WORKING PAPER SERIES 3. Michal Franta The Likelihood of Effective Lower Bound Events

WORKING PAPER SERIES 3 Mihal Franta The Likelihood of Effetive Lower Bound Events

WORKING PAPER SERIES The Likelihood of Effetive Lower Bound Events Mihal Franta 3/2018

CNB WORKING PAPER SERIES The Working Paper Series of the Czeh National Bank (CNB) is intended to disseminate the results of the CNB s researh projets as well as the other researh ativities of both the staff of the CNB and ollaborating outside ontributors, inluding invited speakers. The Series aims to present original researh ontributions relevant to entral banks. It is refereed internationally. The referee proess is managed by the CNB Researh Division. The working papers are irulated to stimulate disussion. The views expressed are those of the authors and do not neessarily reflet the offiial views of the CNB. Distributed by the Czeh National Bank. Available at http://www.nb.z. Reviewed by: Taisuke Nakata (Federal Reserve Board) Tomáš Adam (Czeh National Bank) Projet Coordinator: Volha Audzei Czeh National Bank, May 2018 Mihal Franta

The Likelihood of Effetive Lower Bound Events Mihal Franta Abstrat This paper provides estimates of the probability of an eonomy hitting its effetive lower bound (ELB) on the nominal interest rate and of the expeted duration of suh an event for eight advaned eonomies. To that end, a mean-adjusted panel vetor autoregression with stati interdependenies and the possibility of regime hange is estimated. The simulation proedure produes ELB risk estimates for both the short term, where the urrent phase of the business yle plays an important role, and the medium term, where the ourrene of an ELB situation is determined mainly by the equilibrium values of maroeonomi variables. The paper also disusses the ELB event probability estimates with respet to previous approahes used in the literature. Abstrakt Článek představuje odhady pravděpodobnosti, že nominální úroková míra v ekonomie dosáhne své efektivní dolní hranie, a odhady očekávané doby trvání takové události pro osm rozvinutýh ekonomik. K tomuto účelu je odhadnuta panelová vektorová autoregrese se statikou provázaností, upravená o ustálený stav a s možností režimové změny. Simulační proedura poskytuje odhady rizika efektivní dolní hranie v krátkém horizontu, ve kterém je toto riziko dáno především aktuální fází hospodářského yklu, a ve střednědobém horizontu, kde hraje pro odhad rizika hlavní roli rovnovážná hodnota makroekonomikýh proměnnýh. Článek také diskutuje odhady pravděpodobnosti dosažení efektivní dolní hranie nominálníh sazeb s ohledem na přístupy použité v předhozí literatuře. JEL Codes: C11, E37, E52. Keywords: Effetive lower bound, ELB risk, mean adjustment, panel VAR, regime hange. Mihal Franta, Czeh National Bank, e-mail: mihal.franta@nb.z. I would like to thank Tomáš Adam, Leonardo Gambaorta, Elmar Mertens, Jouhi Nakajima, an anonymous referee and seminar partiipants at the Bank for International Settlements and the Czeh National Bank for valuable omments. I ompleted this projet while visiting the Bank for International Settlements under the Central Bank Researh Fellowship program. The opinions expressed in this paper are those of the author and do not neessarily reflet those of the Czeh National Bank or of the Bank for International Settlements.

2 Mihal Franta Nontehnial Summary The paper provides estimates of the ELB risk, i.e., the probability of an eonomy hitting its effetive lower bound on nominal interest rates. In addition, the expeted duration of suh an event is estimated. From the perspetive of the poliy maker, the ELB risk suggests how aggressive monetary poliy ondut should be. For example, the literature suggests that after a prolonged period of reession with the nominal interest rate stuk at the lower bound, the speed of poliy rate normalization should be lower if the ELB risk is high. The fat that many advaned ountries have reently been through an episode of interest rates being at their lower bound allows us to take an empirial approah. This ontrasts with the majority of the previous studies on the same topi drawing on various alibrated models that in priniple did not need ELB periods present in the data to make onlusions. Even so, data sets that inlude ELB events are rare, so a panel data tehnique is employed to use data from reent ELB events aross ountries within a single model. Furthermore, the model is rearranged so that it treats the steady states of the variables expliitly, allowing us to disuss the effet of alibration on the resulting ELB estimates. Finally, the model allows for regime hange, beause monetary poliy ondut hanged during the Great Reession. Results are provided for eight advaned ountries. They show how the ELB risk stems from the urrent phase of the business yle in the short run and how the estimated values of the interest rate steady state onstitute the ELB risk in the medium run. Zero ELB risk in the short run is thus estimated for Canada, Norway, and the United States. On the other hand, high risk within a horizon of one year is found for the euro area, Japan, and Sweden. In the medium run, the probabilities of an ELB event range from 0.02 (Canada) to 0.17 (Japan). For the United States, the estimated ELB risk is ompared with the existing literature. It turns out that the ELB risks presented in this paper are lose to the estimates provided by estimated non-linear DSGE models and attain values lose to 0.05 in the medium run. Suh estimates are onsiderably lower than those based on alibrated models and assumed steady states, whih range between 0.14 and 0.32. Additional analyses show that estimation of the steady state and aounting for unertainty in the steady state are one of the soures of the differene.

The Likelihood of Effetive Lower Bound Events 3 1. Introdution The reent experiene of a prolonged period of extraordinarily low monetary poliy rates in advaned ountries has rekindled interest in maroeonomi issues relating to the effetive lower bound (ELB) on the nominal interest rate. Before the Great Finanial Crisis (GFC), the only relevant example of an ELB event was Japan, whih, however, was viewed as a peuliar ase. It was not generally believed that advaned ountries ould experiene long and reurrent periods of being stuk at the ELB. This opinion has now hanged, and novel researh relating to the ELB has been published. There are two important poliy questions underlying this researh work: what is the probability of an ELB event at a given point in time, and how often an an ELB event be expeted to our in general? The usual approah to answering the latter involves onduting model-based stohasti dynami simulations and estimating the stationary distribution of the short-term interest rate, with the area below a given lower bound defining the ELB risk. ELB risk estimates of this type are usually based on alibrated linear models and are highly sensitive to the equilibrium values of maroeonomi variables assumed (see, for example, Reifshneider and Williams, 2000, Coenen, 2003, and Kiley and Roberts, 2017). 1 When the examination of the likelihood of an ELB event is related to a speifi point in time, more data-driven approahes are employed. Statistial models are used to estimate a distribution foreast for the interest rate. In ontrast to the other approah, this allows unertainty in model parameters, latent variables and measurement errors to be taken into aount, and enables some simple form of non-linearity (Chung et al., 2012). Nakata (2017a) employs survey data on maroeonomi projetions to bring the standard stohasti simulations to atual data. However, data-driven approahes are not suitable for estimating ELB risk in the medium and long term, beause they often do not possess well-defined unonditional moments. 2 The aim of this paper is to draw on the above-mentioned approahes and to provide estimates of the probability and expeted duration of ELB events. To that end, the paper ombines stohasti simulations around equilibrium values and distribution foreasting refleting urrent observed data. The results are therefore relevant to both the short term, where the ELB likelihood is driven mainly by the urrent phase of the business yle, and the medium term, where the ELB risk is determined mainly by equilibrium values. Dealing with both time sales in one modeling framework makes the estimates for the short and medium term onsistent with eah other. Furthermore, the approah addresses some of the ontentious issues of the previous approahes and should therefore deliver more aurate ELB likelihood estimates. 1 A reent example of an estimated dynami stohasti general equilibrium (DSGE) model used to estimate the ELB risk is Gust et al. (2017). 2 Probit-type models ould, in priniple, be employed to estimate the ELB risk. However, different models have to be estimated for different horizons of interest. Moreover, duration of the ELB spell ould not be examined within these types of model. 3

4 Mihal Franta I adopt an empirial strategy and base ELB likelihood estimates on reent data from advaned ountries, exploiting the fat that many of them have reently experiened, or are still experiening, nominal interest rates at their ELB. As a onsequene, the ELB risk estimates have to rely less on alibrated parameters and an be based more on estimated quantities. The framework thus aounts for parameter unertainty, whih has been found to be important for realistially assessing ELB likelihood (Chung et al., 2012). On the other hand, the time span of the maro data apturing ELB events is still short, suggesting that it is appropriate to exploit the panel nature of that data. Furthermore, the data-driven approah depends less heavily on assumptions about the equilibrium values of maroeonomi variables than do studies based on stohasti simulations. The assumptions about the equilibrium values enter the estimation proedure in the form of priors and are onfronted with the observed data during the Bayesian estimation proess. Finally, as an ELB situation implies a possibility of regime hange, the modeling framework allows for hange in the shok transmission mehanism and shok volatilities. To quantitatively assess the ourrene of ELB events, I employ the mean-adjusted panel vetor autoregression tehnique, whih allows for stati interdependenies and threshold behavior. The model is estimated on data for eight advaned eonomies over the period 1999Q1 2016Q4 and provides estimates of ELB risk in both the short and the medium term. In addition, the modelling framework allows to analyze the impat of various assumptions employed by the tehniques used to estimate ELB risk in previous studies. It turns out that, in the short term, the ELB risk is urrently the highest for Japan and Sweden and the lowest for Canada, Norway and the US. In the medium term, the probability of an ELB situation ranges from 0.01 for Canada to 0.16 for Japan. A supplementary analysis suggests that the alibration of the steady-state values of maroeonomi variables an lead to ELB risk underestimation due to the fat that the alibrated value of the equilibrium interest rate is high and is assumed to be known with ertainty. It is also shown how the empirial approah gains from the panel nature of data by partially pooling ountry-speifi information and by improving the effiieny of estimates. From the poliy maker s perspetive, the estimates should be taken into aount in the ondut of monetary poliy. Higher ELB risk during a reession alls for more aggressive easing of monetary poliy, whereas higher ELB risk during a boom justifies slower normalization of rates (see, for example, Williams, 2014, and referenes therein). Similarly, if there is agreement on the optimal size of the entral bank balane sheet, the speed of balane sheet normalization from urrent levels should reflet the probability of return to the ELB and to unonventional measures that would inrease the size of the sheet again. The struture of the paper is as follows. Setion 2 disusses reent monetary poliy rates in advaned ountries. Setion 3 presents the model. Setion 4 explains the estimation approah and desribes the data. Setion 5 presents the results and onsiders the role of alibration and regime hange in ELB risk estimation. Setion 6 onludes.

The Likelihood of Effetive Lower Bound Events 5 2. The ELB in Advaned Countries Figure 1 shows monetary poliy rates in eight advaned eonomies sine 1999. In general, two yles an be observed, one peaking around the year 2000 and the seond peaking in 2007 just before the GFC hit the world eonomy. The observed monetary poliy rate profiles also suggest a downward trend in the equilibrium interest rate. The short time span of the observations, however, makes any onlusion about the trend unertain. Figure 1: Monetary Poliy Rates in Advaned Countries In the aftermath of the 2008 risis, advaned eonomies approahed or even rossed the zero bound with their interest rates. One a ountry was not able or willing to ease monetary poliy further using the poliy rate, unonventional measures were employed. The United States, the United Kingdom, and the euro area used large-sale asset purhases aimed at lowering long-term interest rates. Japan initially adopted the same approah to quantitative easing, then followed up with additional quantitative and qualitative monetary easing (QQE) in 2013. It also introdued a negative monetary poliy rate in mid-2016. Negative poliy rates have been seen in Switzerland and Sweden reently as 5

6 Mihal Franta well. Furthermore, Switzerland imposed an exhange rate floor to prevent urreny appreiation between 2011 and 2015. Figure 1 suggests some degree of homogeneity and lose interonnetedness aross ountries. On the other hand, signifiant heterogeneity is present in the data, espeially after the start of the GFC. Monetary poliy rules presumably hanged when eonomies hit the ELB, and the hanges differ aross ountries. The shok propagation mehanism and shok volatilities might also be different aross ountries after the GFC. 3. Model The modeling framework is built on the panel vetor autoregression (VAR) struture, whih allows for ross-setional heterogeneity and stati interdependenies. Cross-setional heterogeneity allows for differenes in the dynamis of maroeonomi variables aross ountries. This option is espeially important for the period of unonventional monetary poliy, beause ountries employed different strategies for easing monetary poliy further. Stati interdependenies allow for orrelation of redued-form shoks aross ountries. The GFC spread aross the advaned eonomies primarily through the finanial markets within one quarter, so aounting for stati interdependenies is ruial, probably more so than aounting for dynami ones. 3 Ignoring stati interdependenies, if present, would lead to less effiient estimates and onsequently to less aurate ELB risk estimates. Note also that stati interdependene is the only interonnetion between the ountries in the model. Without that, the model would be effetively a set of single-ountry models. The ase of single-ountry VARs is disussed in Subsetion 5.3. The model is formulated in mean-adjusted form to deal diretly with steady-state maroeonomi variables. Mean adjustment for VAR models is introdued in Villani (2009) and allows the steady state of a maroeonomi variable to be treated as a single parameter. As suh, a prior on the steady state an be formulated and the parameter diretly estimated. The posterior of the steady state then defines the long-run dynamis of the model. Expliit treatment of the steady state leads to welldefined ELB risk in the medium and long term. For N ountries and for 1,,, the model is formulated as follows:,,, 0 0 0 0 0 0 0,,, 0 0, 0 0, 0 0 0 (1), 3 Another reason for not implementing dynami interdependenies in our model is that allowing for them greatly inreases the number of estimated parameters.

The Likelihood of Effetive Lower Bound Events 7,,,. The n 1 vetor,,,,, denotes the vetor of endogenous variables for ountry and l time t. The oeffiient matries A relating to the l-th lag of the vetor of endogenous variables for ountry are of dimension n n. The vetor of endogenous variables ontains real GDP growth, CPI inflation, the short-term interest rate, and the spread between the ten-year government bond yield and the poliy rate. 4 The set of endogenous variables is usually hosen aording to the shoks that are to be identified. We do not perform strutural shok identifiation in our exerise. The main aim of our hoie of variables is to apture both onventional and unonventional monetary poliy rules, i.e., the interest rate rule and the quantitative reation funtion refleted by the spread. The question is whether this minimal set an also represent the other unonventional monetary poliy measures ontained in our data set. Obviously, the vetor of endogenous variables an deal with negative interest rates. There ould be a problem with the Swiss exhange rate floor, as the exhange rate is not inluded in the vetor of endogenous variables. The effet of this type of poliy is aptured to the extent to whih the floor is refleted by long-term government bond yields. The m 1 vetor,,,,, in (1) ontains exogenous variables, whih are the same aross ountries. In our appliation, the vetor of exogenous variables omprises onstant terms only. The oeffiient matries relating to exogenous variable F are of dimension n m. The ountryspeifi matries F inlude the steady states of the endogenous variables, beause it follows from (1) that if the proess for y, t is stationary then,,. (2), Finally, the vetor of error terms,,,, in (1) is distributed independently and normally, with zero mean and ovariane matrix of dimension Nn Nn. The error ovariane matrix is generally non-diagonal, allowing for residual orrelation between ountries. We assume two lags in the benhmark speifiation. Let y denote a NnT 1 data vetor for all ountries: 4 The same vetor of endogenous variables is employed in Baumeister and Benati (2013), who examine the effet of unonventional monetary poliy based on large-sale asset purhases. 7

8 Mihal Franta where the nt 1 vetor ountry and the operator, y is defined as ve Let X denote an where Xi In Xi, with i of Y i, and Z I Z, where Finally, define y Y. Y is a T n matrix of observations for ve denotes the olumnwise transformation of the matrix into a vetor. 2 NnT Nn p data matrix Nn 0 0 0, 0 0 0 X being a np T ountry-speifi data matrix omprising up to p lags.,, and, where, and for 1,..., N. The model (1) an then be written as follows: y X Z ~ N 0,. (4), NnT 1 As disussed in setion 2, the data set presumably inludes hanges in poliy rules when the ELB was hit and unonventional monetary poliies were introdued. Moreover, as suggested by Clark (2011), allowing for hanges in shok volatilities results in more aurate density foreasts, even though the time perspetive of Clark (2011) is longer than in our ase. Translating this into the ontext of our exerise, a more aurate density foreast means a more aurate estimate of ELB risk. Model (4) is therefore extended in suh a way that it allows for two regimes in the form of possible threshold

The Likelihood of Effetive Lower Bound Events 9 behavior driven by the endogenous variables. Due to the short time series available for the estimation, only one regime hange is onsidered. Whether the hange is driven by a hange in dynami oeffiients (e.g., in poliy rules), by a hange in shok volatility, or by both is deided on the basis of model likelihood omparison within the estimation proedure. Combining regime hange with the mean-adjustment proedure fores us to take a stand on the issue of whether the mean-adjustment proedure should be applied to both regimes or to one regime only. Strong prior information exists for the normal times regime with standard monetary poliy ondut, so mean adjustment is imposed in that regime only. For the other regime, a simple onstant term is inluded in the data matries. 5 Allowing for two regimes with mean adjustment applied in one regime only, the model takes the form: y y 1 1 1 1 X 2 2 2 (2) 2 X Z y y TR t TR t r, (5) r 1 1 where ~ N 0,, 2 2 ~ N 0,, and the matries 1 and 2 are in general nondiagonal. Data matries and vetors with supersripts denoting the regime refer to the subsamples of 1 X and y relating to the relevant regime. Note that the data matrix X inludes the onstant term. The swith between the two regimes is driven aording to the threshold variable, y TR t, and the threshold r. The threshold variable is a funtion of the endogenous variables and is defined as the ombination of the short-term interest rate and the spread (Figure 2). More preisely, the threshold variable is the average of the average short-term interest rate aross ountries and the average spread aross ountries, both lagged by one quarter. The definition of the threshold variable reflets the belief that a regime hange is expeted when the world eonomy is around the ELB and unonventional poliies aimed at lowering the spread are employed. The threshold parameter r is estimated. 5 To simplify the notation, we assume that mean adjustment is applied to regime 2. The estimation proedure shows that this is indeed the ase the implied steady state of the threshold variable lies above the estimated threshold. 9

10 Mihal Franta Figure 2: The Threshold Variable, the Average Short-term Interest Rate, and the Average Spread aross Countries 4. Estimation The estimation approah is Bayesian and ombines estimation of a mean-adjusted VAR (Villani, 2009), a hierarhial linear model for VAR (Jaroinski, 2010), and Bayesian estimation of a threshold VAR (Chen and Lee, 1995, and Koop and Potter, 2003). To estimate the model, the ountry-speifi estimates of some parameters exploit information from all ountries by means of an exhangeable prior. We assume that the oeffiients at the lagged values of the endogenous variables staked in (r ) vetor are distributed normally around a regime-speifi ommon mean b : (r) b b, 1,,, 1,2 (6) b. The spread of the ountry-speifi vetors of the dynami parameters ( r ), around the ommon mean is driven by the overall tightness parameter defining the variane Σ (see formula A2). b,( r) where ~ N0, The exhangeable prior is not used for the parameters apturing the steady state, F. Strong prior information on the steady states of the endogenous variables is available, so there is no need to pool information aross ountries through the ommon mean for those parameters. The estimation proedure simulates the posterior distributions based on likelihood, priors, and r onditional priors, respetively. The vetor of model parameters ontains dynami parameters, (r ) b, r hyperparameters b and, error varianes, the steady states inluded in the vetor, and the threshold r that determines the regimes. Using onditionally onjugate priors yields onditional posterior densities that are easy to draw from within the Gibbs sampler. The onditional posterior of

The Likelihood of Effetive Lower Bound Events 11 the threshold annot be expressed by a standard density funtion and a Metropolis step is used to take a draw of the parameter. The speifiation of the prior distributions and a desription of the sampler an be found in Appendix A. The likelihood of an ELB event is estimated during the estimation of the model. For a given draw of model parameters, iterated foreasts for up to 48 quarters are omputed using random draws of shoks from a given distribution. Note that regime swithing is allowed for during the simulation, beause the threshold variable is a funtion of the endogenous variables. Then, for a given period, the proportion of foreasts that are below or at the ELB is omputed to estimate the probability of an ELB event. The medium-term ELB risk is omputed as the average ELB risk for last eight quarters, i.e., the 41 st to 48 th quarters. In addition to the estimate of the ELB risk, the expeted duration of the ELB event is omputed as the average number of quarters for whih the simulated foreast remains at the ELB. During the foreasting, the ELB onstraint is imposed in a straightforward manner, beause the model is bakward looking. The value of the ELB is imposed whenever the one-period-ahead value of the interest rate falls below the ELB. This approah an be interpreted as passive onventional monetary poliy. The monetary authority does not lower the interest rate when the estimated interest rate rule suggests doing so. Even with the interest rate rule swithed off, unonventional monetary poliy still affets the eonomy aording to the estimated spread equation. The numerial value of the ELB is set differently for eah ountry. First of all, it is important to stress that we impose the ELB on the short-term nominal interest rate, beause the poliy rate is not diretly inluded in the vetor of endogenous variables. For those ountries whih have not experiened negative poliy rates, the ELB is set to zero (Norway, the UK, and the US). In addition, the hoie for the US is driven by omparability with other studies. For ountries with negative poliy rates, the ELB is set to the lowest value of the short-term interest rate in the sample. Finally, following Witmer and Yang (2016), the ELB for Canada is set to -0.5. 6 The ELB values are reported in Table 1. Table 1: Imposed Effetive Lower Bounds ELB Canada -0.5 Euro area -0.3125 Japan -0.0376 Norway 0 Sweden -0.78 Switzerland -0.84 United Kingdom 0 United States 0 Note: The ELB imposed on the short-term interest rate. 6 This value is derived from the ost of storing ash. The estimate is originally for the poliy rate; we onsider it as the ELB for the short-term interest rate. 11

12 Mihal Franta The ELB values do not enter the estimation proedure and thus do not affet the estimation of the model parameters. They enter the simulation of the ELB risk and the expeted duration of the ELB spell only. As a onsequene, a different ELB for a ountry only affets the ountry-speifi results (the exeption is regime hange timing, whih relates to all ountries). Different reasoning for ELB values aross ountries therefore does not represent an obstale. 4.1 Data Real GDP and CPI are seasonally adjusted and enter the vetor of endogenous variables as the first differene of their logs. The two series are downloaded from the BIS database. The spread variable is made up of the 10-year bond yield and the monetary poliy rate. The bond yields are downloaded from the OECD MEI database. Euro area long-term bond yields are downloaded from the ECB FM database. They are onstruted from AAA-rated bond yields. 7 Yields for the UK are downloaded from BoE Statistis and yields for the US from the FRED database. Monetary poliy rates are obtained from the IMF IFS. The Swiss National Bank targets the three-month Swiss fran Libor, so the Libor is used as the poliy rate. The poliy rate for the euro area is the interest rate on main refinaning operations. Finally, short-term interest rates are 3-month interbank rates obtained from the OECD MEI database. The data are quarterly and the data set overs the period 1999Q1 2016Q4. The hoie of quarterly data, as opposed to monthly data, is driven by the fat that the panel VAR speifiation with stati and without dynami interdependenies is more reasonable in a setting with quarterly data. 5. Results The results are based on 100,000 iterations, with 5,000 iterations as a burn-in period. Every tenth draw is used for inferene to deal with autoorrelation of draws. Convergene diagnostis of the sampler and additional estimation results are presented in Appendix B. The posterior mean of the threshold r is 1.27, implying that regime 1 overs the period 2012Q1 2016Q4 and regime 2 the period 1999Q2 2011Q4. Regime 1 thus overs the period when advaned eonomies hit or approahed their ELBs, launhed their unonventional monetary poliy measures, and gradually redued the spread between long- and short-term interest rates (Figure 3). 7 An alternative is to obtain yields from OIS rates for details see ECB (2014).

The Likelihood of Effetive Lower Bound Events 13 Figure 3: The Interest Rate and Spreads in Regime 1 (White) and Regime 2 (Grey) The estimated ELB risk is presented in Table 2. The probability of the short-term interest rate being at the ELB in 2017Q1 is driven to a great extent by whether the relevant eonomy was stuk at the ELB in the last quarter used for the estimation, i.e., in 2016Q4. So, for the euro area, Japan, and Sweden, whih are at their assumed ELBs, the ELB risk is in the range of 0.37 0.53. For the other ountries, the ELB risk is lower. For Canada, Norway, and the US, it is virtually zero for 2017Q1. Table 2: ELB Risk 2017Q1 2017Q2 2017Q3 2017Q4 2018Q1 2018Q2 2018Q3 2018Q4 MR Canada (-0.50) 0.00 0.00 0.00 0.02 0.06 0.06 0.09 0.08 0.01 Euro area (-0.3125) 0.40 0.55 0.62 0.35 0.23 0.06 0.06 0.08 0.03 Japan (-0.0376) 0.37 0.49 0.48 0.49 0.30 0.50 0.65 0.53 0.16 Norway (0) 0.00 0.00 0.00 0.07 0.21 0.15 0.17 0.11 0.05 Sweden (-0.78) 0.53 0.37 0.43 0.52 0.47 0.13 0.17 0.20 0.03 Switzerland (-0.84) 0.23 0.52 0.46 0.29 0.31 0.32 0.30 0.21 0.04 United Kingdom (0) 0.02 0.06 0.08 0.16 0.14 0.14 0.09 0.08 0.02 United States (0) 0.00 0.01 0.10 0.24 0.26 0.22 0.19 0.16 0.06 Notes: The imposed ELB is indiated in parenthesis, MR denotes medium-term ELB risk. The ELB risk profile during the first eight forthoming quarters (2017Q1 2018Q4) reflets the urrent phase of, and the outlook for, the business yle. For the euro area, the ELB risk inreases in the first three quarters and then delines due to a delay in its eonomi reovery with respet to other 13

14 Mihal Franta advaned eonomies. 8 For Canada, Norway, and the US, the ELB risk is very low for the first three quarters and rises after that, beause these ountries experiene an expansionary phase of the business yle in the first quarters of 2017. The omparison aross ountries is also influened by the different ELBs. The lower the ELB, the lower the probability of the eonomy being at the ELB, all other things being equal. The medium-term ELB risk is determined mainly by the posterior of the interest rate steady state, the assumed ELB, and the estimated average size of the shoks to maroeonomi variables. It ranges from 0.01 (Canada) to 0.16 (Japan) see the MR olumn in Table 2. The numbers an be interpreted as meaning the perentage of the time (quarters) the ELB is binding. For Canada, the eonomy is at the ELB 1 perent of the time, i.e., for one quarter in 25 years. For Japan, on the other hand, the ELB is estimated to be binding 16 perent of the time, i.e., for four years every 25 years. By ombining the short-run outlook based on the business yle and medium-run simulations drawing on steady-state values, non-monotoniity of ELB risk measures an arise, as doumented in Table 2. The presented ELB risk estimates an be ompared with the results provided by the reent literature. Assuming a steady-state nominal interest rate of 3 perent and imposing an ELB of zero, Kiley and Roberts (2017) estimate the frequeny of ELB events for the US to be 17.4 perent based on the DSGE model of Lindé et al. (2016) and 31.7 perent based on the FRB/US model. The estimate based on the DSGE model is lose to the estimates in Hills et al. (2016) and Nakata (2017b), whih draw on alibrated DSGE models and provide an estimated probability of around 14 perent. If we assume the same ELB as in Kiley and Roberts (2017) and estimate the steady-state nominal interest rate at 2.79 perent, the ELB risk from our model is 5.71, whih is muh lower than the figures obtained by Kiley and Roberts (2017). Lower ELB risk estimates an also be found in the literature based on estimated nonlinear DSGE models. Gust et al. (2017) found for the US that the average probability of hitting the ZLB is 4 perent, while Rihter and Throkmorton (2016) estimated the probability at around 5 perent. Part of the differene an be explained by the fat that our ELB risk estimates inlude the effets of unonventional monetary poliy, whih lowers the probability of hitting/staying at the ELB. Whenever the US eonomy is simulated to be in regime 1, the estimated impat of unonventional monetary poliy is present in the simulation. Unonventional measures are not inluded in Kiley and Roberts (2017). Another reason suggested by the plain omparison is that nonlinear models tend to deliver lower ELB risk the effet of nonlinearity on ELB risk estimates is disussed in Subsetion 5.2. Nakata (2017a) ombines survey-based maroeonomi projetions and stohasti simulations of the FRB/US model and defines the ELB risk as the probability that the federal funds rate will be onstrained by the ELB for at least one quarter in the next three years. For 2016Q4, the ELB risk is estimated to exeed 50 perent for all three survey-based projetions onsidered (Survey of Primary 8 Going bak to the motivation mentioned in the Introdution, the ELB risk is estimated to derease substantially by the beginning of 2018 in the euro area. Suh timing justifies the redution of the monthly pae of asset purhases planned for 2018.

The Likelihood of Effetive Lower Bound Events 15 Dealers, Survey of Professional Foreasters, and Summary of Eonomi Projetions). Exluding the period of elevated maroeonomi volatility in the 1970s and 1980s, whih redues the estimated size of shoks, the ELB risk lies between 40 and 50 perent. Assuming the ELB onsistent with Nakata (2017a), Nakata s ELB risk for the US estimated within our framework is 57 perent. Nakata s ELB risk measures for other ountries are reported in Table 3. Table 3: Probability of an ELB Event in Next Three Years CAN EA JPN NOR SWE SUI UK US Nakata (2017a) US 0.20 0.96 0.97 0.40 0.93 0.77 0.43 0.57 0.40-0.50 The estimated ELB risk an be omplemented with estimates of the expeted duration of the ELB event, i.e., the average number of quarters the eonomy is stuk at the ELB given that it hits the ELB in a given quarter or remains at the ELB from the previous quarter (Table 4). The medium-term duration (olumn MR) is omputed starting with the 11 th quarter to filter out the initial onditions and the business yle phase and ending with the 43 rd quarter to filter out the effet of the maximum time period of 48 quarters used in the simulation proedure. Note that the expeted duration annot be omputed if no simulated foreast hits the ELB (as is the ase for Canada and Norway in 2017Q1). Table 4: Expeted Duration p 2017Q1 2017Q2 2017Q3 2017Q4 2018Q1 2018Q2 2018Q3 2018Q4 MR Canada - 1.00 3.37 3.16 2.11 2.64 2.10 2.13 2.17 Euro area 2.16 2.01 1.81 1.65 1.20 3.78 4.45 4.25 3.04 Japan 2.39 2.78 2.73 1.86 2.65 2.56 2.50 2.55 2.78 Norway - 2.13 2.92 2.74 2.10 2.13 1.93 2.59 2.94 Sweden 2.13 3.00 2.45 1.62 1.46 2.09 2.37 2.31 2.15 Switzerland 3.09 3.13 2.31 2.35 3.08 2.75 2.26 3.12 2.66 United Kingdom 1.77 1.85 2.23 1.79 1.85 1.80 2.29 2.28 2.51 United States 1.00 3.18 3.75 2.82 2.43 2.35 2.30 2.19 2.35 Note: MR denotes medium-term expeted duration defined as the mean duration over the 11 th to the 43 rd quarter. The highest medium-term expeted duration (three quarters) is obtained for the euro area, while the lowest expeted durations are observed for Canada, Sweden, and the US. For the US, the medium-term expeted duration is estimated to be 2.35 quarters. Hills et al. (2016) and Nakata (2017b) estimated the expeted duration at 9 quarters. A muh shorter duration of the ELB spell is estimated by Gust et al. (2017) and Rihter and Throkmorton (2016), who give an average duration of around 3 quarters. Similarly to the estimates of the ELB risk, the estimate of 2.35 15

16 Mihal Franta quarters for the US is loser to the literature dealing with estimated nonlinear DSGE models than alibrated linearized DSGE models. 9 The ELB risk and duration estimates involve unonventional monetary poliy, as it is refleted in the data used for the estimation. So, the estimate for the US assumes a relatively strong unonventional easing of monetary poliy at the ELB, while the estimate for Canada draws on the estimated onventional monetary poliy only, beause the Bank of Canada has never employed unonventional measures in the post-2008 period. The two ountries ELB risk should be ompared with this onsideration in mind. If no unonventional monetary poliy had been onduted in the US after 2008, the ELB risk would be higher and the differene with respet to the ELB risk in Canada would be more profound. 5.1 The Role of Mean Adjustment The extensive literature dealing with equilibrium values of maroeonomi variables in advaned ountries allows us to formulate informative priors on the steady-state parameters. Regarding the informativeness of the priors, two extreme ases an be distinguished. First, imposing a very tight prior is similar to the situation where the steady-state values are alibrated. Seond, imposing a very loose prior resembles the ase where no mean adjustment is onduted and allows us to disuss the role of mean adjustment in the estimation of ELB risk. Examining the two extreme ases an shed some light on, respetively, the effet of alibration and the effet of not imposing any equilibrium values on the ELB risk estimates in the previous literature. Table 5: ELB Risk in the Medium Term for Different Priors on the Steady-state Parameters Benhmark Tight Loose Canada 0.01 0.01 0.03 Euro area 0.03 0.02 0.06 Japan 0.16 0.06 0.17 Norway 0.05 0.03 0.05 Sweden 0.03 0.03 0.06 Switzerland 0.04 0.04 0.07 United Kingdom 0.02 0.02 0.05 United States 0.06 0.04 0.10 Note: The tight prior is given by a 95 perent onfidene band of width 0.1 and the loose prior by a 95 perent onfidene band of width 10. 9 When omparing preditions of ELB spell duration, estimates from alibrated linearized DSGE models seem to be loser to surveys than estimates based on estimated nonlinear models. For example, Survey of Professional Foreasters and Primary Dealers Surveys suggest for the US an expeted duration starting at 5 quarters at the beginning of 2011, inreasing lose to 10 quarters in 2012 and 2013, and then steadily dereasing toward zero at the end of 2015. The expeted duration based on panel VAR does not exeed 2 quarters throughout the ZLB period in the US. In addition to the simplisti dynamis in the panel VAR model, forward guidane not aptured by endogenous variables ould explain the differene between the estimated expeted duration and survey data.

The Likelihood of Effetive Lower Bound Events 17 Table 5 reports the medium-term probabilities of an ELB event for the benhmark ase (a 95 perent onfidene band for the prior mean on the steady-state parameters of width 2 for GDP growth, inflation, and the interest rate and of width 1.5 for the spread), the ase with a very tight prior on the steady state (a 95 perent onfidene interval of width 0.1 for all variables), and the ase with a loose prior (width 10). The tight prior results in the same or a lower ELB risk. The differene is driven to a great extent by the posterior of the interest rate steady state. With the exeption of Norway, the tight prior implies a higher steady-state interest rate (Table 6) and, eteris paribus, a lower ELB risk. The omparison in Table 5 suggests that the alibration of steady states plays an important role in ELB risk estimation. For example, alibrating the interest rate steady state to 1 perent for Japan implies a redution in the ELB risk of more than half. Table 6: Prior Mean and Posterior Means of the Steady-state Interest Rate Prior Posterior Benhmark Tight Loose Canada 4.00 3.38 4.00 2.22 Euro area 3.00 2.81 3.00 2.47 Japan 1.00 0.34 1.00 0.24 Norway 3.00 3.83 3.00 4.03 Sweden 3.00 2.70 3.00 1.62 Switzerland 3.00 2.82 3.00 1.48 United Kingdom 4.00 3.82 3.99 3.16 United States 3.00 2.79 3.00 1.99 Comparing the estimates for the loose prior on the steady states with the benhmark setting demonstrates the role of mean adjustment. It turns out that without treating the steady state expliitly, the estimated ELB risk is higher for some ountries. For example, the estimate for the US suggests that it would be stuk at the ELB 10 perent of the time. With mean adjustment, the ELB situation is estimated to be observed only 6 perent of the time. In the short term, the effet of mean adjustment is negligible. Within a period of one year, the distane between the ELB event probabilities in the benhmark and the ase without mean adjustment does not exeed 0.01 in absolute terms. 17

18 Mihal Franta Table 7: ELB Risk in the Medium Term for the Benhmark and the Tight Prior Centered on the Estimated Steady State Benhmark Tight on SS Canada 0.01 0.01 Euro area 0.03 0.03 Japan 0.16 0.16 Norway 0.05 0.04 Sweden 0.03 0.03 Switzerland 0.04 0.04 United Kingdom 0.02 0.02 United States 0.06 0.05 Note: Tight on SS means that the prior on the steady state is entered on the posterior mean of the benhmark ase and the prior is tight (the width of the 95 perent onfidene band is 0.01). As noted above, the hange in the ELB risk estimates for tight priors with respet to the benhmark is influened by the hange in the posterior of the interest rate. In addition, part of the differene an be explained by the fat that the benhmark ase allows some unertainty of the steady state, whih is not the ase when the steady state is alibrated. The strength of the effet is demonstrated in Table 7. The role of unertainty in the benhmark estimation is filtered out by employing the tight prior entered on the posterior mean of the benhmark speifiation. In other words, the steady state is alibrated to the benhmark posterior values and the unertainty of the steady state is negligible. The table shows that the influene on the ELB risk is of order 0.01 at most. Ignoring unertainty of the steady state leads to underestimation of the ELB risk. 5.2 The Role of Regime Change In addition to the theoretial reasoning for allowing regime hange due to a strutural hange in monetary poliy ondut, the estimation results an justify the hosen modeling framework ex post. A natural question is whether there are hanges in the estimates of the model parameters between regimes. Furthermore, one an ask whether the regime hange is driven by a hange in shok volatilities, a hange in dynami oeffiients, or both.

The Likelihood of Effetive Lower Bound Events 19 Figure 4: The Posterior Distribution of the Parameters at the First Lags of the Endogenous Variable in the Interest Rate Equation for the US Figures 4, 5, and 6 present the posterior distributions of the parameters at the first lag of the endogenous variables in the equation for the interest rate and spread for the US and the posterior distributions of the diagonal elements of the error ovariane matrix for the US. It turns out that the regime hange is driven primarily by a hange in shok volatilities, with the posterior distributions overing mostly different parameter values in the two regimes (Figure 6). The volatility of the shoks to the four equations for the US is muh lower in regime 1, when the interest rate is stuk at, or lose to, the ELB, and the main GFC-related drop in real eonomy variables happened before 2012, i.e., in regime 2. The passive onventional monetary poliy for regime 1 is manifested by posteriors entered on zero for all parameters at the first lag exept for the interest rate (Figure 4). In regime 2, mainly positive numbers are overed by the posterior distributions of the parameters at the first lag of real growth and inflation, whih is reminisent of the standard bakward-looking Taylor rule. In the spread equation, the lagged interest rate is found to have a zero effet on the spread in regime 1, while regime 2 exhibits the standard negative relationship between the two. Lagged real GDP growth and inflation do not affet the spread muh (Figure 5). 19

20 Mihal Franta Figure 5: The Posterior Distribution of the Parameters at the First Lags of the Endogenous Variable in the Spread Equation for the US Figure 6: The Posterior Distribution of the Diagonal Elements of the Shok Volatilities for the US Allowing for regime hange is one possible explanation for the different ELB risks estimated in the literature. The differene between estimates based on linear models (Kiley and Roberts, 2017, Hills et

The Likelihood of Effetive Lower Bound Events 21 al., 2016, and Nakata, 2017b) and those based on nonlinear models (Gust et al., 2017, Rihter and Throkmorton, 2016) exeeds 10 perentage points. Table 8 ompares the medium-term ELB risk in the benhmark model and the linear model that does not allow for regime hange. The ELB event probabilities estimated with the linear model are different but not systematially moved in any speifi diretion. So, the nonlinearity seems not to be the primary reason for the different estimates provided by the literature. However, the differenes may be relevant from the poliy maker s point of view. For example, based on the linear model the euro area medium-term ELB risk is estimated at 0.05, while the model with regime hange suggests that the risk is 0.03. The frequeny of an ELB episode for the euro area dereases from more than one year in 25 to less than one year in 25. Table 8: ELB Risk in the Medium Term for the Benhmark and the Model without Regime Change Regime hange No regime hange Canada 0.01 0.02 Euro area 0.03 0.05 Japan 0.16 0.22 Norway 0.05 0.03 Sweden 0.03 0.04 Switzerland 0.04 0.05 United Kingdom 0.02 0.05 United States 0.06 0.05 5.3 The Role of Panel Struture Exploiting the panel struture of data with observed ELB spells allows for estimation of the whole set of model parameters despite the short time series available for a single ountry. In addition, taking into aount the orrelation of shoks aross ountries leads to more effiient estimates, whih, in turn, result in more aurate simulation of the ELB risk due to the fat that the parameters unertainty enters the simulation proedure. On the other hand, if the heterogeneity between ountries is substantial and the interdependene is weak, the panel approah may not be preferable. 21

22 Mihal Franta Figure 7: The Posterior Distribution of Parameter in the Two Regimes The degree of ommonality of the maroeonomi dynamis aross ountries in the two regimes is ( r ) driven by the overall tightness parameter. As shown in Figure 7, the posterior mean of the ( r ) parameter is higher in regime 1 than in regime 2. A higher suggests more divergent oeffiients aross ountries or more tightly estimated ountry oeffiients. There are 20 observations in regime 1 and 49 in regime 2 and, as suggested by the posteriors presented in Figures 4 and 5, we do not expet the oeffiients in regime 1 to be more tightly estimated. Therefore, the differenes in dynamis aross ountries in regime 1 are probably substantial. This should not ome as a surprise, beause regime 1 ontains different unonventional monetary poliy measures employed during the Great Reession. (1) (2) The posterior mean of is 0.14 and that of is 0.05. Following the disussion in Jaroinski (2010), the square roots of the posterior means are omparable to the usual overall tightness used when setting the Minnesota prior. In our ase, the square roots are 0.38 and 0.22, lose to the interval of 0.1 0.2 overing the usual values used for overall tightness. This gives us some onfidene that the (r ) orresponding prior mean provides valuable information. The estimated suggests that some pooling of information aross ountries is present and justifies the use of a ommon mean when the ountry-speifi dynami parameters are estimated. (r ) The parameter, however, does not apture the possible heterogeneity in the timing of the regime hange, whih is assumed to be the same for all ountries. If there are signifiant differenes in the regime hange timing aross ountries, the ELB measures ould be inaurate. To set up the model, we fae a trade-off between the possibility of pooling information aross ountries in the model with regime hange and inauray due to ountry differenes in the regime hange date. To shed some light on the issue of ountry heterogeneity with respet to regime hange, singleountry mean-adjusted VARs are estimated in this subsetion. The single-ountry VARs are estimated similarly to their panel VAR ounterpart. The Normal-inverse Wishart prior is assumed for the dynami oeffiients and the error ovariane matrix. The prior on the dynami oeffiients is the

The Likelihood of Effetive Lower Bound Events 23 same as that imposed on the ommon mean in the panel VAR ase. The priors on the steady state and the threshold remain the same. The threshold variable is now defined as the single ountry average of the short-term interest rate and spread. Table 9: Swith Date Panel VAR Single-ountry VARs: Canada Euro area Japan Norway Sweden Switzerland United Kingdom United States 2011Q4 2010Q4 2010Q4 2010Q2 2009Q2 2011Q2 2010Q4 2011Q1 2011Q4 Table 9 reports the swith dates the last quarter of regime 2 for eah ountry estimated separately. It turns out that all ountries swith their regimes within four quarters (2010Q4 2011Q4), exept Japan (2010Q2) and Norway (2009Q2). For those two ountries, the additional auray gained by allowing for regime hange may not be high. Espeially for Japan, the medium-run ELB risk found within the panel VAR (0.16) an be viewed as too low, espeially when the fat that the ZLB event is observed for almost the whole data sample is taken into aount. Table 10: ELB Risk in the Medium Term for the Benhmark and the Model without Regime Change panel VAR single-ountry VAR Canada 0.01 0.03 Euro area 0.03 0.13 Japan 0.16 0.35 Norway 0.05 0.07 Sweden 0.03 0.14 Switzerland 0.04 0.15 United Kingdom 0.02 0.26 United States 0.06 0.08 Table 10 shows the hange in the medium-term ELB risk estimates if single-ountry VAR is employed. The omparison with the benhmark panel VAR shows higher probabilities of an ELB 23

24 Mihal Franta event based on single-ountry VAR in the medium term. The differene is not driven by the steadystate estimates. For example, for the UK the medium-term ELB risk inreases from 0.03 to 0.37 while the posterior mean of the interest rate steady state inreases from 3.82 to 3.92 and thus, eteris paribus, should imply a lower ELB risk. The effet of the improved effiieny of the estimates in the panel VAR seems to dominate and justifies the use of the panel VAR even though some evidene of heterogeneity of regime-swithing dates is found. 6. Conlusions This paper provides new estimates of the probability and expeted duration of an ELB event for eight advaned eonomies. Suh estimates are very important for poliy makers, espeially those in entral banks, beause the likelihood of the eonomy being stuk at the ELB is related to monetary poliy ondut and entral bank balane sheet size management. On the one hand, the task is simple, beause the employed methodology always results in a number, and the available data, whih over only a few ELB events, prelude any rigorous ex-post assessment of the auray of the ELB risk estimates. On the other hand, without suh a foreasting auray exerise, the reasoning for the approah underlying the estimates has to be very lear and sound. In this paper, I motivate all the features of the model with the aim of obtaining aurate ELB risk estimates. In addition, the model features are related to the ontentious aspets of previous approahes to estimating ELB risk. So, mean adjustment is present to obtain well-behaved long-run dynamis and to inorporate out-ofdata information on equilibrium values less stritly than during the alibration of the model. As a onsequene, unertainty relating to the equilibrium values is present in the ELB risk estimates. Next, allowing for regime hange seems to be neessary, as I work with a statistial model, not a strutural one. Finally, the panel nature of the data is a partiular advantage in the situation, where only a short time span of data is available and aounting for stati interdependenies is neessary (and suffiient) if finanial markets play an important role in shok transmission between ountries. The multiountry perspetive may enrih the ountry-speifi estimates available so far in the literature. Some methodologial issues are still open. The ELB situation an be viewed as a tail event. As suh, the fous of the modeling proedure should be on aurate modeling of distribution tails, espeially for the interest rate distribution. While the regime hange an be viewed as an attempt to go in this diretion, the assumption of normally distributed errors may not be too realisti. Extending the methodology to inlude shoks exhibiting fat tails represents a natural next step for future researh.