**BIS Working Papers **

**No 703**

**The negative interest rate policy and the yield curve**

by Jing Cynthia Wu and Fan Dora Xia

**Monetary and Economic Department **

February 2018

JEL classification: G2

Keywords: shadow banking, wealth management products (WMPs), structured credit intermediation, investment receivables, entrusted loans, trust loans

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 negative interest rate policy and the yield curve**

Jing Cynthia Wu Chicago Booth and NBER Fan Dora Xia Bank for International Settlements

This draft: February 2018

**Abstract **

We extract the market’s expectations about the ECB’s negative interest rate policy from the euro area’s yield curve and study its impact on the yield curve. To capture the rich dynamics taking place at the short end of the yield curve, we introduce two policy indicators that summarise the immediate and longer-horizon future monetary policy stances. The ECB has cut interest rates four times under zero. We find that the June 2014 and December 2015 cuts were expected one month ahead but that the September 2014 cut was unanticipated. Most interestingly, the March 2016 cut was expected four months ahead of the actual cut.

Keywords: negative interest rate policy, effective lower bound, term structure of interest rates, shadow rate term structure model, regime-switching model

JEL classification codes: E43, E52, E58

We thank Drew Creal, Felix Geiger, Jim Hamilton, Wolfgang Lemke, Eric Swanson, and seminar and conference participants at the UCSD Rady School of Management, the University of Oxford, the 10th Annual Conference of the Society for Financial Econometrics, the Federal Reserve Board’s conference on “Developments in Empirical Monetary Economics,” the Barcelona GSE Summer Forum, the 7th Term Structure Workshop at the Deutsche Bundesbank Bank,” the European Central Bank, the University of Illinois UrbanaChampaign, Texas A&M University, Tilburg University, the Banque de France, the Tinbergen Institute, the University of Copenhagen, the Universit´e Catholique de Louvain, the Bank of Canada, and the BIS Asian office for helpful suggestions. Cynthia Wu gratefully acknowledges financial support from the James S. Kemper Foundation Faculty Scholar at the University of Chicago Booth School of Business. This article was formerly entitled “Time-varying lower bound of interest rates in Europe.” The views expressed herein are those of the authors and do not necessarily represent the views of the BIS. Correspondence: [email protected] and [email protected]

**1 Introduction**

The effective lower bound (ELB) of nominal interest rates is one of the most discussed economic issues of the past decade. The negative interest rate policy (NIRP) is among the latest additions to unconventional monetary policy toolkits, in the hopes of providing further stimulus to the economies facing the ELB. For example, in June 2017, the deposit rate of the Swiss National Bank was set at a record low of -0.75%, while the deposit facility rate of the European Central Bank’s (ECB) was set at -0.4%.

It is important for policy makers and economists to understand the implications of such a new policy tool. First, due to the NIRP, the total value of outstanding government bonds with negative interest rates had reached 10 trillion dollars by the end of 2016 and is still growing. Bearing this in mind, the question is, what is the NIRP’s impact on the yield curve? Second, what are economic agents’ perceptions of this policy and how do they form expectations? Third, because the zero lower bound (ZLB) is no longer binding, the NIRP creates richer shapes at the short end of the yield curve. How do we accommodate them when we model the term structure of interest rates? Understanding these questions is important to euro area countries and Japan, as both areas are currently implementing a NIRP. Such an understanding is also potentially important for the United States, for which the NIRP could be an option should large negative shocks occur.

To address these questions, we propose a new shadow rate term structure model (SRTSM) that we apply to the euro area. At the ELB, the short end of the yield curve displays three different shapes. The first case is flat as seen in the US data when the ZLB prevails but not the NIRP. Second, the yield curve could be downward sloping when agents expect future NIRP-related policy rate cuts. Third, in some days, it is initially flat in the very short end and then downward sloping, implying market participants expect no immediate action from the central bank but nonetheless think that future monetary policy is expansionary overall. To capture these shapes, we introduce two monetary policy indicators: one for the immediate monetary policy stance and another for the stance over longer horizons. We model the discrete movement of the relevant policy rate, the ECB’s deposit facility rate, at the ELB with a simple and intuitive regime-switching model conditioned on the two policy indicators. Our model is able to capture the three different shapes of the yield curve observed in the data. We then build the dynamics of the deposit rate into an SRTSM by using the Black (1995) framework, where the short term interest rate is the maximum of the non-positive deposit rate and a shadow interest rate.

We use our model to extract the market’s expectations of the NIRP. Overall, expectations of financial market participants extracted from our model agree with economists’ expectations from the Bloomberg survey. Importantly, however, our model has an advantage over the Bloomberg survey because we can extract the market’s expectations further into the future, whereas the Bloomberg data are collected only one week before monetary policy meetings. We find that the June 2014 and December 2015 cuts were expected one month before but that the September 2014 cut was entirely unanticipated. Most interestingly, the March 2016 cut was expected four months ahead of the actual cut.

We then evaluate the NIRP’s impact on the yield curve by conducting some counterfac- tual analyses at the end of our sample (June 2017). First, we ask what would happen to the yield curve, if the ECB indicated an easing position at its next meeting but promised that the cut would be the last one in history. In response to such an announcement, the yield curve would shift down by about 0.03% across all maturities. Second, what would happen if the central bank announced that it would not make any move at the next meeting but the overall future policy environment would be expansionary? The one month rate would not decrease, but yields at other maturities would. The change would grow with the maturity up to two years, and then flatten out afterwards at about 0.1%. Third, suppose the ECB communicated with the public about its expansionary plan across all horizons. This action would create the largest impact. The change at the one month horizon would be 0.03%. It would also increase with maturity, with the largest change happening at the two year horizon, amounting to 0.2%. The size of the change would decrease to about 0.16% in the long run.

The term structure model allows us to decompose long term yields into an expectations component and a term premium. Our model-implied 10-year term premium increased between 2005 and 2008. It has trended down since 2009, and became negative at the ELB, potentially due to quantitative easing (QE) purchases. The dynamics of the deposit rate contributes positively to the premium but on a smaller order of magnitude.

We compare our model to various alternatives including several SRTSMs proposed in the literature and the Gaussian affine term structure model (GATSM). We find that our new model performs the best in terms of higher likelihood and lower pricing errors. Other existing models in the literature, on the other hand, do poorly.

After a brief literature review, the rest of the paper proceeds as follows. Section 2 motivates our work and models the dynamics of the deposit rate, and Section 3 sets up the new SRTSM. Section 4 discusses data, estimation, and estimates. Section 5 turns to model implications regarding the NIRP, while Section 6 focuses on implications for the yield curve. Section 7 concludes.

Literature Earlier work has applied the SRTSM mostly to the Japanese and US yield curves. For example, Kim and Singleton (2012) and Ichiue and Ueno (2013) focus on Japan, whereas Krippner (2013), Christensen and Rudebusch (2014), Wu and Xia (2016), and Bauer and Rudebusch (2016) focus on the United States. These papers all keep the lower bound at a constant level.

A few studies have focused on the new development in Europe, where the policy lower bound kept moving down to negative numbers after the NIRP. For example, the online implementation of Wu and Xia (2016) for the euro area, and Lemke and Vladu (2016) and Kortela (2016). However, none of these papers allow agents to be forward-looking in terms of the future movements of the policy rate, which is an important feature of our model. And this feature allows our model to fit the short end of the yield curve much better than other approaches.

Applications of this class of model in the term structure literature include Ang and Bekaert (2002), Bansal and Zhou (2002), and Dai et al. (2007). These papers allow the parameters of the dynamics to take several different values. More closely related to our paper, Renne (2012) allows the monetary policy rate to take discrete values and follow a regime-switching process. Our work differs from his work in that we build the regime-switching model for the policy rate only when the ELB is binding. Otherwise, the state variables follow a Gaussian vector autoregression (VAR) process, as in the literature. The advantages of our model are twofold. First, it significantly reduces the state space for the regime-switching process. Second, when the ELB is not binding, our model is essentially a GATSM, which is the literatures’ preferred model.

**2 NIRP **

*2.1 Rate cuts and yield curve*

Since the deposit rate hit the ELB in July 2012, the ECB has adopted a NIRP and further cut rates four times. To understand whether these cuts were anticipated by the market, Figure 1 plots the yield curves in the months before these cuts. June 2014 was a historical moment during which the ECB cut the deposit rate to a negative value for the first time. The ECB made efforts to communicate with the public prior to the event. By May 2014, the market did expect a rate cut going forward, but it did not fully digest the cut of 0.1% in the following month. Instead, it expected a cut of 0.1% within three months. The second cut was entirely unexpected, and the yield curve was basically flat in August 2014. The December 2015 cut was fully anticipated. Moreover, in November 2015, the market expected further cuts beyond the next meeting, anticipating a total cut of a 0.2% over the next year. In February 2016, the market anticipated further cuts, with the total amounting to over 0.2%.

The NIRP introduces richer dynamics at the short end of the yield curve. See the red solid dots in Figure 2. In July 2013, the front end of the yield curve was flat. This flatness was the basic pattern seen in the data when the US experienced the ZLB. Most of the term structure literature on the ZLB has focused on this feature; see, Christensen and Rudebusch, Wu and Xia (2016) and Bauer and Rudebusch (2016). However, the NIRP introduced additional patterns: in both February 2016 and July 2016, the yield curves were downward sloping, implying future decreases in the policy rate. Interestingly, the very short ends for the two months were different: in February 2016, an easing of the monetary policy stance was expected at all horizons, whereas in July 2016 the very short ends of the curve was flat,

suggesting that no cut would happen over the next month.

We build a simple and intuitive model to capture these shapes at the front end of the yield curve when the ELB is binding. We then will use the model to extract the market’s expectations of the future monetary policy stance. For now, we ignore the difference between the deposit rate and the short end of the yield curve, and we will discuss how this difference is treated in Section 4. We model the risk-neutral Q dynamics of the deposit rate, and use it to capture the three shapes of the yield curve in Figure 2.

First, we summarise some basic data features in Figure 3: (1) the deposit rate is discrete and rfe{0, -0.1, -0.2, -0.3, -0.4,...} percentage point, and (2) the policy rate either stays where it is or moves down by 0.1%, which we formalise as follows.

The simplest model with a1t = implies one shape for the yield curve. See the left panel of Figure 9. This model is a slightly more flexible version of the existing model (see Wu and Xia (2016)), which imposes the restriction = 0. However, it cannot capture the

rich dynamics of the data shown in Figure 2. In particular, it cannot capture both a flat curve (left panel) and a downward sloping curve (middle and right panels).

To separate these two shapes, we introduce a binary variable A_{t}, which captures agents’ forecast of the ECB’s next move. A_{t} = 1 indicates a high probability of a cut next period, whereas A_{t} = 0 implies that monetary policy is more likely to stay put. We augment (2.1) with At:

and a^_{At=1} > a^_{A(=0} grants the interpretation of A_{t}.

We model the dynamics of A_{t} as a two-state Markov chain process:

If these probabilities are time-invariant, that is, a00_{t} = a00, a1_{1t} = a^_{L}, this model implies two different shapes for the yield curve: one for A_{t} =1 and one for A_{t} = 0. The middle panel of Figure 9 provides an example of the two shapes. The blue line captures that

the yield curve in the state At = 0 is flat, which corresponds to a long period of time in the data during which the short end of the yield curve is flat, such as the left panel of Figure 2. The red dashed line is for the state At = 1, which indicates a non-negligible probability of the deposit rate moving down. This can explain the shape in the middle panel of Figure 2. However, this model cannot capture the shape in the bottom panel of Figure 2. In this plot, the market expects no immediate cut but expects a higher probability of cuts at future meetings.

To accommodate this feature, we devise a separation between the immediate monetary policy stance A_{t} and the longer-term monetary policy stance At. At = 1 implies an easier monetary policy at longer horizons, whereas At = 0 implies a lower possibility of future cuts.

We introduce this channel by allowing the dynamics of the state variable At to depend on A[, and (2.3) becomes

Our final model, comprising (2.2), (2.4), and (2.5), can capture various shapes of the yield curve; see the right panel of Figure 9. A_{t} =1 corresponds to the case in which the market highly expects a cut in the next period (see the red dashed line and purple dash-dotted line), whereas A_{t} = 0 corresponds to no immediate cut in the coming month (see the blue solid line and yellow dotted line). At = 1 implies that the market expects cuts not necessarily immediately but in the future (see the yellow dotted and purple dash-dotted lines). When At = 0, agents do not anticipate much further cuts past the next month (see blue solid and red dashed lines). The combination of A_{t} = 0 and At = 1 mimics the shape in the right.

**3 A new shadow rate term structure model**

This section incorporates the deposit rate dynamics introduced in Subsection 2.2 to an SRTSM. Following Black (1995), the short-term interest rate r_{t} is the maximum function of the shadow rate *s _{t}* and a lower bound. The innovation of our paper is that the lower bound

is time varying:

Next, we describe how to model the lower bound and shadow rate, and then discuss bond prices.

**3.1 Deposit rate and lower bound**

The deposit rate is by definition the lower bound of the Euro OverNight Index Average (EONIA), and hence serves naturally as the lower bound of the Overnight Index Swap (OIS) curve based on EONIA. We use a discrete-time model with month-end observations as in much of the term structure literature. However, central banks do not meet at the end of a given month. For our ELB sample, the ECB meets 8 to 12 times a year, at most once a month, and the meeting dates range from the 1^{st} to the 27^{th} day of the month.

We incorporate this calendar effect when we model the lower bound. Suppose that the number of days between the end of the current month t and the next meeting date is a fraction y_{t} of the month from t to t + 1. When the ELB is binding, the monthly lower bound r_{t} is the average of the overnight deposit rate for the month:

Note that we only align the ECB’s meeting schedule with our monthly data for the current month, that is, as of time t,

We assume r_{t} = 0 if the economy is not at the ELB.

**3.2 Shadow rate and factors**

**3.3 Bond prices**

Following Wu and Xia (2016), we model forward rates rather than yields because of the simplicity of the pricing formula. Define the one-period forward rate f_{nt} with maturity n as the return of carrying a government bond from *t* + n to *t* + n +1 quoted at time t, which is

a simple linear function of yields:

**3.3.1 Forward rates with a constant lower bound**

**3.4 Forward rates in the new model**

Next, we derive the pricing formula of our new model. We begin by describing the distribu tion of the lower bound.

**3.4.1 Marginal distribution of the lower bound**

The probability distribution of interest for pricing purposes is the risk-neutral probability distribution of the lower bound n periods into the future Q_{t}(r_{t}+_{n}). It can be written as the sum of the joint distributions of the lower bound and A, A^{1} states:

The last equal sign is based on the assumptions in (2.2), (2.4) and (2.5), and the assumption that no covariances exist between the three variables. The three terms in (3.9) are specified in (2.2), (2.4) and (2.5).

**3.4.2 Pricing formula**

With the results in Section 3.4.1, the pricing formula in (3.5) becomes

where Q(r_{t}±_{n}) is specified in (3.7). Derivations are in Appendix A.1.

The forward rate in (3.10) differs from (3.5) due to the time-varying lower bound. The new pricing formula (3.10) prices in the uncertainty associated with the future dynamics of the lower bound. The forward rate is calculated as an average of forward rates with known r_{t}±_{n}, weighted by the risk-neutral probability distribution of r_{t}±_{n}. If r_{t}±_{n} were a constant, (3.10) would become (3.5). If we had to compute bond prices numerically, the model would not behave as well.

**4 Estimation**

**4.1 Data and estimation details**

Data We model OIS rates on EONIA with data obtained from Bloomberg. Our sample is monthly from July 2005 to June 2017. We date the ELB period when the deposit rate is zero and below starting from July 2012.

Spread The deposit rate is the floor of the EONIA rate. In our model, they are the same for the ELB sample. However, in the data the former is always lower than the latter. To capture this difference, we introduce a spread. The deposit rate is measured overnight. However, the overnight EONIA rate is very volatile due to some month-end effects. Therefore, we define the spread as the difference between the one-week EONIA rate and the overnight deposit rate: sp_{t} — rW^{eek} — r^{f}. Figure 5 plots the time-series dynamics of the one-week EONIA rate and the overnight deposit rate in the top panel and their difference at the bottom to demonstrate a non-zero and time-varying spread.

where /nt is in (4.2). The forward rates we model include /3,3,;, /5,5,;, /12,12,;, /24,12,;, /00,12,;, /84,12,^ ^{and} /108,12,;.

There are a couple of advantages of modeling forwards rates rather than yields. First, forward rates require summing over fewer terms, as per (4.4). Second, forward rates do not involve the “max” operator, which will be included in yields of any maturity. Having the “max” operator is problematic for any gradient-based numerical optimiser.

State space form The state variables X_{t}, A_{t}, and At are latent, whereas r^{f} and *sp _{t}* are observed. Our SRTSM is a nonlinear state-space model. The transition equations include (3.4), and the P version of (2.2), (2.4), (2.5) and (4.1), for which we assume the same process under the physics dynamics P and risk-neutral dynamics Q but with different parameters. The difference between them captures the risk premium.

Adding measurement errors to (4.3) and (4.4), the measurement equations are

where “o” superscript stands for observation, and the measurement errors are i.i.d. normal: nt,nnmt ~ *N*(0,w^{2}).

Normalisation The collection of parameters we estimate consists of four subsets: (1) parameters related to r^{f}, A_{t}, ssand At, including a_{1},A_{t}=o, a_{1},A_{t=1}, a_{00},_{A}£=o, a_{11},A* =o, a_{00},_{A}£_{=1},

^{a}11,A^=1^{, a}00^{, a}11 ^{and a}1^At=0^{, a}1^At = 1^{, a}00,_{A}l =_{0}^{, a}H,_{A}i =_{0}^{, a}00,_{A}l = p * ^{a}u_{A}i_{t}* =

parameters describing the dynamics of sp_{t}, including (^,_{sp}, ^Sp, p_{sp}, pSp, ^{a}_{sp})j (3) parameters related to X_{t}, including (^,, ^, p, p^, S,^_{0},^_{1}); and (4) the parameter for pricing error: w. For identification, we impose «1^= > _{At}=_{0} and «00 a£=_{0} > «00 a£=r The identifying restrictions on the group (3) are similar to Hamilton and Wu (2014): (i) ^_{1} = [1,1,1]', (ii)^ = 0, (iii) p« is diagonal with eigenvalues in descending order, and (iv) £ is lower triangular.

The eigenvalues of p, p^ indicate the factors X_{t} are highly persistent under both measures. This finding is consistent with the term structure literature. Both a_{1},A_{t}=_{0} and ay_{At}=_{0} are zero, which means that when A_{t} = 0, agents do not expect the deposit rate to change in the next period. When A_{t} = 1, the probability of the ECB cutting the deposit rate is much higher: a_{1},_{At}=_{1} = 1 under the physical measure, and «1,a=_{1} = 0-75 under the risk-neutral measure. The difference between the two measures reflects the risk premium. The At = 0 state is very persistent, with the probability of staying in this state (a_{00},A£, «00 a£) being 95% or 100% for At = 0, and 89% or 82% for At = 1. By contrast, the At = 1 state is much less persistent. The spread spt follows a persistent autoregressive process under both measures. Other parameters controlling level and scale are comparable to what we see in the literature.

**4.2 Filtered probabilities**

Figure 6 plots the filtered probabilities for each state. Blue is the dominant state.

**5 NIRP and the yield curve **

**5.1 Extracting the market’s expectations on the NIRP from the yield curve**

In this section, we extract market expectations of the NIRP from our SRTSM. Figure 7 plots the four actual cuts in blue vertical bars together with our model predictions in red crosses and Bloomberg’s survey expectations in black dots. On June 5 2014, the ECB cuts the rate from 0 to -0.1% for the first time. In May, our model predicts this event with more than a 50% probability. As a comparison, over 90% of the respondents to the Blomberg survey expected the cut. The second cut in September 2014 was a surprise to both economists and the market. The next two cuts from -0.2% to -0.3%, and then subsequently to -0.4%, were largely anticipated. For the rest of the meetings, market participants did not price in much of a probability of an immediate cut. This exercise confirms that market participants’

expectations are consistent with economists’ view.

The Bloomberg survey is conducted one week before the meeting. The yield curve, however, contains richer information, which also looks further into the future. In Figure 8, we further inspect for how long the market has anticipated some of the developments. It plots the market’s expectations h months before the four event dates for h = 0,1, 2,..., 6. The blue lines with crosses are the physical expectations E_{t-h}(r^{d}), the red lines with dots are the risk-neutral expectations E^__{h}(r^), and the then deposit rates *rf _{-h}* are in yellow dashed lines. The difference between the yellow lines and the other coloured lines captures an expected future cut. The difference between the blue and red lines captures the risk premium.

Consistent with Figure 7, the June 2014 and December 2015 cuts were anticipated one month ahead, whereas the September 2014 cut was completely unanticipated. The most interesting case is March 2016. A cut to -0.4% was expected four months before, when the actual rate was -0.2% under the risk-neutral expectation. Then agents revised up their

expectations for the next two months. Eventually, when h =1, agents fully priced in -0.4% for the next month.

**5.2 Policy counterfactual analyses**

Much of the existing literature has focused on whether and how much the negative interest rate policy has affected banks’ profitability; see, for example, Borio et al. (2015), Jobst and Lin (2016) and Creure (2016). Our paper evaluates this policy’s impact on the yield curve, which links financial markets to the macroeconomy.

We perform the following experiment in Figure 9: suppose that the central bank could make commitments to change At, and/or At, what would happen to the yield curve? We conduct this exercise at the end of our sample in June 2017, which, according to Figure 6,

has a probability of 99% in the blue state A_{t} = 0, A( = 0, where agents expect the central bank to stay put for both the short and long run. In our exercise, we assume that agents fully internalise the ECB’s announcement and deem it fully credible.

First, suppose that the ECB indicated an easing position at the next meeting, but promised that this cut would be the last one in history. Then the one month rate would decrease by 0.03% (see the red dashed line): A_{t} would move from 0 to 1, making the expected deposit rate one month from now 0.075% lower. In addition, the next meeting happens on 20 July, which is 0.6 of the month from the end of June to the end of July. The current level of the deposit rate would prevail for 60% of the month, and the lower deposit rate would happen for the next 40%. Therefore, 0.075% x 0.6 = 0.03%. The red curve is almost flat.

Second, if the central bank announced it would not make any move at the next meeting but the future environment would be expansionary overall, the change in the yield curve would be as in the yellow dotted line. The one month rate would not move, but yields at other maturities would decrease. The change would grow with the maturity up to two years, and then flatten out afterwards at about 0.1%.

**6 Yield curve implications**

*6.1 Model comparison*

Table 2 compares our model with several alternatives in terms of log likelihood values, information criteria, and measurement errors. The first column is our main model specification. The second columns is our model without A(. Columns 3 to 5 are benchmark shadow rate models commonly found in the literature, and the corresponding lower bounds are specified as the current deposit rate, 0, and -0.4%, respectively. The last column is the GATSM. See details in Appendix C.

Our main model has the highest likelihood value. It also provides the best overall fit to the forward curve with smaller measurement errors. All the evidence points to the conclusion that the data favor our main model over these alternative model specifications.

Figure 10 provides some visual evidence by comparing the observed data in red dots with various model-implied yield curves. When the ELB was not binding, all models fit the data similarly well (see the top left panel). When the yield curve has a flat short end at the beginning of the ELB, our main model and M&t provide a better fit than other models (see the top right panel). In theory, the benchmark shadow rate models Ms-tv, MS-0, and MS-.4 should have exhibited a similar performance. But in practice, because they ignore

the spread between the deposit rate and EONIA, there are discrepancies at the very short end. The GATSM is expected to perform poorly in this case, which is what motivates the entire literature on the SRTSM. Not able to fit the flat short end of the yield curve makes the GATSM one of the worst models; see Table 2.

In the bottom panels, none of the existing shadow rate models are able to generate a downward sloping short end mimicking the data when the ELB is binding. Intuitively, agents in these models are myopic, and do not expect further development of the policy rate. Both our main model and My_{t} are able to generate a downward slope through agents’ expectations that the future deposit rate might decrease further. However, *M^ _{t}* is not flexible enough to match the data for either February or July 2016. Our main model, which is motivated by

various shapes of the yield curve in Figure 2, fits the data well. Although the GATSM is able to fit the downward sloping short end, it does not provide an intuitive interpretation of the market’s expectation on the NIRP.

The term premium is one of the focal points for the term structure literature; see, for example, Duffee (2002), Wright (2011), Bauer et al. (2012, 2014) and Creal and Wu (2016). We compute the 10-year yield term premium for the euro area from our main model, and plot it in the blue solid line in Figure 11.

The term premia have trended down since 2009. At the ELB, we observe some negative term premia. This observation can mainly be attributed to the QE programmes, under which purchases of longer-term government bonds led to a reduction of yields through the term premium channel. For empirical evidence, see Gagnon et al. (2011), Krishnamurthy and Vissing-Jorgensen (2011) and Hamilton and Wu (2012a).

**7 Conclusion**

We have proposed a new shadow rate term structure model that captures the NIRP in the euro area. We model the discrete movement of the deposit rate with a simple and intuitive regime-switching model. To capture the rich dynamics at the short end of the yield curve, we introduce two latent state variables: one captures the immediate monetary policy stance, and the other captures the future monetary policy stance over longer horizons. We illustrate that the two do not always coincide, and that it is therefore useful to have both of the indicators. Compared with alternative models, including the various shadow rate term structure models proposed in the literature and the Gaussian affine term structure model, our new model best fits the data.

We use our model to extract the market’s expectations of the NIRP. Overall, such expectations agree with those of economists surveyed by Bloomberg. Importantly, the expectations extracted from our model are superior to those of the Bloomberg survey because they are available further into the future, whereas the Bloomberg surveys are only collected one week before monetary policy meetings. We find the that June 2014 and December 2015 cuts were expected one month before but that the September 2014 cut was entirely unanticipated. Most interestingly, the March 2016 cut was expected four months before the actual cut.

We then evaluate the NIRP’s impact on the yield curve with some counterfactual analyses.

We find that an immediate monetary policy expansionary in June 2017 would have decreased the one month rate by 0.03%. Taking no immediate action but promising an expansionary environment in the future would lower the yield curve by 0.1% at the two- to ten-year horizon. If the central bank could commit to an expansionary policy in both the short and long run, the impact would be the largest with the two-year yield decreasing by 0.2% and the long term one decreasing by 0.16%.

The 10-year term premium increased between 2005 and 2008. It has since trended down with negative numbers at the ELB, potentially due to QE purchases. The dynamics of the deposit rate contributes positively to the premium, but on a smaller order of magnitude.

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