,1,0) processes. In all cases, the bootstrapped processes are of length equal to the series under investigation. As for the lag order, , since, as it is well known, results from unit root tests may be sensitive to the specific lag order which is being used, for reasons of robustness I consider four alternative lag orders (either 2, 4, 6, or 8 quarters). 10
the present case, a certain fraction of ‘fluke rejections’ of the null should therefore be logically expected. Sure enough, the data I am using herein have not been randomly generatedaspartofaMonteCarloexperiment, 4but the basic logic of this argument should still hold. For example, taking the argument literally–i.e., as if we were here dealing with a Monte Carlo experiment featuring independent random draws–the nine rejections (at the 10 per cent level) of the null of a unit root reported in Table B.1represent 12.5 per cent of the overall number of tests reported in the table, not far from the 10 per cent of ‘fluke rejections’ we would expect from a perfectly sized test with independent Monte Carlo artificial samples. Second, for both the Euro area and New Zealand visual evidence strongly suggests that all nominal interest rate have in fact been non-stationary over the sample period analized herein. For short ratesthisisclearlyapparentfromFigure1; evidence for long rates is–as one would logically expect–even starker. I now proceed to discuss the results from cointegration tests. B.2 Cointegration tests Table 1 in the main text reports, for bivariate systems featuring M1 velocity and a short-term rate, () bootstrapped -values for Johansen’s maximum eigenvalue tests ofthenullhypothesisof0versus 1 cointegration vectors, and () 90%-coverage bootstrapped confidence intervals for the second element of the normalized cointegration vector based on Wright’s (2000) methodology. As for Johansen’s tests, following Benati (2020) and Benati et al. (2021), I bootstrap the tests via the procedure proposed by Cavaliere et al. (2012, henceforth CRT). In a nutshell, CRT’s procedure is based on the notion of computing critical and -values by bootstrapping the model which is relevant under the null hypothesis. This means that, within the present context, the model which is being bootstrapped is a simple, non-cointegrated VAR in differences. All of the technical details can be found in CRT (2012), which the reader is referred to. I select the VAR lag order as the maximum5between the lag orders chosen by the Schwartz and the Hannan-Quinn criteria6for the VAR in levels. Monte Carlo evidence on the performance of CRT’s procedure can be found in CRT (2012), Benati (2015), and especially Benati et al. (2019). Any of three papers documents the excellent performance of the procedure conditional on DataGeneration Processes (DGPs) featuring no cointegration, with the null incorrectly 4In particular, the data for the different countries are not independent random draws, since all countries experienced common events such as the Great Inflation of the 1970s, the disinflation of the early 1980s, the spread of globalization, and the 2008-2009 financial crisis. 5I consider the maximum between the lag orders chosen by the SIC and HQ criteria because the risk associated with selecting a lag order smaller than the true one (model mis-specification) is more serious than the one resulting from choosing a lag order greater than the true one (over-fitting). 6On the other hand, I do not consider the Akaike Information Criterion since, as discussed (e.g.) by Luetkepohl (1991), for systems featuring I(1) series the AIC is an inconsistent lag selection criterion, in the sense of not choosing the correct lag order asymptotically. 11
rejected at close the nominal size irrespective of the sample length. Benati et al. (2019), however, also show that, if the DGP features cointegration,thetestshave a harder and harder time detecting it () the shorter the sample length, and () the more persistent the cointegration residual. This is in line with some of the evidence reported by Engle and Granger (1987) based on the Augmented Dickey-Fuller test, and it implies that if cointegration is not detected, ()and/or() are possible explanations. As for Wright’s (2000) test, since it has been designed to be equally valid for data-generation processes (DGPs) featuring either exact or near unit roots, following Benati (2020) and Benati et al. (2021) I consider two alternative bootstrapping procedures, corresponding to either of the two possible cases. The first procedure involves bootstrapping–as detailed in CRT and briefly described previously–the cointegrated VECM estimated (based on Johansen’s procedure) under the null of one cointegration vector. This bootstrapping procedure is the correct one if the data feature exact unit roots. For the alternative possiblecaseinwhichvelocityandthe short rate are near unit root processes, I proceed as follows. Based on the justmentioned cointegrated VECM estimated under the null of one cointegration vector, I compute the implied VAR in levels, which by construction features one, and only one, eigenvalue equal to 1. Bootstrapping this VAR would obviously be equivalent to bootstrapping the underlying cointegrated VECM, that is, it would be correct if the data featured exact unit roots. Since, on the other hand, here I want to bootstrap under the null of a near unit root DGP, I turn such an exact unit root VAR in levels into its corresponding near unit root VAR by shrinking down the single unitary eigenvalue to =1—0.5×(1/), where isthesamplelength. 7The bootstrapping procedure I implement for the second possible case, in which the processes feature near unit roots, is based on bootstrapping such a near unit root VAR. In practice, as shown by Benati et al. (2021), the two procedures produce near-identical results, and in the present work I therefore uniquely report, as Benati et al. (2021), results based on the first procedure (i.e., based on bootstrapping the VECM estimated conditional on one cointegration vector, as in CRT). As discussed in the main text, based on Wright’s (2000) tests, the null of cointegration is never rejected. Likewise, Johansen’s maximum eigenvalue tests reject the null of 0 cointegration vectors for all countries except Sweden (marginally), and New Zealand (with a -value of 0.1584). 7In particular, I do this via a small perturbation of the parameters of the VAR matrices ’s in the cointegrated VECM representation =+1−1+...+−+−1+,where collects (the logarithms of) M1 velocity and the short rate, and the rest of the notation is obvious. By only perturbating the elements of the VAR matrices ’s–leaving unchanged the elements of the matrix (and therefore both the cointegration vector and the loading coefficients)–I make sure that both the long-run equilibrium relationship between velocity and the short rate, and the way in which disequilibria in such a relationship map into subsequent adjustments in the two series, remain unchanged. 12
B.3 Comparing alternative money demand specifications Table B.2 reports evidence that, in line with Benati et al. (2021), suggests that the data tend to ‘prefer’ the money demand specification proposed by Selden (1956) and Latané (1960)–featuring a linear relationship between velocity and the short rate– to the popular semi-log and log-log specifications proposed by Cagan (1956) and Meltzer (1963), respectively, which have long dominated research on money demand. The table reports results from Johansen’s maximum eigenvalue tests8between M1 velocity and the short rate based on any of the three functional forms: ()theSeldenLatané specification, in which both series enter the system in levels, i.e., =[]0 where and are M1 velocity and the short rate, respectively; ()thesemi-log, with = [ln()]0;and() the log-log, with = [ln()ln()]0. Theevidenceinthetableisquiteclear.Outofninecountries,basedonthe Selden-Latané specification the null of 0 cointegration vectors is rejected six times, wheres in two cases (Norway and Sweden) the lack of rejection is borderline, with -values equal to 0.1121 and 0.1136, respectively. Only for New Zealand the lack of rejection appears as quite solid, with a -value equal to 0.1584. At the other end of the spectrum is Meltzer’s (1963) log-log specification for which, out of six countries featuring a consistently positive nominal short-term interest rate over the sample period, the null of no cointegration is rejected in a single case, Australia. Further, in all other cases the lack of rejection appears as quite solid, with a -value equal to 0.1431 for Canada, and the corresponding -values for the other countries ranging between 0.3005 and 0.6209. The comparison between the results in Table B.2 for the Selden-Latané and log-log specifications provides additional, strong support to Benati et al.’s (2021) point that, for low-inflation (and therefore low-interest rates) countries such as those studies herein, the Selden-Latané specification provides a significantly better characterization of the data than the log-log. Turning to Cagan’s (1956) semilog specification, the null of no cointegration is rejected for just four countries out of nine. Further, only for Sweden, with a -value of 0.1136, the lack of rejection is borderline: for the other four countries the -values range between 0.1444 and 0.6565. Once again, a comparison between these results and those for the Selden-Latané specification clearly suggests that, between the two functional forms, the data quite clearly ‘prefer’ the Selden-Latané. Sure enough, an alternative interpretation of these results is also possible. Instead of interpreting them as suggesting that ()thereis indeed a stable long-run money demand, and that () the correct functional form is the the Selden-Latané, a researcher could alternatively interpret them as suggesting instead that () the correct functional form is (e.g.) the log-log, and ()thatthere is no stable long-run money demand. Admittedly, it is not possible to claim with certainty that the former interpretation (i.e., mine) is correct, whereas the latter is wrong. By Occam’s razor, however, the former interpretation clearly appears (at 8Resultsfromthetracetestsareinlinewiththose from the maximum eigenvalue tests, and they areavailableuponrequest. 13
least, to this author) as the most plausible and logical one. I next turn to the issue of stability of the cointegration relationship. B.4 Testing for stability in the cointegration relationship Table B.3 reports results from Hansen and Johansen’s (1999) Nyblom-type tests for stability in either the cointegration vector, or the vector of loading coefficients, in the estimated VECMs. The -values reported in the table have been computed by bootstrapping, as in Cavaliere et al. (2012), the VECMs estimated conditional on one cointegration vector and no break of any kind, and then performing Hansen and Johansen’s (1999) tests on the bootstrapped series. Before delving into the results, however, it is worth briefly discussing evidence on the performance of the tests. B.4.1 Monte Carlo evidence on the performance of the tests Table G.1 in Online Appendix G of Benati et al. (2021)–see Benati et al. (2019)– reports Monte Carlo evidence on the performance of the tests conditional on bivariate cointegrated DGPs for alternative sample lengths and alternative degrees of persistence of the cointegration residual, which is modeled as an AR(1). The main results can be summarized as follows. The two Nyblom-type tests exhibit an overall reasonable performance, incorrectly rejecting the null of no time variation most of the time at roughly the nominal size. Crucially, this is the case irrespective of the sample length and of the persistence of the cointegration residual. The fluctuation test, on the other hand, exhibits good performance only if the persistence of the cointegration residual is low. The higher the residual’s persistence, however, the worse the performance, so that, for example, when the AR root of the residual is equal to 0.95 for a sample length = 50, the test rejects at twice the nominal size. This result is clearly problematic, since as shown by Benati et al. (2021) cointegration residuals between (log) M1 velocity and (the logarithm of) a short-term nominal interest rate are typically moderately to highly persistent. In what follows I therefore focus on the results from the two Nyblom-type tests, and I instead eschew results from the fluctuation test. B.4.2 Evidence The key finding in Table B.3 is that evidence of breaks in either the cointegration vector or the loading coefficients is nearly non-existent. In particular, based on either the Selden-Latané specification–which, as discussed, appears to be the one preferred by the data for low-inflation (and therefore low-interest rates) countries–or the loglog, not a single break in either the cointegration vector or the loading coefficients is identified. As for the semi-log, no break in the cointegration vector is identified for any country, whereas only for Norway a break in the loading coefficients is identified, although the -value, at 0.0935, is essentially borderline. 14
C Testing for Breaks in the Mean of Inflation Table C.1 reports results from tests for multiple breaks at unknown points in the sample in the mean of inflation based on the methodology proposed by Bai and Perron (1998, 2003). Specifically, the table reports bootstrapped -values for the double maximum test statistics UDmax and WDmax (which test the null hypothesis of no break against the alternative of at least one break). In performing the tests I exactly follow the recommendations of Bai and Perron (2003),9with the only difference that, instead of relying on the asymptotic critical values tabulated in Bai and Perron (1998), I bootstrap both critical and -values via the procedure proposed by Diebold and Chen (1996), setting the number of bootstrap replications to 10,000. I set the maximum allowed number of structural changes to m=2. Asdiscussedinthemain text, for Australia, Canada, New Zealand, Norway, Sweden, and the United Kingdom I focus on the sample period since the introduction of inflation targeting; for the Euro area I consider the period since the start of European Monetary Union; and for the United States I consider the period following the break in the mean of inflation identified by Levin and Piger (2004), in 1992Q2. The evidence in Table C.1 is very clear: the null hypothesis of no breaks cannot be rejected for any country. 9See Bai and Perron (2003) section 5.5, ‘Summary and Practical Recommendations’. 15
References Alvarez, F. and Lippi, F. (2009), “Financial Innovation and the Transactions Demand for Cash”, Econometrica, 77(2), 363-402 Andrews, D.K. and W. Ploberger (1994), “Optimal Tests When a Nuisance Parameter is Present Only Under the Alternative”, Econometrica, 62(6), 1383-1414 Bai, J. and P. Perron (1998), “Estimating and Testing Linear Models with Multiple Structural Changes”, Econometrica, 66(1), 47-78 Bai, J. and P. Perron (2003), “Computation and Analysis of Multiple Structural Change Models”, Journal of Applied Econometrics, 18(1), 1-22 Balke, N. and R.J. Gordon (1986), “Appendix B: Historical Data”, in Robert J. Gordon, ed., “The American Business Cycle: Continuity and Change”, The University of Chicago Press, 781-850 Benati, L. (2015): “The Long-Run Phillips Curve: A Structural VAR Investigation”, Journal of Monetary Economics, 76(November), 15-28 Benati, L. (2020): “Money Velocity and the Natural Rate of Interest”, Journal of Monetary Economics, 116(December), 117-134 Benati, L., R.E. Lucas Jr., J.P. Nicolini, and W. Weber (2019): “Online Appendix for: International Evidence on Long-Run Money Demand”, Federal Reserve Bank of Minneapolis Staff Report 588 (June 2019) Benati, L., R.E. Lucas Jr., J.P. Nicolini, and W. Weber (2021): “International Evidence on Long-Run Money Demand”, Journal of Monetary Economics, 117(January) Cagan, P. (1956), “The Monetary Dynamics of Hyperinflation”,inM.Friedman, ed., “Studies in the Quantity Theory of Money”, The University of Chicago Press, pp. 25-120. Diebold, F., and C. Chen (1996), “Testing Structural Stability with Endogenous Breakpoint: A Size Comparison of Analytic and Bootstrap Procedures”, Journal of Econometrics, 70(1), 221-241 Elliot, G., T.J. Rothenberg, and J.H. Stock (1996), “Efficient Tests for an Autoregressive Unit Root”, Econometrica, 64(4), 813-836 Engle, R.F. and C.W.J. Granger (1987), “Cointegration and Error Correction: Representation, Estimation, and Testing”, Econometrica, 55(2), 251-276 Friedman, M. and A.J. Schwartz (1963), “A Monetary History of the United States, 1867-1960”, Princeton University Press Goldfeld, S.M., and D.M. Sichel (1990), “The Demand for Money”, in Friedman, B.M., and Hahn, F.H., eds., Handbook of Monetary Economics, Vol. I, Amsterdam, North Holland. Haavisto, T. (1992), “Money and Economic Activity in Finland, 1866-1985”, Lund Economic Studies Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton 16
Table B.3 Bootstrapped p-valuesfor Hansen and Johansen’s (1999) tests for stability in the cointegration vector for (log) M1 velocity and (the log of) ashort-termrate Money demand specification: SeldenSemiLogCountry Period Latané log log I: Tests for stability in the cointegration vector Australia 1969Q3-2019Q4 0.7835 0.7880 0.6950 Canada 1967Q1-2019Q4 0.6900 0.7945 0.6070 Euro area 1999Q1-2019Q4 0.4880 0.2915 0.2915 New Zealand 1990Q1-2019Q4 0.5392 0.4726 0.7346 Norway 1978Q1-2019Q4 0.5590 0.1940 0.8560 Sweden 1998Q1-2019Q4 0.2335 0.1690 — United Kingdom 1955Q1-2019Q4 0.5905 0.5480 0.9365 United States 1959Q1-2019Q4 0.5875 0.8030 0.9940 II: Tests for stability in the loading coefficients Australia 1972Q1-2019Q4 0.4430 0.9330 0.8635 Canada 1982Q3-2019Q4 0.6110 0.2865 0.3660 Euro area 1970Q1-2019Q4 0.1250 0.2720 — New Zealand 1990Q1-2019Q4 0.8155 0.4955 0.8525 Norway 1978Q1-2019Q4 0.6975 0.0935 0.5810 Sweden 1998Q1-2019Q4 0.1075 0.2900 — United Kingdom 1963Q1-2019Q4 0.3190 0.2770 0.6370 United States 1959Q1-2019Q4 0.1310 0.4800 0.9235 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 C.1 Tests for multiple breaks at unknown points inthesampleinthemeanofinflation based on Bai and Perron (1998): bootstrapped p-values for double maximum test statistics Country Period UDmax WDmax Australia 1994Q3-2019Q4 0.2348 0.2612 Canada 1991Q1-2019Q4 0.3336 0.2982 Euro area 1999Q1-2019Q4 0.2048 0.2318 New Zealand 1990Q1-2019Q4 0.3558 0.3230 Norway 2001Q2-2019Q4 0.3506 0.3212 Sweden 1998Q1-2019Q4 0.5736 0.6122 United Kingdom 1992Q4-2019Q4 0.3114 0.3548 United States 1992Q2-2019Q4 0.1572 0.1322 Based on 10,000 bootstrap replications.
22 Figures for Online Appendix
23 Figure A.1 Inflation under inflation-targeting regimes; European Monetary Union; and for the United States the period since the break in the mean identified by Levin and Piger (2003)
23 Results obtained by projecting the short rate onto M1 velocity
24 Figure A.2 Estimated deviation of the short rate from the nominal natural rate computed by projecting the short rate on M1 velocity (with 1and 2-standard deviations bootstrapped confidence bands)
25 Figure A.3 Fraction of bootstrap replications for which the deviation of the short rate from the nominal natural rate computed by projecting the short rate on M1 velocity is negative
26 Figure A.4 Fractions of bootstrap replications for which the nominal and real natural rates of interest are estimated to have been negative, computed by projecting the short rate on M1 velocity
27 Figure A.5 Estimated deviation of Wu and Xia’s ‘shadow rate’ from the nominal natural rate computed by projecting the shadow rate on M1 velocity (with 1and 2-standard deviations bootstrapped confidence bands), and fraction of bootstrap replications for which the deviation is negative
28 Results based on cointegrated SVARs identified via long-run restrictions