Time-Varying Lower Bound of Interest Rates in Europe

Time-Varying Lower Bound of Interest Rates in Europe Jing Cynthia Wu Chicago Booth and NBER Fan Dora Xia Bank for International Settlements First draft: January 17, 2017 This draft: February 13, 2017 Abstract We propose a new shadow-rate term structure model to describe the negative interest rate environment in Europe. First, we model the discrete negative policy rate with a regime-switching process. Second, we incorporate a non-constant spread between the overnight policy rate and the short end of the government bond yield curve. These two components together serve as a time-varying lower bound. We price bonds in this model with forward-looking agents, and find both modeling components are crucial to match the yield data and generate sensible economic implications. Keywords: effective lower bound, shadow rate term structure model, regime-switching model, negative interest rate We thank Drew Creal and Jim Hamilton, and seminar and conference participants at Texas A&M University, University of Illinois Urbana-Champaign 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. The views expressed herein are those of the authors and not necessarily the views of the BIS. Correspondence: cynthia.wu@chicagobooth.edu, dora.xia@bis.org. 1

1 Introduction The zero interest-rate policy has been one of the most discussed economic issues of the decade. It has impacted many advanced economies, spreading from Japan to the United States, and then to Europe. The newest addition to enrich this policy toolkit is negative interest rates. That is, the European Central Bank (ECB) and Bank of Japan have further lowered their policy rates to negative numbers. Do negative interest rates further provide the monetary easing economies need? If so, how much? Our paper contributes to this discussion. We propose a new shadow-rate term structure model (SRTSM) to describe the negative interest rate environment and derive bond prices for this new model. The innovation is the time-varying lower bound for interest rates, which consists of two main ingredients. First, we allow a regime-switching process for the dynamics of the discrete negative deposit rate. In our model, agents are fully forward looking and build their future expectations about the policy lower bound into bond prices. We are the first to incorporate the forward-looking aspect of the time-varying lower bound into a model. This is crucial from both the theoretical and empirical perspective. Forward-looking decision-making is arguably the most important building block of modern finance. A model with a time-varying lower bound would be internally inconsistent if agents in the model were naive, never updated their beliefs, and always thought the lower bound would stay where it currently was even after seeing it move repeatedly. Empirically, we find forward-looking models fit the data better. Second, a non-constant spread exists between the overnight policy rate and the short end of the government bond yield curve. This fact has been overlooked in the shadow-rate term structure literature. We show empirically this is an inherent feature of the data, and ignoring this component could lead to a very poor fit, especially for the ELB sample. We apply our model to the Euro area, which has the most data points for negative interest rates. However, when more observations come in, our model can also be applied to other countries or regions. We compare our model with several alternative specifications 2

including the ones popular in the literature. We find our proposed model performs the best in terms of higher likelihood, lower AIC and BIC, and lower measurement errors. Both of the newly proposed modeling components are crucial to fit the forward curve or yield curve. Most pronounced differences are for the short end of the curve and at the ELB. The existing models in the literature, on the other hand, do poorly. Counterfactual policy analyses investigate the effect of a change in the ELB on the yield curve. We find a 10-basis-point drop in the ELB lowers the short end of the curve by the same amount. However, the effect tampers at longer maturities, and it decays to 6-7 basis points at the 10-year horizon. We also show the difference between forward-looking and naive agents. Forward-looking agents expect the lower bound to revert to its unconditional mean in the long run, and therefore the effect of a drop in the ELB is smaller at the longer end of the yield curve. By contrast, naive agents think the change in the lower bound is permanent and the lower bound will stay 10 basis points lower forever, and thus such a change has a more pronounced effect at the longer end. After a brief literature review, the rest of the paper proceeds as follows. Section 2 proposes a regime-switching model for the policy lower bound. Section 3 introduces the spread and develops the pricing formula for government bonds. Section 4 discusses estimation. Section 5 compares our new model with several alternatives and conducts some counterfactual policy analyses. Section 6 concludes. Literature Earlier work has applied the SRTSM mostly to the Japanese and US yield curve. 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 kept the lower bound at a constant level. A few studies have focused on the new development in Europe, where the policy lower bound has kept moving down to negative numbers after it entered the ELB. For example, 3

the online implementation of Wu and Xia (2016) for the Euro zone, and Lemke and Vladu (2016) and Kortela (2016). However, none of these works allows agents to be forward-looking in terms of the lower bound. Moreover, the literature completely overlooks the non-constant spread between the policy rate and the government bond yield curve. We show both of these components are crucial to fit the yield curve as well as generate sensible economic implications. 2 A regime-switching model for the policy lower bound In this section, we focus on the features of the data for the policy rate, and use them to inform a regime-switching model for the policy lower bound. Figure 1 shows that since July 2012, the ECB has lowered the deposit rate in blue to zero, and subsequently to negative numbers. The deposit rate r d t is a natural candidate for the policy lower bound r t. Before the economy was binding at the ELB, this lower bound was a constant at 0. The policy lower bound is marked by the black dotted line in Figure 1, and formalized as follows: r t = 0, if not ELB r t = rt d, if ELB. (2.1) Simple facts The first empirical observation from Figure 1 is that the policy lower bound is discrete and takes multiples of -10 basis points. In the data, we have observed it takes the value of 0, -10, -20, -30, and -40 basis points. In the model, we assume its maximum value is 0, and its minimum is R: r t {0, 10,..., R} basis points. (2.2) 4

Figure 1: Lower bound 4 3 deposit rate policy lower bound 2 1 0-1 2005 2007 2009 2011 2013 2015 Notes: Blue solid line: rate for deposit facility in Euro area. Black dotted line: policy lower bound. X-axis; time; Y-axis: interest rates in percentage points. Sample spans from September 2004 to November 2016. Another observation from Figure 1 is the policy lower bound only moves 10 basis points at a time, if it moves. Moreover, for most of the time, it stays constant. In the data, we only see the lower bound decrease. In the future, we expect the lower bound to eventually move back up to zero. Hence, r t r t 1 {r t 1 10, r t 1, r t 1 + 10}. (2.3) Momentum and directional state Equation (2.3) captures the persistence in the policy lower bound in the sense that the lower bound could stay where it was. Another type of persistence exists in terms of whether the policy lower bound or deposit rate is moving downward or upward. For example, when a broad downward trend occurs, as we experienced up until the end of our sample in November 2016, it is more likely to keep moving down than up. This demonstrates the momentum of the lower-bound movements. We formalize this idea by introducing t, which describes the direction in which the lower bound is moving. t = ±1, where t = +1 is the up state and t = 1 is the down state. The lower bound r t might stay constant for a period of time between two downward movements, and we still 5

Table 1: Conditional transition probability P (r t r t 1, t ) r t r t 1 R < rt 1 < 0 R 0 t = +1 r t 1 10 0 0 0 r t 1 π π 1 r t 1 + 10 1 π 1 π 0 t = 1 r t 1 10 1 π 0 0 r t 1 π 1 1 r t 1 + 10 0 0 0 refer to this period as the down state 1. More specifically, we define it as follows: Definition 1 When r d t > R, t = 1 if n > 0 such that r d t < r d t n and r d t = r d t j 0 < j < n; otherwise, t = +1. When r d t = R, t = ±1. Figure 1 demonstrates the down state is persistent or has momentum. This can be extend to the up state as well. We capture this persistence with the following dynamics: P ( t = t 1 t 1 ) = p, (2.4) where p 1 describes a down state is most likely followed by another down state, which is consistent with the data. P ( t t 1 t 1 ) = 1 p is the probability of changing the direction. Conditional dynamics of the lower-bound state Equation (2.3) and Definition 1 imply that if we are in the t = 1 state, the lower bound can only stay at where it was r t = r t 1 or move down by 10 basis points r t = r t 1 10. Similarly, if t = +1, then r t = r t 1 or r t = r t 1 + 10. We formalize this idea by specifying the conditional transition probability P (r t r t 1, t ) in Table 1. Per (2.3), the lower bound does not always move. We allow for a probability π that the lower bound stays the same, and π 1. Next, per (2.2), the lower bound is not allowed to go above 0 or below R. Therefore, when t = 1 and 6

r t 1 = R, all the mass concentrates at staying. We make r t = 0 an absorbing state, that is, r t = 0 if r t 1 = 0, to capture the long history of the lower bound at zero. Joint dynamics of r t, t The joint dynamics of r t, t follows: P (r t, t ) = r t 1, t 1 P (r t, t r t 1, t 1 )P (r t 1, t 1 ), (2.5) where the transition probability can be decomposed as follows: P (r t, t r t 1, t 1 ) = P (r t r t 1, t, t 1 )P ( t r t 1, t 1 ) = P (r t r t 1, t )P ( t t 1 ), (2.6) where the two components are defined in (2.4) and Table 1. 3 Government bond yield curve 3.1 Spread The main interest of the paper is to model the government bond yield curve, or equivalently forward curve, in the Euro area. Section 2 proposes a model for the time-varying deposit rate, serving as the policy lower bound. Using the deposit rate directly as the lower bound for the government bond yield curve might not be appropriate due to some institutional details. For example, short-dated government bonds have been traded below the deposit rate in the Euro area because government bonds issued by countries such as Germany and France are often posted as collateral for short-term borrowing in repo markets, and excessive cash in these economies leads to scarcity of these assets. Figure 2 plots the time series dynamics of the one-month-ahead one-month forward rate and deposit rate in the top panel, and their difference in the middle. At the beginning of the ELB period, they were very similar to each other. However, the difference grew over time. 7

Figure 2: Spread 0-0.5-1 1m forward rate deposit rate 2013 2014 2015 2016 0.2 0-0.2-0.4 spread 2013 2014 2015 2016 1.5 1 0.5 0-0.5 forward curve deposit rate -1 0 20 40 60 80 100 120 maturity Notes: Top panel: time series dynamics of the one-month-ahead one-month forward rate in blue solid line and deposit rate in red dashed line; middle panel: the difference between the forward rate and deposit rate. X-axis for these two panels: time. Sample spans from July 2012 to November 2016. Bottom panel: forward curve in November 2016 in blue solid line and deposit rate in dashed red line. X-axis for this panel: maturity in months. 8

It went as high as 20 basis points and as low as -40 basis points. The bottom panel plots the cross section of the forward curve in November 2016. The front end of the curve was at -80 basis points, whereas the deposit rate was only -40 basis points. Note the discrepancy between government bond interest rates and the deposit rate is not limited to the very short end. Rather, it extends to all the maturities. To capture this fundamental feature of the data, we introduce a spread into our model, and model the lower bound for the government bond as the sum of the policy lower bound and a spread: 1 r G t r G t = 0, if not ELB = r t + sp t, if ELB, (3.1) where sp t follows a random walk according to the feature of the data: sp t = sp t 1 + e t, e t N(0, σ 2 sp). (3.2) 3.2 Short rate, shadow rate, and factor dynamics Following Black (1995), the short-term interest rate r t is the maximum function of the shadow rate s t and a lower bound. Our paper differs in that the government bond lower bound r G t is time varying: r t = max ( s t, r G t ). (3.3) The shadow rate is an affine function of the latent yield factors, often labeled as level, slope, and curvature : s t = δ 0 + δ 1X t, 1 We estimated a constant lower bound using the sample from September 2004 to May 2014, and the estimate is essentially 0. 9

whose physical dynamics follow a first-order vector autoregression (VAR): X t = µ + ρx t 1 + Σε t, ε t N(0, I). (3.4) Similarly, the Q dynamics are X t = µ Q + ρ Q X t 1 + Σε Q t, ε Q t Q N(0, I). 3.3 Bond prices The pricing equation for the n-period zero-coupon bond at time t is P nt = E Q t [exp( r t )P n 1,t+1 ]. The n-period yield relates to the price of the same asset as follows: y nt = 1 n log(p nt). Following Wu and Xia (2016), we model forward rates rather than yields for 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: f nt = (n + 1)y n+1,t ny nt. Therefore, modeling forward rates is equivalent to modeling yields. In fact, their short ends are identical: f 0t = y 1t = r t. 10

The model implied forward rate is f nt rt+n ( P Q t (r t+n ) sp t + r t+n + σn Q g ( an + b nx t sp t r t+n σ Q n )), (3.5) where P Q t (r t+n ) is the risk-neutral probability distribution for the policy lower bound n- j=0 periods later, g(z) = zφ(z) + φ(z), (σn Q ) 2 = n 1 j=0 δ 1(ρ Q ) j ΣΣ (ρ Q ) j δ 1 + nσsp, 2 a n = δ 0 + ( n 1 ( δ ) ) ( 1 ρ Q j n 1 ( µ Q ) ) ( 1 2 δ 1 ρ Q j n 1 ( ΣΣ ) ) ρ Q j ( δ1, and b n = δ ) 1 ρ Q n. See Appendix B for the derivation. j=0 Next, we need to specify P Q t (r t+n ). The risk-neutral dynamics and the physical dynamics for the lower bound follow the same process as described in Section 2, but with different parameters. For the risk-neutral dynamics, we replace the probability in (2.4) with p Q and in Table 1 with π Q. Following similar logic as in Section 2, the risk-neutral probability that j=0 the lower bound is in state r t+n n-period in the future is P Q t (r t+n ) = P Q t (r t+n, t+n ) (3.6) t+n and = P Q t (r t+n, t+n ) P Q t (r t+n, t+n r t+n 1, t+n 1 )P Q t (r t+n 1, t+n 1 ), (3.7) r t+n 1, t+n 1 where the transition probability can be decomposed as follows: P Q t (r t+n, t+n r t+n 1, t+n 1 ) = P Q t (r t+n r t+n 1, t+n, t+n 1 )P Q t ( t+n r t+n 1, t+n 1 ) = P Q (r t+n r t+n 1, t+n )P Q ( t+n t+n 1 ). (3.8) The matrix form of the joint dynamics is in Appendix A. 11

3.4 Comparison with pricing formula in Wu and Xia (2016) When the lower bound is a constant r, the model-implied forward rate in (3.5) becomes the same as in Wu and Xia (2016): ( ) f nt = r + σ n Q an + b g nx t r, (3.9) σ Q n where ( σ Q n ) 2 = n j=0 δ 1(ρ Q ) j ΣΣ (ρ Q ) j δ 1. Equations (3.5) and (3.9) have the same basic structure: the forward rate equals a lower bound plus a conditional standard deviation times the g function. Inside the g function is the n-period forward rate from the Gaussian Affine Term Structure Model a n + b nx t minus the lower bound and normalized by the same conditional standard deviation. The differences between (3.5) and (3.9) are threefold. First, (3.5) replaces the constant lower bound in (3.9) with two components: the policy lower bound in the future r t+n and the spread sp t. As the spread follows a random walk, the expected value n-periods later is the same as today s realization. Second, the conditional standard deviation is different because of the uncertainty associated with the future spread. Third, the new pricing formula (3.5) also prices in uncertainty associated with the future policy lower bound. Because the policy lower bound is a discrete state variable, the forward rate is calculated as a weighted average of forward rates with known r t+n, weighted by the risk-neutral probability distribution of r t+n. 4 Estimation We construct 3- and 6-month-ahead and 1-, 2-, 5-, 7-, and 10-year-ahead 1-month forward rates for AAA-rated bond yields in the Euro area, the dataset is available on the ECB s website. 2 Our sample is monthly from September 2004 to November 2016. We date the ELB period when the deposit rate is zero and below, and it starts from July 2012. 2 These forward rates map into f nt in (3.5) with n = 3, 6, 12, 24, 60, 84, 120. 12

We express our SRTSM with a time-varying lower bound as a nonlinear state-space model. The transition equations include (2.5), (3.2), and (3.4), which describe the dynamics of the policy lower bound, spread, and the latent yield factors, respectively. (3.5) relates the model-implied forward rate f nt to the state variables. Adding measurement errors, the measurement equation for the observed forward rate f o nt becomes f o nt = rt+n ( P Q t (r t+n ) sp t + r t+n + σn Q g ( )) an + b nx t sp t r t+n + η nt, (4.1) σ Q n where the measurement error η nt is i.i.d. normal, η nt N(0, ω 2 ). We use maximum likelihood estimation with the extended Kalman filter. See Appendix C for details. We treat r t, t, and sp t as observables. r t and t are measured according to (2.1) and Definition 1. sp t is the difference between the one-month-ahead forward rate and deposit rate since the beginning of the ELB period. In our implementation, we set R = 100 basis points. The collection of parameters we estimate consists of three subsets: (1) parameters related to r t and t, including (p, π, p Q, π Q ); (2) σ sp describing the dynamics of sp t ; and (3) parameters related to X t, including (µ, µ Q, ρ, ρ Q, Σ, δ 0, δ 1 ). For identification, we impose restrictions on parameters in group (3) similar to Hamilton and Wu (2014): (i) δ 1 = [1, 1, 1], (ii)µ Q = 0, (iii) ρ Q is diagonal with eigenvalues in descending order, and (iv) Σ is lower triangular. Maximum likelihood estimates and robust standard errors (see Hamilton (1994)) are reported in Table 2. The eigenvalues of ρ and ρ Q indicate the factors X t are highly persistent under both measures, more so under the risk-neutral measure than the physical measure. This is consistent with the term-structure literature. The lower-bound dynamics are also persistent, and more persistent under Q than P. Under the physical measure, t has a 97% chance of staying the same, for example, from a down state to another down state. This probability is numerically 100% under the risk-neutral measure because in the observed data, agents only see the lower bound moving downward from bond prices. By contrast, in the 13

Table 2: Maximum likelihood estimates 1200µ -0.6282 0.6094-0.6988 1200µ Q 0 0 0 (0.4231) (1.4012) (1.2405) ρ 0.9413-0.0218-0.0707 ρ Q 0.9972 0 0 (0.0384) (0.0412) (0.0463) (0.0007) 0.0746 1.0820 0.2195 0 0.9694 0 (0.1376) (0.1563) (0.1990) (0.0047) -0.0841-0.1233 0.7449 0 0 0.9082 (0.1193) (0.1382) (0.1845) (0.0414) eig(ρ) 0.9681 0.9681 0.8322 1200δ 0 11.9915 (1.5819) p 0.9728 p Q 1.0000 (0.0134) (0.0077) π 0.8966 π Q 0.9651 (0.0566) (0.0061) 1200Σ 0.7586 0 0 (0.1321) -0.6381 1.0248 0 (0.1670) (0.4606) 0.1598-1.0080 0.3547 (0.2940) (0.4640) (0.1244) 1200σ sp 0.0461 (0.0056) 1200ω 0.0688 (0.0055) Notes: Maximum likelihood estimates with quasi-maximum likelihood standard errors in parentheses. Sample: September 2004 to November 2016. physical dynamics, the deposit rate had periods of moving upward prior to the ELB. The chance for the lower bound to stay where it was is 90% under the physical measure, and 97% under the risk-neutral measure. Other parameters controlling level and scale are comparable to what we see in the literature. 5 Lower bound and yield curve This section investigates the relationship between the lower bound and yield curve. We first show the importance of correctly specifying the lower bound process in Subsection 5.1. We 14

short description Table 3: Model specifications full description M1 full model The model is specified in Sections 2-3. The government bond lower bound r G t is time varying, consisting of the dynamics policy lower bound r t and a spread sp t as in (3.1). Agents are fully forward looking. M2 model w/o spread Everything else is the same as in the full model, but sp t = 0. M3 model w/o forward Everything else is the same as in the full model, but the agents looking M4 model w/o spread or at t think r G t+n is deterministically known and r G t+n = r G t n. forward looking This model combines the restrictions imposed in M2 and M3. The policy lower bound r t is still time varying. This is similar to the models used in the literature. See the online implementation of Wu and Xia (2016) for the Euro area, and Lemke and Vladu (2016), and Kortela (2016). M5 constant lower bound This model replaces the time-varying lower bound in M4 with a constant lower bound at -20 basis point. This is another benchmark model in the literature. For example, see Christensen and Rudebusch (2014), Wu and Xia (2016), and Bauer and Rudebusch (2016). compare our benchmark model with alternative specifications for the lower bound. Subsection 5.2 conducts counterfactual policy analyses to see the effect of changing ELB on the yield curve. 5.1 Model comparison We first show the importance of our new modeling ingredients by comparing our full model M1 with alternative model specifications described in Table 3. M2, M4, and M5 do not allow any spread between the policy rate and the yield curve. M3 - M5 exclude the forward-looking aspect of the lower bound, although M3 and M4 still allow the lower bound to dynamically track the deposit rate. M4 is one benchmark, consistent with the online implementation of Wu and Xia (2016) for the Euro zone, and similar to Lemke and Vladu (2016) and Kortela (2016). M5 is another benchmark, consistent with most SRTSM papers on the US yield curve. For example, see Christensen and Rudebusch (2014), Wu and Xia (2016), and Bauer and Rudebusch (2016). The observation equations and estimates for these models can be 15

Table 4: Model comparison M1 M2 M3 M4 M5 full sample likelihood 773 605 677 483 266 AIC -1490-1155 -1302-914 -480 BIC -1346-1011 -1168-781 -347 3m 3.4 6.5 3.8 8.7 13.1 6m 4.6 5.0 4.9 8.3 12.5 1y 5.8 6.5 6.3 9.7 14.0 2y 5.4 4.7 5.2 8.1 12.2 5y 7.5 7.6 7.6 7.5 8.6 7y 4.8 4.2 4.9 4.3 5.2 10y 8.7 8.6 8.7 8.3 9.0 ELB 3m 3.9 9.0 4.4 13.0 20.5 6m 5.4 6.0 6.1 12.4 19.7 1y 6.2 4.8 6.9 12.5 19.9 2y 6.0 4.5 5.8 11.9 18.7 5y 5.6 7.1 5.7 6.7 8.8 7y 3.6 4.2 3.7 4.0 5.5 10y 6.6 7.8 6.9 7.6 9.5 Notes: Top panel: full sample from September 2004 to November 2016; bottom panel: ELB sample from July 2012 to November 2016. Maturities span from 3 months to 10 years. First column: full model M1 with both spread and fully forward-looking agents; second column: M2 without spread; third column: M3 where agents are not forward looking in terms of the future dynamics of the lower bound; forth column: M4 without spread or forward looking; last column: benchmark model M5 with a constant lower bound at -20 basis points. Measurement errors are in basis points. We highlight the smallest measurement errors, AIC and BIC, and the highest log likelihood value. found in Appendix C.3. Table 4 compares these models in terms of measurement errors, likelihood values, and information criteria. Our full model M1 yields the highest likelihood value and the smallest AIC and BIC, where the information criteria penalize overparameterization. All the evidence points to the conclusion that the data favor our full model over these alternative model specifications. The full model also provides the best overall fit to the forward curve with the smaller measurement errors, and the difference is more prominent in the ELB sample. Figure 3 provides some visual evidence by comparing the observed data in blue dots with various model-implied curves. The left panel plots the average forward rates in 2006 when the ELB 16

Figure 3: Forward curves 4.5 4 3.5 data M1 M2 M3 M4 M5 2006 0.2 0.1 0-0.1-0.2 2016 3-0.3-0.4 2.5-0.5-0.6 2 10 20 30 40 50 60-0.7 10 20 30 40 50 60 Notes: Left panel: average observed and fitted forward curve in the year 2006; right panel: average observed and fitted forward curve in year 2016; blue dots: observed data; red solid line: full model M1 with both spread and fully forward-looking agents; black dotted line: M2 without spread; cyan dotted-dashed line: M3 where agents are not forward looking in terms of the future dynamics of the lower bound; magenta dashed line: M4 without spread or forward looking; green dotted line: benchmark model M5 with a constant lower bound. X-axis: maturity in month; Y-axis: interest rates in percentage points. was not binding, and all the models fit the data equally well. The right panel shows that in 2016, when the ELB was binding and the spread was 24 basis points on average, the full model M1 fits the short end of the term structure significantly better. Both Table 4 and Figure 3 show that compared to M2, M1 is doing better in the short end of the forward curve (also the long end for the ELB period). This finding demonstrates the importance of the time-varying spread, which allows the short end of the forward curve to be decoupled from the policy lower bound. This data feature is evident in Figure 2. For example, in 2016, the policy lower bound was at -0.4%, whereas the average three-monthahead forward rate was less than -0.6%. Without the spread, models M2 (black dotted line), M4 (magenta dashed line), and M5 (green dotted line) in Figure 3 all have difficulty fitting the three-month maturity. The advantage of M1 over M3 is consistent across all the maturities. The difference might 17

seem less. However, M1 with agents being fully forward-looking is a much better and more useful model than M3, because in M3, agents do not learn from experience and always think the lower bound will stay at the current level indefinitely even after seeing it move repeatedly. The importance of fully forward-looking agents can also be demonstrated as the difference between M2 and M4. The right panel of Figure 3 shows that although M2 is not able to fit the three-month-ahead forward rate, it does a better job fitting the six-month-ahead and one- to two-year-ahead forward rates, because the forward-looking aspect allows the curve to go down when maturity increases. This is not feasible in the benchmark model M4 or M5. Hence, the implied forward curves in magenta and green stay flat, missing the entire short end. Both of the benchmarks in the literature M4 and M5 are not doing nearly as well as our full model M1, in terms of all the metrics we are evaluating. They tend to have lower likelihoods, higher values fror information criteria, and larger fitting errors. In particular, measurement errors for the short end at the ELB reduced substantially, moving from M4 or M5 to M1-3. The overall conclusion is that both modeling components are crucial to model the term structure of the Euro area government bond yield curve. 5.2 Counterfactual policy analyses In this section, we conduct counterfactual policy analyses to study the impact of changing ELB on the yield curve. Specifically, we ask how the yield curve reacts if we lower the ELB by 10 basis points. Figure 4 plots such reactions across three different dates represented by different colors: blue represents changes in June 2014, red in June 2015, and yellow in June 2016. The left panel is the implied changes in our benchmark model M1, and the right panel is those in M3. By lowering the ELB by 10 basis points, the short end of the curve subsequently decreases by 10 basis points as well. This effect tampers when maturity increases. At the 10-year horizon, the change decays to about 6-7 basis points in our benchmark model in the left 18

Figure 4: Impact of lowering ELB on the yield curve -0.05-0.06 June 2014 June 2015 June 2016 M1-0.05-0.06 M3-0.07-0.07-0.08-0.08-0.09-0.09-0.1 2y 4y 6y 8y 10y -0.1 2y 4y 6y 8y 10y Notes: Changes in the yield curve by lowering the ELB by 10 basis points. Left: implied changes in M1; right: implied changes in M3. Blue: June 2014; red: June 2015; yellow: June 2016. X-axis: maturity in years; Y-axis: interest rates in percentage points. panel. The comparison between M1 on the left and M3 on the right demonstrates the importance of forward-looking agents. The effect on the longer end of the yield curve is less when agents are forward looking, because they expect the lower bound to revert to its unconditional mean in the long run, whereas naive agents in M3 think the 10-basis-point change in the lower bound is permanent. 6 Conclusion We have proposed a new shadow-rate term structure model, which captures the time-varying feature of the lower bound that has emerged recently in Europe and Japan. Two main aspects contribute to this time-varying lower bound. First, the deposit rate has moved in a discrete negative grid, and we capture this with a regime-switching model. In our model, agents are forward looking and anticipate future changes in the deposit rate. Consequently, they price bonds taking expected future changes in the policy lower bound into account. Second, a time-varying spread exists between the policy lower bound and the government yield curve capturing some institutional details. We find both ingredients crucially contribute to the fit 19

of the forward curve and yield curve. Compared to alternative specifications for the lower bound, our model fits the data the best. Our counterfactual policy analyses show a 10-basispoint decrease in the ELB lowers the short end of the yield curve by the same amount. The effect decays as the maturity increases, and it is 6-7 basis points at the 10-year horizon. 20

References Bauer, Michael D. and Glenn D. Rudebusch, Monetary Policy Expectations at the Zero Lower Bound, Journal of Money, Credit and Banking, 2016, 48 (7), 1439 1465. Black, Fischer, Interest Rates as Options, Journal of Finance, 1995, 50, 1371 1376. Christensen, J. H. E. and Glenn D. Rudebusch, Estimating shadow-rate term structure models with near-zero yields, Journal of Financial Econometrics, 2014, 0, 1 34. Hamilton, James D., Time Series Analysis, Princeton, New Jersey: Princeton University Press, 1994. and Jing Cynthia Wu, Testable Implications of Affine Term Structure Models, Journal of Econometrics, 2014, 178, 231 242. Ichiue, Hibiki and Yoichi Ueno, Estimating Term Premia at the Zero Bound : an Analysis of Japanese, US, and UK Yields, 2013. Bank of Japan Working Paper. Kim, Don H. and Kenneth J. Singleton, Term Structure Models and the Zero Bound: an Empirical Investigation of Japanese Yields, Journal of Econometrics, 2012, 170, 32 49. Kortela, Tomi, A shadow rate model with time-varying lower bound of interest rates, 2016. Bank of Finland Research Discussion Paper. Krippner, Leo, A Tractable Framework for Zero Lower Bound Gaussian Term Structure Models, August 2013. Australian National University CAMA Working Paper 49/2013. Lemke, Wolfgang and Andreea L Vladu, Below the zero lower bound: A shadow-rate term structure model for the euro area, 2016. Deutsche Bundesbank Discussion Paper. Wu, Jing Cynthia and Fan Dora Xia, Measuring the macroeconomic impact of monetary policy at the zero lower bound., 2016, 48 (2-3), 253 291. 21

Appendix A Joint Q dynamics of the lower bound This section derives the matrix form of (3.7) - (3.8). Define Its dynamics is given by ξ t+n t = P Q t ( t+n = +1; r t+n = 0) P Q t ( t+n = +1; r t+n = 10). P Q t ( t+n = +1; r t+n = R) P Q t ( t+n = 1; r t+n = 0) P Q t ( t+n = 1; r t+n = 10). P Q t ( t+n = 1; r t+n = R) ξ t+n t = Πξ t+n 1 t,. where [ Π = p Q Π +1 (1 p Q )Π +1 (1 p Q )Π 1 p Q Π 1 ] and Π +1 = Π 1 = 1 1 π Q 0 0... 0 0 0 π Q 1 π Q 0... 0 0 0 0 π Q 1 π Q... 0 0 0 0 0 π Q... 0 0...... 0 0 0 0... π Q 1 π Q 0 0 0 0... 0 π Q 1 0 0 0... 0 0 0 π Q 0 0... 0 0 0 1 π Q π Q 0... 0 0 0 0 1 π Q π Q... 0 0....... 0 0 0 0... π Q 0 0 0 0 0... 0 1 Appendix B Deriving pricing formula First, P Q t (s t+n sp t+n ) N ( ā n + b nx t sp t, (σ Q n ) 2), 22

where ā n δ 0 + δ 1 The first term is ( n 1 ( j=0 ρ Q ) ) j µ Q. As shown in Wu and Xia (2016), the forward rate is ( f nt E Q t [r t+n] 1 [ n ] Var Q t r t+j Var Q t 2 j=1 E Q t [r t+n] = E Q t [max(r t+n + sp t+n, s t+n )] = E Q t [max(r t+n, s t+n sp t+n ) + sp t+n ] [ n 1 ] ) r t+j. = r t+n P Q t (r t+n)e Q t [max(r t+n, s t+n sp t+n ) r t+n ] + E Q t (sp t+n) = r t+n P Q t (r t+n) ( r t+n + σ Q n g j=1 (ān + b nx t sp t r t+n σ Q n )) + sp t, where the derivation for the last equal sign follows Wu and Xia (2016). The second term is ( [ n ] [ n 1 1 ] ) Var Q t r t+j Var Q t r t+j 2 j=1 j=1 ( P Q t (s t+n sp t+n r t+n ) 1 [ n ] [ n 1 ] ) Var Q t s t+j Var Q t s t+j 2 j=1 j=1 = ( P Q t (r t+n)p Q t (s t+n sp t+n r t+n r t+n ) 1 [ n ] Var Q t s t+j Var Q t 2 r t+n j=1 = (ān P Q t (r + b ) nx t sp t r t+n t+n)φ (ā n a n ), r t+n σ Q n [ n 1 ] ) s t+j j=1 where the first approximation sign and last equal sign follow Wu and Xia (2016). Adding them together yields (3.5): f nt r t+n P Q t (r t+n) = r t+n P Q t (r t+n) ( r t+n + σ Q n g ( sp t + r t+n + σ Q n g where the approximation follows Wu and Xia (2016). ( an + b nx t sp t r t+n σ Q n )) + sp t ( an + b nx t sp t r t+n σ Q n )), 23

Appendix C Estimation Appendix C.1 Likelihood for observed state variables The observed state variables include (r t, t, sp t ). The likelihood for r t and t is P (r 1:T, 1:T r 0, 0 ) = Π T t=1p (r t, t r 0:t 1, 0:t 1 ) = Π T t=1p (r t, t r t 1, t 1 ) = Π T t=1p (r t t, r t 1 )P ( t t 1 ) = p N 1 (1 p) T N 1 π N 2 (1 π) ( T N 2 ), where N 1 is the number of periods that t does not change, and N 2 is the number of periods that r t does not change; T is the total number of periods, T is the number of periods rt 0. The log likelihood value is L 1 = log(p (r 1:T, 1:T r 0, 0 )) = N 1 log(p) + (T N 1 )log(1 p) + N 2 log(π) + ( T N 2 )log(1 π). Taking first-order derivatives of the log likelihood with respect to p and π and setting them to 0 yields The log likelihood for sp t is ˆp = N 1 T ˆπ = N 2 T. L 2 = log(p (sp 1:T )) ( = T τ log(2π) + log(σ 2 2 sp) + (T ) τ) 1 T t=τ (sp t sp t 1 ) 2 σsp 2, where τ marks the beginning of the ELB. Appendix C.2 Extended Kalman filter The transition equation for latent yield factors is in (3.4). Stack the observation equation for all seven maturities, and express it as follows: F o t = F (X t, r t, t, sp t ) + η t η t N(0, ω 2 I 7 ). 24

Approximate the conditional distribution of X t with X t F o 1:t, r 1:t, 1:t, sp 1:t N( ˆX t t, P t t ), update ˆX t t and P t t as follows: with the matrices defined as ˆX t t = ˆX t t 1 + K t (F o t ˆF o t t 1 ), P t t = ( I 3 K t H t) Pt t 1, ˆX t t 1 = µ + ρ ˆX t 1 t 1, P t t 1 = ρp t 1 t 1 ρ + ΣΣ, ˆF t t 1 o = F ( ˆX t t 1, r t, t, sp t ), ( F (X t, r H t = t, t, sp t ) X t, Xt= t t 1) ˆX K t = P t t 1 H t ( H t P t t 1 H t + ωi 7 ) 1. Given the initial values ˆX 0 0 and P 0 0, we can update { ˆX t t, P t t } T t=1 algorithm. The log likelihood is recursively with the above L 3 = log(p (F1:T o r 1:T, 1:T, sp 1:T )) = 7T 2 log2π 1 T log H 2 tp t t 1 H t + ω 2 I 7 1 2 t=1 T (Ft o F ( ˆX t t 1, r t, t, sp t )) ( H tp t t 1 H t + ω 2 ) 1 I 7 (F o t F ( ˆX t t 1, r t, t, sp t )). t=1 The likelihood of the model is the sum of the three components: Appendix C.2.1 L = log(p (F o 1:T, r 1:T, 1:T, sp 1:T r 0, 0 )) = L 1 + L 2 + L 3 Application to our full model For our full model (M1), F (X t, r t, t, sp t ) is obtained by stacking (3.5) for all seven maturities, and H t stacks ( ) r P Q t+n t (r a n+b t+n)φ ˆX n t t 1 sp t r t+n b n for all maturities. Appendix C.3 M2-M5 σ Q n Observation equations and their derivatives for estimating M2-M5 specified in Table 3 are described in Table C.1, where ( σ n Q ) 2 = n j=0 δ 1 (ρq ) j ΣΣ (ρ Q ) j δ 1 and r = 20 basis points. Estimates and standard errors for M2 to M5 are in Table C.2. 25

Table C.1: Observation equations for M2-M5 model observation equation ( ( )) M2 r t+n P Q t (r t+n ) r t+n + σ n Q an+b g n Xt r t+n ( σ n Q ) M3 sp t + r t + σ n Q an+b g n Xt spt r t ( σ n Q ) M4 r t + σ n Q an+b g nx t r t ( σ n Q ) M5 r + σ n Q g an+b n Xt rc model σ Q n derivative ( an+b n ˆX t t 1 r t+n M2 r t+n P Q t (r t+n )Φ ( σ n Q ) an+b M3 Φ n X t t 1 sp t r t b σ n Q n ( ) an+b M4 Φ nx t t 1 r t b σ n Q n ( ) an+b M5 Φ n X t t 1 r b n σ Q n ) b n Notes: Top panel: observation equations for M2-M5, which are components of F (X t, r t, t, sp t ); bottom panel: derivatives, components of H t. 26

Table C.2: Maximum likelihood estimates for M2-M5 M2 M3 M4 M5 1200µ -0.1361 0.3941-0.4169-0.2628-0.8834 1.0317-0.1005-1.7796 2.1569-0.0899-0.5583 0.6505 (0.1626) (1.5434) (1.5532) (0.3301) (1.9501) (3.0911) (0.3856) (3.0016) (2.9599) (0.2202) (2.0606) (2.1997) ρ 0.9817-0.0230-0.0338 0.9764 0.0349 0.0134 0.9810-0.0006-0.0103 0.9760-0.0094-0.0204 (0.0235) (0.0283) (0.0320) (0.0284) (0.0626) (0.0784) (0.0403) (0.0252) (0.0307) (0.0336) (0.0282) (0.0305) 0.0930 1.4783 0.6230-0.0743 1.0326 0.1214-0.1932 1.0988 0.1592-0.0343 1.2312 0.3019 (0.2097) (0.4990) (0.5224) (0.2073) (0.2729) (0.3587) (0.4730) (0.2766) (0.3087) (0.3227) (0.3603) (0.3929) -0.0890-0.4738 0.3909 0.0902-0.0713 0.8447 0.2557-0.0914 0.8572 0.0612-0.2404 0.6932 (0.2089) (0.4903) (0.5193) (0.3243) (0.3055) (0.4117) (0.5023) (0.2955) (0.3313) (0.3522) (0.3672) (0.4036) eig(ρ) 0.9866 0.9640 0.9002 0.9882 0.9337 0.9337 1.0130 0.9637 0.9637 1.0013 0.9506 0.9506 ρ Q 0.9974 0 0 0.9975 0 0 0.9982 0 0 0.9980 0 0 (0.0005) (0.0013) (0.0016) (0.0006) 0 0.9584 0 0 0.9676 0 0 0.9576 0 0 0.9590 0 (0.0024) (0.0075) (0.0041) (0.0023) 0 0 0.9448 0 0 0.9180 0 0 0.9458 0 0 0.9494 (0.0100) (0.0439) (0.0065) (0.0038) 1200δ0 9.0073 12.4287 10.3400 9.0957 (1.2126) (3.4151) (5.2913) (1.2376) p 0.9728 0.9728 0.9728 0.9728 (0.0134) (0.0134) (0.0134) (0.0134) π 0.8966 0.8966 0.8966 0.8966 (0.0566) (0.0566) (0.0566) (0.0566) p Q 1.0000 (0.0076) π Q 0.6932 (0.0595) 1200Σ 0.4630 0 0 0.6974 0 0 0.3851 0 0 0.3322 0 0 (0.1471) (0.2205) (0.0700) (0.0430) -1.5732 2.6851 0-0.5593 1.1630 0-1.3431 2.8642 0-1.4416 2.6457 0 (0.9777) (1.5979) (0.3099) (0.7845) (1.3876) (2.1027) (0.8133) (1.0708) 1.3356-2.6554 0.3425 0.1102-1.1611 0.3476 1.1028-2.8782 0.3336 1.1830-2.6583 0.2976 (0.9959) (1.6150) (0.0866) (0.3319) (0.7938) (0.1247) (1.3410) (2.0966) (0.0774) (0.8663) (1.1037) (0.0532) 1200σsp 0.0456 0.0456 0.0456 0.0456 (0.0054) (0.0054) (0.0054) (0.0054) 1200ω 0.0778 0.0703 0.0959 0.1307 (0.0059) (0.0053) (0.0079) (0.0135) Notes: Maximum likelihood estimates with quasi-maximum likelihood standard errors in parentheses. Sample: September 2004 to November 2016. 27