Exiting from quantitative easing

Hayashi, Fumio; Koeda, Junko Article Exiting from quantitative easing Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Hayashi, Fumio; Koeda, Junko (2019) : Exiting from quantitative easing, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 10, Iss. 3, pp. 1069-1107, https://doi.org/10.3982/QE1058 This Version is available at: https://hdl.handle.net/10419/217163 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 10 (2019), 1069–1107 1759-7331/20191069 Exiting from quantitative easing Fumio Hayashi National Graduate Institute for Policy Studies Junko Koeda School of Political Science and Economics, Waseda University We propose an empirical framework for analyzing the macroeconomic effects of quantitative easing (QE) and apply it to Japan. The framework is a regimeswitching structural vector autoregression in which the monetary policy regime, chosen by the central bank responding to economic conditions, is endogenous and observable. QE is modeled as one of the regimes. The model incorporates an exit condition for terminating QE. We find that higher reserves at the effective lower bound raise inflation and output, and that terminating QE may be contractionary or expansionary, depending on the state of the economy at the point of exit. Keywords. Effective lower bound, structural vector autoregression, monetary policy, Taylor rule, impulse responses, Bank of Japan. JEL classification. C13, C32, C54, E52, E58. 1. Introduction and summary Quantitative easing (QE) combines forward guidance, positive excess reserves held by depository institutions at the central bank, and targeted asset purchases at an effective lower bound (ELB), where the difference between the nominal policy rate and the ELB (hereinafter referred to as the net policy rate) is very close to zero. We study the macroeconomic effects of QE and exiting from QE using a regime-switching structural vector autoregression (SVAR) model. The model incorporates forward guidance in the form of an exit condition for terminating QE, the supply of excess reserves by the central bank, and an ELB. The data are drawn from Japan, a country with a relatively long history of QE. Japan’s experience of multiple QE spells allows us to study exits from QE. Note that during our sample period, which ends in 2012, targeted asset purchases were not a focus of the Bank of Japan (BOJ).1 Fumio Hayashi: [email protected] Junko Koeda: [email protected] We are grateful to Toni Braun for consultation and guidance, and to Kosuke Aoki, Gauti Eggertsson, Martin Eichenbaum, Yuzo Honda, Tatsuyoshi Okimoto, Stephanie Schmitt-Grohe, and Etsuro Shioji for their comments and suggestions. The paper benefited from presentation at the Federal Reserve Bank of Atlanta, Keio University, Kyoto University, and Waseda University. We thank Kenwin Maung for his excellent research assistance. This research was supported by grants-in-aid from the Ministry of Education, Culture, Sports, Science, and Technology of the Japanese government (Grants 25285097 and 26870124). 1We end our sample period in 2012 because the BOJ under Governor Kuroda since 2013 appears to have embarked on a regime that is very different from that observed in our sample period. Under the pre-2013 ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1058 1070 Hayashi and Koeda Quantitative Economics 10 (2019) We start by documenting that the net policy rate was effectively zero whenever the BOJ’s stated policy was to guide the policy rate to “as low a level as possible.” Thus, the ELB regime of a zero net policy rate is observable. We identify the following three ELB spells: March 1999 to July 2000, March 2001 to June 2006, and December 2008 to date. During these spells, the BOJ made a stated commitment of not exiting QE if inflation remained below a certain threshold. Our baseline SVAR has two monetary policy regimes: an ELB regime and a regime of positive net policy rates. There are four variables: inflation, output (measured by the output gap), the policy rate, and excess reserves. The model’s first two equations are reduced-form equations describing inflation and output dynamics. The reduced-form coefficients can be regime dependent. The third equation is the Taylor rule. The policy rate cannot be set to the Taylor rate (the rate prescribed by the Taylor rule) if it lies below the ELB. The fourth equation specifies the central bank’s supply of excess reserves under QE. The exit condition requires that the central bank end QE only if the Taylor rate is positive and inflation exceeds a given threshold. Thus, regime endogeneity (where the regime’s occurrence depends on inflation and output) arises not only from the ELB, but also from the exit condition. We conduct nonlinear impulse-response and counterfactual analyses, in which nonlinearity arises from multiple regimes and the nonnegativity constraints on excess reserves. The regime and associated inflation and output dynamics change endogenously over the horizon. We find the following: •QE is expansionary. When the current regime is an ELB regime, the response of output and inflation to an increase in excess reserves is positive. However, the statistical significance of this result depends on our measure of the output gap. •Policy-induced exits from QE can be expansionary or contractionary, depending on the history.2We consider an alternative and counterfactual history for the July 2006 exit. Because the alternative history is chosen judiciously such that it differs from the baseline history solely in terms of policy shocks, the response is policy induced. An exit would have been expansionary for May or June 2006, and nearly so for April 2006. However, it would have been contractionary until March 2006 because of the higher level of excess reserves and the weaker macroeconomic conditions at exit. The remainder of the paper is organized as follows. Section 2reviews the related literature and states the paper’s contributions. Section 3documents the case for the monetary policy regime’s observability. Section 4describes the SVAR model. Section 5ex- plains the estimation strategy and the results. Section 6provides the nonlinear impulseresponse and counterfactual analyses. Section 7provides extension and robustness checks. Section 8concludes the paper. regime, the BOJ held mostly short- and medium-term Japanese government bonds. It was not until April 2013 that the BOJ under Governor Kuroda started to address the maturity structure of long-term Japanese government bond holdings. As a result, reserves began to increase at a far higher rate in 2013. 2To address possible concerns of spurious causality, where the monetary policy regime appears to cause the output (if the inflation–output dynamics is not adequately captured by our model), we have estimated a hidden-state Markov-switching model of Hamilton (1989) with a censored Taylor rule. We find that such a model does not generate the sort of impulse response profiles found in Section 6.1. Quantitative Economics 10 (2019) Exiting from quantitative easing 1071 2. Relation to the literature Theoretical explanations of the QE effect, as surveyed, for example, in Woodford (2012, Section 3), include the signaling and portfolio-balance channels.3As discussed in Section 7, we do not find strong empirical evidence for the macroeconomic effects of these channels over our sample period, which ends in 2012. However, other transmission channels could exist through central bank purchases of a risky asset in the presence of collateral constraints (Araújo, Schommer, and Woodford (2015)) and capital requirements for commercial banks (Ennis (2014)).4 An exit can be expansionary if it triggers the economy to move to a “better” economic state. The severity of the ELB is discussed in the seminal work of Eggertsson and Woodford (2003). Benhabib, Schmitt-Grohe, and Uribe (2001) showed that the Taylor rule with the ELB has multiple equilibria, one of which is a liquidity trap. Aruoba, Cuba- Borda, and Schorfheide (2018) computed the sunspot equilibrium of a nonlinear New Keynesian model with the ELB, in which an economy can move between a targeted inflation regime and a deflation regime. Lansing (2017) developed a New Keynesian model with learning about regime transitions, and illustrated a case in which exit is expansionary. Gust, Herbst, López-Salido, and Smith (2017) quantified the cost of the ELB by estimating a nonlinear dynamic stochastic general-equilibrium model with the ELB, though their model does not converge to a deflationary steady state for the U.S. economy. However, the triggers for a shift from one regime to another vary; for example, it can be a rise in the equilibrium real rate (Eggertsson and Woodford (2003)), a sunspot (Aruoba, Cuba-Borda, and Schorfheide (2018)), or changes in perceived transition probabilities (Lansing (2017)). In our case, the trigger is a combination of macroeconomic conditions and policy shocks. The empirical literature on the macroeconomic effects of QE is growing rapidly. Relevant to our study are those that use quantities as a QE measure.5The impulse-response analyses in all studies reviewed here find positive QE effects, albeit with varying magnitudes and statistical significance. The identification assumption employed in most of these studies is that inflation and output are predetermined.6 Several works exploit the observability of the ELB regime by estimating singleregime SVAR models for a sample deemed to be under the regime. Honda, Kuroki, and 3See Gagnon, Raskin, Remache, and Sack (2010) for a discussion on these channels and Chen, Cúrdia, and Ferrero (2012) and Farmer and Zabczyk (2016) for general-equilibrium models of the portfolio-balance channel. 4Ennis’s (2014) explanation of the QE effect can be aptly called the “quantity theory of bank capital.” Commercial banks’ assets expand as the central bank increases the supply of reserves. A leverage requirement forces commercial banks to provide more bank capital, but the real supply of bank capital is fixed. Therefore, prices must rise. 5Studies with price-based QE measures include those of Kapetanios, Mumtaz, Stevens, and Theodoridis (2012) and Baumeister and Benati (2013), who use the yield spread in their vector autoregressions (VARs), and Wu and Xia (2014), who use a properly defined shadow policy rate. They all report expansionary QE effects. 6Weale and Wieladek (2016) reported an expansionary QE effect under four different identification assumptions, including that of predetermined inflation and output. 1072 Hayashi and Koeda Quantitative Economics 10 (2019) Tachibana (2013)7and Schenkelberg and Watzka (2013) found a positive QE effect for Japan. Gambacorta, Hofmann, and Peersman (2014) utilized a panel of countries for the period January 2008 to June 2011. They overcome the shortness of the sample by utilizing data from eight advanced economies, including that of Japan. Weale and Wieladek (2016) used U.S. and U.K. data from March 2009. The shortness of their sample is addressed through a Bayesian method. Consistent with studies for Japan, the authors find that asset purchases by the central bank raise inflation and output. This is also consistent with the findings of event studies of large-scale asset purchases in the two countries, surveyed in Woodford (2012), where announcements of such purchases raised asset prices. Another way of addressing the small-sample problem is to include a period of positive policy rates, but to allow the model parameters to vary over time in specific ways. One strand of the literature employs regime switching. The regime-switching SVAR that has been used to study U.S. monetary policy (Bernanke and Mihov (1998); Sims and Zha (2006)) assumes the regime to be unobservable and exogenous. Fujiwara (2006)andInoue and Okimoto (2008) apply this line of approach with Japanese monthly data, and find that the probability of one of the regimes becomes very high from the late 1990s onward.8For those months, the impulse response of output to an increase in the base money is positive and persistent. Another strand of literature applies a time-varying parameter approach. Kimura and Nakajima (2016) use quarterly Japanese data from 1981, and assume two QE spells (2001:Q1–2006:Q1 and 2010:Q1 on). Their time-varying parameter VAR takes the ELB into account by forcing the variance of the coefficient in the policy rate equation to shrink during QEs. With the ELB, the policy rate is a censored variable, which renders the regime endogenous. Therefore, restricting the sample to the ELB period entails a sample selection bias. Iwata and Wu (2006) add the period of positive policy rates to the sample and estimate their SVAR while treating the policy rate as a censored variable. They assume that the inflation and output dynamics under positive policy rates is the same as that under the ELB regime. Relative to the literature reviewed here, our study makes three contributions. •Our analysis of the QE effect is more general in two respects. First, unlike the studies cited above, we allow the regime and associated inflation–output dynamics to change endogenously over the horizon of the impulse response function. Second, our estimation takes into account regime endogeneity that is due not only to the ELB, but also to the exit condition. 7The work of Honda et al. (2013), which was originally written in Japanese in 2007, was perhaps the earliest QE study for Japan. Their QE measure is reserves, which was the target used by the BOJ during the ELB period of 2001 through 2006. Their recursive VAR of prices, output, and reserves, estimated on monthly data for the ELB period, shows that the impulse response of output to an increase in reserves is positive. 8Aruoba et al. (2018) can be viewed as providing an underlying model to this type of regime-switching model. Their structural model is the (nonlinear) New Keynesian model with an ELB on the nominal interest rate. It has two equilibria, one of which is the liquidity trap with deflation. A nonlinear filtering technique reveals that the inflation–output dynamics during the period of near-zero policy rates in Japan was very likely generated by the deflationary equilibrium. Quantitative Economics 10 (2019) Exiting from quantitative easing 1073 •Our estimated SVAR can generate ELB spells comparable in length to those experienced in Japan and the United States, whereas recent structural New Keynesian models with an ELB have difficulty doing so. Thus, our study fills a gap in the literature by providing a model-free characterization of how inflation and output interact with the ELB. •Our study is (to the best of our knowledge) the first to analyze the macroeconomic effects of exiting from QE. We provide a detailed discussion in Section 6.2. 3. Identifying the ELB regime Three Spells of the ELB Regime If the policy rate is the overnight interbank rate, its ELB is the interest rate on reserves (IOR), that is, the interest rate paid on reserves held by depository institutions at the central bank.9We say that the monetary policy regime stin period tis the ELB regime, denoted by st=Z,ifthenet policy rate rt−rt, defined as the excess of the policy rate rtover the IOR rt, is “effectively zero.” We can identify the months under regime Zbased on the monetary-policy statements made by the BOJ that guide the policy rate to a level near the lower bound.10 There are three spells of ELB regimes in Japan: March 1999 to July 2000 (Spell 1), March 2001 to June 2006 (Spell 2), and December 2008 onward (Spell 3). As it turns out, during those spells, the average over the reserve maintenance period (from the 16th of the month to the 15th of the next month) of the net rate rt−rtis less than five annual basis points (bps), thus providing an operational meaning to the phrase “effectively zero.”11 Outside these spells, with a net rate above 5bps, the regime stis the positive-net-policy-rate regime (P). We define the excess reserve rate (mt) as the logarithm of the ratio of the actual to required levels of reserves for month t. Therefore, excess reserves are positive whenever mt>0. Because this definition involves required reserves, the excess reserve rate mtapplies to reserve maintenance periods, from the 16th of month tto the 15th of month t+1. Figure 1plots mtsince 1997. The shaded areas in the figure indicate the three spells of Z. Barring unusual events (e.g., financial crises), excess reserves would be zero under P. Outside the shaded areas in Figure 1and before 1997, incidents of positive excess reserves are rare. Among those incidents, the largest moccurs, not surprisingly, in September 2008. It is true that the months before September 2008 and after the second Zspell have positive excess reserves (see the thin bars in Figure 1), but there is good reason to 9The IOR in Japan was zero until November 2008, when it was raised to 01%. 10The relevant statements are the announcements (“Statement on Monetary Policy”) made after the BOJ’s Monetary Policy Meetings (the Japanese equivalent of the U.S. Federal Open Market Committee, held every month and, sometimes, more often). They can be accessed at https://www.boj.or.jp/en/mopo/ mpmdeci/index.htm/. 11The policy announcement made on February 12, 1999, includes a passage, reproduced in Table 1, that an ELB regime will be initiated. However, this passage is followed by the statement: “To avoid excessive volatility in the short-term financial markets, the Bank of Japan will.... initially aim to guide the.... call rate to move around 015%, and subsequently induce further decline...” Indeed, for the first reserve maintenance period after the announcement (February 16 to March 15, 1999), the average policy rate was above 5bps, at 0075%. 1074 Hayashi and Koeda Quantitative Economics 10 (2019) Figure 1. Excess reserve rate, 1997–2012. ignore these months of positive excess reserves.12 After setting mt=0for those months with thin bars, there are only a handful of months of positive excess reserves outside the three Zspells. QE The excess reserve rate mtin the second and third Zspells, which is far higher than in September 2008, would be supply-determined. Indeed, during the second spell (between March 2001 and June 2006), the BOJ set a target range for reserves (see “Guideline for Money Market Operations” in their statements on monetary policy). More debatable is the first Zspell (March 1999–July 2000). Our reading of the minutes of the BOJ policy meetings at that time is that it supplied just enough reserves to prevent the interbank rate from rising above zero.13 The Japanese financial crisis of the late 1990s had not sub- 12Industry sources indicate that, after several years of a near-zero interbank rate with large excess reserves, the response by smaller-scale banks when the policy rate turned positive from essentially zero was to delay reentry to the interbank market. A breakdown of excess reserves by type of financial institution since 2005, available from the BOJ’s homepage, shows that large banks quickly reduced their excess reserves after the termination in July 2006 of the ELB policy, whereas other banks (regional banks, foreign banks, and trust banks) were slow to adjust. The average of excess reserves for the period from July 2006 to August 2008 is only 01% of the average for the period from January 2005 to June 2006 for large banks and 54% for other banks. To exploit the arbitrage opportunity presented by the positive interbank rates, banks need to train their employees anew. The reason commonly cited for the slow adjustment (e.g., Kato (2010)) is that medium- to small-scale banks, after several years of near-zero overnight rates, did not find it profitable to return immediately to the interbank market by incurring this re-entry cost. 13Proposals by one board member to supply far more reserves were repeatedly voted down. The minutes can be accessed at https://www.boj.or.jp/en/mopo/mpmsche_minu/. Quantitative Economics 10 (2019) Exiting from quantitative easing 1075 sided during the first Zspell. Reserves had to exceed required reserves for the policy rate to stay at zero. If we use the term QE to refer to the special case of the ELB regime Zin which reserves are supply determined, the discussion thus far can be summarized as follows: QE is in place during the second and third Zspells, whereas reserves are demand determined not only under regime P, but also during the first Zspell. Exit condition Several authors have noted that the BOJ’s zero-interest-rate policy is a combination of the net zero-interest-rate policy and a stated commitment to a condition on inflation for exiting from the ELB regime.14 The BOJ statements collected in Table 1indicate that during the three Zspells, the BOJ repeatedly expressed its commitment to an exit condition, stated in terms of the year-on-year (i.e., 12-month) consumer price index (CPI) inflation rate. For example, in the very first reserve maintenance period under Z(March 16, 1999– April 15, 1999), the BOJ governor pledged to continue the zero-interest-rate policy rate “until the deflationary concern is dispelled” (see the April 13, 1999, announcement in Table 1). 4. Regime-switching SVAR This section presents our regime-switching SVAR model for four variables: the inflation rate p,outputgapx,policyrater, and excess reserve rate m, defined as the logarithm of the actual-to-required reserve ratio. Strictly for expositional clarity, the baseline model presented here makes the following two simplifying assumptions: (i) mis zero under the regime of positive net policy rates (P); and (ii) mis supply-determined by the central bank under the ELB regime (Z) (so that the ELB regime Zcan be equated with the QE). A more general model without these assumptions is developed in Appendix A. The standard three-variable SVAR We start on familiar ground by considering the standard three-variable SVAR in the review conducted by Stock and Watson (2001). The three variables are the monthly inflation rate from months t−1to t(pt), the output gap (xt), and the policy rate (rt).15 The inflation and output gap equations are reduced-form equations, where the regressors are the (constant and) lagged values of all three variables. The third equation is the Taylor rule, which relates the policy rate to the contemporaneous values of the year-on-year inflation rate and the output gap. As in a standard block-recursive SVAR (see Christiano, Eichenbaum, and Evans (1999)), the identifying assumption is that inflation and output are predetermined. This assumption is plausible with our data alignment. We measure reserves for the month as 14See, for example, Okina and Shiratsuka (2004) and Ueda (2012). 15The slack measure in Stock and Watson (2001) is the unemployment rate, not the output gap. We use the output gap because Okun’s Law does not seem to apply to Japan. 1076 Hayashi and Koeda Quantitative Economics 10 (2019) Table 1. Policy statements by the Bank of Japan, 1999–2012. Date Quotes 1999.2.12 “The Bank of Japan will provide moreample funds and encourage the uncollateralized overnight call rate to move as low as possible.” 1999.4.13 “(The Bank of Japan will) continue to supply ample funds until the deflationary concern is dispelled.” (A remark by governor Hayami in a Q & A session with the press. Translation by authors.) http://www.boj.or.jp/announcements/press/kaiken_1999/kk9904a.htm/ 1999.9.21 “The Bank of Japan has been pursuing an unprecedented accommodative monetary policy and is explicitly committed to continue this policy until deflationary concerns subside.” 2000.8.11 “... the downward pressure on prices... has markedly receded.... deflationary concern has been dispelled, the condition for lifting the zero interest rate policy.” 2001.3.19 “The main operating target for money market operations be changed from the current uncollateralized overnight call rate to the outstanding balance of the current accounts at the Bank of Japan. Under the new procedures, the Bank provides ample liquidity, and the uncollateralized overnight call rate will be determined in the market ... The new procedures for money market operations continue to be in place until the consumer price index (excluding perishables, on a nationwide statistics) registers stably a zero percent or an increase year on year.” 2003.10.10 “The Bank of Japan is currently committed to maintaining the quantitative easing policy until the consumer price index (excluding fresh food, on a nationwide basis) registers stably a zero percent or an increase year on year.” 2006.3.9 “... the Bank of Japan decided to change the operating target of money market operations from the outstanding balance of current accounts at the Bank to the uncollateralized overnight call rate... The Bank of Japan will encourage the uncollateralized overnight call rate to remain at effectively zero percent.... The outstanding balance of current accounts at the Bank of Japan will be reduced towards a level in line with required reserves.... the reduction in current account balance is expected to be carried out over a period of a few months.... Concerning prices, year-on-year changes in the consumer price index turned positive. Meanwhile, the output gap is gradually narrowing.... In this environment, year-on-year changes in the consumer price index are expected to remain positive. The Bank, therefore, judged that the conditions laid out in the commitment are fulfilled.” 2006.7.14 “... the Bank of Japan decided... to change the guideline for money market operations... The Bank of Japan will encourage the uncollateralized overnight call rate to remain at around 025 percent.” 2008.12.19 “... it (author note: meaning the policy rate) will be encouraged to remain at around 01 percent (author note: which is the rate paid on reserves)...” 2009.12.18 “The Policy Board does not tolerate a year-on-year rate of change in the CPI equal to or below 0percent.” 2010.10.5 “The Bank will maintain the virtually zero interest rate policy until it judges, on the basis of the ‘understanding of medium- to long-term price stability’ that price stability is in sight...” 2012.2.14 “The Bank will continue pursuing the powerful easing until it judges that the 1percent goal is in sight...” Note: Accessible from https://www.boj.or.jp/en/mopo/mpmdeci/index.htm/ except for the 13 April 1999 statement. the average of daily values over the reserve maintenance period (from the 16th of the month to the 15th of the following month). We measure the policy rate for the month in a similar manner. If there is a half-month lag for the policy instruments to have macroeconomic effects, the inflation and output of the month cannot respond to the policy instruments of the same month. Another issue is whether there exist underlying models Quantitative Economics 10 (2019) Exiting from quantitative easing 1083 •The Mieno disinflation. It is widely agreed that the rapid rate hike from December 1989, when Yasushi Mieno became the BOJ governor, to June 1991, when the policy rate peaked, was specifically to respond to asset bubbles.20 We view this as a prolonged deviation from the Taylor rule. •Variable equilibrium real interest rates. We have been treating the intercept in the desired Taylor rate r∗ t(α∗ rin (4.1)) as a constant because of the assumption of the constant real interest rate. This assumption does not seem appropriate for Japan, given the well-documented decline in the trend growth rate since around 1990.21 If we ignore these issues and estimate the Taylor rule on the full sample from 1988 to 2012, then the estimate of γr(the coefficient of the lagged policy rate in the Taylor rate; see equation (4.1)) is about one. This yields a very imprecise estimate of the inflation and output coefficients in the desired Taylor rate (β∗ rin equation (4.1)). Therefore, we exclude the so-called bubble period of 1988–1991, which includes the Mieno disinflation and the sharp initial decline in the trend growth rate shown in Figure 3.22 Because trend growth kept declining after 1991, it is included in the desired Taylor rate to control for movements in the equilibrium real interest rate.23 In Section 7,wedescribehowour results change if trend growth is dropped from the model. Table 3reports our ML estimates for the sample period from 1992 to 2012. The positive trend-growth coefficient estimate of 062 implies that the equilibrium real rate indeed kept declining with the trend growth after the bubble period. The estimated speed of adjustment per month, 1−γr,isabout10%. The target inflation rate πis mere 053% per year. The inflation coefficient is estimated to be 069. The relatively small output coefficient of 005 means that inflation, rather than output stability, was the BOJ’s primary concern. The Taylor principle is violated because the inflation coefficient is less than one. As shown by Lubik and Schorfheide (2004), if the underlying model is the New Keynesian model with forward-looking agents, the violation means that the dynamics described by the SVAR could contain sunspot fluctuations. Sunspots, however, would not affect the equilibrium of the underlying model if agents are backward-looking. This point is illustrated by the second example in Appendix D in the Online Supplementary Material. Even if agents are forward-looking, Davig and Leeper (2007)andFarmer, Waggoner, and Zha (2009) showed that the prospect of monetary policy becoming active (in the sense of the inflation coefficient exceeding one) sometime in the future could eliminate sunspot equilibria. More recently, Hagedorn (2016) showed that violating the Taylor principle does not lead to price indeterminacy in a large class of incomplete market models. 20See, for example, the booklet on popular consumption by Okina (2013), who was a director of the BOJ’s research arm. 21For example, Hayashi and Prescott (2002) document that both the total factor productivity and the rate of return on capital declined in the early 1990s. 22If we include the bubble period of 1988–1991, the parameter estimates are similar to those reported in Table 3, provided that both the trend growth rate and a dummy for the Mieno disinflation are included. The inflation and output coefficients in the desired Taylor rate are 064 (t-value =36) and 008 (t-value =13), respectively. 23Okina and Shiratsuka (2002) and Braun and Waki (2006) used trend growth to control for the equilibrium real interest rate. 1084 Hayashi and Koeda Quantitative Economics 10 (2019) Table 3. Taylor rule, January 1992–December 2012 (sample size =252). Coefficients in the Desired Taylor Rate Trend Growth Rate Inflation Output Gap 062 [15]069 [25]005 [07] Speed of Adjustment (1−γr), % per Month Std. Dev. of Error (σr), % per Year Target Inflation (π), %perYear Std. Dev. of Threshold (σπ), % per Year 98 (28)0113 (00073)053 (045)033 (027) Note: Estimation by the ML (maximum likelihood) method described briefly in the text and more fully in Appendix 2. tvalues in brackets and standard errors in parentheses. The Taylor rule is the regime-dependent rule defined in (4.4), with the desired Taylor rate r∗ tgiven in (4.1)andtheregimestdefined in (4.7). The intercept α∗ rin the desired Taylor rate is a linear function of the trend growth rate. Excess reserve supply equation (θC) As already noted, the ML estimation is by Tobit on subsample Zthat excludes the first ELB spell. Since mis well above zero on the subsample, Tobit reduces to OLS. The estimates are presented in Table 4. Both the inflation and output coefficients pick up the expected signs. Inflation and output reduced-form equations (θA) The ML estimate of the reduced form is OLS for two separate subsamples: the “lagged” subsample P(i.e., those t’s with st−1=P) and the “lagged” subsample Z(with st−1=Z), excluding the first ELB spell. We include the trend growth rate in the set of regressors. For the lagged subsample P, we exclude lagged min order to be consistent with the model’s current assumption that m=0under regime P. The Bayesian information criterion (BIC) instructs us to set the lag length to one for both subsamples.24 Table 4. Excess reserve supply equation. Coefficient of tis in Const πtxtmt−1R2σs(%) QE (113 obs.)−0009 [−01]−0007 [−02]−0018 [−22] 098 (0033)0.94 0132 (00088) Note: Estimation by OLS. t-values in brackets and standard errors in parentheses. The sample of 113 observations is the QE months, which consists of the second and third ELB (effective lower bound) spells (March 2001–June 2006 and December 2008–December 2012). mtis the excess reserve rate, πtis the 12-month inflation rate to month tin percent, xtis the output gap in percent, σs(standard deviation of the error) is estimated as σs=SSR/n where nis the sample size. The standard error of σsis calculated as σs √2n. 24Given the moderate sample size, we set the maximum lag length to six and start the sample from July 1988 when we choose the lag length. Under the Akaike information criterion (AIC), we choose a lag length of two for the lagged subsample Pand one for Z. Quantitative Economics 10 (2019) Exiting from quantitative easing 1085 Table 5. Inflation and output reduced form. Coefficient of t−1is in Dependent Variable Const. gtpt−1xt−1rt−1mt−1R2 Lagged Subsample P P (85 obs.)inflation (pt)−011 [−02] 025 [04]−005 [−04] 016 [18]004 [01]004 output (xt)−070 [−23] 083 [21]−004 [−05] 094 [16]−031 [−09] 077 Lagged Subsample Z Z/QE (112 obs.) inflation (pt)−057 [−11]−026 [−08]−002 [−02] 010 [13]−003 [−00] 055 [22]009 output (xt)−107 [−28] 001 [00]002 [03]080 [15]−072 [−04] 046 [25]079 Note: Estimation by OLS. t-values in brackets. The “lagged subsample P” of 85 observations refers to t’s for which t≥March 1995 and t−1is not in the three ELB (effective lower bound) spells. The “lagged subsample Z”of112 observations refers to t’s for which t−1is in the second or third ELB spells (March 2001–June 2006 and December 2008–December 2012). pis the monthly inflation rate in percent per year, xis the output gap in percent, ris the policy rate in percent per year, mis the excess reserve rate (defined as the log of the ratio of actual to required reserves), and gis the trend growth rate in percent (the 12-month growth rate of potential output). The value of rt−1is 0% during the second ELB spell and 01% during the third ELB spell. The reduced-form estimates are shown in Table 5.First,weconsiderthereduced form for the lagged subsample P.Andrews’(1993)supF-test finds no structural break for the inflation equation, but does find a structural break for the output equation in March 1995.25 Therefore, we show the reduced-form estimates for the sample beginning March 1995. The monthly inflation (p) equation exhibits two notable features. First, inflation-persistence is nonexistent, as indicated by the lagged p-coefficient of almost zero. Second, the lagged r-coefficient has the wrong sign, but its magnitude is very small. We turn now to the lagged ELB subsample Z, excluding the first Zspell. The regressors include rt−1because, although it is constant in each of the two QE spells, it differs across spells. The positive lagged m-coefficients under QE imply that both inflation and output rise as excess reserves are increased. These effects are statistically significant. The coefficient of 046 in the output (x) equation, for example, means that the impulse response of xto a unit increase in mis 046 percentage points in the subsequent period. 6. Nonlinear impulse-response and counterfactual analyses For counterfactual analyses involving nonlinear models, such as ours, we find it more transparent to designate histories in terms of model variables (as in Gallant, Rossi, and Tauchen (1993)) rather than in terms of shocks. The shock-based translations of all the histories considered in this section, which are far more cumbersome to write down, are provided in Appendix C in the Online Supplementary Material. The model we use for the impulse-response and counterfactual analyses in this section is the baseline model with two regimes, Pand Z. This means that the other ELB 25The trimming parameter is 15%.TheF-statistic peaks twice, in October 1994 and in March 1995. We choose the latter because it is the recent months of the sample that are relevant in the counterfactual analysis. Miyao’s (2002) VAR analysis finds that a break occurs in 1995. Fujiwara (2006) and Inoue and Okimoto (2008) estimated the regime-switching VAR and find that the probability of one of the regimes becomes very high after the late 1990s. 1086 Hayashi and Koeda Quantitative Economics 10 (2019) regime, with demand-determined excess reserves, is assumed not to arise in the underlying simulations. Section 7will show that incorporating this other ELB regime with active excess reserve demand does not substantially alter the results presented here. 6.1 Transmission channels of monetary policy at and away from the ELB Policy-rate effect We start with the familiar case where the only difference between the two histories lies in the Taylor-rate shock vrt. Our counterfactual analysis asks what the response of the variables would be if the shock were of a different size. The response is given by the difference Eyt+k|st=P(p txtrt+δr0)   (ptxtrtmt)in the alternative history   lagged information −Eyt+k|st=P(p txtrt0)   (ptxtrtmt)in the baseline history   lagged information y=pxr m (6.1) where the lagged information indicated by “...” consists of the lagged values of (spx r m), and the conditional expectations are defined by the mapping given in equation (4.8).26 Not only the history up to t−1butalsothecurrent values of (ptxt)arethe same in both the baseline and the alternative histories. Thus, we control for the reducedform shocks for inflation and output. The Taylor-rate shock vrt is the only relevant policy shock here, because both mtand stare the same across the histories as well. This point is made more fully in Appendix C in the Online Supplementary Material (see equation (C.22)). The response profile, that is, the difference in the conditional expectation at various horizons (k), reduces to the standard impulse-response function if the two histories differ in the value for the base period of only one policy variable (as here) and the model is linear with only one regime.27 Figure 4shows the policy-rate effect, that is, the response profiles given in (6.1)for horizons k=01260 months.28 The interest-rate shock is δr=−1,thatis,apolicy rate cut of 100 bps. In contrast to the linear case, the difference in conditional expectations depends not only on how the alternative history differs from the baseline, but also 26Because the mapping is time-dependent, the expectations operator should have a subscript (Etrather than E). However, we omit this subscript tfor notational simplicity. We compute the conditional expectations by simulating 10000 sample paths of (s pxrm) generated by the mapping, and then taking the average of the simulated sample paths. 27There are two exogenous variables in the system: r(the IOR rate paid on reserves) and the trend growth rate (the 12-month growth rate of the potential GDP). Each simulated sample path of (s px rm)fromthe base period tdepends on the projected path from tonward for those exogenous variables. This affects the difference (6.1) because our model is nonlinear. We assume static expectations, in that the projected path from tonward is constant at the value at t. 28The error bands are obtained by drawing parameter vectors from the asymptotic distribution and picking the 84 and 16 percentiles for each horizon (such that the coverage rate is 68%, corresponding to one standard-error bands). For further detail, see Appendix E in the Online Supplementary Material. Quantitative Economics 10 (2019) Exiting from quantitative easing 1087 Figure 4. The policy-rate effect, the base period is March 1995. on the baseline history itself. Therefore, the base period needs to be specified. However, in order to calculate the response profiles of the policy rate cut, the base period has to be June 1995 or before, when the policy rate was above 1percent. We take the base period t to be the earliest month after the structural break, March 1995, when the policy rate, at rt=20%, was comfortably above zero. That the rate cut is 100 bps (i.e., 1percentage point) can be read from the intercept of the profile in the lower-left panel of Figure 4, which shows the response profile of rto r. The response profile of p, shown in the upper-left panel, is not significantly different from zero in that the error band includes the horizontal axis. The same is true for the profile for xin the upper-right panel. Because of the high initial policy rate of 20%,the system rarely switches to Zin the simulations, which explains the almost no response of m, as shown in the lower-right panel of the figure. The QE effect We now turn to the response to supply-determined changes in the excess reserve rate m. To study the effect of changes in the excess reserve rate munder the ELB regime, we set st=Zand rt=rtin both the baseline and the alternative histories. The response to the reserve-supply shock vst is given by Eyt+k|st=Z(p txtrtmt+δm)   (ptxtrtmt)in the alternative history   lagged information 1088 Hayashi and Koeda Quantitative Economics 10 (2019) Figure 5. The QE effect, the base period is February 2004. −Eyt+k|st=Z(p txtrtmt)   (ptxtrtmt)in the baseline history   lagged information y=px r m (6.2) The only difference between the two histories is that the reserve-supply shock vst is higher in the alternative history by δm. This point is made formally in equation (C.23). Figure 5shows the QE effect, that is, the profiles of the response to mfor the base period of February 2004 (the peak QE month) when mt=1849,ofabout64(=exp(1849)) times the required reserves, or about 6% of the GDP. The lower-right panel shows the response profile of mto m, so its intercept at horizon k=0(the base period) is equal to the perturbation δm.Wesetδm=10.29 The perturbation is about 10% of the GDP. Theresponseprofileforxis shown in the upper-right panel of Figure 5.Itsnextperiod response (the response at k=1)isabout046% (the lagged m-coefficient in the output equation of 046 showninTable5multiplied by δm=1). Because of the persistence in the output dynamics exhibited in the estimated reduced form, the response feeds into the next-period response and increases to about 14% in about 10 months. For p, the next-period response (at k=1) is greater, at 055, but the effect tapers off owing to the lack of persistence in monthly inflation. 29The perturbation size is chosen so that its ratio to the estimated standard deviation of the reservesupply shock vst (0.132 in Table 4)isroughlyequaltotheratioof−δr(the size of the interest-shock) to the estimated standard deviation of the Taylor-rate shock vrt (0.113 in Table 3). We have already set δr=−1 (100 bps). Quantitative Economics 10 (2019) Exiting from quantitative easing 1089 Table 6. Months leading up to the exit in July 2006. March April May June July regime Z ZZZP m,log of actual-to-required reserve ratio 151 100 055 046 0 actual reserves relative to GDP (%) 4226161409 π, year-on-year inflation rate (% per year) 01−01000202 x, output gap (%) −07−04−07−04−09 r, the policy rate (% per year) 00000000026 g, trend growth rate (% per year) 0909090909 r∗, desired Taylor rate (% per year) 0504040505 Note: The desired Taylor rate r∗is defined in (4.1). Because both output and inflation increase, the initial regime of Zis more likely to switch to Pearlier under the alternative scenario. This is why the response of rgradually rises from zero, whereas m’s response turns negative. The mean duration of the initial regime Zis indeed shorter with the positive shock to reserves: about 26 months under the baseline, and about nine months under the alternative. Interestingly, for the QE spell from March 2001 to June 2006, the mean duration at the time of QE entry is about 64 months, which is approximately equal to the actual duration of the spell. Thus, our estimated SVAR generates ELB spells comparable in length to those experienced in Japan and the United States, whereas recent structural New Keynesian models with the ELB have difficulty doing so. Gust et al. (2017) found the average duration for a lower bound spell is just over 35quarters for the U.S. economy. Boneva, Braun, and Waki (2016)further discussed the failure of the New Keynesian models in this respect. 6.2 Timing of exit A more interesting analysis can be conducted by allowing the two histories to differ in more than one policy shock. To illustrate this, we examine the exit from the March 2001– July 2006 QE spell. The relevant statistics are shown in Table 6. The last row shows that the desired Taylor rate. Hence, the Taylor rate30 was already positive before the exit. What kept the BOJ from exiting was the low inflation rate. What would the difference have been if the exit had occurred a month earlier? We can answer this question by setting the base period to t=June 2006 (when the regime was st=Z) and then considering the difference31 Eyt+k|st=P(ptxtrt0)−Eyt+k|st=Z(ptxt rtmt)(6.3) 30Recall that the Taylor rate is (1−γr)r∗ t+γrrt−1+vrt. During ELB spells, the net rate is zero: rt−rt=0. During the second ELB spell, rt=rt=0, which means the Taylor rate is proportional to the desired rate plus noise: (1−γr)r∗ t+vrt. 31Under P,rt>rt. We define E(yt+k|st=P(ptxtrt0)) as limr↓rtE(yt+k|st=P(ptxtr0)).Alternatively, we could set rtto some low value (e.g., 025%, which is the policy rate for July 2006) instead of rt; however, the profiles would look very similar. 1090 Hayashi and Koeda Quantitative Economics 10 (2019) Figure 6. The effect of exiting QE in June 2006. Thus, the perturbation occurs for not just one, but two variables: mtand st. Because the regime is different between the baseline and alternative histories, the inflation– output dynamics in the following period will be different as well. The response profiles of the difference (6.3) are shown in Figure 6.Theperturbationstomof δm=046 (the value of min June 2006, see Table 6) can be read from the intercepts in the lower-right panel. Surprisingly, despite the decrease in m, both inflation and output increase. Thus, exiting from QE in June 2006 would have been expansionary. To see why, decompose the (overall) response (6.3) as the difference between two components: (6.3)=Eyt+k|st=P(ptxt rt0)−Eyt+k|st=Z(ptxt rt0)   transitional effect of an exit from Zto P −Eyt+k|st=Z(ptxt rtmt)−Eyt+k|st=Z(ptxt rt0)   the QE effect (6.4) The culprit is the first component, which we call the “transitional effect.” The configuration of the three monetary policy shocks (vrtvπtvst) underlying the difference is shown in equation (C.24). Its response profile (not shown here) exhibits an expansionary effect for inflation and output. The second component is the QE effect, which (as can be surmised from Figure 5)isexpansionaryfort=June 2006. Whether or not the overall response (6.3) is positive depends on the relative strength of the second component, which, in turn, depends on the size of mt.Thethresholdvalue Quantitative Economics 10 (2019) Exiting from quantitative easing 1091 Figure 7. Actual and threshold m, March 2001–June 2006. of mtunder which an exit would be expansionary32 is plotted for all months of the second Zspell in Figure 7, along with the actual value of mt,asapercentageofGDP.The figure shows that an exit would have been expansionary for May 2006 as well, and nearly so for April 2006. However, an exit would have been contractionary for March 2006, or earlier. Turning to the other end of the spell, the March 2001 value of the actual mis less than the threshold m, implying that continuing Pwould have been expansionary. The central bank, if it wishes to stimulate the economy by entering the ELB regime, needs to expand reserves aggressively upon entry. When is the transitional effect expansionary? The positive transitional effect from Zto Pfor t=June 2006 arises because the reduced form is more conducive to inflation and output under P. For both inflation and output, the intercept in the reduced-form equation is higher under Pthan it is under Z.The difference in the intercept is 09(t-value =17) for inflation and 11(t=30) for output. Here, the intercept includes the effect of trend growth. The trend growth rate for June 2006 is 09% (see Table 6). The difference in the intercept for output, for example, can be calculated for June 2006 from Table 5as (−070 +083 ×09)−(107 +001 ×09)= 11. Thus, a higher trend growth (which represents the time-varying real rate) is more conducive to output under P. This positive transitional effect from Zto Pis consistent 32An exit is deemed “expansionary” if the impulse response of output adds up to a positive number, that is,ifthesumof(6.3)overk=1260 is positive for y=x. 1092 Hayashi and Koeda Quantitative Economics 10 (2019) with Lansing’s (2017) finding that the output response to the real rate gap is larger in a deflation equilibrium.33 The transitional effect can be negative with low trend growth, depending on the initial conditions. The average trend growth has been around zero during the 2000s in Japan. To illustrate this point, consider an exit for the following two cases with zero trend growth: πt=pt=2xt=0rt=0(Case 1), and πt=pt=0xt=2rt=0(Case 2). The only difference between the two cases arises from the macroeconomic conditions upon exit; that is, Case 1 has higher inflation, but lower output than Case 2. We find that an exit is contractionary in Case 1, but it is expansionary in Case 2. This is mainly because the policy rate is more sensitive to inflation than output after exit (because the inflation coefficient in the Taylor rule is higher than the output coefficient). As a result, Case 1 leads to policies that are more contractionary than in Case 2 after the exit. In either case, the probability of the economy remaining under Pfor at least 12 months after the exit is less than 10 percent. In other words, the economy cannot enjoy the benefit of conducive inflation and output reduced-form dynamics under Pfor long. 7. Alternative specifications In this section, we examine how the results from our counterfactual analysis are affected by various alterations of the baseline model. We show that (a) failure to control for the equilibrium real interest rate in the Taylor rule results in the price puzzle, (b) the QE effect is no longer statistically significant if the measure of potential GDP is the HP- filtered GDP, (c) allowing for active excess reserve demand and two types of the ELB regime hardly changes our results, and (d) the macroeconomic effects of signaling and portfolio rebalance channels are weak during our sample period. Dropping trend growth The baseline model includes the trend growth rate of potential GDP in the desired Taylor rate (defined in (4.1)), as well as in the inflation and output reduced forms. If trend growth is dropped from the model, the inflation coefficient in the desired Taylor rate rises from 069 (as shown in Table 3)to096 (t-value =36). This can be understood as an omitted-variable bias. Recall from Figure 3the simultaneous fall in inflation, trend growth, and the policy rate in the early to mid-1990s. The effect on the policy rate of the fall in the equilibrium real rate is now picked up by inflation, which is correlated with trend growth. For the reduced form, the break date detected by the sup F-statistic for the output equation, which was March 1995 when trend growth was included, is now May 1993. If the reduced form is estimated without trend growth on the postbreak sample, the lagged r-coefficient in the inflation equation, which was virtually zero in Table 5,risesto039 (t=26). This is because the longer sample period for the reduced form now includes the period of the rapid fall of both the policy rate and inflation. 33For theoretical examples of expansionary policy-induced exits, see Hayashi (2019). Quantitative Economics 10 (2019) Exiting from quantitative easing 1099 where b(·;Ω)is the density of the bivariate normal distribution with mean 0 (2×1)and variance-covariance matrix Ω (2×2). The third term, Prob(st|y1txtxt−1Zt−1) This is the transition probability matrix for {st}. The probabilities depend on (re tπtrt) (which in term can be calculated from (y1txtZt−1),see(A.3)and(A.13)). They are easy to derive from (A.4): st st−1PW S PPrt (1−Prt)q (1−Prt)(1−q) WPrtPπt 1−PrtPπt 0 SPrtPπt 01−PrtPπt Here, Prt ≡Probre t+vrt > rt|re trt=re t−rt σr(A.15) Pπt ≡Prob(πt≥π+vπt|πt)=πt−π σπ(A.16) where (·)is the cdf of N(01). The first term, p(mt|rtsty1txtxt−1Zt−1) mtis given by (A.5)wheremdt and mst are defined in (A.6)and(A.7). The right-hand side variables in those definitions, including max[mdt−10]and mt−1, are functions of (rtsty1txtZt−1). So this term is the Tobit distribution-density function given by hjt ≡1 σj φmt−me jt σj1(mt>0) ×1−me jt σj1(mt=0)  j=dif st=Por W;j=sif st=S(A.17) where 1(·)is the indicator function, φ(·)and (·)are the density and the cdf of N(01). The second term, p(rt|sty1txtxt−1Zt−1) If st=Wor S, then rt=rtwith probability 1, so this term can be set to 1.Ifst=P,there are two cases to consider due to the exit condition. In either case, the second term turns out to be the same, as shown below: •For st−1=P, p(rt|st=Py1txtxt−1Zt−1) =pre t+vrt|re t+vrt > rtre trt 1100 Hayashi and Koeda Quantitative Economics 10 (2019) by (A.4)and(A.5), and since re trtis a function of (y1txtZt−1) =pre t+vrt|re t Probre t+vrt > rt|re trt = 1 σr φvrt σr Probre t+vrt > rt|re trtb/c re t+vrt ∼Nre tσ2 r = 1 σr φrt−re t σr Prt by definition (A.15)ofPrt(A.18) •For st−1=Wor S, p(rt|st=Py1txtxt−1Zt−1) =pre t+vrt|re t+vrt > rtπt≥π+vπtre trtπt by (A.4)and(A.5), and since re trtπtis a function of (y1txtZt−1) =pre t+vrt|re t+vrt > rtre trt(b/c vrt and vπt are independent) = 1 σr φrt−re t σr Prt (as above)(A.19) Putting all pieces together Putting all those pieces together, the components of the likelihood for date t, defined in (A.12), can be written as (with Xthere denoting (xtxt−1)for brevity) st|st−1p(mt|rtsty1tXtZt−1)p(r t|sty1tXtZt−1)Prob(st|y1tXtZt−1)p(y1t|XtZt−1) P|Phdt gt Prt Prt fPt P|Whdt gt Prt PrtPπt fWt P|Shdt gt Prt PrtPπt fSt W|Phdt 1(1−Prt)q fPt W|Whdt 11−PrtPπt fWt S|Phst 1(1−Prt)(1−q) fPt S|Shst 11−PrtPπt fSt Here, fjt ≡by1t−c(j) −A(j)xt−(j)yt−1;Ω(j)for j=PWS gt≡1 σr φrt−re t σrP rt ≡re t−rt σras in (A.15) Pπt ≡πt−π σπas in (A.16) hjt is defined in (A.17)andb(·;Ω)is the density function of the bivariate normal distribution with mean 0 (2×1)and variance-covariance matrix Ω (2×2). Quantitative Economics 10 (2019) Exiting from quantitative easing 1101 Dividing it into pieces Taking the log of both sides of (A.10) while taking into account (A.11)and(A.12)and substituting the entries in the table, we obtain the log likelihood of the sample: L≡log(L)= T  t=1 logp(styt|xtxt−1Zt−1)=LA+L1+L2+LD+Lq where LA= st−1=P log[fPt]+  st−1=W log[fWt]+  st−1=S log[fSt](A.20) L1= st=P log[Prt]+  st|st−1=P|WP|S log[Pπt] + st|st−1=W|PS|P log[1−Prt]+  st|st−1=W|WS|S log[1−PrtPπt](A.21) L2= st=Plog(gt)−log(Prt)+ st=S log[hst](A.22) LD= st=PW log[hdt](A.23) Lq= st|st−1=W|P log[q]+  st|st−1=S|P log[1−q](A.24) The terms in L1+L2can be regrouped into LBand LC,asin L=LA+LB+LC   =L1+L2 +LD+Lq(A.25) where LB= st=P log[gt]+  st|st−1=P|WP|S log[Pπt] + st|st−1=W|PS|P log[1−Prt]+  st|st−1=W|WS|S log[1−PrtPπt](A.26) LC= st=S log[hst](A.27) LALBLC,LD,andLqcan be maximized separately, because Lj(j =ABCD) depends only on θj(j =ABCD) ((θAθBθCθD)was defined in (A.8)above)andLq depends only on q. As a special case, consider simplifying step (ii) of the mapping above by dropping the exit condition “πt≥π+vπt”. This is equivalent to constraining Pπt to be 1,soLB 1102 Hayashi and Koeda Quantitative Economics 10 (2019) becomes LB= st=P log[gt]+  st=WS log[1−Prt](A.28) which is the Tobit log likelihood function. A.3 Parameter estimates For estimation, we need to designate the set of months for each regime. As we argued in Section 3, excess reserves were demand-determined during the first ELB (effective lower bound) spell (March 1999–July 2000) and supply-determined during the second and third ELB spells (March 2001–June 2006 and December 2008–December 2012). Thus: •st=Wif March 1999 ≤t≤July 2000 (17 months), •st=S(i.e., QE) if March 2001 ≤t≤June 2006 or December 2008 ≤t≤December 2012 (113 months), •st=Pfor all other months since January 1988 (170 months). The model’s parameters are listed in (A.8). Of these, •the ML estimate of θAfor st−1=S(on the sample of t’s such that st−1=S; 112 months) is shown in the lower panel of Table 5(Zin the table corresponds to Sor QE), •the ML estimate of θB(on the sample of t’s such that st=PWor S,t≥May 1995; 252 months) in Table 3, •the ML estimate of θC(for st=S; 113 months) in Table 4. The ML estimate of θAfor st−1=Pis in the upper panel of Table 5,butmt−1is excluded as a regressor because in the text mis constrained to be zero under P. With active excess reserve demand, we need to include mt−1, which is occasionally positive even under P, as a regressor. Thus, the remaining parameters to be estimated are: θAfor st−1=P,θA for st−1=W,andθD(for the excess reserve demand equation). θAfor st−1=P: The upper panel of Table A.1 has the reduced-form estimates for the post-break period beginning March 1995. The lagged mcoefficient comes in with a negative sign in both the inflation and output equations, perhaps because positive excess reserves act as a signal of shocks disrupting financial intermediation that would have a contractionary effect in the next period. θAfor st−1=W: The sample, which is composed of t’s for which st−1=W,hasonly17 observations. Because ris constant (at zero), the lagged rcoefficient cannot be identified. We constrain it to be zero. There is not much variation in the trend growth rate, which creates near multicollinearity between trend growth and the constant. We subsume the effect of trend growth in the constant by dropping it from the reduced form. The lower panel of Table A.1 has the reduced-form estimates for W. The positive lagged mcoefficient in the output equation is not consistent with our interpretation, given above for the negative lagged mcoefficients in the upper panel, that a Quantitative Economics 10 (2019) Exiting from quantitative easing 1103 Table A.1. Reduced form, with occasionally positive excess reserve demand. Coefficient of t−1is in Dependent Variable Const. gtpt−1xt−1rt−1mt−1R2 Lagged subsample P P (85 obs.)inflation (pt)004 [01]009 [01]−005 [−04] 013 [14]007 [01]−33 [−06] 005 output (xt)−023 [−07] 033 [08]−003 [−04] 087 [14]−023 [−07]−102 [−30] 080 Lagged subsample W W (17 obs.)inflation (pt)062 [08]−029 [−11] 012 [06]−53 [−22] 031 output (xt)−20 [−26] 026 [09]049 [25]64 [26]063 Note: Estimation by OLS. t-values in brackets. The “lagged subsample P”of85 observations refers to t’s for which t≥March 1995 and t−1is not in the three ELB spells. The “lagged subsample W”of17 observations refers to t’s for which t−1is in the first ELB spell (March 1999–July 2000). pis the monthly inflation rate in percent per year, xis the output gap in percent, ris the policy rate in percent per year, mis the excess reserve rate (defined as the log of the ratio of actual to required reserves), and g is the trend growth rate (the 12-month growth rate in percent of potential output). For lagged subsample W, the trend growth rate gtis excluded to avoid near-multicollinearity with the constant, and rt−1is excluded because its value is 0for all months. positive excess reserve is a signal of a disintermediation shock. This positive coefficient will not materially affect our counterfactual analysis because regime Wdoes not occur very often and also because excess reserves are positive only rarely under W. θD:The specification of mdt isgivenin(A.6). The equation is to be estimated on those months for which mtis demand-determined, that is, on the sample of t’s such that stis either P(170 months) or W(17 months). So the sample size is 187. The lagged term in (A.6) is not the lagged dependent variable mdt−1but its censored value max[mdt−10]. If the lagged term were the former, then the likelihood function would be very difficult to evaluate; see Lee (1999). When the lagged regime is not S,the lagged term equals mt−1because excess reserves are demand-determined (see (A.5)). When the lagged regime is S, which in our data occurs only for t=July 2006 when the regime changed from Sto P, the lagged term is unobservable because excess reserves were supply-determined in t−1=June 2006. We set the lagged term to zero, which amounts to assuming that the excess reserve demand in June 2006 was nonnegative. The estimation method is Tobit because of the censoring of max[mdt0](recall that mt=max[mdt0]when st=Por W;see(A.5)). We define the limit observations as the months for which m<05%.Thereare144 such months in the sample of 187 months. Recall that we have set mt=0for months between the second and the third ELB spells (except the Lehman crisis months of September to November 2008), on the ground that banks postponed reentry to the interbank market and held on to excess reserves. So those months, indicated by the thin bars in Figure 1,arelimitobservations. 1104 Hayashi and Koeda Quantitative Economics 10 (2019) The estimated excess reserve demand equation is (t-values in brackets) mdt =0003 [01]+0008 [04]πt−0013 [−21]xt−014 [−26](rt−rt)+059 [41]max[mdt−10] estimated standard deviation of the error =0055 (s.e. =00061) sample size =187number of limit observations =144(A.29) The output coefficient is negative, probably because commercial banks desire more excess reserves in proportion to the severity of recessions. References Andrews, D. W. K. 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