**BIS Working Papers **

**No 731**

**The likelihood of effective lower bound events **

by Michal Franta

**Monetary and Economic Department **

June 2018

JEL classification: E37, E52, C11

Keywords: effective lower bound, ELB risk, mean adjustment, panel VAR, regime change

This publication is available on the BIS website (www.bis.org).

*© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.*

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)

**The likelihood of effective lower bound events **

Michal Franta

**Abstract **

This paper provides estimates of the probability of an economy hitting its effective lower bound (ELB) on the nominal interest rate and of the expected duration of such an event for eight advanced economies. To that end, a mean-adjusted panel vector autoregression with static interdependencies and the possibility of regime change is estimated. The simulation procedure produces ELB risk estimates for both the short term, where the current phase of the business cycle plays an important role, and the medium term, where the occurrence of an ELB situation is determined mainly by the equilibrium values of macroeconomic variables. The paper also discusses the ELB event probability estimates with respect to previous approaches used in the literature.

JEL codes: E37, E52, C11

Keywords: effective lower bound, ELB risk, mean adjustment, panel VAR, regime change

- Michal Franta, Czech National Bank, e-mail: [email protected] I would like to thank Tomáš Adam, Leonardo Gambacorta, Elmar Mertens, Jouchi Nakajima, an anonymous referee and seminar participants at the Bank for International Settlements and the Czech National Bank for valuable comments. I completed this project while visiting the Bank for International Settlements under the Central Bank Research Fellowship program. The opinions expressed in this paper are those of the author and do not necessarily reflect those of the Czech National Bank or of the Bank for International Settlements.

**1. Introduction**

The recent experience of a prolonged period of extraordinarily low monetary policy rates in advanced countries has rekindled interest in macroeconomic issues relating to the effective lower bound (ELB) on the nominal interest rate. Before the Great Financial Crisis (GFC), the only relevant example of an ELB event was Japan, which, however, was viewed as a peculiar case. It was not generally believed that advanced countries could experience long and recurrent periods of being stuck at the ELB. This opinion has now changed, and novel research relating to the ELB has been published.

There are two important policy questions underlying this research work: what is the probability of an ELB event at a given point in time, and how often can an ELB event be expected to occur in general? The usual approach to answering the latter involves conducting model-based stochastic dynamic 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 calibrated linear models and are highly sensitive to the equilibrium values of macroeconomic variables assumed (see, for example, Reifschneider and Williams, 2000, Coenen, 2003, and Kiley and Roberts, 2017).

When the examination of the likelihood of an ELB event is related to a specific point in time, more data-driven approaches are employed. Statistical models are used to estimate a distribution forecast for the interest rate. In contrast to the other approach, this allows uncertainty in model parameters, latent variables and measurement errors to be taken into account, and enables some simple form of non-linearity (Chung et al., 2012). Nakata (2017a) employs survey data on macroeconomic projections to bring the standard stochastic simulations to actual data. However, data-driven approaches are not suitable for estimating ELB risk in the medium and long term, because they often do not possess well-defined unconditional moments.

The aim of this paper is to draw on the above-mentioned approaches and to provide estimates of the probability and expected duration of ELB events. To that end, the paper combines stochastic simulations around equilibrium values and distribution forecasting reflecting current observed data. The results are therefore relevant to both the short term, where the ELB likelihood is driven mainly by the current phase of the business cycle, and the medium term, where the ELB risk is determined mainly by equilibrium values. Dealing with both time scales in one modeling framework makes the estimates for the short and medium term consistent with each other. Furthermore, the approach addresses some of the contentious issues of the previous approaches and should therefore deliver more accurate ELB likelihood estimates.

I adopt an empirical strategy and base ELB likelihood estimates on recent data from advanced countries, exploiting the fact that many of them have recently experienced, or are still experiencing, nominal interest rates at their ELB. As a consequence, the ELB risk estimates have to rely less on calibrated parameters and can be based more on estimated quantities. The framework thus accounts for parameter uncertainty, which has been found to be important for realistically assessing ELB likelihood (Chung et al., 2012). On the other hand, the time span of the macro data capturing ELB events is still short, suggesting that it is appropriate to exploit the panel nature of that data.

Furthermore, the data-driven approach depends less heavily on assumptions about the equilibrium values of macroeconomic variables than do studies based on stochastic simulations. The assumptions about the equilibrium values enter the estimation procedure in the form of priors and are confronted with the observed data during the Bayesian estimation process. Finally, as an ELB situation implies a possibility of regime change, the modeling framework allows for change in the shock transmission mechanism and shock volatilities.

To quantitatively assess the occurrence of ELB events, I employ the mean-adjusted panel vector autoregression technique, which allows for static interdependencies and threshold behavior. The model is estimated on data for eight advanced economies 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 impact of various assumptions employed by the techniques used to estimate ELB risk in previous studies.

It turns out that, in the short term, the ELB risk is currently 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 calibration of the steady-state values of macroeconomic variables can lead to ELB risk underestimation due to the fact that the calibrated value of the equilibrium interest rate is high and is assumed to be known with certainty. It is also shown how the empirical approach gains from the panel nature of data by partially pooling country-specific information and by improving the efficiency of estimates.

From the policy maker’s perspective, the estimates should be taken into account in the conduct of monetary policy. Higher ELB risk during a recession calls for more aggressive easing of monetary policy, whereas higher ELB risk during a boom justifies slower normalization of rates (see, for example, Williams, 2014, and references therein). Similarly, if there is agreement on the optimal size of the central bank balance sheet, the speed of balance sheet normalization from current levels should reflect the probability of return to the ELB and to unconventional measures that would increase the size of the sheet again.

The structure of the paper is as follows. Section 2 discusses recent monetary policy rates in advanced countries. Section 3 presents the model. Section 4 explains the estimation approach and

describes the data. Section 5 presents the results and considers the role of calibration and regime change in ELB risk estimation. Section 6 concludes.

**2. The ELB in advanced countries**

Figure 1 shows monetary policy rates in eight advanced economies since 1999. In general, two cycles can be observed, one peaking around the year 2000 and the second peaking in 2007 just before the GFC hit the world economy. The observed monetary policy rate profiles also suggest a downward trend in the equilibrium interest rate. The short time span of the observations, however, makes any conclusion about the trend uncertain.

In the aftermath of the 2008 crisis, advanced economies approached or even crossed the zero bound with their interest rates. Once a country was not able or willing to ease monetary policy further using the policy rate, unconventional measures were employed. The United States, the United Kingdom, and the euro area used large-scale asset purchases aimed at lowering long-term interest rates. Japan initially adopted the same approach to quantitative easing, then followed up with additional quantitative and qualitative monetary easing (QQE) in 2013. It also introduced a negative monetary policy rate in mid-2016. Negative policy rates have been seen in Switzerland and Sweden recently as well. Furthermore, Switzerland imposed an exchange rate floor to prevent currency appreciation between 2011 and 2015.

Figure 1 suggests some degree of homogeneity and close interconnectedness across countries. On the other hand, significant heterogeneity is present in the data, especially after the start of the GFC. Monetary policy rules presumably changed when economies hit the ELB, and the changes differ across countries. The shock propagation mechanism and shock volatilities might also be different across countries after the GFC.

**3. Model**

The modeling framework is built on the panel vector autoregression (VAR) structure, which allows for cross-sectional heterogeneity and static interdependencies. Cross-sectional heterogeneity allows for differences in the dynamics of macroeconomic variables across countries. This option is especially important for the period of unconventional monetary policy, because countries employed different strategies for easing monetary policy further. Static interdependencies allow for correlation of reduced-form shocks across countries. The GFC spread across the advanced economies primarily through the financial markets within one quarter, so accounting for static interdependencies is crucial, probably more so than accounting for dynamic ones. Ignoring static interdependencies, if present, would lead to less efficient estimates and consequently to less accurate ELB risk estimates. Note also that static interdependence is the only interconnection between the countries in the model. Without that, the model would be effectively a set of single-country models. The case of single-country VARs is discussed in Subsection 5.3.

The model is formulated in mean-adjusted form to deal directly with steady-state macroeconomic variables. Mean adjustment for VAR models is introduced in Villani (2009) and allows the steady state of a macroeconomic variable to be treated as a single parameter. As such, a prior on the steady state can be formulated and the parameter directly estimated. The posterior of the steady state then defines the long-run dynamics of the model. Explicit treatment of the steady state leads to well-defined ELB risk in the medium and long term.

For N countries and for t = p + 1,..., T, the model is formulated as follows:

c and time t. The coefficient matrices Alc relating to the l-th lag of the vector of endogenous variables for country c are of dimension n x n.

The vector of endogenous variables contains real GDP growth, CPI inflation, the short-term interest rate, and the spread between the ten-year government bond yield and the policy rate. The set of endogenous variables is usually chosen according to the shocks that are to be identified. We do not perform structural shock identification in our exercise. The main aim of our choice of variables is to capture both conventional and unconventional monetary policy rules, i.e., the interest rate rule and the quantitative reaction function reflected by the spread.

The question is whether this minimal set can also represent the other unconventional monetary policy measures contained in our data set. Obviously, the vector of endogenous variables can deal with negative interest rates. There could be a problem with the Swiss exchange rate floor, as the exchange rate is not included in the vector of endogenous variables. The effect of this type of policy is captured to the extent to which the floor is reflected by long-term government bond yields.

The mx1 vector xct = (xc1t,..., xcmt)' in (1) contains exogenous variables, which are the same across countries. In our application, the vector of exogenous variables comprises constant terms only. The coefficient matrices relating to exogenous variable Fc are of dimension n x m.

The country-specific matrices Fc include the steady states of the endogenous variables, because it follows from (1) that if the process for yc ,t is stationary then

Finally, the vector of error terms (£1jt,£N t)' in (1) is distributed independently and normally, with zero mean and covariance matrix E of dimension Nn x Nn . The error covariance matrix is generally non-diagonal, allowing for residual correlation between countries. We assume two lags in the benchmark specification.

Let y denote a NnT x1 data vector for all countries:

As discussed in section 2, the data set presumably includes changes in policy rules when the ELB was hit and unconventional monetary policies were introduced. Moreover, as suggested by Clark (2011), allowing for changes in shock volatilities results in more accurate density forecasts, even though the time perspective of Clark (2011) is longer than in our case. Translating this into the context of our exercise, a more accurate density forecast means a more accurate estimate of ELB risk. Model (4) is therefore extended in such a way that it allows for two regimes in the form of possible threshold behavior driven by the endogenous variables. Due to the short time series available for the estimation, only one regime change is considered. Whether the change is driven by a change in dynamic coefficients (e.g., in policy rules), by a change in shock volatility, or by both is decided on the basis of model likelihood comparison within the estimation procedure.

Combining regime change with the mean-adjustment procedure forces us to take a stand on the issue of whether the mean-adjustment procedure should be applied to both regimes or to one regime only. Strong prior information exists for the “normal times” regime with standard monetary policy conduct, so mean adjustment is imposed in that regime only. For the other regime, a simple constant term is included in the data matrices.

where £(1) ~ N(0,2(1)) , s(2) ~ N(0,2(2)), and the matrices 2(1) and 2(2) are in general non

diagonal. Data matrices and vectors with superscripts denoting the regime refer to the subsamples of X and y relating to the relevant regime. Note that the data matrix X(1) includes the constant term.

**4. Estimation**

The estimation approach is Bayesian and combines estimation of a mean-adjusted VAR (Villani, 2009), a hierarchical linear model for VAR (Jarocinski, 2010), and Bayesian estimation of a threshold VAR (Chen and Lee, 1995, and Koop and Potter, 2003). To estimate the model, the country-specific estimates of some parameters exploit information from all countries by means of an exchangeable prior. We assume that the coefficients at the lagged values of the endogenous variables stacked in vector ftp are distributed normally around a regime-specific common mean

where b(cr) ~ n(q, 'Lb/r)). The spread of the country-specific vectors of the dynamic parameters around the common mean is driven by the overall tightness parameter A(r) defining the variance (seeformula A2).

The exchangeable prior is not used for the parameters capturing the steady state, Fc. Strong prior information on the steady states of the endogenous variables is available, so there is no need to pool information across countries through the common mean for those parameters.

The estimation procedure simulates the posterior distributions based on likelihood, priors, and conditional priors, respectively. The vector of model parameters contains dynamic parameters vector 8, and the threshold r that determines the regimes. Using conditionally conjugate priors yields conditional posterior densities that are easy to draw from within the Gibbs sampler. The conditional posterior of the threshold cannot be expressed by a standard density function and a Metropolis step is used to take a draw of the parameter. The specification of the prior distributions and a description of the sampler can 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 forecasts for up to 48 quarters are computed using random draws of shocks from a given distribution. Note that regime switching is allowed for during the simulation, because the threshold variable is a function of the endogenous variables. Then, for a given period, the proportion of forecasts that are below or at the ELB is computed to estimate the probability of an ELB event. The medium-term ELB risk is computed as the average ELB risk for last eight quarters, i.e., the 41st to 48th quarters. In addition to the estimate of the ELB risk, the expected duration of the ELB event is computed as the average number of quarters for which the simulated forecast remains at the ELB.

During the forecasting, the ELB constraint is imposed in a straightforward manner, because the model is backward looking. The value of the ELB is imposed whenever the one-period-ahead value of the interest rate falls below the ELB. This approach can be interpreted as passive conventional monetary policy. The monetary authority does not lower the interest rate when the estimated interest rate rule suggests doing so. Even with the interest rate rule switched off, unconventional monetary policy still affects the economy according to the estimated spread equation.

The numerical value of the ELB is set differently for each country. First of all, it is important to stress that we impose the ELB on the short-term nominal interest rate, because the policy rate is not directly included in the vector of endogenous variables. For those countries which have not experienced negative policy rates, the ELB is set to zero (Norway, the UK, and the US). In addition, the choice for the US is driven by comparability with other studies. For countries with negative policy 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. The ELB values are reported in Table 1.

The ELB values do not enter the estimation procedure and thus do not affect the estimation of the model parameters. They enter the simulation of the ELB risk and the expected duration of the ELB spell only. As a consequence, a different ELB for a country only affects the country-specific results (the exception is regime change timing, which relates to all countries). Different reasoning for ELB values across countries therefore does not represent an obstacle.

4.1 Data

Real GDP and CPI are seasonally adjusted and enter the vector of endogenous variables as the first difference 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 policy 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 constructed from AAA-rated bond yields. Yields for the UK are downloaded from BoE Statistics and yields for the US from the FRED database. Monetary policy rates are obtained from the IMF IFS. The Swiss National Bank targets the three- month Swiss franc Libor, so the Libor is used as the policy rate. The policy rate for the euro area is the interest rate on main refinancing 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 covers the period 1999Q1-2016Q4. The choice of quarterly data, as opposed to monthly data, is driven by the fact that the panel VAR specification with static and without dynamic interdependencies 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 inference to deal with autocorrelation of draws. Convergence diagnostics 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 covers the period 2012Q1- 2016Q4 and regime 2 the period 1999Q2-2011Q4. Regime 1 thus covers the period when advanced economies hit or approached their ELBs, launched their unconventional monetary policy measures, and gradually reduced the spread between long- and short-term interest rates (Figure 3).

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 economy was stuck at the ELB in the last quarter used for the estimation, i.e., in 2016Q4. So, for the euro area, Japan, and Sweden, which are at their assumed ELBs, the ELB risk is in the range of 0.37-0.53. For the other countries, the ELB risk is lower. For Canada, Norway, and the US, it is virtually zero for 2017Q1.

The ELB risk profile during the first eight forthcoming quarters (2017Q1-2018Q4) reflects the current phase of, and the outlook for, the business cycle. For the euro area, the ELB risk increases in the first three quarters and then declines due to a delay in its economic recovery with respect to other advanced economies. For Canada, Norway, and the US, the ELB risk is very low for the first three quarters and rises after that, because these countries experience an expansionary phase of the business cycle in the first quarters of 2017. The comparison across countries is also influenced by the different ELBs. The lower the ELB, the lower the probability of the economy 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 shocks to macroeconomic variables. It ranges from 0.01 (Canada) to 0.16 (Japan) - see the MR column in Table 2. The numbers can be interpreted as meaning the percentage of the time (quarters) the ELB is binding. For Canada, the economy is at the ELB 1 percent 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 percent of the time, i.e., for four years every 25 years. By combining the short-run outlook based on the business cycle and medium-run simulations drawing on steady-state values, non-monotonicity of ELB risk measures can arise, as documented in Table 2.

The presented ELB risk estimates can be compared with the results provided by the recent literature. Assuming a steady-state nominal interest rate of 3 percent and imposing an ELB of zero, Kiley and Roberts (2017) estimate the frequency of ELB events for the US to be 17.4 percent based on the DSGE model of Linde et al. (2016) and 31.7 percent based on the FRB/US model. The estimate based on the DSGE model is close to the estimates in Hills et al. (2016) and Nakata (2017b), which draw on calibrated DSGE models and provide an estimated probability of around 14 percent. If we assume the same ELB as in Kiley and Roberts (2017) and estimate the steady-state nominal interest rate at 2.79 percent, the ELB risk from our model is 5.71, which is much lower than the figures obtained by Kiley and Roberts (2017). Lower ELB risk estimates can 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 percent, while Richter and Throckmorton (2016) estimated the probability at around 5 percent.

Part of the difference can be explained by the fact that our ELB risk estimates include the effects of unconventional monetary policy, which lowers the probability of hitting/staying at the ELB. Whenever the US economy is simulated to be in regime 1, the estimated impact of unconventional monetary policy is present in the simulation. Unconventional measures are not included in Kiley and Roberts (2017). Another reason suggested by the plain comparison is that nonlinear models tend to deliver lower ELB risk - the effect of nonlinearity on ELB risk estimates is discussed in Subsection 5.2.

Nakata (2017a) combines survey-based macroeconomic projections and stochastic simulations of the FRB/US model and defines the ELB risk as the probability that the federal funds rate will be constrained by the ELB for at least one quarter in the next three years. For 2016Q4, the ELB risk is estimated to exceed 50 percent for all three survey-based projections considered (Survey of Primary Dealers, Survey of Professional Forecasters, and Summary of Economic Projections). Excluding the period of elevated macroeconomic volatility in the 1970s and 1980s, which reduces the estimated size of shocks, the ELB risk lies between 40 and 50 percent. Assuming the ELB consistent with Nakata (2017a), Nakata’s ELB risk for the US estimated within our framework is 57 percent. Nakata’s ELB risk measures for other countries are reported in Table 3.

The highest medium-term expected duration (three quarters) is obtained for the euro area, while the lowest expected durations are observed for Canada, Sweden, and the US.

For the US, the medium-term expected duration is estimated to be 2.35 quarters. Hills et al. (2016) and Nakata (2017b) estimated the expected duration at 9 quarters. A much shorter duration of the ELB spell is estimated by Gust et al. (2017) and Richter and Throckmorton (2016), who give an average duration of around 3 quarters. Similarly to the estimates of the ELB risk, the estimate of 2.35 quarters for the US is closer to the literature dealing with estimated nonlinear DSGE models than calibrated linearized DSGE models.

The ELB risk and duration estimates involve unconventional monetary policy, as it is reflected in the data used for the estimation. So, the estimate for the US assumes a relatively strong unconventional easing of monetary policy at the ELB, while the estimate for Canada draws on the estimated conventional monetary policy only, because the Bank of Canada has never employed unconventional measures in the post-2008 period. The two countries’ ELB risk should be compared with this consideration in mind. If no unconventional monetary policy had been conducted in the US after 2008, the ELB risk would be higher and the difference with respect 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 macroeconomic variables in advanced countries allows us to formulate informative priors on the steady-state parameters. Regarding the informativeness of the priors, two extreme cases can be distinguished. First, imposing a very tight prior is similar to the situation where the steady-state values are calibrated. Second, imposing a very loose prior resembles the case where no mean adjustment is conducted and allows us to discuss the role of mean adjustment in the estimation of ELB risk. Examining the two extreme cases can shed some light on, respectively, the effect of calibration and the effect of not imposing any equilibrium values on the ELB risk estimates in the previous literature.

Table 5 reports the medium-term probabilities of an ELB event for the benchmark case (a 95 percent confidence 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 case with a very tight prior on the steady state (a 95 percent confidence interval of width 0.1 for all variables), and the case with a loose prior (width 10). The tight prior results in the same or a lower ELB risk. The difference is driven to a great extent by the posterior of the interest rate steady state. With the exception of Norway, the tight prior implies a higher steady-state interest rate (Table 6) and, ceteris paribus, a lower ELB risk. The comparison in Table 5 suggests that the calibration of steady states plays an important role in ELB risk estimation. For example, calibrating the interest rate steady state to 1 percent for Japan implies a reduction in the ELB risk of more than half

Comparing the estimates for the loose prior on the steady states with the benchmark setting demonstrates the role of mean adjustment. It turns out that without treating the steady state explicitly, the estimated ELB risk is higher for some countries. For example, the estimate for the US suggests that it would be stuck at the ELB 10 percent of the time. With mean adjustment, the ELB situation is estimated to be observed only 6 percent of the time. In the short term, the effect of mean adjustment is negligible. Within a period of one year, the distance between the ELB event probabilities in the benchmark and the case without mean adjustment does not exceed 0.01 in absolute terms.

As noted above, the change in the ELB risk estimates for tight priors with respect to the benchmark is influenced by the change in the posterior of the interest rate. In addition, part of the difference can be explained by the fact that the benchmark case allows some uncertainty of the steady state, which is not the case when the steady state is calibrated. The strength of the effect is demonstrated in Table 7. The role of uncertainty in the benchmark estimation is filtered out by employing the tight prior centered on the posterior mean of the benchmark specification. In other words, the steady state is calibrated to the benchmark posterior values and the uncertainty of the steady state is negligible. The table shows that the influence on the ELB risk is of order 0.01 at most. Ignoring uncertainty of the steady state leads to underestimation of the ELB risk.

5.2 The role of regime change

In addition to the theoretical reasoning for allowing regime change due to a structural change in monetary policy conduct, the estimation results can justify the chosen modeling framework ex post. A natural question is whether there are changes in the estimates of the model parameters between regimes. Furthermore, one can ask whether the regime change is driven by a change in shock volatilities, a change in dynamic coefficients, or both.

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 covariance matrix for the US. It turns out that the regime change is driven primarily by a change in shock volatilities, with the posterior distributions covering mostly different parameter values in the two regimes (Figure 6). The volatility of the shocks to the four equations for the US is much lower in regime 1, when the interest rate is stuck at, or close to, the ELB, and the main GFC-related drop in real economy variables happened before 2012, i.e., in regime 2.

The passive conventional monetary policy for regime 1 is manifested by posteriors centered on zero for all parameters at the first lag except for the interest rate (Figure 4). In regime 2, mainly positive numbers are covered by the posterior distributions of the parameters at the first lag of real growth and inflation, which is reminiscent of the standard backward-looking Taylor rule. In the spread equation, the lagged interest rate is found to have a zero effect 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 affect the spread much (Figure 5).

Allowing for regime change is one possible explanation for the different ELB risks estimated in the literature. The difference between estimates based on linear models (Kiley and Roberts, 2017, Hills et al., 2016, and Nakata, 2017b) and those based on nonlinear models (Gust et al., 2017, Richter and Throckmorton, 2016) exceeds 10 percentage points. Table 8 compares the medium- term ELB risk in the benchmark model and the linear model that does not allow for regime change.

The ELB event probabilities estimated with the linear model are different but not systematically moved in any specific direction. So, the nonlinearity seems not to be the primary reason for the different estimates provided by the literature. However, the differences may be relevant from the policy 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 change suggests that the risk is 0.03. The frequency of an ELB episode for the euro area decreases from more than one year in 25 to less than one year in 25.

5.3 The role of panel structure

Exploiting the panel structure 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 country. In addition, taking into account the correlation of shocks across countries leads to more efficient estimates, which, in turn, result in more accurate simulation of the ELB risk due to the fact that the parameters’ uncertainty enters the simulation procedure. On the other hand, if the heterogeneity between countries is substantial and the interdependence is weak, the panel approach may not be preferable.

The degree of commonality of the macroeconomic dynamics across countries in the two regimes is driven by the overall tightness parameter A(r). As shown in Figure 7, the posterior mean of the parameter is higher in regime 1 than in regime 2. A higher A(r) suggests more divergent coefficients across countries or more tightly estimated country coefficients. 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 expect the coefficients in regime 1 to be more tightly estimated. Therefore, the differences in dynamics across countries in regime 1 are probably substantial. This should not come as a surprise, because regime 1 contains different unconventional monetary policy measures employed during the Great Recession.

The posterior mean of A(1) is 0.14 and that of A(2) is 0.05. Following the discussion in Jarocinski (2010), the square roots of the posterior means are comparable to the usual overall tightness used when setting the Minnesota prior. In our case, the square roots are 0.38 and 0.22, close to the interval of 0.1-0.2 covering the usual values used for overall tightness. This gives us some confidence that the corresponding prior mean provides valuable information. The estimated A(r) suggests that some pooling of information across countries is present and justifies the use of a common mean when the country-specific dynamic parameters are estimated.

The parameter A( r), however, does not capture the possible heterogeneity in the timing of the regime change, which is assumed to be the same for all countries. If there are significant differences in the regime change timing across countries, the ELB measures could be inaccurate. To set up the model, we face a trade-off between the possibility of pooling information across countries in the model with regime change and inaccuracy due to country differences in the regime change date.

To shed some light on the issue of country heterogeneity with respect to regime change, singlecountry mean-adjusted VARs are estimated in this subsection. The single-country VARs are estimated similarly to their panel VAR counterpart. The Normal-inverse Wishart prior is assumed for the dynamic coefficients and the error covariance matrix. The prior on the dynamic coefficients is the same as that imposed on the common mean in the panel VAR case. The priors on the steady state and the threshold remain the same. The threshold variable is now defined as the single country average of the short-term interest rate and spread.

Table 10 shows the change in the medium-term ELB risk estimates if single-country VAR is employed. The comparison with the benchmark panel VAR shows higher probabilities of an ELB event based on single-country VAR in the medium term. The difference is not driven by the steady-state estimates. For example, for the UK the medium-term ELB risk increases from 0.03 to 0.37 while the posterior mean of the interest rate steady state increases from 3.82 to 3.92 and thus, ceteris paribus, should imply a lower ELB risk. The effect of the improved efficiency of the estimates in the panel VAR seems to dominate and justifies the use of the panel VAR even though some evidence of heterogeneity of regime-switching dates is found.

**6 Conclusions**

This paper provides new estimates of the probability and expected duration of an ELB event for eight advanced economies. Such estimates are very important for policy makers, especially those in central banks, because the likelihood of the economy being stuck at the ELB is related to monetary policy conduct and central bank balance sheet size management.

On the one hand, the task is simple, because the employed methodology always results in a number, and the available data, which cover only a few ELB events, preclude any rigorous expost assessment of the accuracy of the ELB risk estimates. On the other hand, without such a forecasting accuracy exercise, the reasoning for the approach underlying the estimates has to be very clear and sound. In this paper, I motivate all the features of the model with the aim of obtaining accurate ELB risk estimates. In addition, the model features are related to the contentious aspects of previous approaches to estimating ELB risk.

So, mean adjustment is present to obtain well-behaved long-run dynamics and to incorporate out- of-data information on equilibrium values less strictly than during the calibration of the model. As a consequence, uncertainty relating to the equilibrium values is present in the ELB risk estimates. Next, allowing for regime change seems to be necessary, as I work with a statistical model, not a structural one. Finally, the panel nature of the data is a particular advantage in the situation, where only a short time span of data is available and accounting for static interdependencies is necessary (and sufficient) if financial markets play an important role in shock transmission between countries. The multi-country perspective may enrich the country- specific estimates available so far in the literature.

Some methodological issues are still open. The ELB situation can be viewed as a tail event. As such, the focus of the modeling procedure should be on accurate modeling of distribution tails, especially for the interest rate distribution. While the regime change can be viewed as an attempt to go in this direction, the assumption of normally distributed errors may not be too realistic. Extending the methodology to include shocks exhibiting fat tails represents a natural next step for future research.

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