Emanuele Franschi

Emanuele Franceschi
email: emanuele.franceschi@psemail.eu

M Ranaldi

I Iodice

Taylor Rules and liquidity in financial markets

We study the parameter instability in the monetary policy rule followed by the US Federal Reserve Bank since WWII. We find evidence across a variety of econometric methods of fundamental instability, in particular on the parameter governing the reaction to inflation expectations -the Taylor Principle. We augment the monetary policy rule to account for liquidity conditions and find consistent violations of the Taylor Principle without sunspot inflation episodes. We study the presence of multiple regimes and find that when uncertainty and economic slowdown are looming the Fed reacts passively to expected inflation.

Introduction

Before the Global Financial Crisis (GFC), there was wide consensus that the central bank could stabilize inflation and the real economy by following a Taylor-rule type monetary policy [START_REF] Taylor | Discretion versus policy rules in practice[END_REF]. Such rule, as emphasized by the literature on determinacy of equilibria and feedback interest rate rules, would also insulate the economy from erratic expectations unrelated to market fundamentals [START_REF] Gali | Monetary Policy, Inflation, and the Business Cycle[END_REF][START_REF] Walsh | Monetary Theory and Policy[END_REF][START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF]. Even though there were some criticisms about the theoretical foundations of these views [START_REF] Cochrane | Determinacy and identification with Taylor rules[END_REF], economists and policymakers were confident on the virtues of the Taylor rule. Furthermore, financial stability, liquidity of financial assets, or financial conditions in general, including quantitative measures of money and credit, had no relevance in the monetary policy conduct, and their consideration was discouraged because of the difficult identification of bubbles or excessive risk taking [START_REF] Bernanke | Asset-price "bubbles" and monetary policy[END_REF][START_REF] Greenspan | Economic volatility[END_REF]. The GFC fundamentally challenged the existing consensus. Once interest rates hit the zero lower bound, the Taylor principle was violated, as interest rates could no longer respond to the behaviour of the economy. The US Federal Reserve Bank (and subsequently other major central banks in the world) resorted to unconventional monetary tools, such as liquidity injections in the financial system (Quantitative Easing rounds, QE). Interestingly, in spite of the a violation of the Taylor Principle, inflation did not show an erratic behaviour as implied by workhorse New Keynesian Dynamic Stochastic General Equilibrium models (NKDSGEs) (Cochrane, 2018). Moreover, the assumption in NKDSGEs that the policy interest rate is essentially an interest on a pure, illiquid, bond began to appear awkward [START_REF] Calvo | Macroeconomics in times fo liquidity crises: searching for economic essentials[END_REF]B. Diba and Loisel, 2017;[START_REF] Hall | Achieving price stability by manipulating the Central Bank's payment on reserves[END_REF]. Introducing liquidity considerations and the fact that the policy rate might be an interest rate on money or liquid assets rather than pure bonds, drastically modifies the properties of the model and especially the implications of the Taylor rule for determinacy. Specifically, the fact that the Taylor principle is necessary to induce determinacy and thus that it is necessary and sufficient to anchor inflationary expectations, does not apply anymore. In light of the fundamental issues raised by the GFC, the revisitation of the actual implementation of the Taylor rule appear warranted and compelling.

In this paper we investigate the behaviour of the Fed in the whole post WWII period in the framework of the Taylor rule. We evaluate the monetary policy conduct focusing on the interplay between inflation response -embodied in the Taylor Principle (TP) -and liquidity in the financial system, measured as a spread between safe assets and less liquid assets. In a nutshell, we find that financial conditions affect monetary policy conduct of the Fed, since Central Bank consistently deviates from the Taylor Principle when financial liquidity is included in the Taylor rule. Moreover, we report instability in the policy rule and therefore investigate the presence of multiple regimes. When considering a standard Taylor rule with real time data, we do not find deviations from the TP even in periods of high inflation. On the other hand, when we include proxies for liquidity we find regimes in which the Fed lowers the weight of expected inflation, hence in violation of the TP, in favour of financial liquidity.

We contribute and relate to a vast literature. On the empirical part we confront the exogenous regime definition of [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF], we also make preferably use of real-time data, whose importance has been shown by [START_REF] Orphanides | Monetary policy rules based on real-time data[END_REF]Orphanides ( , 2004)). In studying how the monetary policy rule has evolved over time we consider the insights from [START_REF] Boivin | Has US monetary policy changed? Evidence from drifting coefficients and real-time data[END_REF], [START_REF] Canova | Back to square one: identification issues in DSGE models[END_REF], [START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF], and [START_REF] Sullivan | Qualitative Taylor rule[END_REF], and finally follow [START_REF] Davig | Monetary-fiscal policy interactions and fiscal stimulus[END_REF][START_REF] Davig | Monetary-fiscal policy interactions and fiscal stimulus[END_REF], [START_REF] Murray | Markov switching and the Taylor principle[END_REF], and [START_REF] Sims | Were there regime switches in US monetary policy?[END_REF] in adopting a Markov Switching model approach to let the data speak as freely as possible. Concerning the specifications we utilise to look at the data, we also consider the contributions on leaning against the wind policies (LAW, see as a general reference Svensson (2017a,b)), although this avenue is out of the scope of this work.

The inclusion of financial variables in the monetary policy rule is motivated on one hand by the leaning against the wind policies: in this framework Central Banks reacts to excessive risk-taking and deflate bubbles before their bust, and symmetrically revive languishing financial conditions. Svensson (2017a,b) offers an excellent summary of this literature. On the other hand, we relate to works that analyse official reports and documents to assess whether the Fed takes into account financial factors. [START_REF] Oet | Does financial stability matter for the Fed in setting US monetary policy?[END_REF] and [START_REF] Peek | Does Fed policy reveal a ternary mandate?[END_REF] find that discussing financial stability in FOMC meetings affects policy decisions, [START_REF] Wischnewsky | Financial stability and the Fed: evidence from congressional hearings[END_REF] reinforce this evidence analysing congressional hearings of Fed Chairmen. We restrict our focus to liquidity factors because QE injections addressed this specific issue through bal-ance sheet expansions. To capture this effect, we proxy liquidity in financial markets with two spreads. The first one is the difference between the Standard & Poor 500 quarterly returns and the 3-month Treasury Bill, the second is the gap between the average yield of BAA corporate bonds1 and the US 10-year Treasury note. These two spreads capture the liquidity premium between safe and totally liquid assets and risky, less liquid ones.

Section 2 presents the dataset we compile and ; Section 3 presents the specifications we estimate, with results for the full sample analysis and the exogenous subsampling ones; Section 4 presents stability tests on the specifications that motivate a Markov Switching model estimation. Section 5 concludes the paper.

Data

To investigate the US monetary policy conduct, we build a database of the most relevant time series on the US economy aggregates. It includes historical and real time data at the macroeconomic level, as well as statistics from specific microeconomic data to account for expectations. At a broad level, we collect data on inflation, interest rates,2 real output slack, monetary aggregates, government debt and deficit, financial market indicators, and finally (measures of) expectations of these variables. For each one of these aggregates, we collect a set of more specific measures that differ in the exact definition or computation: the clearest examples are the GDP deflator, the CPI, and the CPE for inflation, or the capacity utilization, lay-off rate, unemployment, and Fed's own calculations for the output gap. A side contribution of this paper is to provide a collected database as well as the tools to maintain, fine-tune, and customize it. We also source microdata from the Greenbook dataset on forecasts and expectations, as well as the University of Michigan Survey of Consumers. The former includes the Survey of the Professional Forecasters, which provides expectations on economic aggregates at several time horizons. The financial information we employ relies on two measures: the quarterly returns of the S&P 500 index and the weighted average return of BAA corporate bonds. These series are then treated and used as proxies for the liquidity in the economy.

The vast majority of the series are retrieved from the Federal Reserves of St. Louis and Philadelphia: a complete list is provided in the Appendix, Table (9). 3For a subset of the data we performed some transformations and manipulations to isolate precise information. We briefly introduce the transformed data below.

Output gap

We included several different measures of output slack, among which two are worth detailing. First, official output gap nowcast. This extrapolation uses data from the Greenbook database on the real time estimates on the GDP level and implements the methodology mentioned in [START_REF] Murray | Markov switching and the Taylor principle[END_REF]. For each available date t, we regress the time series against a quadratic time trend and finally take the residual of the latest available data point, ε t , as output gap observation for date t. We label the resulting time series as real time output gap. 4 It is closer to the signal policymakers receive at the time of the decision.

Second, we compute the percentage difference between installed capacity and actual GDP, both provided by St. Louis Fed. We call this series ex post output gap since it relies on historical, revised data, not necessarily those available to policy makers at the time of their contingent decisions.

All measures are intended to encompass the advances in the related literature, including contributions due to [START_REF] Benhabib | The perils of Taylor rules[END_REF], [START_REF] Bilbiie | Asset market participation, monetary policy rules, and the Great Inflation[END_REF], [START_REF] Boivin | Has US monetary policy changed? Evidence from drifting coefficients and real-time data[END_REF], [START_REF] Cochrane | Determinacy and identification with Taylor rules[END_REF], [START_REF] Gali | Inflation dynamics: a structural econometric analysis[END_REF], [START_REF] Gali | European inflation dynamics[END_REF], and [START_REF] Orphanides | Monetary policy rules based on real-time data[END_REF]Orphanides ( , 2004)).

Inflation and expectations

Concerning the measures of inflation, we include revised time series like the indexes of GDP deflator, the Consumer Price Index (CPI), and the Personal Consumption Expenditure (PCE). For the last two, we also include two reduced versions that exclude food and energy prices, dubbed Core CPI and PCELFE.

The Greenbook database from the Philadelphia Fed provides information on last, current and expected future values for three of the aforementioned indexes, namely CPI, Core CPI, and GDP deflator. In particular, forecasts are available up to eight quarters ahead from time t. The same database also offers nowcasts on current and last value of these variables. These expectations are part of the information set of the policy-maker at the time of decision and thus represent a reliable tool to gauge the policy rule.

Liquidity proxies: financial indexes After the financial markets collapse that triggered the Global Financial Crisis in 2008, liquidity in the economy gained momentum as research topic (alongside with safe assets and thus risk) especially following the massive injections carried out by the Federal Reserve, see among others [START_REF] Caballero | Safe Asset Scarcity and Aggregate Demand[END_REF][START_REF] Caballero | Safe Asset Scarcity and Aggregate Demand[END_REF], M. B. [START_REF] Canzoneri | Interest rate rules and price determinacy: the role of transactions services of bonds[END_REF], M. Canzoneri et al. (2008a,b), Del [START_REF] Negro | Safety, liquidity, and the natural rate of interest[END_REF][START_REF] Hall | Achieving price stability by manipulating the Central Bank's payment on reserves[END_REF][START_REF] Hall | Achieving price stability by manipulating the Central Bank's payment on reserves[END_REF].

We employ the financial condition of the economy to parse out information on its liquidity. Financial market prices embody plenty of different information, so the risk of picking up the wrong signal or incur in plain endogeneity is high. Considering these threats we compute our indicators as premia over safe assets of comparable maturity, which are subject to "fire-purchases" in times of uncertainty or economic turmoil.

The simple intuition goes as follows: on the edge of a recession or slowdown, publicly traded assets are fully liquid5 and smoothly traded; when uncertainty kicks in or expectations turn pessimistic, these assets become second choice to more reliable, safer assets. Therefore, the spread between the former and the latter factors in the variation in liquidity of the the economy.

We include these spreads in the decision rule of the Central Bank in order to test whether policy makers are also attentive to the liquidity in the economy and act accordingly. In a sense, these specifications might report biased results when the Central Bank resorts to unconventional monetary policy tools. The main threat comes in particular from forward guidance announcements that substantially affect the financial market. To this end, we control for equivalent maturity Treasuries yields to isolate liquidity factors.

Throughout the rest of the paper we use the GDP deflator as measure of inflation, since it maps closely the price dynamics emerging from the US economy. Moreover, its time series is the longest and affords estimation in the pre-Volcker period. To reproduce the information set of the Fed, we use our real time output gap measure. Both series strongly correlate with the alternatives and bear little differences in trends.

For completeness, Table (11) in the Appendix presents correlations among selected variables in our dataset over the period 1986Q1:2013Q4, when all series overlap.

Empirical results

In the literature there is a broad consensus on the empirical validity of the Taylor rule at least since the chairmanship of Volcker. Volcker, the consensus goes, induced a switch in policy from a regime of indeterminacy (accommodation policy) to one of determinacy. One of the first attempts to verify such break in policy is [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF], who exogenously divide their sample in two periods according to chairmanship and obtain estimates for each subsample. They find that the Fed was following a passive monetary policy during the first part of the sample, whereas an active policy emerged after Volcker chairmanship, engineering the ensuing stable inflation path over the following decades.

By contrast, [START_REF] Boivin | Has US monetary policy changed? Evidence from drifting coefficients and real-time data[END_REF] uses a different approach and lets the real time data speak, estimating a Taylor rule with drifting parameters over the sample. He finds that inflation response was weak in the second half of the 1970s, but strong in the rest of the sample. The response to real activity, measured as the deviation of the unemployment from its natural rate, decreased significantly and permanently after the 1970s. Starting from the mid-1980s, he finds that stability in the policy parameters increased. These changes happened in an unsynchronised way, with the most important changes occurring between 1980 and 1982, the transition years of Volcker chairmanship. In the same vein, [START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF] employs a Bayesian SVAR to estimate how the Fed reacted dynamically to inflation and unemployment, finding that it complies to the Taylor Principle even prior to Volcker.

Another approach to endogenise policy changes is proposed in [START_REF] Murray | Markov switching and the Taylor principle[END_REF], where [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF] algorithm for Markov processes estimation is applied to monetary policy rules. They explore the two-state case, finding two periods of undetermined policy, precisely 1973:1975 and 1979:1985, again in coincidence of the transition period before inflation was "conquered." The point estimates from [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF] and [START_REF] Murray | Markov switching and the Taylor principle[END_REF] are summarized in the tables below.

Exogenous break µ γ ω ρ

Pre -Volcker 4.24 -.17 .27 .68

(1.09) (.07) (.08) (.05)

Post -Volcker 3.58 1.15 .93 .79 Motivated by these results, we study the stability of the parameters of the Taylor rule in the post-WWII period. We propose a set of estimates of the decision rule followed by the monetary authority. After analysing the full sample with OLS estimators6 , we exogenously split the sample in three sub-samples and compare the parameters. Then, we investigate possible structural breaks over the full sample via stability diagnosis. Third, we let the sub-sampling be endogenous with a Markov Switching estimation for two possible states.

(.5) (.4) (.42) (.04)

Specifications

We assess the robustness of the traditional specification in comparison to the alternatives with liquidity proxies. Our interest lies particularly in the parameters stability over dif-ferent methods, sample cuts, and when we include financial conditions in the information set.

Throughout this Section, we will estimate equation (1): r is the effective federal fund rate, π t+h is h-period ahead expected inflation (mapped to forecasts, up to 8-quarter), ŷ is output gap in percentage deviation, and X collects any additional variables used in the study, as detailed in Section (2).

r t = (1 -ρ) µ + (1 + γ) E t π t+h + ω ŷt + X t β + ρr t-1 + ε t (1)
In equation ( 1), we assume that the Central Bank smooths its policy decision putting a weight 0 < ρ < 1 on past interest rate level. Therefore, we need to recover estimates and confidence intervals from the estimated ρ. Moreover, we allow for inflation targeting including an intercept µ. ε t captures the exogenous shock the monetary authority impulses to the rate path.

We focus our attention on a subset of parameters, β i and γ. The sign and the magnitude of the former will tell how relevant other factors are for the Central Bank; on the other hand, γ will shed light on the robustness of the Taylor Principle. Established consensus points to a value close to γ = .8 following the onset of the Great Moderation and the inflation conquest carried out by Volcker.

Before exposing the results, we precise the specifications we estimate throughout this Section and briefly motivate their utilization. We select and present these specifications because other combinations, although conceptually appealing, do not necessarily add interesting insights or fall outside the scope of this work.

Spec. I r t = (1 -ρ) [µ + (1 + γ) E t π t+1 + ω ŷt ] + ρr t-1 + ε t : the standard specification as
in [START_REF] Taylor | Discretion versus policy rules in practice[END_REF] and many other works. We employ one period ahead forecasts of GDP deflator as expected inflation and real-time gap for output slack. Considerably, we employ uniquely real time data so to track as closely as possible the information set available to the Central Bank.

Spec. II r t = (1 -ρ) [µ + (1 + γ) E t π t+1 + ω ŷt + βBAA t ] + ρr t-1 + ε t :
we introduce in this specification a proxy for financial distress, namely the spread between BAA corporate bonds and 10 years Treasury bonds.

Spec. III r t = (1 -ρ) [µ + (1 + γ) E t π t+1 + ω ŷt + βS&P500 t ] + ρr t-1 + ε t :
we test a second proxy for liquidity with this specification. We exploit quarterly returns on the stock market to obtain a spread with 3-month Treasury Bills. In this case we want to avoid the possible influence of default risk present in bonds yields.

Specifications I is a traditional formulation of the Taylor rule, Spec. II and III embed financial condition proxies.

Full sample analysis

The first step is to estimate our specifications on the full sample, overlooking the possibility of structural breaks or fluctuations in the parameters. The period covered varies with the series included in the specification, from a maximal of about 185 observations to a minimum of slightly less than 115. 7 Such approach simply yields an average of the estimated parameters over the set of possible subsamples. Assuming parameter instability of any form and correct specification, this approach produces simple averages over the possible parameter values. In this sense, regimes that are more frequently in place will be more represented in the final estimate. These results are interesting in a number of aspects. First, the sample encompasses a variety of regimes: from the pre-Volcker era to the ZLB period, with the Great Moderation dwarfing other regimes in terms of observations. Therefore, it blends together different rules and behaviours with diverse weights.

Second, the addition of liquidity in the policy rule of the Central Bank marks a consistent violation of the Taylor Principle, with γ estimates below 1 for Specifications II and III. The inclusion of liquidity proxies significantly lowers the weight on inflation expectations. This finding is even more surprising when considering that the BAA spread series starts in 1986Q1, at the end of Volcker's mandate. These results suggest that financial conditions enter the decision set of the Central Bank with the expected sign.

As mentioned before, the length of the sample blends multiple regimes and stances. In this light it is not surprising to find substantial parameters instability, as Table (3) presents. Under the hypothesis of µ as inflation target, its estimates greatly vary according to the sample and the specification. The same, with less variability, applies for ω and ρ, the smoothing factor. To address this issue, we split the sample in three sub-samples, upon which we cast the assumption of different regimes.8

Exogenous breaks

The estimates above might result from a variety of underlying regimes, smooth [START_REF] Boivin | Has US monetary policy changed? Evidence from drifting coefficients and real-time data[END_REF][START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF] or discrete [START_REF] Murray | Markov switching and the Taylor principle[END_REF]. To account for such regime change, a straightforward approach is to look for relevant historical events that mark a discontinuity and estimate the policy rule before and after such dates. In our framework it corresponds to splitting the sample in three subsamples: post-WWII period, the Great Moderation, and the GFC. The latter in particular encompasses the liquidity injections of the Quantitative Easing in correspondence of the Zero Lower Bound period. This last condition distinguishes the first from the third subsample. In fact, in the postwar period the federal fund rate averaged around 5%, with a minimum value of .93% for one quarter only. The downside is that at date this subsample presents a small number of observations, which makes inference rather hard. We split the sample to obtain three phases: Historically, the period covered by sub-sample (i) reports high inflation and federal fund rate, as well as volatile cyclical fluctuations. According to [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF], among others, the Fed carried out an accommodative monetary policy in that period, following inflation instead of aggressively responding to its expectations. Hence, we expect to see values close to those presented in Table (1). Unfortunately, data availability limits the estimation of our Spec. II.

(i) pre-Volcker regime [-:1981Q4], (
The second chunk of data covers the inflation conquest and the miraculously steady and sustained growth that followed, with mild recessions and inflation in check. Supposedly, this conditions were brought about by a Central Bank eventually fighting back inflation aggressively, adjusting the FFR more than one-to-one with respect to expected inflation.

The third period starts right before the GFC. Data are still scarce: to date, we have about 10 years of quarterly data with hardly enough variation, mainly because of the FFR hitting the zero lower bound and hovering in its neighbourhood. Therefore, the estimates here shall be considered cum grano salis.

Skimming through Table (4) it is interesting to compare the regimes in place. Although the heterogeneity in available observations restricts significantly the econometric robustness of such exercise, a number of regularities emerges.

In particular, γ estimates post a high volatility, both across and within subsamples. Contrary to the consensus (but in line with [START_REF] Boivin | Has US monetary policy changed? Evidence from drifting coefficients and real-time data[END_REF] and [START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF]), the Fed broadly complied to the Taylor Principle, although Spec. II and III present values statistically close to 0 for γ. In the second period, in line with the narrative of a committed and credible Central Bank, γ complies to the Taylor Principle in all specifications, but present a significant volatility in the estimates. Strikingly, inflation expectations seem to disappear from the relevant set for monetary policymaking, as overall its estimates are not significant.

Likewise, output gap measures seem less relevant in the wake of the GFC than previous periods, when ω estimates take expected values and signs without much volatility. Implicit inflation target, µ, display an erratic behaviour and provide a less than convincing picture over the periods. Focusing on cross samples comparison, the BAA spread holds robust estimates over time and across specifications, like the S&P500 spread, pointing to our intuition that financial conditions are key. When liquidity dries up because of financial or real turmoil (and hence spreads increase) the monetary authority puts in place accommodative policies by decreasing the reference interest rate. On top of that, there is a noticeable increase in the federal fund rate persistence over time. The first subsample present a quite volatile policy rate, whilst subsample (ii) reports significant increases in ρ. The most recent subsample, with scarce observations, includes a key policy rate that barely moves, with other variables displaying more variability. This explains why in all regression the most significant variable is the lagged interest rate, while all other variables present odd estimates. Nevertheless, some results are suggestive of fundamental parameters instability, consistently with the Fed switching to QE policies (de facto liquidity injections).

These early results point towards an unstable behaviour of the Central Bank -if we assume its only behaviour takes the form of a strictly parametrised Taylor rule. In particular the great deal of volatility in the estimates across sub samples motivates a deeper investigation in the stability of the parameters. This in turns mirrors a changing policy stance for the monetary authority, as encapsulated by the Taylor rule specifications. The next step, therefore, is to diagnose possible sources of structural breaks or parameter variability and to address these pathologies with proper, flexible tools. sampling, this approach is more data driven, as it makes use of the information contained in the sample to check for breaks and eventually propose the most likely break date(s). We run these diagnostics on the full sample to elicit precise dates. Following the established consensus, we expect to find breaks in correspondence of the Volcker chairmanship and the GFC.

To this purpose, we take the models estimated on the full sample and run first a simple CUSUM diagnostic test, then a [START_REF] Chow | Tests of equality between sets of coefficients in two linear regressions[END_REF] test. The latter in particular is flexible enough to provide an optimal segmentation of the sample based on parameters stability. Basing on the Bayes Information Criterion, the Chow test can produce the most likely date of break, provided that we require only one single date; or it can provide the most likely number of breaks in the the sample. We exploit these properties in our analysis.

As straightforward check, CUSUM tests do not report significant fluctuations in the empirical process, meaning that the cumulative sum of the residuals eventually levels off to 0 without significant erratic deviations. It is nevertheless interesting to remark how persistent over time the deviations are and how consistently the CUSUM statistics builds up towards the end of the sample. Plots of this diagnostic are presented in the Appendix (B.3). The F-Test derived from [START_REF] Chow | Tests of equality between sets of coefficients in two linear regressions[END_REF] points in another direction, though. The output of the test actually reports multiple breaks along the sample, some of which occur with unexpected timing. 2): these specifications, especially those including the 3-month spread variable, report F-statistics hovering above the threshold for well more than one observation. The results of optimal segmentation of the sample pave the way to the Markov switching estimation below as it motivates further inquiry into the existence of multiple policy regimes. Table ( 6) summarises the analysis on optimal segmentation. It highlights, also, that most of the specifications likely involve more than one structural break. On top of Volcker's regime change, one would reasonably expect that the mix of ZLB and unconventional policies would be sufficient to mark an additional break. Moreover, a striking result is that in two cases (specifications I and III) the end of Volcker's two terms is also a candidate point for a strctural discontinuity.9 These results suggest that there has been a structural break when Volcker left the Chair, with Greenspan chairmanship introducing an additional new regime. Hence, there might be one or more than one discrete regimes of monetary policy, among which the Federal Reserve switches back and forth. Therefore, it is clearly worth pursuing additional insights into these structural breaks with adequate techniques.

Markov Switching estimation

We further unconstrain the data via a Markov Switching model: we adopt [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF]Hamilton ( , 1994) ) and [START_REF] Murray | Markov switching and the Taylor principle[END_REF] approach to our extended sample and only assume it comprises k discrete states. We restrict our analysis to k = 2, in line with the discussion on the determinacy or indeterminacy regimes at the beginning and at the end of our sample. As aforementioned, pre-Volcker and post-GFC periods yield deeper insight on the functioning of the Federal Reserve monetary policy conduct away from the Great Moderation "steady state."

r t = 1 -ρ S µ S + 1 + γ S E t π t+h + ω S ŷt + X t β S + ρ S r t-1 + ε S t var ε S t = σ S with S = k ∈ {1, . . . , K} (2)
Hamilton (1989) provides the algorithm to estimate our specifications with k states, generating also transition matrices and smoothed probabilities to pick the prevailing regime in any date t. For every specification, we allow for the variation of every parameter and the variance: in k different states, all parameters are freely estimated, with no constraint posed by other states' estimates.10 Table ( 7) presents estimates for the two state Markov switching model. In contrast to the results of [START_REF] Murray | Markov switching and the Taylor principle[END_REF], Specification I -mirroring those of the cited work -finds two states complying to the Taylor Principle.11 This traditional Taylor rule presents two distinct states that differ in terms of reaction intensity. S 2 estimates present larger coefficients in absolute values for virtually all variables. Focusing on γ estimates, both states comply to the Taylor principle, pointing towards a more aggressive reaction in the second state. This evidence does not align with the established consensus, but rather corroborate [START_REF] Orphanides | Monetary policy rules based on real-time data[END_REF]Orphanides ( , 2004) ) findings. Interestingly, S 1 is more likely to be in place during downturns, with a lower persistence of the policy rate. This last piece of evidence suggests that the Fed might react asymmetrically to evolving economic conditions: rapidly in light of downturns, cautiously when the recovery materializes. With respect to the timing, it appear to precede economic turmoil periods: it covers roughly the years of the oil shock, the high inflation that followed, the dot-com crash, and the early stages of the GFC. It is suggestive to connect this regime with a cautious approach of the Federal Reserve: as soon as slowdown factors build up, the policy intervention intensity need to adjust accordingly. On the contrary, when risk, uncertainty, and sources of slowdown are a weak threat, the Central Bank acts with decision to steer the economy.

Turning to the financial specifications (II and III), the picture is less clear. Estimates for γ display both regimes of determinacy and indeterminacy. Across these two specifications estimates for the weight of the spread are rather consistent with previous results: to a higher relevance of the financial condition corresponds a lower reaction to expected inflation and real activity (summarized in S 1 for both Specifications), putting in place an accommodative policy regime. Again, the interest rate persistence is lower in the undeter-mined regime. The prevailing period for the accommodative monetary policy corresponds roughly to the first Volcker term and the first half of the 1970s: this evidence is at odds with the established narrative that sees Volcker to put swiftly in place an aggressive regime to tame inflation. Looking at the time dimension of these regimes, the picture is complementary to the one drawn from the traditional Taylor rule. In both cases, the indeterminate regime is in place at times of economic and financial distress. One difference is worth noting: in Specification III the ZLB period is described by an active monetary policy regime, while the opposite is found for Spec. II. This difference depends on the fact that bonds are more exposed to the liquidity risk, as well as incorporate a quantity of default risk. This latter factor is less present in the most capitalized companies on the financial market12 and might drive the result.

The panels collected in Fig. (3) depict the prevailing state along the sample for the estimates of Table (7). We also propose the transition matrices for the two estimated states. In general, every state appear to be rather sticky: virtually all states will persist to the following period with a probability greater than .8. This collection of evidences points towards a fundamental instability in parameters of the Taylor rule. Moreover, we retrieved periods with -theoretically -destabilising monetary rules and inflation under control at the same time, once liquidity is included in the decision set of the monetary authority. These results suggests that the Taylor Principle might not play a fundamental role in anchoring the inflation expectations: agents could -and did, according to our findings -form expectations about the future paths of prices that are non-degenerate even when the Central Bank deviates from the prescriptions of the New Keynesian workhorse. Other factors are therefore at play in anchoring inflation expectations: while the influence of Central Bank actions remains relevant for the determination of the inflation behaviour, the Taylor Principle appear to be less than granitic.

Spec

In our battery of proposed specifications for the monetary rules, proxies for liquidity and for financial conditions have a sizeable and robust role across different methodologies. As soon as liquidity dries up, financial conditions worsen, or uncertainty builds up, the Federal Reserve Bank acts and reacts lowering the reference rate. This finding is consequential since these violations of the Taylor Principle are not accompanied by degenerate behaviour of inflation, at least in the most recent cases.

Conclusion

In this paper we collect and aggregate an important number of data on the US economy, both in real time and revised series. Moreover, we also provide a flexible and easy to tune database including granular information on expectations and now-/forecasts, from policymakers and economic agents. We use this extensive database to empirically investigate the stability of the monetary policy rule in the US postwar period and to test whether the Federal Reserve Bank considers financial markets liquidity in its policymaking.

Using a wide variety of econometric techniques, we study the robustness of the standard Taylor rule required in modern NKDSGE models and cast upon the Central Banks behaviour. A standard rule reveal compliance -with varying intensity -of the Fed to the Taylor principle over the whole period considered, contrary to the established consensus. On the other hand, the inclusion of liquidity spreads reveals that the Fed also considers financial conditions in its interest rate setting. This inclusion also implies consistent violations of the Taylor Principle, without ensuing sunspot inflation episodes. We then provide statistical evidence for the presence of structural breaks in the monetary policy rule.

We analyse the existence of multiple monetary policy regimes and investigate the periods in which each state prevails. We find that when considering financial markets liquidity, the Fed react less than one to one to expected inflation in periods of looming economic uncertainty and distress, when it weights more the liquidity conditions in the economy. Conversely, it switches back to a compliant Taylor rule in tranquil times. Our findings challenge the narrative that sees the US Central Bank accommodating expected inflation until Volcker's regime change and shed new light on the functioning of the US monetary authority and its information set.

Across all estimates we also find evidence of parameters instability, whose effects on realised inflation dynamics and other aggregates are yet to be thoroughly analysed.

On top of this parameter instability, we also find that the Taylor Principle, warranting uniqueness and stability in NKDSGE models, is violated when we include measures of financial markets liquidity. Nevertheless, this violation does not bring about degenerate occurrences of sunspot equilibria, with hyper-inflation or deep disinflation episodes.

B Empirical appendix B.1 Output gap

We detail the econometric derivation of the real time output gap from the Greenbook dataset. This dataset offers data and forecasts prepared by the Federal Reserve System staff for the Federal Open Market Committee meetings, where policy rate decisions are discussed. To gauge the output gap at any date t, we select all observations up to time t, run a simple quadratic time trend and keep the last observation residual. This term is then normalized as percentage deviation from the computed trend and used as output gap observation. The algorithm is the following:

1 set a number j of observations sufficiently high to compute precise estimates 2 for all t in [j, T] run the following routine i sub-select observations in the [j,

FIGURE 1 :

 1 FIGURE 1: VARIABLES EMPLOYED IN THE ESTIMATES. HISTORICALLY REVISED GDP DEFLATOR (ACT. INFL., DASHED); GREENBOOK ONE-QUARTER GDP DEFLATOR FORECAST (EXP. INFL., DOT-DASHED); FEDERAL FUND RATE (FFR, SOLID); GREENBOOK ESTIMATED OUTPUT GAP (GAP, DOTTED).

  ii) the Great Moderation [1982Q1:2007Q2], (iii) and finally the GFC [2007Q3:-].

  FIGURE 2: F-STATISTIC PLOTS FOR SPECIFICATIONS I TO VIII. DASHED LINE INDICATES THE STATISTICS VALUE, SOLID HORIZONTAL LINE MARKS THE SIGNIFICANCE AREA AT 95%. INDIVID-UAL CAPTIONS OFFERS MOST LIKELY DATE FOR A singular STRUCTURAL BREAK IN THE SPECIFI-CATION POINTED BY THE VERTICAL DASHED LINE.

FIGURE 3 :

 3 FIGURE 3: MARKOV STATES: ABOVE SHADED AREAS CORRESPOND TO S 1 PREVAILING OVER S 2 , BELOW SMOOTHED PROBABILITIES FOR S 1 .

Fig.( 4

 4 Fig.(4) plots the residuals generated from the regression on the full sample. As OLS sort of averages over the full sample, sudden and ample fluctuations in the residuals point to observations where the model underperforms. This occurs typically in the late '70s, late '80, and around the GFC period.

FIGURE 4 :FIGURE 5 :

 45 FIGURE 4: RESIDUALS PLOT FOR THE THREE SPECIFICATIONS. DASHED LINE DEPICTS SINGLE RESIDUALS AS TIME SERIES, SOLID HORIZONTAL LINES CONTOUR THE 2-SDS AREA AROUND ZERO, THE EXPECTED -AND EMPIRICAL -RESIDUALS MEAN.

TABLE 1 :

 1 CLARIDA, GALI, AND[START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF]

	Markov State	µ	γ	ω	ρ
	S 1	9.44 -.3 (3.07) (.34) (.26) (.15) .46 .49
	S 2	.59 (.66)	.85 .58 (.23) (.09) (.05) .8

TABLE 2 :

 2 MURRAY

, NIKOLSKO-RZHEVSKYY, AND

[START_REF] Murray | Markov switching and the Taylor principle[END_REF]

  Table (3) summarises the results of OLS estimates on the full length sample for the different specifications.

	Var.		I	Specification II	III
	µ		.1327 (.7766)	8.1884 * * * (1.6854)	.3355 (.5268)
	E t (GDP defl.) t+1	1.5811 * * * (.2736)	.9522 * (.4673)	.9528 * * * (.1883)
	Real time	ŷ	.48 * * (.1439)	.2496 + (.1437)	.2361 * (.0998)
	BAAspr.			-2.9673 * * * (.4939)
	SPspr.				-.4349 * * * (.0951)
	FFR t-1		.7782 * * * (.0369)	.8643 * * * (.0289)	.6877 * * * (.0402)
	Obs.		188	112	188
	R 2		.8949	.9733	.9052
	BIC		628	157	612

TABLE 3 :

 3 OLS ESTIMATES IN THE FULL SAMPLE, UP TO 2013 Q2. SIGNIFICANCE CODES: 0 * * * 0.001 * * 0.01

* 0.05 + 0.1; SE IN PARENTHESES.

TABLE 4 :

 4 EXOGENOUS SPLITS, THREE SAMPLES. SIGNIFICANCE CODES: 0 * * * 0.001 * * 0.01 * 0.05 + 0.1; SE IN PARENTHESES. (i) RUNS FROM THE EARLIEST AVAILABLE OBSERVATION TO 1981Q4; (ii) RUNS FROM 1981Q4 TO 2007Q2; (iii) GOES FROM 2007Q3 TO THE LATEST OBSER-VATION AVAILABLE, CURRENTLY 2013Q2, AS SOME DATA ARE PUBLISHED WITH A FIVE YEARS LAG.

TABLE 5

 5

			Specification
		I	II	III
	Sing. break 1980Q3 1990Q3 1982Q3
	N. of breaks	2	3	2
	Date 1	1980Q3 1989Q4 1980Q3
	Date 2	1987Q3 2000Q4 1987Q3
	Date 3	-	2008Q3	-

TABLE 6 :

 6 OPTIMAL SEGMENTATION AND BREAK DATES: FIRST ROW PRESENTS THE MOST LIKELY BREAK ADMITTING ONLY A SINGLE ONE. THIRD TO FIFTH ROWS PRESENT BREAK DATES WHEN UP TO 5 BREAKS ARE ALLOWED.

TABLE 7 :

 7 ESTIMATES FOR k = 2 MARKOV SWITCHING MODEL. SIGNIFICANCE CODES: 0 * * * 0.001 * * 0.01 * 0.05 + 0.1; SE IN PARENTHESES. (-) STANDS FOR NON-CONVERTIBLE SE: SIGNIFICANCE IS HENCE DERIVED FROM THE MAIN REGRESSION.

	Spec. State	µ	E t (GDP defl.) t+1 Real time ŷ BAA spr.	SP spr.	FFR t-1 Adj.R 2 BIC
	I	S 1 S 2	-.7759 (2.025) -.4052 (1.199)	1.693 * * * (.4816) 2.012 * * * (.4131)	.3554 (.329) .6329 * * (.2274)			.6128 * * * (.1118) .9208 * * * (.0253)	.7853 .9758	518
	II	S 1 S 2	2.74 * (-) 11.01 * * * (1.264)	-.7505 + (-) .3199 (.3338)	.0564 (-) .442 * * (.1367)	-.3956 (-) -4.066 * * * (.3692)		1.099 * * * (.0234) .8385 * * * (.0234)	.9987 .9765	142
	III	S 1 S 2	1.45 (1.322) -.8855 (.9915)	.5299 * * * (.1596) 1.561 * * * (.3917)	-.2855 + (.1458) .4535 * (.2005)		-.6061 * * * -.0642 (.0938) (.1416) -.2201 .8982 * * * (.1712) (.0286)	.8505 .9623	518

TABLE 8 :

 8 TRANSITION MATRICES: TRANSITION PROBABILITIES FOR TWO-STATE MARKOV PRO-CESS. COLUMNS ARE CURRENT STATE, HENCE CONDITIONAL ON IT NEXT STATE IS ONE OF THE ROWS.

	.I	S 1	S 2
	S 1	.945	.171
	S 2	.055	.829
	Spec.II	S 1	S 2
	S 1	.9086	.2399
	S 2	.0914	.7601
	Spec.III	S 1	S 2
	S 1	.8451	.0511
	S 2	.1549	.9489

TABLE 11 :

 11 CORRELATIONS: r IS THE FFR, E t π i IS NOW-OR ONE PERIOD AHEAD FORECAST FOR INFLATION MEASURE i, E t ŷt IS REAL TIME OUTPUT GAP, ŷt IS CAPACITY UTILISATION, ût IS FLUC- TUATIONS AROUND THE NATURAL UNEMPLOYMENT RATE, SP AND BAA ARE THE SPREADS.

		r t	r t-1	E t π	de f l t	E t π	de f l t+1	E t π	cpi t	E t π	cpi t+1	E t π	pce t	E t π	pce t+1
	r t	1	0.982	0.547	0.693	0.393	0.518	0.729	0.822
	r t-1	0.982	1		0.549	0.682	0.356	0.484	0.735	0.817
	E t π E t π E t π de f l t de f l t+1 cpi t E t π cpi t+1 E t π pce t E t π pce t+1	0.547 0.693 0.393 0.518 0.729 0.822	0.549 0.682 0.356 0.484 0.735 0.817	1 0.543 0.476 0.206 0.667 0.629	0.543 1 0.410 0.571 0.727 0.813	0.476 0.410 1 0.493 0.500 0.431	0.206 0.571 0.493 1 0.469 0.582	0.667 0.727 0.500 0.469 1 0.861	0.629 0.813 0.431 0.582 0.861 1
	E t ŷt	0.616	0.612	0.304	0.218	0.223	0.214	0.295	0.357
	ŷt	0.614	0.575	0.145	0.192	0.223	0.204	0.153	0.222
	ût	-0.735	-0.723	-0.311	-0.347	-0.260	-0.314	-0.366	-0.449
	SP	-0.622	-0.629	-0.499	-0.406	-0.179	-0.292	-0.466	-0.533
	BAA	-0.623	-0.552	-0.311	-0.431	-0.471	-0.414	-0.399	-0.461
				E t ŷt	ŷt		ût		SP		BAA
		r t		0.616	0.614	-0.735	-0.622	-0.623
		r t-1	0.612	0.575	-0.723	-0.629	-0.552
		E t π E t π E t π de f l t de f l t+1 cpi t E t π cpi t+1 E t π pce t E t π pce t+1	0.304 0.218 0.223 0.214 0.295 0.357	0.145 0.192 0.223 0.204 0.153 0.222	-0.311 -0.347 -0.260 -0.314 -0.366 -0.449	-0.499 -0.406 -0.179 -0.292 -0.466 -0.533	-0.311 -0.431 -0.471 -0.414 -0.399 -0.461
		E t ŷt	1		0.736	-0.881	-0.551	-0.355
		ŷt		0.736	1		-0.889	-0.393	-0.557
		ût		-0.881	-0.889	1		0.543	0.533
		SP		-0.551	-0.393	0.543	1		0.235
		BAA	-0.355	-0.557	0.533	0.235	1

2 Full sample regression: residuals

  t] interval ii estimate y s = α 1 s + α 2 s 2 + s , s = j, j + 1, . . . , t; recover fitted values ŷs iii compute gap t = y t -ŷt y t × 100 = ˆ t y t × 100; keep the latest observation/nowcast prior to the FOMC meeting 3 stack and date all gap t to construct the real time output gap variable

B.

This series is provided by Moody's.

We also include[START_REF] Wu | Measuring the macroeconomic impact of monetary policy at the zero lower bound[END_REF] shadow rate for the FFR, although it is not used in the work as it does not add any further insight.

The resulting dataset, as well as the code to compile and maintain it, are available in this Git repository. The code itself might undergo significant improvement and extension over time.

Further details are presented in the Data Appendix (B.1).

A narrow ask/bid gap.

[START_REF] Carvalho | Taylor Rule Estimation by OLS[END_REF] provide motivation for using OLS in estimating monetary policy rules and quantify the bias of such method compared to GMM or IV.

Greenbook real time data, in particular, are released to the public with a 5-year lag with respect to the estimates or forecasts and thus presents data until 2014. Moreover, data on BAA spreads are available only from 1986Q1.

Appendix (B.2) offers the residuals plot for the full sample regression. Eyeballing these plots provides sufficient motives to carry out additional analyses on model instability.

Diagnostics on structural breaks and Markov SwitchingSubsampling according to historical events is an appealing device, but it does not provide a robust statistical motivation. To address this shortcoming, we exploit a number of tests that diagnose the stability of the estimated parameters. Compared to the exogenous sub-

Volcker was nominated July 25 th , 1979, sworn shortly after August 6 th , and left the Chair in August 11 th , 1987. In our quarterly dataset it translates in 1979Q3:1987Q3.

Alternatively, a subset of parameters can be optionally estimated across all regimes, so its estimate is invariant to the prevailing regime.

Most likely this discrepancy arises from our longer sample and slightly different method employed in the estimation.

While it is true that during the GFC some systemic banks underwent actual bailouts, it is disputable for the rest of the companies listed in the S&P 500.

This section presents the CUSUM plots for the eight specifications of the Taylor rule studied in Section (3). The solid black line marks the cumulative sum of residuals, whilst the red lines define the significance areas, in which the residuals sum signals a change in the underlying data generating process.


Emanuele Franceschi
email: emanuele.franceschi@psemail.eu@web.

I Eryzhenskiy

I Iodice

J M Monta Ña

M Ranaldi

A simple model of liquidity

We introduce liquidity motives in an otherwise standard monetary model. The Central Bank's policy rule is adapted to target the interest rate on liquid bonds. These deviations are sufficient to relax the requirement for active monetary policy and warrant determinacy in both passive and active policy regimes. We compare this model of liquidity with workhorse models and find that it can substantially replicate usual dynamics. By means of stochastic simulations, we also study how monetary policy stance affect inflation dynamics and find evidence of increased persistence for passive monetary policy.

Introduction

Liquidity and finance have risen to the central stage of macroeconomic discussion in the wake of the 2008 Global Financial Crisis. As financial and interbank markets froze and interest rates fell to zero, major Central Banks deployed credit and liquidity facilities to unlock the flow of assets and revive their trade, de facto injecting unprecedented amounts of liquidity in the markets. Fed's Chairman Ben Bernanke famously commented on the ex ante effectiveness of Quantitative Easing policies saying that "The problem with QE is [that] it works in practice, but it doesn't work in theory."

This paper contributes to the study of liquidity on monetary policy in general, and how it affects the restrictions on interest rate rules in particular. We introduce liquid bonds and total liquidity in an otherwise standard business cycle model from the New Keynesian tradition. We equip the monetary authority with a simple rule that fixes total liquidity and sets the interest rate on liquid assets looking at expected inflation. This framework yields a stable equilibrium as long as the Central Bank responds positively to expected inflation, regardless of the magnitude of the interest rate adjustment.

This model thus provides a unified system to analyse how monetary policy rules affect the response of the economy under a supposedly indeterminacy regime. We compare this liquidity model to the basic standard model that is at the core of the New Keynesian Dynamic Stochastic General Equilibrium class, with nominal frictions, technological shocks, and rational expectations. The inclusion of liquidity and a simpler monetary policy rule are sufficient to broadly match and reproduce the behaviour of the NKDSGE baseline model. This effort is twofold. First, it allows us to move on known territory, presenting the results in a transparent way and keeping contact with the enormous literature flourished around this class of models. Second, it can be easily implemented in existing theoretical structures with negligible adjustments, making possible a direct test against other models.

We further analyse by means of stochastic simulations how the two policy stances affect directly inflation dynamics. We find that an accommodative monetary policy induces more persistence in the inflation process, making it harder to steer toward any target.

Lastly, in close connection to the 2008 crisis, we experiment with a large drop in liquidity and analyse how different policy regimes affect economic responses.

Related literature

This paper studies the role and consequences of liquidity in monetary policy and formulates a setup encompassing both aggressive and passive monetary policy interest setting rules. This theoretical model builds on [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] and relates in spirit with [START_REF] Michaillat | A New Keynesian model with wealth in the utility function[END_REF][START_REF] Michaillat | A New Keynesian model with wealth in the utility function[END_REF], B. [START_REF] Diba | Pegging the interest rate on bank reserves[END_REF], and M. B. [START_REF] Canzoneri | Interest rate rules and price determinacy: the role of transactions services of bonds[END_REF] and M. Canzoneri et al. (2008a,b). We further detail [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] model by fully specifying the supply side of the economy and studying its behaviour in discrete time. Throughout the exposition, our aim is to propose a small deviation from the standard NKDSGE framework that constitutes the core of modern fluctuations theory in macroeconomics [START_REF] Gali | Monetary Policy, Inflation, and the Business Cycle[END_REF][START_REF] Walsh | Monetary Theory and Policy[END_REF][START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF].

This paper is also related to studies on conditions for determinacy in several vintages of New Keynesian models. For example, [START_REF] Benhabib | The perils of Taylor rules[END_REF] include money in the production function and studies under which fiscal and monetary policy regimes the model displays indeterminacy. Similarly, [START_REF] Leeper | Equilibria under 'active' and 'passive' monetary and fiscal policies[END_REF] studies how fiscal and monetary policy interact and establishes parametrical regions for determinacy. Along the same lines of fiscal interaction, Cochrane further develops a price level theory grounded on debt and taxes, working around the limitations and shortcomings of the monetary NKDS-GEs [START_REF] Cochrane | The Fiscal Theory of the Price Level[END_REF]. In this model, we abstract from fiscal considerations and focus exclusively on the liquidity factor.

The remainder of the paper is structured as follows: Section 2 introduces liquidity in the NKDSGE framework; Section 3 compares the liquidity model with a baseline one; Section 4 studies the effects of a severe liquidity shock; Section 5 analyses how monetary rules influence inflation dynamics; Section 6 concludes.

A model of liquidity

We introduce liquid bonds in an otherwise standard, small scale NKDSGE model. This extension is motivated by the utility they provide in being liquid assets, much in line with money holdings, which provide utility services. The idea is to model agents' liquidity portfolio allocation between plain money and liquid assets, interchangeably bonds or bank deposits.

Accordingly, we modify the monetary policy behaviour. We postulate a skeletal rule that feedbacks expected inflation to the interest rate paid on liquid assets.

Consumer

We postulate an economy with an infinitely-lived representative agent, who works, consumes, and holds money alongside with two types of assets. Hence, total wealth is divided between a liquid bond B, cash M, and an illiquid bond X. This latter serves the only purpose of smoothing consumption over time and allocate wealth intertemporally. All assets mature after one period. We assume that the agent is willing to hold B and M because they provide transaction services, and therefore utility. In addition, B bonds pay s nominal interest rate, X ones pay nominal interest rate i, while money pays no interest and is carried on to next period. While money balances in the utility function is not new, bond holdings are more exotic. In this respect, our model is close to [START_REF] Michaillat | A New Keynesian model with wealth in the utility function[END_REF], although the resulting Euler equation at the steady state is not modified by bonds but rather money holdings. In fact, it is completely homomorphic to introduce a cash-in-advance (or rather, liquidity-in-advance) constraint, as in [START_REF] Calvo | Disinflation and interest-bearing money[END_REF].

Under these assumptions, the utility maximization program of the consumer is structured as follows.

max

{C s ,M s ,B s ,N s } ∞ s=0 E t ∞ ∑ s=0 β s (u(C t+s ) + h (B t+s ) + v (M t+s ) -g (N t+s )) s.t. C t + M t + X t + B t = W t N t + (1 + s t-1 ) B t-1 + (1 + i t-1 ) X t-1 + M t-1 (1)
Where we assume additively separable (dis)utilities for consumption, cash, liquid bonds, and hours worked N. The intertemporal budget constraint summarises expenditures, allocations and income sources: interests promised in t -1, carried-on money, and labour income. Moreover, c is the result of aggregating a measure one of differentiated goods via a Dixit-Stiglitz aggregator with constant elasticity of substitution θ. This also implies that P is the price index of the underlying goods.

It is useful to reformulate the budget constraint in terms of total wealth and real quantities before deriving the system of first order conditions. Therefore, let D = X + M + B be the total wealth held by the consumer. Replacing X = D -M -B in the budget constraint and appropriately dividing through P gives the following real budget constraint.

c t + d t = w t N t + s t-1 -i t-1 1 + π t b t-1 + (1 + r t-1 ) d t-1 - i t-1 1 + π t m t-1 (2)
Where π = P t P t-1 -1 is the inflation rate and lower-case indicates real quantities. With this reformulation, the FOCs system implies the equilibrium conditions in (3).

u (c t ) = E t β (1 + r t ) u (c t+1 ) g (N t ) u (c t ) = w t h (b t ) = E t βλ t+1 i t -s t 1 + π t+1 v (m t ) = E t βλ t+1 i t 1 + π t+1 (3)
where λ t is the Lagrangian multiplier, the first equation is the usual Euler equation for intertemporal consumption, the second equates marginal cost and benefits of work, and the last two equations govern the allocation decision between liquid bonds b and real money balances m. In particular, last equation discounts money holdings by the real interest rate, that is the loss from inflation; the equilibrium condition on bonds, instead, regulates liquid asset holdings on the spread between nominal interest rate i and yield of such bonds, both in real terms.

Firms

The production side of this economy is straightforward and assumes a measure one of infinitesimal firms, indexed by j ∈ [0, 1]. Each firm is embedded with a technology that employs only labour, so that the production function is

Y jt = A t N a jt . ( 4
)
The term A captures the stochastic productivity of the economy and follows a simple AR(1) process, a ∈ (0,1) represents the decreasing returns to scale, and N jt the individual employment of each firm. As the consumption good results from the CES aggregator, every firm j faces a demand schedule (5), relative to aggregate production, with θ being the elasticity of substitution and P jt the firm's price.

Y jt = P jt P t -θ Y t (5)
We assume nominal rigidity à la [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF]: 1 every period (1α) % of firms are given the chance to update their price, while the remaining share will stick to previous prices. This entices a forward-looking behaviour in firms when they optimise their expected discounted profits, as they take into account the duration of their price. Firm j's marginal cost is MC jt = W t /P t aY jt /N jt , whence the expected discounted profits in eq.( 6). max

P * j E t ∞ ∑ s=0 α s Q t,
Where α is the Calvo pricing parameter, Q t,t+s is the stochastic discount factor between periods t and t + s the consumer uses to weight future profits, and P * j is the optimal price chosen by the firm. Factoring in the constraint and solving the program with a symmetry argument gives two results. First, firms price with a constant markup over the marginal cost, and second that the optimal price is specified as a function of expected future marginal costs, price index levels and economic activity, as shown in eq.( 7). This same equation also presents the inflation dynamic as a AR(1) process, depending on past prevailing prices and current updated prices.

P * P t = θ θ -1 E t ∑ ∞ s=0 (βα) s Y t+s P t+s P t θ MC t+s E t ∑ ∞ s=0 (βα) s Y t+s P t+s P t θ-1 P 1-θ t = (1 -α) P * 1-θ t + αP 1-θ t-1 (7)

Monetary authority and market clearing

A second exotic ingredient is the Central Bank, its policy, and total liquidity management.

We assume the existence of a monetary authority that operates in two ways in the economy. First, it sets the total amount of liquidity in circulation, namely eq.( 8). Importantly, the Central Bank does not determine the allocation between cash and liquid bonds, but only the sum of the two, irrespectively of the portfolio composition. For the sake of simplicity, we will also assume a fixed nominal amount of liquidity in the economy, so that

Z t = B t + M t (8) Z t = Z ∀t (9)
A fixed amount of liquidity implies no issuance of liquid bonds, which are instead traded and circulated among agents. The same principle applies to money, which is injected or withdrawn to seamlessly counteract the movements in liquid bonds trade. In this light, B closely resembles bank deposits, for both can be readily exchanged for cash. One can think of Z as a fixed money supply (M2), although a full characterisation would have B issued by government in the form of liquid, risk-free treasury bonds. Along the same lines, the Central Bank would influence the circulation of these bonds to manage total liquidity and, more importantly, to coordinate with fiscal policy. 2The Central Bank also sets the policy interest rate s t paid on liquid bonds following a constant rule. It is useful to think in this context to the link between the Federal fund rate and the interest rate on the shortest-maturity Treasury Bill. In detail, the rule responds solely to inflation expectations and does ignore any level of economic slack contrary to more usual Taylor feedback rules:

exp (s t ) = E t exp (γπ t+1 ). ( 10
)
This skeletal structure is clearly a simplified Taylor rule, but the values for γ are crucial in showing that an accommodative Central Bank does not necessarily drive the economy down a hyper-inflation spiral, specifically when it takes values smaller than one. 3 It can easily be extended to a full fledged Taylor Rule including real slack without impairing or modifying the final results. For example, this specification of the Taylor Rule relates to that of the European Central Bank, which in its mandate contemplates explicitly only prices stability and not employment or economic slack.

The rationale behind this specification is that keeping under control the liquidity in circulation the Central bank assures that inflation follows a specified path. This particular specification of the monetary policy links the inflation rate and the return yield on the liquid bonds. Therefore, the Central Bank in our model levers the liquidity allocation via interest rate setting.

Lastaking eq.( 9) and dividing through by the prices level, one can obtain the values for liquidity allocation in real terms, as well as a backward-looking expression for real liquidity depending on current inflation.

z t = m t + b t ⇐⇒ z t = z t-1 1 + π t (11)
This last equation, with the market clearing condition C t = Y t , closes the model.

Linearised model

In this section we briefly present the system of equations resulting from loglinearising eqs.

(3), ( 7), ( 10), and ( 11) around a zero-inflation steady state. To this end, we assume precise functional forms for the utility functions, namely

u (c) = c 1-σ 1 -σ h (b) = b φ φ v (m) = m ψ ψ g (N) = χ N 1+η 1 + η (12)
With 1 ≥ φ > ψ > 0, which implies that the consumer is more sensitive to bonds rather than money, in line with everyday financial decisions. The other functional forms assumed are consistent with more traditional exercises and well settled in the NKDSGE literature. This shared ground highlights how consequential liquid assets can be, once modelled as a complement to cash, especially on the policy rule.

Once we log-linearise the model we obtain a system of linear equations whose properties can be easily and extensively studied. We perform a comparison with the simple 3-equation model presented in [START_REF] Gali | Monetary Policy, Inflation, and the Business Cycle[END_REF], for example.

4 ŷt = 1 -ψ σ mt + E t ŷt+1 + 1 σ E t π t+1 ( 13
)
π t = βE t π t+1 + κ ŷt - ŷ f t (14) 1 -ψ m 2 + m φ-ψ 1-ψ (1 -ψ) mt + βy -σ m 1-ψ s t -(1 -φ) 1 + m φ-ψ 1-ψ ẑt = 0 (15) ẑt = ẑt-1 -π t (16) s t = γE t π t+1 (17)
Briefly describing the system above, eq.( 13) is the Euler equation augmented with the real money balances, which affect positively the contemporaneous output gap. Eq.( 14) is the Phillips curve of this economy, in line with more classical models, eq.( 15) is the money demand function (depending on liquid bonds interest rate s, total real liquidity z), eq.( 16) captures intertemporal changes in total real liquidity, and finally eq.( 17) represents the monetary policy rule. We adopt the convention that x is the percentage deviation of x from its steady state, while all lower-case, unhatted, and time independent variables are steady state values. Moreover, we use y f for flexible prices output. To add more details:

κ = (1 -α) (1 -αβ) a α (a + θ (1 -a)) 1 + η + a (σ -1) a ŷ f t = η + 1 1 + η + a (σ -1) Ât m = y σ 1 -β 1 1-a y = η + 1 1 + η + a (σ -1) (18)
To finalise the model, we introduce two classic shock to perturb the model around its steady state to display convergence dynamics. Both shocks follow a stationary AR (1) pro- cess:

Ât = (1 -ρ A ) Ā + ρ A Ât-1 + A t ν t = ρ ν ν t-1 + ν t A t ∼ N (0, σ A ) ν t ∼ N (0, σ ν )
Both shocks are simply added to their respective equations and throughout the whole simulations we calibrate their persistence parameters ρ to the same value.

As this approximation of the full model can be easily simulated, we exploit it to check for which calibration sets the model generates a unique and stable equilibrium. In addition to the real, technological disturbance, we add a policy shock impulsed by the Central Bank. The former induces a change in the total factor productivity A from its steady-state value, set to 1. The latter is a shock to the monetary rule detailed in eq.( 10). These two shocks allow the comparison with the aforementioned standard models, so to perform a horse race and check consistency of our augmented model. Assessing whether our setup replicates the classic reactions of well-known models is paramount to validate its structure and internal workings.

Calibration and IRFs

We calibrate the model to the values presented in Table (1), taking the most common values used in the literature. The only restriction involved in the model concerns the exponents of bonds and real balances utility functions. For the "Taylor Principle" parameter of our policy rule, we explore two values: the first is supposed to violate the [START_REF] Blanchard | The solution of linear difference models under rational expectations[END_REF] condition and thus produce an unstable solution, whilst the second is the one most commonly found in both empirical studies and theoretical exercises.

Parameter

Descr First, our calibrated model generates a unique, stable equilibrium for both values of γ, meaning that sunspot equilibria are ruled out even if the Central Bank reacts passively the inflation expectations of the economy. Second, our model behaves as expected once compared with the 3-equation NKDSGE counterpart. 5 In fact, under the same calibration the two models show the same behaviour in terms of reactions to shock, as it is possible to see in the next figures. We first compare side-by-side the effects of a technological shock (Fig.

Technological shock

From Fig.

(1), a one-standard-deviation, positive shock to total factor productivity produces the same response in our model and in the standard NKDSGE one. This is due to the very same structure of the production side of the two modelled economies. We focus on three common aggregates. Following a TFP shock, in both cases the output gap turns negative, inflation falls, and the reference interest rate falls in response. While this reaction is common across models, magnitudes mark little differences. In particular, while the output gap and inflation falls more in the NKDSGE model, our proposal implies less movement in these aggregates: for inflation and output gap, the response in our model is broadly half on impact. This milder response in our model might be driven by the stripped-down version of our monetary policy rule, which prevents output gap movements to percolate into the variations in s. Alternatively, this dampened propagation might result from our assumptions in the consumption and financial blocks of the model, where the TFP shock is dissipated smoothly through two complete financial markets.

Monetary policy shock

When the modelled economies are hit by a monetary policy shock and the liquidity model complies to the Taylor Principle, they generate the IRFs pictured in Fig.

(2), again corresponding to common aggregates across models. The IRFs produce the same profiles in both cases, but the liquidity models displays greater impacts on output gap and inflation. In both cases the magnitude is roughly thrice that of the baseline NKDSGE, while direction and adjustment correspond closely. The skeletal policy rule engrained in the liquidity model abstracts from economic activity level, thus the Central Bank is not facing trade-offs between economic activity and inflation and focuses solely on the latter. This monetary policy rule, moreover, impedes the feedback loops between policy rate, inflation and economic activity, designing a different propagation mechanism: adjustments in the market for money and liquid bonds affect consumption and in turn production.

Focusing on the policy rate, our model responds roughly twice less than the NKDSGE model to the very same shock, pointing again to a propagation mechanisms that is substantially different and occurs through the financial position of the agent. In addition, our monetary shock is well more persistent in the interest rate, converging back to zero only when inflation levels off, too. The absence of economic activity in the rule makes that the policy rate essentially mirrors the inflation profile. In comparison to the baseline model, the liquidity one presents a slower convergence path to the steady state. This model, therefore, entails a higher degree of persistence in its design.

Liquidity with an accommodative Central Bank

We turn now to showing how the liquidity model reacts to the previous shocks when the Central Bank does not comply to the Taylor principle. In this case, we set γ = .5 and expose the behaviour of endogenous aggregates present in the economy, plus the behaviour of the liquid bond holdings, recovered form the FOC for a complete picture. In the aggressive regime, output, inflation and the policy rate s fall. Interestingly, s and π slightly overshoot before converg-ing to zero from above from the fifth quarter, roughly. This behaviour is reflected in real liquidity and its components: the technological shock doubles the impact on total real liquidity, the interest rate drop triggers a reallocation away from bonds towards cash m, until s overshoots and subsequently balances the overall effect on liquidity reshuffling. All in all, the main discrepancy between the two regimes is in the asset block: the inflation path influences more dramatically and on a shorter period the amount of real liquidity when the Central Bank is passive. Conversely, the path back to the steady state is significantly quicker in this regime. Fig. 4 summarises the response to a monetary policy shock that raises the interest rate s on bonds of 1%. First off, the output gap moves in the same direction and is affected similarly to the γ > 1 case. Secondly, inflation responds as expected and after 15 quarters the shock is fully absorbed, without unexpected evolutions of the prices.

Total real liquidity z and its components, m and b, comove in reaction to a monetary policy shock. Two forces are at work in this case, the inflation effect and the reallocation towards the more remunerative asset. On impact, inflation decreases: this path influences real liquidity as future inflation will be higher than today, increasing current liquidity until inflation overshoots its steady state level and turns slightly positive. This happens in the first five quarters, approximately. When inflation turns positive (although extremely close to zero) real liquidity peaks and decreases smoothly.

In the meantime, the agent adjusts its portfolio of assets profiting from the increased return on the liquid asset, b. In this sense, the IRF for m mirrors that of b, with a reallocation away from money holdings to liquid bonds.

Under the passive monetary policy regime the economy experiences different magnitudes in the aggregates. When the Central Bank is accommodative, output, inflation, and the key interest rate double their impact change, and the latter becomes less persistent, overall. This last outcome falls in line with [START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF], and sheds light on how monetary policy stances can affect aggregate dynamics in general, and inflation dynamics in particular. Taking this implication to the data would imply that prior to the Great Moderation period inflation was reporting higher degrees of inertia, whilst it would decrease afterwards. We analyse the effects of different monetary policy regimes on inflation dynamics in Section (5).

The 2008 crisis: severe liquidity shortage

This model lends itself to an interesting experiment: although nominal liquidity is assumed to be constant at Z and real liquidity z moves with the inflation rate, we hit z with a negative shock and study the behaviour of our model, as in eq.( 19). Although not orthodox, this is a practical short-cut: neglecting where that missing liquidity goes physically -and in which proportion money and bonds are affected -lets us focus on the dynamics of convergence to the steady state. One could think of this experiment as a sudden drop in the combined liquidity of bonds and money, closely related to the liquidity dry up triggering the 2008 Global Financial Crisis. In the aftermath of such recession, major Central Bank rapidly hit the zero lower bound on policy interest rates, thus in effect adopting an accommodative monetary policy stance, equivalent to that admitted by the liquidity model. In this respect, analysing how aggregates react to a sudden liquidity dry up sheds light on how different interest rate rules interact and affect the overall dynamics.

We produce IRFs and discuss their economic interpretations under the two regimes of monetary policy. ẑt -z t = ẑt-1π t , with z t /z 10 (19)

Following an abrupt and violent liquidity dry-up the model shows a general reaction broadly independent from the behaviour of the monetary authority, but substantial differences emerge in magnitude and reversion to the steady state levels.

The shock impacts at first money demand m, for a given policy rate s, which spikes up. Conversely the representative agent disinvests in the liquid bond, proportionally more than the missing liquidity. This results from the preference for money with respect to bonds that assumed in the calibration.

At this stage, the Phillips curve ( 14) and the IS equation ( 13) propagate the shock to the rest of the economy. The money term in the Euler equation ( 13) transmits the shock to current output that spikes as well on impact. Under a passive monetary policy this translates into a limited effect of the shock on impact, and a degradation in the following quarters. The Phillips curves then squares the expected inflation with current π and widened output gap. Inflation expectations subsequently drive the monetary policy decisions which translate into two distinct paths for realised inflation.

The money term in the IS curve is the telling point between the two regimes, together with the path for liquid bonds, b. Under both regimes money converges back to the steady state relatively fast, showing that real liquidity is deeply intertwined with liquid bonds. Most notably, under an accommodative Central Bank real liquidity recovers more rapidly, thanks to a deflation that accelerates the recovery of z but impedes a quick rebound in output. Interestingly, when the money authority conducts an active policy, the inflation path -contained deflation with slow recovery -turns into persistently low rates, well beyond the case of a passive Central Bank.

The sharp difference in the reaction of the two regimes lies in the severity of the impact and the duration of the recovery. Inflation and output, in particular, show starkly different behaviours: when the Central Bank passively follows inflation expectations a liquidity dryup triggers a deep recession with a relatively long recovery (more than 30 quarters); the same applies to inflation, too. An active monetary stance against a liquidity shock tames the damage and facilitates the recovery, somewhat controlling the effects within the financial sector of the economy.

To sum up, and combining these results with previous information, we suggest that Central Banks complying to the Taylor Principle have a firmer control on contagion when a liquidity crisis hits. According to our stylised model, in fact, active monetary policy helps containing and limiting the damage to the sole financial sector of the economy, with reduced impact and consequence on real activity. The stark difference in the set of IRFs, clearly, lies in the fall of the output gap: it widens under passive monetary policy while it marks a mild recessions under active policy, although duration is similar. As we already remarked, recovery speeds are substantially different.

Dissecting simulated inflation dynamics

One interesting experiment involves the inflation dynamics generated by these two models: does an passive Central Bank produce unexpected inflation dynamics? In this framework, it is straightforward to generate abundant time series and hence conduct some expost econometric exploration. To offer a more comprehensive comparison, we simulate and analyse two other DSGE model, namely [START_REF] Ascari | The Macroeconomics of Trend Inflation[END_REF] and [START_REF] Smets | Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach[END_REF]. The former study how trend inflation affects aggregate dynamics and policy in a generalised New Keynesian model, while the latter build a rich environment with nominal frictions and indexation for wages and prices, investment adjustment cost, and a large number of shocks and outperform VARs in short term forecasting.

We generate for each model 500000 observations, or 125000 years of synthetic history: this should assure convergence of the estimators and tight confidence intervals. By the very structure of the models, the Data Generating Processes of these series is a linear system shocked by AR(1) normal innovations: one should not be surprised that the data generated are also Gaussian. Comparability across models is meaningful because of the close parametrisation and the same sequence of shocks used.

For clear reasons, we limit our interest to the global autoregressive properties of the inflation series, which is generated by a single, stable, and well behaved DGP for each case. As we calibrate the persistence of all shocks to ρ = .65, we expect to find values in this neighbourhood. Any difference between the simulated series is due to the propagation mechanism and, most importantly, the monetary response function. Moreover, when comparing our model in its two policy regimes, we will be able to pick up the different inflation dynamics enticed by the passive monetary policy stance.

We estimate first an AR (5): our model complying to the Taylor Principle, the same violating it, and the workhorse NKDSGEs. Secondly, we set an upper bound on the lags to 120, and pick the optimal lag number minimising the Bayesian Information Criterion. Table (2) presents the results for an AR (5).

Looking at the coefficients on the first lag, we see our expectations confirmed, as all coefficients are tightly close to the calibrated parameter, no constant is statistically different from zero, and significance decreases after the first lag for classic NKDSGE models. A notable exception is the [START_REF] Smets | Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach[END_REF] model, which reports five significant lags with relatively large coefficients. The most remarkable feature of such model, though, is that it displays a quasi unit-root in inflation, as the first lag is statistically close to one.

Interestingly, as we depart from the NKDSGE models to the liquidity model with accommodating Central Bank, the coefficient on the first lag moves away from the calibrated value, downwards. This is interesting because it confirms the consensus that a passive monetary authority has less command over inflation path and fails at taming its dynamics back on target. The flip side of this latter aspect is that, when first lag becomes less relevant, previous ones acquire more weight. Overall, hence, inflation seems to become more persistent when Central Banks do not follow an aggressive Taylor rule.

On the other hand, the magnitude of the significant coefficients of the two parametrisations of the liquidity model are comparatively small. While for an active Central Bank (model (3)) the autoregressive coefficients quickly approach zero, for a passive one these still become smaller but remain roughly ten times bigger than those of the other models, and still significant.

Dependent variable:

Simulated Inflation (1) (2) (3) (4) (5)
Const.

- These results point to a role for monetary policy stance in substantially influence the inflation dynamics, along the same lines traced by [START_REF] Cogley | Inflationgap persistence in the US[END_REF], Cogley and[START_REF] Cogley | Drift and volatilities: monetary policies ad outcomes in the post World War II US[END_REF], and [START_REF] Primiceri | Time varying structural vector autoregressions and monetary policy[END_REF].

A more interesting exercise is to compare the optimal lags for an AR (k) process. This procedure finds that the optimal number of lags for our model with liquidity and Taylor principle (model (3) in Tab.( 3)) is around 70, whilst for the version parametrised in accordance to the Taylor Principle (model (2), ibid) it is around 50, for the standard NKDSGE it is merely 2. An optimal lag number does not imply that all regressor lags are significant. It only implies that all significant lags are part of the regression, likely a subset of the total lags. With this in mind, our model of liquidity with an accommodative monetary authority shows that today's inflation depends on a long sequence of lags. The number of optimal lags decreases when we let the Central Bank respond aggressively to expected inflation. On the other hand, the baseline NKDSGE model produces a process with extremely short memory, likewise for the model of trend inflation. In line with the result from Tab.(2), the medium scale [START_REF] Smets | Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach[END_REF] 6) plots the estimates, confidence bands, and p-values6 for the 71 optimal lags of the liquidity model with accommodative policy rule. Of all lags included, only 21 are strongly significant (±30%); interestingly, these are mostly negative in sign and relatively small in magnitude. This last evidence might suggest that some erratic, although not degenerate, dynamic is at play in this policy regime.

Fig.( 7) plots the same information for an aggressive policy rule. This model features liquidity and an accommodative Central Bank. What is striking is the length of lags deemed relevant and the share of significant ones (±80%), as opposed to the aggressive policy rule. As remarked in Table (2), coefficients are greater and all significant at 1% up to the 38th one. These features point toward a higher persistence in inflation, something compatible with a Central Bank that, for instance, targets monetary aggregates (prior to Volcker's chairmanship) or finds itself short of conventional monetary tools (QE at the ZLB), as it has been for long periods recently. Nonetheless, this setting does not imply necessarily sunspot equilibria or spiralling aggregates, the system eventually converges back to steady state even when the monetary policy stance is accommodative.

This result offers a framework to analyse inflation dynamics in light of diverse monetary policy regimes. Higher levels of persistence -as long as it is measured in number of significant lags -are expected in periods like the US hyperinflation of the 70's, when indeed prices skyrocketed out of control. Conversely, tranquil periods results from aggressive monetary stances, like the Great Moderation. The inclusion of liquidity is extremely helpful to systematise the 2008 crisis, while also accounting for the inactive interest rate policy. For this latter case, though, other factors need to be considered, like the level of liquidity and the wide range of unconventional policies put in place.

Conclusion

Since the Global Financial Crisis, liquidity has gained a central role in the general macroeconomic discussion and in monetary policy in particular. It has been the main concern for major Central Banks in engaging in unconventional policies. We adapt a parsimonious framework with minor departures from the core model of monetary policy and derive relevant results for Central Banks' mandate of price stabilisation. First and foremost, we show that the simple addition of a liquid asset -and the consequent modification of the intertemporal Euler equation -pins down a solution with an accommodative, stripped down policy rate rule. The latter needs only to positively correlate the policy rate and expected inflation to rule out degenerate, multiple equilibria. These latter sunspot equilibria would arise in the baseline New Keynesian Dynamic Stochastic General Equilibrium model when the Central Bank does not react to inflation aggressively.

We compare the responses of our liquidity model with those of the baseline NKDSGE and confirm that they broadly match for technological and monetary policy shocks, under identical calibration. We then study how the model respond to such shocks when the Central Bank is allowed to under-react to expected inflation. We find no evidence of degenerate behaviour for the model aggregates, contrary to the predictions of the baseline NKDSGE.

In fact, all aggregates, and inflation especially, broadly react in the same way under the two regimes: we signal though a change in the dynamics, rather than direction. We find that an accommodative Central Banks generates more persistent inflation. We test such hypothesis on simulated data, including two other workhorse models from the field.

We find that the inclusion of liquidity and liquid bonds generates overall more persistence in inflation. Within the sets of calibration parameters, inflation displays more persistence when the monetary policy stance is passive, in line with evidence from the US hyperinflation period. Concerning the Phillips curve, it remains unchanged from traditional New Keynesians models and derives from the use of equations (7). The output gap it includes results from the comparison to the flexible prices version of the model. Other relations do not need further manipulation.

A.3 Extreme policies

This section investigates what are the consequences of two extreme calibrations for the textbook NKDSGE models. We let the reaction to expected inflation be negligibly close to its determinacy limit, such that γ + → 1, and then test for extremely high levels of reaction. The IRFs for these two cases are presented in Figures below. Interestingly, while calibrations close to indeterminacy do not display dramatic changes but only a higher effect on impact, an extremely reactive Central Bank can substantially suppress a sizeable share of volatility in the economy. Indeed, it appear to curb inflation and economic gap in a much more effective way when a fundamental productivity shock hits the economy -at the cost of considerable volatility in the policy rate, though. On the monetary policy shock, the decrease in volatility is even sharper.

  (1)), then compare the effects of a monetary policy shock under two regimes for our model (Fig.(2) and Fig.(4)).

FIGURE 1 :FIGURE 2 :

 12 FIGURE 1: LIQUIDITY MODEL (DASHED, γ = 1.8) AND GALI (2015) (SOLID): IMPULSE RESPONSE FUNCTIONS FOLLOWING A 1-σ TFP SHOCK. VALUES IN PERCENTAGES.

FIGURE 3 :

 3 FIGURE 3: IMPULSE RESPONSE FUNCTIONS FOR OUR MODEL: WITH ACCOMMODATIVE (SOLID) AND AGGRESSIVE (DASHED) REACTION TO EXPECTED INFLATION AFTER A PRODUCTIVITY SHOCK (ONE STANDARD DEVIATION) -ALL ENDOGENOUS VARIABLES, VALUES IN PERCENTAGES.

FIGURE 4 :

 4 FIGURE 4: IMPULSE RESPONSE FUNCTIONS FOR OUR MODEL: WITH ACCOMMODATIVE (SOLID) AND AGGRESSIVE (DASHED) REACTION TO EXPECTED INFLATION AFTER A MONETARY POLICY SHOCK (ANNUALISED 1% POLICY RATE HIKE) -ALL ENDOGENOUS VARIABLES, VALUES IN PER-CENTAGES.

FIGURE 5 :

 5 FIGURE 5: REAL LIQUIDITY SHOCK: AGGRESSIVE (γ = 1.8, DASHED) VS ACCOMMODATIVE (γ = .5, SOLID) POLICY RULES, VALUES IN PERCENTAGES.

Fig.(

  Fig.(6) plots the estimates, confidence bands, and p-values 6 for the 71 optimal lags of the liquidity model with accommodative policy rule. Of all lags included, only 21 are strongly significant (±30%); interestingly, these are mostly negative in sign and relatively small in magnitude. This last evidence might suggest that some erratic, although not degenerate, dynamic is at play in this policy regime.Fig.(7) plots the same information for an aggressive policy rule. This model features liquidity and an accommodative Central Bank. What is striking is the length of lags deemed relevant and the share of significant ones (±80%), as opposed to the aggressive policy rule. As remarked in Table(2), coefficients are greater and all significant at 1% up to the 38th one. These features point toward a higher persistence in inflation, something compatible with a Central Bank that, for instance, targets monetary aggregates (prior to Volcker's chairmanship) or finds itself short of conventional monetary tools (QE at the ZLB), as it has been for long periods recently. Nonetheless, this setting does not imply necessarily sunspot equilibria or spiralling aggregates, the system eventually converges back to steady state even when the monetary policy stance is accommodative.This result offers a framework to analyse inflation dynamics in light of diverse monetary policy regimes. Higher levels of persistence -as long as it is measured in number of significant lags -are expected in periods like the US hyperinflation of the 70's, when indeed prices skyrocketed out of control. Conversely, tranquil periods results from aggressive monetary stances, like the Great Moderation. The inclusion of liquidity is extremely

FIGURE 9 :FIGURE 10 :

 910 FIGURE 9: AUTOREGRESSIVE ON OPTIMALLY SELECTED LAGS: LIQUIDITY MODEL WITH ACCOM-MODATIVE CENTRAL BANK (γ = .5). TOP PANEL PLOTS THE ESTIMATED COEFFICIENTS IN SOLID, BANDS ARE TWICE THE ESTIMATED STANDARD ERRORS. BOTTOM PANEL PLOTS THE p-VALUES FOR EVERY COEFFICIENT AND A 5% THRESHOLD. FULL LAGS, ONLY THE INTERCEPT IS OMITTED.

TABLE 1 :

 1 CALIBRATION FOR MODEL SIMULATIONS

TABLE 2 :

 2 AR (5) ESTIMATES ON SIMULATED DATA FROM FIVE MODELS: (1) GALI (2015), (2) LIQ-UIDITY MODEL γ = 1.8, (3) LIQUIDITY MODEL γ = .5, (4) ASCARI AND SBORDONE (2014), (5) SMETS AND WOUTERS (2007). ONLY TECHNOLOGICAL AND MONETARY POLICY SHOCKS ARE AL-LOWED, EACH MODEL IS SIMULATED FOR 500000 PERIODS, AFTER DISCARDING THE FIRST 100000 ITERATIONS. ALL SHOCKS ARE SET TO HAVE ZERO MEAN, EQUAL VARIANCE, AND ARE IID. SEC-OND AND THIRD COLUMNS PRESENT ESTIMATES FOR OUR MODEL WITH LIQUIDITY, COMPLYING TO THE TAYLOR PRINCIPLE AND VIOLATING IT, RESPECTIVELY.

  Table (3) offers more details on this result.

	Models		Stats
		Opt. lags Sign. lags adj. R 2	BIC
	(1) Gali (2015)	2	50%	.421	-280676.9
	(2) Liq. γ = 1.8	51	80%	.348	-413028.9
	(3) Liq. γ = .5	71	32%	.401	-1033066
	(4) Ascari and Sbordone (2014)	2	50%	.421	-422629.6
	(5) Smets and Wouters (2007)	13	38%	.719	-486558.7

TABLE 3 :

 3 OPTIMAL LAGS FOR AR PROCESS OF INFLATION FOR STANDARD (1) BASELINE NKDSGE (GALI, 2015), (2) LIQUIDITY MODEL COMPLYING TO THE TAYLOR PRINCIPLE (γ = 1.8), (3) LIQUIDITY MODEL VIOLATING IT (γ = .5), A MODEL WITH TIME-VARYING TREND INFLA-TION (ASCARI AND SBORDONE, 2014), AND (5) A MEDIUM SCALE WORKHORSE DSGE (SMETS AND WOUTERS, 2007). OPTIMAL LAGS ARE THOSE MINIMISING THE BIC. ALL MODELS ARE FED THE SAME SEQUENCE OF SHOCKS OF THE SAME VARIANCE, GENERATING 500000 QUARTERLY OBSERVATIONS.

  model displays longer endogenous lags, due to numerous frictions and feedback engrained in this model. AUTOREGRESSIVE ESTIMATES ON OPTIMALLY SELECTED LAGS: LIQUIDITY MODEL WITH ACCOMMODATIVE CENTRAL BANK (γ = .5). TOP PANEL PLOTS THE ESTIMATED COEFFI-CIENTS IN SOLID, BANDS ARE TWICE THE ESTIMATED STANDARD ERRORS. BOTTOM PANEL PLOTS THE p-VALUES FOR EVERY COEFFICIENT AND A 5% THRESHOLD. NOTE: FIRST LAG EXCLUDED FROM THE PLOT FOR SCALE READABILITY.

		0.000
	Coefficient est.	-0.020 -0.015 -0.010 -0.005
		-0.025
		0	10	20	30	40	50
		1.00
	Coefficient est. Significance	-0.005 0.000 0.005 0.00 0.25 0.50 0.75
		0	10	20	30	40	50
		-0.010		Lag
		0	20		40	60
		FIGURE 7:
		1.00
	Significance	0.25 0.50 0.75
		0.00
		0	20		40	60
				Lag
		FIGURE 6: AUTOREGRESSIVE ESTIMATES ON OPTIMALLY SELECTED LAGS: LIQUIDITY MODEL
		WITH AGGRESSIVE CENTRAL BANK (γ = 1.8). TOP PANEL PLOTS THE ESTIMATED COEFFICIENTS
		IN SOLID, BANDS ARE TWICE THE ESTIMATED STANDARD ERRORS. BOTTOM PANEL PLOTS THE
		p-VALUES FOR EVERY COEFFICIENT AND A 5% THRESHOLD. NOTE: FIRST LAG EXCLUDED FROM
		THE PLOT FOR SCALE READABILITY.

  v (m t ) = E t βλ t+1 i t 1 + π t+1 i t = v (m t ) E t 1 + π t+1 βλ t+1 plug this result into h (b t ) h (b t ) = E t βλ t+1 i ts t 1 + π t+1 ⇒ h (b t ) = E t ¨¨¨β + π t+1 exploit the fact that λ t+1 = u (c t+1 ) and b t = z tm t to obtain h (z tm t )v (m t ) = E t -βs t u (c t+1 ) 1 + π t+1Setting this equation to its steady state and using a first Taylor approximation generates equation (15) in the text.The backward dependence of real liquidity (11) results as follows:Z t = Z = M t + B tM t = P t m t B t = P t b t Z = P t m t + P t b t

	λ t+1 1 + π t+1 1 Z v (m t ) ¨¨¨1 + π t+1 βλ t+1 βλ t+1 s t -P t-1 (m t + b t ) P t = P t-1
	z t-1 = z t (1 + π t )
	⇒ z t =	z t-1 1 + π t

  BASELINE NKDSGE (GALI, 2015) WITH VERY LOW REACTION TO EXPECTED INFLA-TION BY THE CENTRAL BANK: γ + → 1. VALUES IN PERCENTAGES. BASELINE NKDSGE (GALI, 2015) WITH EXTREMELY HIGH REACTION TO EXPECTED INFLATION BY THE CENTRAL BANK. VALUES IN PERCENTAGES.

							TFP shock
			Policy Rate				Inflation TFP shock				Output Gap
	-0.20 -0.15 -0.10 -0.05 0.00 0.00 0.05 0.10 0.15 -0.6 -0.4 -0.2 0.0 0.000 0.001 0.002 0.003	0 0 0 0	10 10 10 FIGURE 11: Policy Rate 20 Policy Rate 20 20 Policy Rate 10 20	30 30 30 30	40 40 -0.08 -0.20 -0.15 -0.10 -0.05 0.00 -0.06 -0.04 -0.02 0.00 40 -0.004 -0.003 -0.002 -0.001 0.000 -1e-03 -5e-04 0e+00 40	0 0 0 0	10 10 Interest Rate Shock 20 30 Inflation 20 30 Quarters Inflation 10 20 30 Inflation Interest Rate Shock 10 20 30 Quarters	40 40 -0.10 -0.4 -0.3 -0.2 -0.1 0.0 -0.05 0.00 40 -0.006 -0.004 -0.002 0.000 -0.0020 -0.0015 -0.0010 -0.0005 0.0000 40	0 0 0 0	10 10 10 10	20 Output Gap 20 Output Gap 20 Output Gap 20	30 30 30 30	40 40 40 40
			FIGURE 12:

The precise modelling of the nominal rigidity is inconsequential, as the main novelties are in the consumers' side of the model. Hence, quadratic adjustment costs as in[START_REF] Rotemberg | Sticky prices in the United States[END_REF] might be added without loss of fundamental insights.

Notable examples are the CARES Act in the US, or the combined SURE and Next Generation EU in Europe, both in response to Covid-19 crisis. These were preceded by ECB's APP-PSPP and Fed's Open Market Operations as measures for the 2008 crisis.

Although it is worth recalling that a Central Bank complying to the Taylor principle actually destabilizes the economy[START_REF] Benhabib | The perils of Taylor rules[END_REF].

A more detailed walk-through for obtaining this system is provided in the Appendix(A.1)

[START_REF] Gali | Monetary Policy, Inflation, and the Business Cycle[END_REF], Chapter III, version with interest rate rule.

For the sake of readability, we do not plot the first lag. All-inclusive plots are in Appendix A.2.

Appendix A Model appendix A.1 Obtaining the system of equations

We combine the equations presented in the body of the paper so to obtain the system of equations that will be later loglinearised and fed to Dynare for simulations.

The first target is the augmented Euler equation. The version proposed in the paper includes real money balances on top of the usual terms. Take the first intertemporal FOC from the consumer utility maximisation program as the starting point:

Now recall the condition on marginal utility of real money balances, v (m t ), and em- ploy it to define the nominal interest rate, i t :

Turning back to the Euler equation, plug in the nominal interest rate relation just recovered:

This last expression is then loglinearised to obtain equation (13).

To condense the money equation start with the last two relations in (3):