Bond Risk Premiums at the Zero Lower Bound

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Bond Risk Premiums at the Zero Lower Bound Martin M. Andreasen, Kasper Jørgensen, and Andrew Meldrum 2019-040 Please cite this paper as: Andreasen, Martin M., Kasper Jørgensen, and Andrew Meldrum (2019). “Bond Risk Premiums at the Zero Lower Bound,” Finance and Economics Discussion Series 2019-040. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.040. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Bond Risk Premiums at the Zero Lower Bound Martin M. Andreasen, Kasper Jłrgensen, and Andrew Meldrum (cid:3) May 14, 2019 Abstract This paper documents a signi(cid:133)cantly stronger relationship between the slope of the yield curve and future excess bond returns on Treasuries from 2008-2015 than before 2008. Thisnewpredictabilityresultisnotmatchedbythestandardshadowratemodel with Gaussian factor dynamics, but extending the model with regime-switching in the (physical) dynamics of the factors at the lower bound resolves this shortcoming. The modelisalsoconsistentwiththedownwardstrendinsurveysonshortrateexpectations at long horizons, but requires a break in the level of its factors to closely (cid:133)t the low level of these surveys since 2015. Keywords: Dynamic term structure model, bond return predictability, shadow rate model, structural break, regime-switching. JEL: E43, E44, G12. Andreasen, mandreasen@econ.au.dk, Aarhus University, CREATES, and Danish Finance Institute; (cid:3) Jłrgensen, kasper.joergensen@frb.gov, Board of Governors of the Federal Reserve System; Meldrum, andrew.c.meldrum@frb.gov, Board of Governors of the Federal Reserve System. We thank Joachim Grammig, Thomas B. King, and Cynthia Wu for useful comments and discussions. The analysis and conclusions are those of the authors and do not indicate concurrence by the Board of Governors of the Federal Reserve System or other members of the research sta⁄of the Board. 1

1 Introduction One of the most widely used dynamic term structure models (DTSMs) in recent years is the shadow rate model (SRM) proposed by Black (1995) (see Kim and Singleton (2012), Christensen and Rudebusch (2015), Bauer and Rudebusch (2016), Wu and Xia (2016), Andreasen and Meldrum (2018), among others). The popularity of the SRM stems from the fact that it enforces the zero lower bound (ZLB) on nominal bond yields, while preserving a (near-)linear relationship between bond yields and the pricing factors away from the ZLB. However, in this paper we show that the standard SRM when applied to the U.S. has two important shortcomings, which we address by proposing a new SRM. The (cid:133)rst shortcoming of the standard SRM that we document is an inability to match a shift in bond return predictability at the ZLB. Beginning with the seminal studies of Fama and Bliss (1987) and Campbell and Shiller (1991), regressions of excess bond returns on the slope of the yield curve have received substantial attention in the (cid:133)nance literature. Dai and Singleton (2002) suggest to use such return regressions as a diagnostic test for DTSMs by requiringthatawell-speci(cid:133)edmodelshouldimplypopulationregressioncoe¢ cientsthatmatch the regression coe¢ cients in the data. We document a shift in these bond return regressions at the ZLB and therefore propose a modi(cid:133)ed diagnostic test for DTSMs. Speci(cid:133)cally, we arguethatawell-speci(cid:133)edZLB-consistentDTSMshouldmatchtheslopecoe¢ cientsinthese predictability regressions when the short rate is close to the ZLB and when it is away from the lower bound. We show that the standard SRM matches these slope coe¢ cients away from the lower bound (as is well known), but that the model is unable to match the shift in these slope coe¢ cients at the ZLB. This suggests that simply enforcing the ZLB by truncating the short rate at zero in a Gaussian model is not su¢ cient to properly capture the change in bond yield dynamics that occurred at the ZLB. We then consider whether two extensions of the standard SRM can match the shift in bond return predictability at the ZLB. The (cid:133)rst extension introduces regime-switching 2

in the SRM by allowing the (physical) dynamics of the pricing factors to change at the lower bound. This extension is motivated by a desire to accommodate the e⁄ects of various unconventional monetary policy measures that are largely expected to have a temporary e⁄ect on bond yields. The second extension we consider allows for a permanent break in the (physical) dynamics of the pricing factors in 2008 to accommodate long-lasting e⁄ects of recentdevelopments, suchasapossible"secularstagnation"asproposedbySummers(2015). The SRM with regime-switching is able to match the shift in bond return predictability at the ZLB, whereas the SRM with a permanent break does not improve on the performance of the standard SRM. The second shortcoming of the standard SRM that we document is its inability to match the low level of long-horizon short rate expectations in surveys since 2015. If we include surveys when estimating DTSMs, as suggested by Kim and Orphanides (2012), we (cid:133)nd that the standard SRM broadly matches the downward trend in short rate expecations at long horizons since 1990, but that the model cannot explain the low level of these surveys from 2015 and onwards. This could indicate that the standard SRM fails to match a recent decline in the natural rate of interest, as documented by Laubach and Williams (2016), Del Negro, Giannone, Giannoni and Tambalotti (2017), Christensen and Rudebusch (2019), among others. Unfortunately, neither of the two extensions that we consider are able to improve on the performance of the standard SRM in terms of matching these low short rate expectations. However, a further extension to the regime-switching model that incorporates a permanent break in the level of the (physical) pricing factors in 2015 is able to match the recent low level of short rate expecations in surveys. The remainder of this paper proceeds as follows. Section 2 documents the shift in bond return predictability during the recent ZLB period. Section 3 shows that the standard SRM cannot match this new empirical (cid:133)nding. The two modi(cid:133)cations of the standard SRM are presented in 4. Section 5 studies the ability of the considered SRMs to match long-horizon short rate expectations from surveys. Section 6 discusses the timing of a permanent break in 3

the SRM and introduces a permanent break in the level of the pricing factors for the SRM with regime-switching. Concluding comments are provided in Section 7. 2 Shift in Bond Return Predictability at the ZLB This section documents a signi(cid:133)cant shift in the ability of the yield spread to predict future excess bond returns during the recent ZLB period. We proceed by showing instability in the classic bond return predictability regressions in Section 2.1, while Section 2.2 presents our new empirical (cid:133)nding about the nature of this instability. Various robustness checks are provided in Section 2.3. 2.1 Instability in the Classic Return Predictability Regression The risk premium implied by a long-maturity bond is often measured by its expected return in excess of the return on a short-maturity bond. These expected returns are not directly observable and therefore are typically estimated by regressing realized excess bond returns rx(n) on a set of predictors z . That is, by running the regression t;t+h t rx(n) = (cid:12)(n) +(cid:12)(n)z +"(n) ; (1) t;t+h 0 0 t t;t+h where "( t; n t ) +h is a residual. Here, rx( t; n t ) +h (cid:17) p( t+ n (cid:0) h h) (cid:0) p( t n)+p( t h) denotes the excess return on an n-month bond relative to an h-period bond between time t and t+h, with p(n) being the log t price at time t of a zero-coupon bond with n periods to maturity. A natural benchmark is the expectations hypothesis, which implies (cid:12)(n) = 0 and hence no bond return predictability. One of the most prominent and robust predictors of excess bond returns is the slope of the yield curve, as used in the seminal work of Campbell and Shiller (1991).1 Letting y(n) t denote the yield at time t on an n period zero-coupon bond, the speci(cid:133)cation in Campbell 1A number of other yield curve and macroeconomic variables have recently been shown to predict excess returns (see, for example, Cochrane and Piazzesi (2005), Ludvigson and Ng (2009), Joslin, Priebsch and Singleton (2014), Cieslak and Povala (2015), and Bauer and Rudebusch (2017)). 4

and Shiller (1991) is equivalent to rx(n) = (cid:12)(n) +(cid:12)(n) y(n) y(h) +"(n) ; (2) t;t+h 0 1 t (cid:0) t t;t+h (cid:16) (cid:17) where the yield spread y(n) y(h) measures the slope of the yield curve. t t (cid:0) However, the recent (cid:133)nancial crisis has generated several unusual developments in the bond market that are likely to have a⁄ected the relationship between excess bond returns and the yield spread. The most obvious is probably that U.S. short-term interest rates were constrained by the ZLB from late 2008 to late 2015. This introduced an obvious nonlinearity in the yield curve that most likely caused the yield spread y(n) y(h) to be smaller than it t t (cid:0) would otherwise have been, because the short-term yield y(h) was constrained from below. t Such a slope compression is likely to have increased (cid:12)(n) because a given yield spread during 1 the ZLB period carried a stronger signal about future bond returns than before the (cid:133)nancial crisis. A second important development was the Large-Scale Asset Purchase (LSAP) programs that the Federal Reserve used during the ZLB period to stimulate the U.S. economy. These programs were mainly designed to temporarily lower long-term yields by reducing bond risk premiums, as discussed in D(cid:146)Amico, English, Lopez-Salido and Nelson (2012), Li and Wei (2013),andBonis,IhrigandWei(2017). Thus,theLSAPprogramsarelikelytohavea⁄ected both the level of excess bond returns and the yield spread, implying that the coe¢ cients in equation (2) may be a⁄ected. A third development is that the large (cid:133)nancial shock in 2008 may permanently have reduced expectations about the long-term level of the short rate due to concerns about a secular stagnation in the U.S. (see, for instance, Summers (2015)). Such a shift in expectations seems also capable of a⁄ecting the coe¢ cients in equation (2). Given these considerations, we formally test whether the coe¢ cients in equation (2) are constantacrosstimeusingamonthlysampleofU.S.bondyieldsfromG(cid:252)rkaynaketal.(2007) 5

Figure 1: Chow Tests and Historical Bond Yields The top chart reports the Chow test statistic for breaks in equation (2) using at least 15 percent of the observations for the pre- and post-break sample. The reported 95 percent critical value is for the maximum Chow test of Andrews (1993). The bottom chart shows selected historical bond yields. The 5- and 10-year yields are end-month from G(cid:252)rkaynak, Sack and Wright (2007) starting in January 1990 and ending in December 2018. Rolling Chow Test Statistics 60 3-year 5-year 7-year 40 10-year Andrews (1993) 95% Critical Value 20 0 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Percent Selected U.S. Interest Rates 10 Effective federal funds rate 5-year 10-year 5 0 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 between January 1990 and December 2018, adopting a 1-year holding period (h = 12). The top chart in Figure 1 shows the Chow test statistics for the 3-, 5-, 7-, and 10-year bond yield for all possible break points after January 1990, reserving at least 15 percent of the observations for the pre- and post-break sample. The reported 95 percent critical value is for the maximum Chow test of Andrews (1993). We (cid:133)nd that the Chow test statistics increase strongly around 2008 for all bond yields and generally reach their highest level between 2008 and 2009. The bottom chart in Figure 1 shows that this break in equation (2) coincides with the short rate approaching the ZLB during 2008. Thus, the classic predictability regressions in equation (2) display evidence of a break around 2008 when the short rate in the U.S. approached the ZLB. 6

2.2 Formalizing the Shift in Bond Return Predictability The next step is to examine the nature of this break in the classic return predictability regression. The (cid:133)rst two of the unusual bond market developments mentioned above (that is, the ZLB episode and the LSAP programs) were speci(cid:133)c to the period between 2008 and late 2015. These developments seem therefore likely to only have had an e⁄ect on bond yields that is largely contained to that period. Hence, a natural extension of equation (2) is to allow for a threshold e⁄ect depending on the level of the short rate. That is, to consider a regression of the form rx(n) = (cid:12)(n) +(cid:12)(n) + (cid:12)(n) +(cid:12)(n) y(n) y(h) +"(n) ; (3) t;t+h 0;1If rt (cid:21) c g 0;2If rt<c g 1;1If rt (cid:21) c g 1;2If rt<c g t (cid:0) t t;t+h (cid:16) (cid:17)(cid:16) (cid:17) where is an indicator function that takes a value of 1 at time t if the short rate r c If rt (cid:21) c g t (cid:21) and 0 otherwise. As a result, equation (3) allows the regression coe¢ cients in equation (2) to shift when r becomes su¢ ciently low. t On the other hand, concerns about secular stagnation and a lower long-term level for the short rate are not speci(cid:133)c to the ZLB period and may therefore a⁄ect bond yields for several years after the ZLB period. Hence, an alternative extension of equation (2) is to allow for a permanent break at time (cid:28). That is, to consider a regression of the form rx(n) = (cid:12)(n) +(cid:12)(n) + (cid:12)(n) +(cid:12)(n) y(n) y(h) +"(n) ; (4) t;t+h 0;1If t<(cid:28) g 0;2If t (cid:21) (cid:28) g 1;1If t<(cid:28) g 1;2If t (cid:21) (cid:28) g t (cid:0) t t;t+h (cid:16) (cid:17)(cid:16) (cid:17) where is anindicatorfunctionthat takes avalueof 1at timet if t < (cid:28) and0otherwise. t<(cid:28) If g Given that there is only one ZLB period for the postwar U.S. economy in our sample, it is hard to tell which of the nonlinear speci(cid:133)cations in equations (3) and (4) we should prefer. To avoid taking a view on the best speci(cid:133)cation at this stage, we simply estimate the standard predictability regression in equation (2) using two separate samples. The (cid:133)rst sample ("Regime 1") runs from January 1990 to September 2008, excluding December 2003 7

Figure 2: Bond Return Predictability (n) (n) (n) (n) (h) (n) The top chart reports the slope coe¢ cient in rx = (cid:12) + (cid:12) y y + " with t;t+h 0 1 t (cid:0) t t;t+h h = 12 when estimated from January 1990 to September 2008 (exclud(cid:16)ing Decemb(cid:17)er 2003) where r 0:01, and when estimated from October 2008 to November 2015 where r < 0:01. The middle t t (cid:21) chart reports the di⁄erence in the slope estimates at a given maturity. The 95 percent con(cid:133)dence interval for these di⁄erences are computed using a block bootstrap with 5,000 repetitions and a block window of 24 months. In each bootstrap sample it is required that there are at least 50 observations where the short rate is above and below 1 percent, respectively. The bottom chart showstheincreaseintheR2 whenallowingtheinterceptandslopecoe¢ cientstochangeduringthe ZLB period. The x-axes reports maturity in years, and all yields are end-month from G(cid:252)rkaynak et al. (2007) with r measured by the e⁄ective federal funds rate. t Slope Coefficients (n) in Sub-Samples 1 15 r 1% (Jan. 1990-Sept. 2008) t 10 r < 1% (Oct. 2008-Nov. 2015) t 5 0 -5 2 3 4 5 6 7 8 9 10 Differences in (n) between Sub-Samples 1 15 Difference 10 95% Confidence Interval 5 0 -5 2 3 4 5 6 7 8 9 10 R2 Statistics 0.3 No break Break in Oct. 2008 0.2 0.1 0 2 3 4 5 6 7 8 9 10 when the short rate was slightly below 1 percent. The second sample ("Regime 2") runs from October 2008 to November 2015, which is the period where the short rate was close to the ZLB. Because the (cid:133)rst sample includes only observations before October 2008 with a short rate above 1 percent, and the second sample includes only observations after October 2008 with a short rate below 1 percent, the estimates are consistent with either equation (3) with c = 0:01 or equation (4) with (cid:28) = 225 (October 2008 is the 225th month of our sample). 8

We estimate the regressions for both regimes using monthly observations of yields and excess returns on bonds with maturities from 2 to 10 years using a 1-year holding period as above. The gray circles in the top chart of Figure 2 for the pre-ZLB period reveal the usual empirical pattern that the slope coe¢ cients in equation (2) are positive and increase with maturity. Our new empirical (cid:133)nding is that these slope coe¢ cients are larger and increase faster with maturity during the recent ZLB period (the gray triangles) when compared to the pre-ZLB period. The middle chart in Figure 2 shows that these di⁄erences in the slope coe¢ cients are signi(cid:133)cant at the 5 percent level for maturities beyond 3 years. We also note from the bottom chart in Figure 2 that this shift in the regression coe¢ cients in equation (2) is accompanied by sizable boosts in the explained variation, as measured by the R2 statistic, that increase gradually with maturity. For the 10-year bond yield, the R2 doubles from about 10 to 20 percent, showing that the shift in the regression coe¢ cients between the two regimes explains a substantial proportion of the variation in excess bond returns. Thus, recent developments in the bond market imply a shift in bond return predictability with a much stronger relationship between the yield spread and excess bond returns than observed in the pre-ZLB period. 2.3 Robustness We examine the robustness of this shift in bond return predictability in Table 1, where (cid:1)(cid:12)(n) (cid:12)(n) (cid:12)(n) refers to the change in the slope coe¢ cient in equation (2). The (cid:133)rst 1 (cid:17) 1;2 (cid:0) 1;1 column considers biannual bond returns (h = 6) instead of the annual horizon used so far. The shift in the slope coe¢ cients is again positive, and it is signi(cid:133)cant beyond the 4-year maturity. Our new (cid:133)nding is also robust to measuring the slope of the yield curve by the second principal component (PCA2) of bond yields, as shown by the second column in Table 1. The next column shows that the shift in bond return predictability is also robust to replacing the yield spread by the forward spread f(n;h) y(h) , as in Fama and Bliss (1987), t t (cid:0) where the forward rate is given by f(n;h) = log P(n+h)=P(n) . The fourth robustness check t t t (cid:16) (cid:17) 9

reportedinTable1addstheChicagoFedNationalActivityIndex(CFNAI)asinJoslinetal. (2014) and the in(cid:135)ation trend factor of Cieslak and Povala (2015) as additional regressors in equation (2) to control for the e⁄ect of macro variables. Table 1 shows that we again (cid:133)nd a positive and signi(cid:133)cant shift in the ability of the yield spread to predict future bond returns during the ZLB period. Table 1: Bond Return Predictability: Robustness Column(1)estimatesequation(2)withh=6. Column(2)replacesy(n) y(h) inequation(2)bythesecond t (cid:0) t principal component of bond yields. Column (3) replaces y(n) y(h) in equation (2) by the forward spread t (cid:0) t f(n;h) y(h). Column (4) adds CFNAI and the in(cid:135)ation trend factor of Cieslak and Povala (2015) as two t (cid:0) t additional macro control variables to equation (2). Column (5) adds i to equation (2) to control for the e⁄ect of the equilibrium nominal short rate i , when measured b I yf r t t h < e c ge (cid:3)t xpected 3-month Treasury bill (cid:3)t yieldfrom5to10yearsaheadintheBlueChipEconomicIndicatorssurvey. Thesealternativespeci(cid:133)cations are estimated from January 1990 to September 2008 (excluding December 2003) where r 0:01 and from t (cid:21) October 2008 to November 2015 where r <0:01. Standard errors are provided in parenthesis using a block t bootstrap with 5,000 repetitions and a block window of 24 months. In each bootstrap sample it is required that there are at least 50 observations where the short rate is above and below 1 percent, respectively. Signi(cid:133)cance at the 10, 5, and 1 percent level is denoted by *, **, and ***. All yields are end-month from G(cid:252)rkaynak et al. (2007) with r measured by the e⁄ective federal funds rate. t (1) (2) (3) (4) (5) h = 6 PCA2 f(n;h) y(h) Macro Trend t t (cid:0) n (cid:12)(n) (cid:1)(cid:12)(n) (cid:12)(n) (cid:1)(cid:12)(n) (cid:12)(n) (cid:1)(cid:12)(n) (cid:12)(n) (cid:1)(cid:12)(n) (cid:12)(n) (cid:1)(cid:12)(n) 1 1 1 1 1 1 1 1 1 1 2 0:64 0:79 0:03 0:46? 0:16 0:78 0:04 1:81? 0:32 1:18 (0:57) (0:69) (0:25) (0:28) (0:50) (0:52) (1:00) (1:06) (0:97) (0:98) 3 0:83 1:13 0:17 1:12?? 0:33 1:05? 0:41 2:24? 0:64 1:63 (0:63) (0:82) (0:46) (0:52) (0:57) (0:61) (1:08) (1:21) (1:05) (1:10) 4 0:95 1:53 0:38 1:97??? 0:46 1:50?? 0:68 2:77?? 0:87 2:22 (cid:3) (0:64) (0:98) (0:62) (0:69) (0:59) (0:68) (1:08) (1:28) (1:06) (1:21) 5 1:02 1:93? 0:62 2:90??? 0:56 2:06??? 0:92 3:39??? 1:07 3:00 (cid:3)(cid:3) (0:63) (1:12) (0:74) (0:82) (0:60) (0:69) (1:06) (1:30) (1:03) (1:32) 6 1:08? 2:33? 0:89 3:83??? 0:64 2:67??? 1:12 4:08??? 1:24 4:00 (cid:3)(cid:3)(cid:3) (0:61) (1:20) (0:83) (0:98) (0:61) (0:70) (1:04) (1:33) (1:00) (1:45) 7 1:12? 2:70?? 1:17 4:72??? 0:71 3:31??? 1:30 4:80??? 1:39 5:21 (cid:3)(cid:3)(cid:3) (0:60) (1:25) (0:89) (1:19) (0:61) (0:86) (1:02) (1:41) (0:98) (1:57) 8 1:16?? 3:04?? 1:45 5:52??? 0:77 3:99??? 1:46 5:54??? 1:52 6:63 (cid:3)(cid:3)(cid:3) (0:59) (1:32) (0:94) (1:45) (0:63) (1:13) (1:00) (1:59) (0:96) (1:66) 9 1:19?? 3:35?? 1:72? 6:22??? 0:83 4:72??? 1:59 6:26??? 1:63 8:22 (cid:3) (cid:3)(cid:3)(cid:3) (0:59) (1:41) (0:99) (1:75) (0:64) (1:46) (1:00) (1:85) (0:95) (1:71) 10 1:22?? 3:64?? 1:98? 6:81??? 0:88 5:54??? 1:71? 6:97??? 1:73 9:93 (cid:3) (cid:3)(cid:3)(cid:3) (0:59) (1:54) (1:03) (2:07) (0:67) (1:83) (1:00) (2:17) (0:95) (1:73) The (cid:133)fth column in Table 1 controls for the trend in the natural (or equilibrium) nominal rate i , which Bauer and Rudebusch (2017) show has a strong e⁄ect on bond yields. We use (cid:3)t the approach in Christensen and Rudebusch (2019) among others and measure the natural ratebyitslong-horizonexpectationsfrom5to10yearsintothefuture. Specially, wemeasure 10

i by the average expected 3-month Treasury bill yield from 5 to 10 years ahead, as reported (cid:3)t biannually in the Blue Chip Economic Indicators survey.2 Given that i operates as a level (cid:3)t factor a⁄ecting all yields equally, it should cancel out in the yield spread away from the ZLB but not close to the bound, where short-term interest rates cannot respond one-to-one with i . This suggests that i should only a⁄ect excess bond returns close to the ZLB, and (cid:3)t (cid:3)t we therefore add the regressor i to equation (2). The last column in Table 1 shows If rt<c g (cid:3)t that controlling for i cannot explain our new empirical result, as we also in this case (cid:133)nd a (cid:3)t positive and signi(cid:133)cant shift in the ability of the yield spread to predict bond returns.3 Thus, the shift in bond return predictability documented in Section 2.2 is robust to several modi(cid:133)cations and extensions of the classic predictability regression in (2). 3 A Shortcoming of the Standard Shadow Rate Model ThissectionshowsthatthestandardSRMwithGaussianfactordynamicscannotexplain the shift in bond return predictability documented in Section 2. We proceed by describing the model in Section 3.1, our estimation method in Section 3.2, and the inability of the standard SRM to generate more return predictability from the yield spread at the ZLB in Section 3.3. 2The forecast horizons are not the same in each edition of the survey, so we linearly interpolate the mean expectations across respondents to get short rate expectations at the desired horizon. This biannual series is then extended to a monthly frequency by linear interpolation. The data source is Wolters Kluwer Legal and Regulatory Solutions U.S., Blue Chip Economic Indicators. 3Unreported results show that we also (cid:133)nd a positive and signi(cid:133)cant shift in (cid:12)(n) if we simply add our 1 measure of i as an extra regressor in equation (2) to account for the e⁄ect of i away from the ZLB and (cid:3)t (cid:3)t close to the lower bound. 11

3.1 A Shadow Rate Model Suppose that the U.S. economy can be described by a set of factors collected in x of t dimension n 1 that evolves as x (cid:2) x = h(x )+(cid:6)"P : (5) t+1 t t+1 The function h(x ) is potentially nonlinear and "P is an n 1 vector of independent stant t+1 x (cid:2) dard Gaussian shocks under the physical probability measure P, denoted "P (0;I). t+1 (cid:24) NID Throughout, we consider models with the standard 3 pricing factors (that is, n = 3). A x stochastic discount factor M is assumed to have the (cid:135)exible form t+1 M = exp r (cid:21)(x ) (cid:21)(x ) (cid:21)(x )"P ; t+1 (cid:0) t (cid:0) t 0 t (cid:0) t t+1 (cid:8) (cid:9) where r is the one-period nominal short rate and (cid:21)(x ) is a potentially nonlinear function t t for the market prices of risk. Many macroeconomic models relate M to consumption t+1 and in(cid:135)ation. The precise structural underpinnings of M are, however, not provided in t+1 reduced-form DTSMs to reduce the risk of model misspeci(cid:133)cation, although the factors in x at an overall level are linked to the U.S. economy. The short rate is given by t r = max 0;s ; (6) t t f g where the use of the maximum function following Black (1995) constrains r from below t at zero and s (cid:11) + (cid:12) x is an unconstrained "shadow rate." To make the bond pricing t 0 t (cid:17) tractable, it is assumed that the market prices of risk are given by (cid:21)(x ) = (cid:6) 1(h(x ) (cid:8)(cid:22) (I (cid:8))x ); (7) t (cid:0) t t (cid:0) (cid:0) (cid:0) 12

because it implies a (cid:133)rst-order vector auto-regression (VAR) under the risk-neutral probability measure Q. That is, x = (cid:8)(cid:22)+(I (cid:8))x +(cid:6)"Q ; (8) t+1 (cid:0) t t+1 where "Q (0;I). t+1 (cid:24) NID The standard SRM with Gaussian factor dynamics is obtained by letting h(x ) = h + t 0 H x , implying that equation (5) reduces to x t x = h +H x +(cid:6)"P : (9) t+1 0 x t t+1 Yields do not have closed-form expressions in this version of the SRM, and we therefore use the second-order approximation of Priebsch (2013). For identi(cid:133)cation, we impose the standard restrictions that (cid:12) = 1, (cid:22) = 0, (cid:6) is lower triangular, and (cid:8) is in Jordan formwith 0 increasing diagonal elements (see Joslin, Singleton and Zhu (2011)). A preliminary analysis reveals that the (cid:133)rst eigenvalue of (cid:8) is often indistinguishable from zero, meaning that the pricing factors have a near unit root under Q. This implies that the unconditional mean of the shadow rate under Q is badly identi(cid:133)ed, and we therefore impose (cid:8)(1;1) = (cid:11) = 0 to ensure identi(cid:133)cation (see Hamilton and Wu (2012)).4 Finally, the P transition parameters h and H are unrestricted. 0 x 3.2 Estimation Method and Data MostpreviousstudiesthatestimateDTSMsusequasi-maximumlikelihood(QML)methods. However, the joint optimization of all parameters required by standard QML is computationally challenging and the asymptotic properties of these QML estimators are unknown in nonlinear models such as the SRM. We overcome these limitations by using the sequential 4Previous studies that have imposed these restrictions on (cid:8)(1;1) and (cid:11) include Christensen and Rudebusch (2015). 13

regression (SR) approach of Andreasen and Christensen (2015), which is computationally simpler and provides asymptotically Gaussian parameter estimates.5 To apply the SR approach to the standard SRM, it is convenient to de(cid:133)ne two vectors with partly overlapping sub-sets of parameters. First, (cid:18) 1 collects the "Q parameters" in equation (8) that determine the cross-sectional relationship between the pricing factors and bond yields. Second, (cid:18) 2 collects the "P parameters" in equation (9) that determine the timeseries dynamics of the pricing factors. Because (cid:6) appears in both (cid:18) and (cid:18) , it is convenient 1 2 0 0 to partition these vectors further as (cid:18) = (cid:18) vech((cid:6)) and (cid:18) = (cid:18) vech((cid:6)) , 1 011 0 2 022 0 (cid:20) (cid:21) (cid:20) (cid:21) 0 0 where (cid:18) = (cid:8)(2;2) (cid:8)(3;3) and (cid:18) = h vec(H ) . 11 22 00 x 0 (cid:20) (cid:21) (cid:20) (cid:21) The SR approach proceeds in three steps. At step 1, (cid:18) and the pricing factors x are 1 t jointlyestimatedusingnon-linearregressionsfromn yieldswithmaturitiesm ;m ;:::;m y;t 1 2 ny;t in period t. The observed yield with maturity m at time t is denoted y (mj) = g (x ;(cid:18) )+ j t mj t 1 v , where g (x ;(cid:18) ) is the model-speci(cid:133)c function that relates the pricing factors to the mj;t mj t 1 cross section of yields and v is a measurement error. We assume that these measurement mj;t errors have zero means and (cid:133)nite, positive-de(cid:133)nite covariance matrices. For a given value of (cid:18) , we obtain the pricing factors in each period by 1 1 ny;t 2 x ((cid:18) ) = argmin y (mj) g (x ;(cid:18) ) : (10) t 1 2n t (cid:0) mj t 1 xt 2 Rnx y;t X j=1 (cid:16) (cid:17) b The estimate of (cid:18) is then given by minimizing the sum of squared residuals from equation 1 (10). That is, (cid:18) step1 = argmin 1 T ny;t y (mj) g (x ((cid:18) );(cid:18) ) 2 ; (11) 1 2N t (cid:0) mj t 1 1 (cid:18)1 2 (cid:2)1 X t=1 X j=1 (cid:16) (cid:17) b b where (cid:18) step1 denotes the step 1 estimate of (cid:18) , N = T n , and (cid:2) is the feasible domain 1 1 t=1 y;t 1 step1 of (cid:18) . Andreasen and Christensen (2015) show thatP(cid:18) is consistent and asymptotically 1 b 1 5The SR approach has been applied to SRMs previously by Andreasen and Meldrum (2018). b 14

Gaussian for n for all t given standard regularity conditions. y;t ! 1 At step 2 of the SR approach, the time-series parameters (cid:18) are estimated using the 2 step1 estimated factors x (cid:18) from step 1 with a correction for estimation uncertainty in t 1 (cid:16) (cid:17) these factors. As shown in Andreasen and Christensen (2015), this procedure corresponds to b b running a modi(cid:133)ed VAR, where all second moments are corrected for estimation uncertainty step1 in x (cid:18) . The details are provided in Appendix A. t 1 (cid:16) (cid:17) At step 3 of the SR approach, the estimates of (cid:6) from step 1 and 2 can be combined b b optimally and, conditional on the optimal estimate of (cid:6), the remaining Q parameters (cid:18) 11 and the P parameters (cid:18) 2 are re-estimated. Preliminary results for our applications reveal that (cid:6) is estimated very imprecisely at step 1 compared with step 2.6 So, for simplicity, we settle by conditioning on the estimate (cid:6)step2 at step 2 and re-estimate (cid:18) as 11 (cid:18) step3 = argmin 1 T ny;t y b (mj) g x (cid:18) ;(cid:6)step2 ;(cid:18) ;(cid:6)step2 2 ; (12) 11 2N t (cid:0) mj t 11 11 (cid:18)11 2 (cid:2)11 X t=1 X j=1 (cid:16) (cid:16) (cid:16) (cid:17) (cid:17)(cid:17) b b b b step3 where (cid:18) denotes the step 3 estimate of (cid:18) and (cid:2) is the feasible domain of (cid:18) . Finally, 11 11 11 11 we update the estimate of (cid:18) by re-running step 2 using the estimated factors from step 3, b 2 step3 that is x (cid:18) ;(cid:6)step2 . t 11 (cid:16) (cid:17) The SR approach is designed for a setting with a large cross section, and we therefore b b b include more yields than typically used when estimating DTSMs. That is, we represent the yield curve by 25 points, using the 3-month yield, the 6-month yield, yields in the 1-year to 3-year range at 3-month intervals, and yields in the 3- to 10-year range at 6-month intervals. Forthe3- and6-monthyields, weuseTreasurybill yields, whileyieldswithlongermaturities are obtained from the G(cid:252)rkaynak et al. (2007) data set. Our monthly sample covers the period from January 1990 through December 2018, where the starting point is chosen to reduce the possibility of a structural break associated with the shift in U.S. monetary policy 6This (cid:133)nding is similar to the result of Joslin et al. (2011) for a¢ ne DTSMs, because their estimates of (cid:6) from the time-series dynamics of their (observed) factors hardly change when taking account of the cross section of bond yields. 15

during the 1980s (see Rudebusch and Wu (2007)). 3.3 Matching the Shift in Bond Return Predictability We next examine whether the standard SRM can explain the shift in bond return predictability when the short rate approaches the ZLB. Given that the standard SRM has constant parameters, it is obvious that it cannot match this shift by generating a permanent break in the regression coe¢ cients, as implied by equation (3). However, this model may be able to match the shift in bond return predictability through a threshold e⁄ect due to its nonlinear mapping between the pricing factors and bond yields. We therefore simulate a sample of 1 million observations from the standard SRM and estimate equation (3) with a threshold of 1 percent. When the short rate is above 1 percent, the topchart in Figure 3 shows that the standard SRM (the black circles) matches closely the corresponding data moment (the gray circles). This (cid:133)nding is not too surprising, given that the standard SRM away from the ZLB reduces to the Gaussian a¢ ne model, which is able to match these moments (see, for instance, Dai and Singleton (2002)). Turning to the more interesting case when the short rate is below 1 percent, we (cid:133)nd that the standard SRM (the black triangles) is indeed able to generate largerslopecoe¢ cients whencomparedtotheloadings awayfromtheZLB. Butthisincrease in the slope coe¢ cients is insu¢ cient to match the shift in the data. This is shown by the bottom chart in Figure 3, as the di⁄erences between the model-implied slope coe¢ cients lie outside the 95 percent con(cid:133)dence interval for the corresponding data moment for maturities exceeding 3 years.7 Thus, it appears that simply enforcing the ZLB by the standard SRM is insu¢ cient to capture the shift in bond return predictability that occurred during the recent ZLB period. 7This result supports the (cid:133)ndings of Andreasen and Meldrum (2018), who show that once the ZLB periodisincludedintheestimationsample,thestandardSRMcannotmatchthedesiredslopecoe¢ cientsin equation (2) conditional on yields being away from the ZLB. This is because the estimated factor dynamics change once the ZLB period is included in the estimation sample, and this distorts the time-series dynamics of yields when they are away from the ZLB. 16

Figure 3: Bond Return Predictability: Standard SRM The top chart shows the ability of the standard SRM to match the shift in the slope coe¢ cients in equation (3) with h = 12 and c = 0:01 using a simulated sample of 1 million observations. The corresponding data moments are from equation (2) with h = 12 when estimated from January 1990 to September 2008 (excluding December 2003) where r 0:01, and when estimated from t (cid:21) October 2008 to November 2015 where r < 0:01. The bottom chart shows the di⁄erence in the t slope coe¢ cients in equation (3) at a given maturity as implied by the standard SRM and the data. All x-axes show the maturity in years. Slope Coefficients (n) in Each Regime 1 15 Data: r 1% (Jan. 1990-Sept. 2008) t Data: r < 1% (Oct. 2008-Nov. 2015) t 10 Model: r 1% t Model: r < 1% 5 t 0 -5 2 3 4 5 6 7 8 9 10 Difference in (n) between Regimes 1 15 Data 95% Confidence Interval for Data 10 Model 5 0 -5 2 3 4 5 6 7 8 9 10 4 Two Extended Shadow Rate Models This section proposes two extensions of the standard SRM by introducing nonlinearities in the P dynamics of the pricing factors. Section 4.1 describes the two extensions, which allow for i) regime-switching in the pricing factors or ii) a permanent break in the factor dynamics. Section 4.2 shows that it is impossible to distinguish between these two models (or the standard SRM) from the in-sample (cid:133)t to the cross-section of yields, but that the models have substantially di⁄erent implications for the time-series dynamics of bond yields. Section 4.3 shows that the SRM with regime-switching goes a long way in explaining the 17

shift in bond return predictability documented in Section 2, whereas the extension with a permanent break performs even worse than the standard SRM. 4.1 Nonlinear Factor Dynamics in the SRM As discussed in Section 2, there are at least two interpretations of the recent shift in bond return predictability. If we believe that the shift is temporary, as implied by the threshold regression in equation (3), it seems natural to accommodate regime-switching in the P dynamics of the pricing factors. That is, to replace equation (9) with x = h(1) +h(2) + H(1) +H(2) x +(cid:6)"P ; (13) t+1 0 If rt (cid:21) c g 0 If rt<c g x If rt (cid:21) c g x If rt<c g t t+1 (cid:0) (cid:1) where both the level and the persistence of the factors may change when r falls below the t threshold c. In our application, we consider a threshold of 1 percent, which means that the two regimes implied by the model are consistent with the two regimes for the predictability regression in equation (2). The short rate and the Q dynamics remain given by equations (6) and(8), respectively, meaningthatbondpricesalsoforthisextendedmodelcanbecomputed using the second-order approximation of Priebsch (2013). We refer to this modi(cid:133)ed shadow rate model with regime-switching dynamics as the R-SRM. Another possibility is to interpret the shift in bond return predictability as being permanent, as impliedbyequation(4). Inthis case, it seems more natural to allowforapermanent break in the P dynamics of the pricing factors. That is, to replace equation (9) with x = h(1) +h(2) + H(1) +H(2) x +(cid:6)"P ; (14) t+1 0 If t<(cid:28) g 0 If t (cid:21) (cid:28) g x If t<(cid:28) g x If t (cid:21) (cid:28) g t t+1 (cid:0) (cid:1) where both the level and the persistence of the factors are allowed to change after the (cid:28)th month in the sample. In our application, we set (cid:28) = 225 to obtain a break in October 2008, which implies that the two regimes in the model are consistent with the two regimes for the predictability regression in equation (2). We again compute bond yields using the second- 18

order approximation of Priebsch (2013), as the short rate and the Q dynamics remain given by equations (6) and (8), respectively. We refer to this modi(cid:133)ed shadow rate model with a permanent break as the B-SRM. These two modi(cid:133)cations of the standard SRM do not change the cross-sectional relationship between the pricing factors and yields, but correspond to introducing nonlinearities in the market prices of risk given by equation (7). To guide intuition for why such an extension of the standard SRM has the potential to explain the shift in equation (2), consider a the well-known a¢ ne DTSM where r = s with closed-form expressions for bond yields and t t returns. The slope coe¢ cient in equation (2) with h = 1 is then (cid:12)(n) = (n (cid:0) 1)b n (cid:0) 1 (cid:21) xV[x t ] n 1b n +(cid:12) 0 0 ; (15) 1 n 1b n +(cid:12) 0 V[x t ] n 1 (cid:0) b n +(cid:12) 0 0(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) where (cid:21) = H (I (cid:8)) are the loadings of the market prices of risk on x and yields are x x t (cid:0) (cid:0) given by y(n) = 1 (a +b x ) (see Du⁄ee (2002)). Thus, allowing for a shift in H without t (cid:0)n n 0n t x changing (cid:8) corresponds to a re-pricing of risk that may explain the observed shift in the predictability regression in equation (2). An even more (cid:135)exible extension of the standard SRM than implied by the R-SRM and the B-SRM might also allow for regime-switching in i) the short rate in equation (6), ii) the Q dynamics in equation (8), and iii) the conditional covariance matrix (cid:6), as considered in the regime-switching model of Dai, Singleton and Yang (2007). Such a model would display evengreater(cid:135)exibilityin(cid:133)ttingthecrosssectionofbondyieldsthanimpliedbythestandard SRM and the two extensions we propose. However, previous studies have shown that the standard SRM with (cid:133)xed parameters is able to (cid:133)t the cross-section of bond yields closely, both when yields are away from and close to the ZLB (see, for example, Christensen and Rudebusch (2015) and Andreasen and Meldrum (2018)). Hence, any improvements to the cross-sectional (cid:133)tinamore(cid:135)exibleSRMseemslikelytobeeconomicallymarginal andwould increase the risk of over-(cid:133)tting the data. We therefore prefer to consider more parsimonious 19

models that only allow for structural changes in the P dynamics. Estimation of the R-SRM and B-SRM proceeds as described in Section 3.2, except that step 2 of the SR approach is extended by running either a threshold vector autoregression (for the R-SRM) or an autoregression with a break (for the B-SRM). This implies that the additional parameters introduced in the R-SRM and the B-SRM in comparison to the standard SRM are estimated in closed form and without any additional computational cost. The details are described in Appendix B. 4.2 In-Sample Results WestartourcomparisonofthestandardSRM,theR-SRM,andtheB-SRMbyexamining their in-sample (cid:133)t to bonds yields. The results reported in Table 2 show that the root mean squared errors (RMSEs) between observed and model-implied yields are almost identical for thethreemodelsandbelow10basispointsatallmaturities. Thisresultisfairlyunsurprising because all the models have the same short rate speci(cid:133)cation and Q dynamics, meaning that the cross-sectional mapping between the pricing factors and bond yields only di⁄ers slightly due to di⁄erent estimates of (cid:6). Table 2: In-Sample Fit to Bond Yields Thistablereportstherootmeansquarederrors(RMSEs)inannualizedbasispointsbetweenactual and model-implied yields at selected maturities at step 3 of the SR approach. Maturity (in months) 6 12 24 60 120 SRM 8.28 9.93 2.60 5.37 9.81 R-SRM 8.28 9.93 2.59 5.37 9.78 B-SRM 8.28 9.93 2.59 5.37 9.75 However, the models have di⁄erent implications for the time-series dynamics of bond yields due to the nonlinearities introduced in the P dynamics. For the R-SRM, we (cid:133)nd a signi(cid:133)cant shift in the persistence of the pricing factors, as we reject the null hypothesis of H(1) = H(2) in a Wald test (p-value = 0.000). On the other hand, the intercepts in (13) do x x not appear to shift when we approach the ZLB (p-value = 0.34). For the B-SRM, we get 20

qualitatively the same results, as we (cid:133)nd a signi(cid:133)cant shift in the persistence of the pricing factors but not in their intercepts after the break in October 2008.8 One way to illustrate these di⁄erences in P dynamics is to compare the model-implied term premiums TP(n) , which are de(cid:133)ned as t n 1 TP t (n) = y t (n) (cid:0) n 1 (cid:0) Et [r t+i ]: (16) i=0 X Giventhatthethreemodelsprovidethesameclose(cid:133)ttobondyields,anydi⁄erencesinTP(n) t must primarily re(cid:135)ect di⁄erent model-implied projections of the short rate as determined by the P dynamics of the pricing factors. The top chart in Figure 4 shows the 10-year term premiums implied by the three models. Over much of the sample, the three term premium estimates are highly correlated. However, there are also important di⁄erences. For instance, the term premium implied by the R-SRM before 2008 is somewhat higher when compared to the term premium implied by the B-SRM. This is because the R-SRM assigns the last three years of the sample (that is, December 2015 to December 2018) with low interest rates to the regime away from the ZLB, whereas these observations in the B-SRM are in the post-break regime. This implies that the R-SRM for the pre-ZLB regime has slightly lower expected short rate paths than the B-SRM, which then generates higher term premiums in the R-SRM than in the B-SRM. Another interesting di⁄erence appears toward the end of the sample, where the B-SRM implies higher term premiums than either the standard SRM or the R-SRM. This result arises because the model with the permanent break implies a permanent downward shift in the long-run mean of the short rate, which generates a lower expected short rate path in the B-SRM and hence higher term premiums than in the two other models. 8A table with the full set of estimates is provided in the online appendix. 21

Figure 4: Model-Implied 10-Year Term Premiums This (cid:133)gure reports model-implied 10-year term premiums. For the R-SRM, we compute the expected average short rate over the next 10 years in a given period using Monte Carlo integration with 1 million draws. Percent Models without Surveys 8 6 4 2 0 -2 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Percent Models with Surveys 8 6 4 2 0 -2 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 SRM R-SRM B-SRM 4.3 Matching the Shift in Bond Return Predictability We next consider whether the two extensions of the standard SRM can match the shift in bond return predictability documented in Section 2. For the R-SRM, it is natural to view this shift as being generated by the proximity of the short rate to the ZLB. Thus, we adopt the same procedure as for the standard SRM in Section 3.3 and estimate equation (3) with a threshold of 1 percent using a simulated sample of 1 million observations from the R-SRM. The upper left chart in Figure 5 shows that the R-SRM preserves the satisfying ability of the standard SRM to generate the desired degree of return predictability in the pre-ZLB period when r 0:01. Even more encouraging is the ability of the R-SRM to generate a t (cid:21) much stronger relationship between the yield spread and excess bond returns close to the ZLB with r < 0:01 than in the pre-ZLB period. This shift in return predictability is not as t 22

Figure 5: Bond Return Predictability: Extended SRMs TheleftchartsshowtheabilityoftheR-SRMtomatchtheshiftintheslopecoe¢ cientsinequation (3) with h = 12 and c = 0:01 using a simulated sample of 1 million observations. The right charts show the ability of the B-SRM to match the shift in the slope coe¢ cients in equation (4) with h = 12 using a simulated sample of 1 million observations with (cid:28) = 500;001. The corresponding data moments are from equation (2) with h = 12 when estimated from January 1990 to September 2008 (excluding December 2003) where r 0:01, and when estimated from October 2008 to t (cid:21) November 2015 where r < 0:01. All x-axes show the maturity in years. t R-SRM B-SRM Data: r 1% (Jan. 1990-Sept. 2008) t Data: r 1% (Jan. 1990-Sept. 2008) 15 15 t Data: r < 1% (Oct. 2008-Nov. 2015) e m Model: t r 1% Data: r t < 1% (Oct. 2008-Nov. 2015) ig 10 t 10 Model: Pre-Break e R Model: r t < 1% Model: Post-Break h c 5 5 a E n )n ( i 1 0 0 -5 -5 2 4 6 8 10 2 4 6 8 10 15 15 n e e 10 10 w te s e b m e c ig 5 5 n e e R r e 0 0 ffiD -5 -5 2 4 6 8 10 2 4 6 8 10 Data 95% Confidence Interval for Data Models strong as seen in the data, but the lower left chart in Figure 5 shows that the increase in the slope coe¢ cients lie within the 95 percent con(cid:133)dence interval for the sample moment at all maturities. For the B-SRM, it seems natural to view the shift in bond return predictability as the result of a permanent break. Thus, we obtain the model-implied predictability coe¢ cients for the B-SRM by estimating equation (4) on a simulated sample of 1 million observations with a break at (cid:28) = 500;001 to generate two regimes of 500;000 observations. The top right chart in Figure 5 shows that the B-SRM matches the degree of bond predictability in the pre-ZLB sample, but the model is simply unable to explain the shift in bond predictability 23

after the break in 2008. In summary, the R-SRM goes a long way in explaining the stronger link between the yield spread and bond returns during the recent ZLB period, meaning that this shift in return predictability can be attributed to a temporary re-pricing of risk. The results are less encouraging for the B-SRM with a permanent break, as this model is unable to generate stronger bond predictability after the break in 2008. 5 Matching Survey Expectations This section examines the ability of the three SRMs to match surveys on short rate expectations at long horizons. In Section 5.1 we show that none of the three models in Sections 3 and 4 can match these surveys when the models are estimated using only data on bond yields as done above. Section 5.2 explains how survey expectations can be included in the data set for estimating the three SRMs using the SR approach. Section 5.3 shows that this modi(cid:133)cation allows the three models to match survey expectations reasonably well and that our key conclusion from Section 4 is una⁄ected, that is, only the R-SRM is able to match the documented shift in bond return predictability. 5.1 Out-of-Sample Fit to Surveys We explore the ability of the three SRMs to match the expected short rate in 5 years and the average expected short rate from 5 to 10 years ahead. That is, we focus on longhorizonsurveyexpectationsanddisregardmoreshort-termexpectations,becauseitisunclear whether surveys represent the mean or the mode of respondents(cid:146)probability distributions and the model-implied mean and mode are likely to be materially di⁄erent near the ZLB.9 The distinction between mean and mode matters less at long horizons, where the ZLB is 9The SRM and extensions of it are only able to address di⁄erences between the mean and mode of the conditional short rate distribution to a limited extent because the model implies that the mean must always lie above the mode. 24

likelytohaveamuchsmallere⁄ectontheconditionalmodel-impliedshortratedistributions. Anotherbene(cid:133)tofincludinglong-termshortrateexpectationsisthattheyserveasobservable proxies for i , and hence allow us to explore whether the models match the evolution in the (cid:3)t natural nominal short rate. Our empirical source for these long-horizon expectations is the Blue Chip Economic Indicators survey, where repondents every 6 months report a point expectation for the 3month Treasury bill yieldat horizons up to 11 years ahead. The forecast horizons are not the same in each edition of the survey, so we linearly interpolate the mean expectations across respondents at di⁄erent horizons to get a measure of the average expected 3-month Treasury bill yield at a constant horizon of 5 years and from 5 to 10 years ahead - that is, Et y t ( + 3) 59 and Et 6 1 0 1 i= 19 60 y t ( + 3) i , respectively. These expectations are evaluated exactly in th h e SRM i h i andB-SRMPandbyMonte Carlo integrationinthe R-SRMs. We exploit the close correlation betweenEt y t ( + 3) i andEt [r t+i ]atlongforecasthorizonstoapproximateEt y t ( + 3) 59 byEt [r t+59 ] and Et 6 1 0 h 1 i= 19 6 i 0 y t ( + 3) i by Et 6 1 0 1 i= 19 60 r t+i . This substantially reduces h the co i mputational h i burden, becPause we avoid to (cid:2)repePatedly sol(cid:3)ve for the three-month yield in the SRMs. The left charts in Figure 6 show the survey-based measure at the 5 year horizon (top left) and at the 5-to-10 year ahead (bottom left) along with the corresponding expectations in the three SRMs. The survey-based measures (the black plus signs) generally decline at both horizons throughout the sample, from about 6.5 percent to about 3 percent. In contrast, the short rate expectations implied by the standard SRM (the black line) are typically lower than 3 percent, have no clear downwards trend, and display only a weak correlation with the two surveys. The short expectations from the R-SRM (the dark gray line) are generally even further below the survey-based measure but are otherwise very similar to those from the standard SRM. Finally, the B-SRM (the light gray line) implies long-horizon expectations that are relatively stable (with the obvious exception of the drop at the time of the break in October 2008), and these expectations di⁄er therefore also substantially from the surveys. Insummary,weconcludethatnoneoftheconsideredmodelsareabletoproduceplausible 25

Figure 6: Matching Long-Horizon Short Rate Surveys This (cid:133)gure shows the ability of the SRMs to match the expected short rate in 5 years and the averageexpectedshortratefrom5to10yearsaheadasimpliedbyresponsestoBlueChipEconomic Indicatorssurveys. Theleftchartsshowtheresultsformodelsestimatedusingonlyapanelofbond yields, while the right charts show the results for models estimated using a data set that includes a panel of bond yields and surveys. For the R-SRM, we compute the expected average short rate at each month using Monte Carlo integration with 1 million draws. Models Estimated without Surveys Models Estimated with Surveys Percent Percent 8 8 6 6 r a e y 4 4 - 5 2 2 0 0 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Percent Percent 8 8 r a 6 6 e y - 0 1 4 4 o t - 5 2 2 0 0 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Surveys SRM R-SRM B-SRM short rateexpectationsat longhorizons, andhencematchthedownwardtrendinthesurveybased measure of the natural nominal short rate, when these models are estimated solely based on a panel of bond yields. 5.2 Incorporating Surveys into the SR Approach The reason that the SRMs do not produce short rate expectations at long horizons that match those from surveys is most likely explained by the fairly short span of our sample (from 1990 to 2018). Given the high persistence in bond yields, it is well-known that short samples imply estimates of the P dynamics that may be biased and subject to substantial estimation uncertainty (see Bauer, Rudebusch and Wu (2012), Kim and Orphanides (2012), 26

and Wright (2014)). One way to mitigate these problems is to incorporate survey data when estimating DTSMs, as proposed by Kim and Orphanides (2012). We incorporate surveys at step 2 of the SR approach when estimating the P parameters in (cid:18) 2 , but not at step 1 and 3 when determining the Q parameters and the pricing factors, because they are already accurately identi(cid:133)ed from the considered panel of bond yields. Although this assumption implies a degree of simpli(cid:133)cation, it means that we can continue to separate the Q and P parameters, which is the key computational advantage of the SR approach. Moreformally, atstep2oftheSRapproach, weallowsurveystobemeasuredwitherrors, which we denote by (cid:17)( t 5y) and (cid:17) t (5 (cid:0) 10y) for the short rate expectations in 5 years and from 5 to 10 years ahead, respectively. Each of these errors are assumed to be independent and identically distributed with zero mean and standard deviation (cid:27) . For the standard SRM (cid:17) and the R-SRM, we augment the moment conditions for the P parameters by the (cid:133)rst and second moments of (cid:17)( t 5y) and (cid:17)( t 5 (cid:0) 10y) when r t 0:01 and when r t < 0:01. That is, a total of (cid:21) 8 additional moment conditions. For the B-SRM, we include the (cid:133)rst and second moments of (cid:17)( t 5y) and (cid:17) t (5 (cid:0) 10y) both before and after the break point, which also imply 8 additional moment conditions. Further details are provided in Appendix C. For our application, we assume that the measurement errors in surveys have a standard deviation of 10 basis points, i.e. (cid:27) = 0:1, which seems reasonable given the fairly stable (cid:17) evolution in the considered surveys.10 5.3 Estimation Results with Surveys The in-sample (cid:133)t to bond yields is basically una⁄ected by the inclusion of surveys, and the three SRMs therefore display the same satisfying (cid:133)t as reported in Table 2. For the R-SRM we once again (cid:133)nd a signi(cid:133)cant shift in the persistence of the pricing factors but not 10A preliminary analysis also considered (cid:27) = 0:75 as in Kim and Orphanides (2012), but this implied a (cid:17) verylowweighttosurveysinourGMMestimation, andhencefairlysmalle⁄ectsofincludingsurveysatthe second step of the SR approach. The intention of this paper is not to suggest that (cid:27) =0:1 is the optimum (cid:17) calibration but to show that the models are capable of (cid:133)tting the broad movements in the surveys without changing our key conclusions about return predictability. 27

in the intercepts. The shift in the persistence of the pricing factors is also signi(cid:133)cant in the B-SRM (p-value = 0.000), but now we also (cid:133)nd a signi(cid:133)cant shift in the intercepts (p-value = 0.047) whereas it was insigni(cid:133)cant without surveys.11 The right charts in Figure 6 show that the models track the overall evolution in surveys reasonably well once they are included in the data set for estimation. That is, the standard SRM now matches the downward trend in long-term short rate expectations, and hence this empirical proxy for the equilibrium short rate. But it is also evident from Figure 6 that this model generates too high short rate expectations at the end of our sample, up to 1 percentage point higher than in the data. Figure 6 further shows that the R-SRM and the B-SRM are also able to match the downward trend in surveys, and that these models produce less elevated short rate expectations at the end of our sample when compared to the standard SRM. The bottom panel of Figure 4 reveals that the inclusion of surveys in the estimation reduces the level of the 10-year term premium substantially in all three SRMs. The main di⁄erences between the three models now only appear at the end of our sample, where the R-SRM implies a slightly higher term premium than seen in the B-SRM. Figure 7 (cid:133)nally explores the ability of the SRMs to broadly match the shift in bond return predictability once surveys are included in the estimation. The encouraging (cid:133)nding is that the R-SRM remains able to match the shift, whereas both the standard SRM and the B-SRM fail to generate a su¢ ciently large shift in bond return predictability. In summary, we conclude that the models are capable of matching long-term short rate expectations reasonably closely without changing our fundamental conclusion that only the R-SRM is able to match the documented shift in bond return predictability at the ZLB. 11A table with the full set of estimates is provided in the online appendix. 28

syevruS htiw sMRS :ytilibatciderP nruteR dnoB :7 erugiF gnisu 10:0 = c dna 21 = h htiw )3( noitauqe ni stneic ¢eoc epols eht ni tfihs eht hctam ot MRS-R eht fo ytiliba eht wohs strahc tfel ehT stneic ¢eoc epols eht ni tfihs eht hctam ot MRS-B eht fo ytiliba eht wohs strahc thgir ehT .snoitavresbo noillim 1 fo elpmas detalumis a era stnemom atad gnidnopserroc ehT . 100;005 = (cid:28) htiw snoitavresbo noillim 1 fo elpmas detalumis a gnisu 21 = h htiw )4( noitauqe ni dna ,10:0 r erehw )3002 rebmeceD gnidulcxe( 8002 rebmetpeS ot 0991 yraunaJ morf detamitse nehw 21 = h htiw )2( noitauqe morf t (cid:21) .sraey ni ytirutam eht wohs sexa-x llA .10:0 < r erehw 5102 rebmevoN ot 8002 rebotcO morf detamitse nehw t 51 01 5 0 5- 01 8 6 4 2 )n(1 emigeR hcaE ni MRS-B MRS-R MRS )8002 .tpeS-0991 .naJ( %1 r :ataD )8002 .tpeS-0991 .naJ( %1 r :ataD )8002 .tpeS-0991 .naJ( %1 r :ataD t t t )5102 .voN-8002 .tcO( %1 < t r :ataD 51 )5102 .voN-8002 .tcO( %1 < t r :ataD 51 )5102 .voN-8002 .tcO( %1 < t r :ataD kaerB-erP :ledoM %1 t r :ledoM %1 t r :ledoM 01 01 kaerB-tsoP :ledoM %1 < t r :ledoM %1 < t r :ledoM 5 5 0 0 5- 5- 01 8 6 4 2 01 8 6 4 2 51 01 5 0 5- 01 8 6 4 2 semigeR neewteb ecnereffiD 51 51 01 01 5 5 0 0 5- 5- 01 8 6 4 2 01 8 6 4 2 sledoM ataD rof lavretnI ecnedifnoC %59 ataD 29

6 Alternative Speci(cid:133)cations This section considers two alternative speci(cid:133)cations of the models considered above. We (cid:133)rst show in Section 6.1 that the performance of the B-SRM does not improve by changing the timing of the break point. Section 6.2 improves the ability of the R-SRM to match the recent low level of long-term short rate expectations by allowing for a break in the intercepts of the pricing factors after the ZLB period. 6.1 Alternative Timing of a Permanent Break We have so far assumed that the break in the B-SRM coincides with the start of the ZLB period based on the instability of the predictability regression in equation (2). But, a di⁄erent timing of the break point may be more suitable for the P dynamics in the B-SRM and this may improve its performance. We therefore brie(cid:135)y explore whether the results for the B-SRM are robust to determining the break point directly from the historical evolution of the estimated pricing factors. Here, we exploit a convenient property of the SR approach that the pricing factors are estimated non-parametrically without any assumption about the P dynamics, unlike typical QML estimators based on Kalman (cid:133)ltering.12 This property of the SRapproachimplies that we candirectlytest forbreakinthe P dynamics byusingChow tests applied to the estimated pricing factors in step 1 of the SR approach. Given that the timing of the break is unknown, we compute Chow tests using the same range of potential break points as considered in Section 2.1. The solid line in Figure 8 shows the Chow test statistic when estimating equation (9) using the pricing factors from the standard SRM. The test statistic is generally higher during the ZLB period than before, but it does not peak until November 2013, when it is well above the 95 percent critical value of Andrews (1993). This result suggests that a break in the P dynamics during the fall of 2013 may be more 12Here, we refer to the step 1 estimates of the pricing factors in the SR approach. 30

appropriate than during the fall of 2008.13 Figure 8: Chow Tests for a Break in the Pricing Factors ThischartreportstheChowteststatisticforbreaksintheestimatedpricingfactorsinthestandard SRM ((cid:133)rst step estimates) using at least 15 percent of the observations for the pre- and post-break sample. The reported 95 percent critical value is for the maximum Chow test in Andrews (1993). 125 Test statistic Andrews (1993) 95% critical value 100 75 50 25 0 1995 1998 2001 2004 2007 2010 2013 Given this (cid:133)nding, we brie(cid:135)y consider whether a version of the B-SRM with a break in November2013((cid:28) = 287)isabletomatchthedocumentedshiftinbondreturnpredictability. To give the model the best possible chance of matching this shift, we also re-estimate the regression coe¢ cients in the data using the same sample split. Hence, the gray markers on the top chart in Figure 9 show the slope estimates in equation (2) for the sub-sample from January 1990 through October 2013 ("Regime 1") and the sub-sample from November 2013 through December 2018 ("Regime 2"). The black markets are for the B-SRM estimated using surveys on short rate expectations as described in Section 5. The clear message from Figure 9 is that this alternative timing for the break point does not change our conclusion 13Onepossibleexplanationforthisresultcouldbethatseveralbondyieldsdidnotappeartobeconstrained by the ZLB until after 2010, as shown in Swanson and Williams (2014). 31

fromabove, as the B-SRMalso in this case is unable to match the recent shift in bond return predictability. Figure 9: Bond Return Predictability: Di⁄erent Break Point in the B-SRM The top chart shows the slope coe¢ cient in equation (4) with h = 12 using a simulated sample of 1 million observations with (cid:28) = 500;001 using an estimated version of the B-SRM with a break in November 2013. The corresponding data moments are from equation (2) with h = 12 when estimated from January 1990 to October 2013, and when estimated from November 2008 to December 2018. The bottom chart shows the di⁄erence in the slope estimates at a given maturity. The 95 percent con(cid:133)dence interval for these di⁄erences are computed using a block bootstrap with 5,000 repetitions and a block window of 24 months. In each bootstrap sample it is required that there are at least 50 observations in each regime. The x-axes reports maturity in years, and all yields are end-month from G(cid:252)rkaynak et al. (2007) Slope Coefficients (n) in Each Regime 1 15 Data: r 1% (Jan. 1990-Sept. 2008) t Data: r < 1% (Oct. 2008-Nov. 2015) t 10 Model: Pre-Break Model: Post-Break 5 0 -5 2 3 4 5 6 7 8 9 10 Differences in (n) between Regimes 1 15 Data 95% Confidence Interval for Data 10 Model 5 0 -5 2 3 4 5 6 7 8 9 10 6.2 The Recent Low Short Rate Expectations in Surveys The empirical evidence presented so far clearly favors the R-SRM compared to the standard SRM and the B-SRM. Hence, considering the recent (cid:133)nancial crisis as having a temporary impact on U.S. Treasury yields goes a long way in explaining the observed shift in bond return predictability documented in Section 2. We have also seen that the R-SRM is 32

able to match long-term short rate expectations in surveys reasonably well, but that all the consideredmodels (including the R-SRM) struggle to explain the lowshort rate expectations since 2015.14 Although the magnitude of these (cid:133)tting errors is not larger than seen prior to the (cid:133)nancial crisis, they could indicate more long-lasting e⁄ects of the (cid:133)nancial crisis, as argued in Summers (2015). We explore this possibility by considering an extension of the R-SRM that allows for a permanent break in the level of the P dynamics as follows x = h(1;1) +h(2) + h(1;2) t+1 0 If rt (cid:21) c gIf t<(cid:28) g 0 If rt<c gIf t<(cid:28) g 0 If t (cid:21) (cid:28) g pre ZLB period ZLB period post ZLB period + H(1) +H(2) x +(cid:6)"P : (17) | x I {z f rt (cid:21) c g } x| If rt<c g {z t } t+1| {z } (cid:0) (cid:1) with c = 0:01. That is, we allow for a separate intercepts for the post ZLB-period to accommodate a permanent e⁄ect on the level of the pricing factors from the recent (cid:133)nancial crisis. Asforthetimingofthebreak,welet(cid:28) = 302tointroduceabreakinMarch2015,when the R-SRM starts to generate large (cid:133)tting errors for the long-term short rate expectations in Section 5.3. The short rate and the Q dynamics remain given by equations (6) and (8), respectively, meaning that bond prices also for this extended R-SRM can be computed using the second-order approximation of Priebsch (2013). We estimate this extended R-SRM using surveys as described in Section 5.2. The shift in thepersistenceofthepricingfactorsremainssigni(cid:133)cantasinSection5.3. Wealso(cid:133)ndashift in the intercepts away from the ZLB period, as we reject the null hypothesis of h(1;1) = h(1;2) 0 0 in a Wald test (p-value = 0.000). That is, the level of the pricing factors appears to change from h(1;1) to h(1;2) when exiting the ZLB period, which may be interpreted as evidence of 0 0 non-stationarity in the P dynamics. Our results are therefore in line with those obtained in BauerandRudebusch(2017), whichalso(cid:133)ndevidenceofnon-stationarityintheP dynamics, although they use a di⁄erent formulation of non-stationarity within a DTSM. ToconsiderwhetherthisextendedR-SRMcanmatchtheshiftinbondreturnpredictabil- 14Unreported results show that reducing the value of (cid:27) below does not materially improve the (cid:133)t to (cid:17) surveys since 2015. 33

Figure 10: Bond Return Predictability: The Extended R-SRM The left charts show the ability of the extended R-SRM to match the shift in the slope coe¢ cients in equation (3) with h = 12 and c = 0:01 using a simulated sample of 1 million observations. The corresponding data moments are from equation (2) with h = 12 when estimated from January 1990 to September 2008 (excluding December 2003) where r 0:01, and when estimated from t (cid:21) October 2008 to November 2015 where r < 0:01. The x-axes show the maturity in years. The t right charts show the ability of the extended R-SRMs to match the expected short rate in 5 years and the average expected short rate from 5 to 10 years ahead as implied by responses to Blue Chip Economic Indicators surveys. The model-implied short rate expectations are computed using Monte Carlo integration with 1 million draws. Slope Coefficients (n) in Each Regime 5-Year-Ahead Expected Short Rate 1 Percent 15 Data: r t 1% (Jan. 1990-Sept. 2008) 8 Data: r < 1% (Oct. 2008-Nov. 2015) Surveys t Extended R-SRM 10 Model: r 1% 6 t Model: r < 1% t 5 4 0 2 -5 0 2 4 6 8 10 1990 1995 2000 2005 2010 2015 Differences in (n) between Regimes 5-to-10-Year-Ahead Expected Short Rate 1 Percent Data 15 8 95% Confidence Interval for Data Surveys Model Extended R-SRM 10 6 5 4 0 2 -5 0 2 4 6 8 10 1990 1995 2000 2005 2010 2015 ity, we compute the model-implied slope coe¢ cients in equation (3) using only the pre-break intercepts in the simulation.15 The left charts in Figure 10 show that the extended R-SRM preserves the satisfying ability of the R-SRM to match bond predictability away from the ZLB,andthatitalsogoesalongwayinmatchingthedocumentedshiftinbondpredictability at the lower bound. The right charts in Figure 10 show that the extended R-SRM generates a close (cid:133)t to long-term short rate expectations from 2015 and hence improves upon the 15ThisexerciseisthereforeonlyabletoexplorewhethertheextendedR-SRMcanmatchtheabilityofthe yield spread to predict excess bond returns before the break in March 2015. The period after the break is likely too short to draw any (cid:133)rm conclusions. 34

performance of the R-SRM. In summary, we conclude that a new level for the pricing factors after 2015 allows the R-SRMto explain the recent low level of survey-based measures of the natural nominal short rate, while at the same time generating a notable shift in bond return predictability during the ZLB period. 7 Conclusion This paper documents a shift in the predictability of excess bond returns during the recent ZLB period that a standard SRM cannot replicate. This shows that simply enforcing the ZLB by truncating the short rate at zero in a Gaussian model is insu¢ cient to properly capture the change in bond yield dynamics that occurred at the ZLB. We (cid:133)nd that this new predictability result is consistent with a SRM that allows the P dynamics of the pricing factors to change at the lower bound. In contrast, a SRMthat introduces a permanent break in the pricing factors in 2008 is not able to explain this shift in bond return predictability at the ZLB. None of the considered models is able to match the low level of long-term short rateexpectationsinsurveyssince2015. However, afurtherextensiontotheregime-switching model that incorporates a permanent break in the level of the pricing factors in 2015 is able to address this shortcoming of the R-SRM. 35

References Andreasen, M. M. and Christensen, B. J. (2015), (cid:145)The SR Approach: A New Estimation Procedure for Non-Linear and Non-Gaussian Dynamic Term Structure Models(cid:146), Journal of Econometrics 184(2), 420(cid:150)451. Andreasen, M. M., Engsted, T., M(cid:246)ller, S. V. and Sander, M. (2016), (cid:145)Bond Market Asymmetries across Recessions and Expansions: New Evidence on Risk Premia(cid:146), Working paper . Andreasen, M. M. and Meldrum, A. C. (2018), (cid:145)A Shadow Rate or a Quadratic Policy Rule? the Best Way to Enforce the Zero Lower Bound in the United States(cid:146), Journal of Financial and Quantitative Analysis Forthcoming. Andrews, D. W. K. (1993), (cid:145)Tests for Parameter Instability and Structural Change With Unkown Change Point(cid:146), Econometrica 61(4), 821(cid:150)856. Bauer, M. D. and Rudebusch, G. D. (2016), (cid:145)Monetary Policy Expectations at the Zero Lower Bound(cid:146), Journal of Money, Credit, and Banking 48(7), 1439(cid:150)1465. Bauer, M. D. and Rudebusch, G. D. (2017), (cid:145)Interest Rates Under Falling Stars(cid:146), Working Paper . Bauer, M. D., Rudebusch, G. D. and Wu, J. C. (2012), (cid:145)Correcting Estimation Bias in Dynamic Term Structure Models(cid:146), Journal of Business and Economic Statistics 30(3), 454(cid:150)467. Black, F. (1995), (cid:145)Interest Rates as Options(cid:146), Journal of Finance 50(5), 1371(cid:150)1376. Bonis,B.,Ihrig,J.andWei,M.(2017),(cid:145)ProjectedEvolutionoftheSOMAPortfolioandthe10-year Treasury Term Premium E⁄ect(cid:146), FEDS Notes, Washington: Board of Governors of the Federal Reserve System, . Campbell, J. Y. and Shiller, R. J. (1991), (cid:145)Yield Spreads and Interest Rate Movements: A Bird(cid:146)s Eye View(cid:146), Review of Economic Studies 58(3), 495(cid:150)514. 36

Christensen, J. H. E. and Rudebusch, G. D. (2015), (cid:145)Estimating Shadow-Rate Term Structure Models with Near-Zero Yields(cid:146), Journal of Financial Econometrics 13(2), 226(cid:150)259. Christensen, J. H. E. and Rudebusch, G. D. (2019), (cid:145)A New Normal for Interest Rates? Evidence from In(cid:135)ation-Indexed Debt(cid:146), Federal Reserve Bank of San Francisco Working Paper Series . Cieslak, A. and Povala, P. (2015), (cid:145)Expected Returns in Treasury Bonds(cid:146), Review of Financial Studies 28(10), 2859(cid:150)2901. Cochrane, J. H. and Piazzesi, M. (2005), (cid:145)Bond Risk Premia(cid:146), American Economic Review 95(1), 138(cid:150)160. Dai, Q. and Singleton, K. J. (2002), (cid:145)Expectation Puzzles, Time-Varying Risk Premia, and A¢ ne Models of the Term Structure(cid:146), Journal of Financial Economics 63(3), 415(cid:150)441. Dai, Q., Singleton, K. J. and Yang, W. (2007), (cid:145)Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields(cid:146), Review of Financial Studies 20(5), 1669(cid:150)1706. D(cid:146)Amico, S., English, W., Lopez-Salido, D. and Nelson, E. (2012), (cid:145)The Federal Reserve(cid:146)s Large- Scale Asset Purchase Programmes: Rationale and E⁄ects(cid:146), Economic Journal 122, F415(cid:150)F446. DelNegro,M.,Giannone,D.,Giannoni,M.P.andTambalotti,A.(2017),(cid:145)Safety,Liquidity,andthe National Rate of Interest(cid:146), Federal Reserve Bank of New York, Sta⁄ Reports No. /812 pp. 1(cid:150)52. Du⁄ee, G. (2002), (cid:145)Term Premia and Interest Rate Forecast in A¢ ne Models(cid:146), Journal of Finance 57(1), 405(cid:150)443. Fama, E. F. and Bliss, R. R. (1987), (cid:145)The Information in Long-Maturity Forward Rates(cid:146), American Economic Review 77(4), 680(cid:150)692. G(cid:252)rkaynak, R. S., Sack, B. and Wright, J. H. (2007), (cid:145)The U.S. Treasury Yield Curve: 1961 to the present(cid:146), Journal of Monetary Economics 54(8), 2291(cid:150)2304. Hamilton, J. D. and Wu, J. C. (2012), (cid:145)Identi(cid:133)cation and Estimation of Gaussian A¢ ne Term Structure Models(cid:146), Journal of Econometrics 168(2), 315(cid:150)331. 37

Joslin, S., Priebsch, M. and Singleton, K. J. (2014), (cid:145)Risk Premiums in Dynamic Term Structure Models with Unspanned Macro Risk(cid:146), Journal of Finance 69(3), 453(cid:150)468. Joslin, S., Singleton, K. J. and Zhu, H. (2011), (cid:145)A New Perspective on Gaussian Dynamic Term Structure Models(cid:146), Review of Financial Studies 24(3), 926(cid:150)970. Kim, D. H. and Orphanides, A. (2012), (cid:145)Term Structure Estimation with Survey Data on Interest Rate Forecasts(cid:146), Journal of Financial and Quantitative Analysis 47(1), 241(cid:150)272. Kim, D.H.andSingleton, K.J.(2012), (cid:145)TermStructureModelsandtheZeroBound: AnEmpirical Investigation of Japanese Yields(cid:146), Journal of Econometrics 170(1), 32(cid:150)49. Laubach, T. and Williams, J. C. (2016), (cid:145)Measuring the National Rate of Interest Redux(cid:146), Business Economics 51(2), 57(cid:150)67. Li, C. and Wei, M. (2013), (cid:145)Term Structure Modeling with Supply Factors(cid:146), International Journal of Central Banking 9(1), 3(cid:150)39. Ludvigson, S. C. and Ng, S. (2009), (cid:145)Macro Factors in Bond Risk Premia(cid:146), Review of Financial Studies 22(12), 5027(cid:150)5067. Priebsch, M. A. (2013), (cid:145)Computing Arbitrage-Free Yields in Multi-Factor Gaussian Shadow-Rate Term Structure Models(cid:146), Board of Governors of the Federal Reserve System (U.S.) Finance and Economics Discussion Series 2013-63 . Rudebusch, G. D. and Wu, T. (2007), (cid:145)Accounting for a Shift in Term Structure Behavior with No-Arbitrage and Macro-Finance Models(cid:146), Journal of Money, Credit, and Banking 39(2-3), 395(cid:150) 422. Summers, L. H. (2015), (cid:145)Demand Side Secular Stagnation(cid:146), American Economic Review: Papers and Proceedings 105(5), 60(cid:150)65. Swanson, E. T. and Williams, J. C. (2014), (cid:145)Measuring the E⁄ect of the Zero Lower Bound on Medium- and Longer-Term Interest Rates(cid:146), American Economic Review 104(10), 3154(cid:150)3185. 38

Wright, J. H. (2014), (cid:145)Term Premia and Iin(cid:135)ation Uncertainty: Empirical Evidence from an International Panel Dataset: Reply(cid:146), American Economic Review 104(1), 338(cid:150)341. Wu, J. C. and Xia, F. D. (2016), (cid:145)Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound(cid:146), Journal of Money, Credit, and Banking 48(2-3), 253(cid:150)291. 39

Appendix A: Step 2 in the SR Approach To estimate (cid:18) in Step 2 of the SR approach we follow Andreasen and Christensen (2015) 2 and use the moment conditions 0 Et "P t+1 2 3 2 6 6 6 6 4 vec v (cid:16) e E ch t h (cid:16) b " V h P tb+ t 1 h x b " t P t i + ( 1 (cid:18) i 1 (cid:17) ) 0 i(cid:17) 3 7 7 7 7 5 = 6 6 6 6 6 6 6 vech 0 B (cid:0) v V e C t c (cid:2) t ( " [ C u P t+ t t+ 1 [u (cid:3) 1 ; t + + u 1 t V ; ] u H t [ t u ] 0x (cid:0) t (cid:0) ]+ H H H x x V C x t V t [u [ t u [ t u t ]) ; t u ]H t+1 0x ] 1 C 7 7 7 7 7 7 7 ; (18) b 4 @ A 5 step1 where "P denotes the estimated values of "P and u x (cid:18) x . Consistent estimates t t t (cid:17) t 1 (cid:0) t (cid:16) (cid:17) of the b second moments Vt [u t+1 ], Ct [u t+1 ;u t ], and Ct [ b u t ;u bt+1 ] follow from step 1 of the SR approach. Thus, (cid:18) can be estimated using the generalized method of moments (GMM), 2 which has a closed-form solution as shown in Andreasen and Christensen (2015) Appendix B: Estimating the Extended SRMs 0 Theparameterstobeestimatedatstep2canagainbewrittenas(cid:18) = (cid:18) vech((cid:6)) , 2 022 0 (cid:20) (cid:21) but now we have 0 (cid:18) 22 = h( 0 1) 0 vec H( x 1) 0 h( 0 2) 0 vec H( x 2) 0 : (cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) 40

For the R-SRM, consider the revised set of moment conditions are Et "P t+1If rt (cid:21) c g 0 2 h i 3 2 3 6 Et b "P t+1If rt<c g 7 6 0 7 6 6 6 vec Et h b "P t+1 x( t 1)((cid:18) i 1 ) 0 7 7 7 = 6 6 6 vec Ct u( t+ 1) 1 ;u( t 1) (cid:0) H( x 1) Vt u( t 1) 7 7 7 ; (19) 6 6 6 vec (cid:16) Et h b "P t+1 x b ( t 2)((cid:18) 1 ) 0 i(cid:17) 7 7 7 6 6 6 vec (cid:16) Ct h u( t+ 2) 1 ;u( t 2)i (cid:0) H( x 2) Vt h u( t 2)i(cid:17) 7 7 7 6 7 6 7 6 6 v (cid:16) ech h b Vt b "P t+1 i(cid:17) 7 7 6 6 (cid:16) ve h ch Vt "P t i +1 +(cid:10) t+1 h i(cid:17) 7 7 6 7 6 7 4 (cid:16) h i(cid:17) 5 4 5 (cid:0) (cid:2) (cid:3) (cid:1) b wherex(1)((cid:18) ) = x (cid:18) ,x(2)((cid:18) ) = x (cid:18) ,u(1) = u ,u(2) = u , t 1 t 1 If rt (cid:21) c g t 1 t 1 If rt<c g t t If rt (cid:21) c g t t If rt>c g (cid:16) (cid:17) (cid:16) (cid:17) and b b b b b b (cid:10) t+1 = Vt [u t+1 ]+H( x 1) Vt u( t 1) H( x 1) 0 +H( x 2) Vt u( t 2) H( x 2) 0 h i h i (cid:0) Ct u( t+ 1) 1 ;u( t 1) H( x 1) 0 (cid:0) H( x 1) Ct u( t 1);u( t+ 1) 1 h i h i (cid:0) Ct u( t+ 2) 1 ;u( t 2) H( x 2) 0 (cid:0) H( x 2) Ct u( t 2);u( t+ 2) 1 : h i h i Andreasen, Engsted, M(cid:246)ller and Sander (2016) provide the closed-form solution for (cid:18) using 2 these moment conditions. It is easy to verify that a similar result applies for the B-SRM, where regimes are de(cid:133)ned based on the time point in the sample instead of the value of r . t Appendix C: The SR Approach with Surveys The model parameters to be estimated at step 2 are the same as when surveys are not included. For the standard SRM and the R-SRM, we consider the following revised set of 41

moment conditions Et "P t+1If rt (cid:21) c g 0 2 3 h i 6 Et b "P t+1If rt<c g 7 2 0 3 6 6 6 vec Et h b "P t+1 x( t 1)((cid:18) i 1 ) 0 7 7 7 6 6 0 7 7 6 7 6 7 6 6 vec (cid:16) Et h b "P t+1 x b ( t 2)((cid:18) 1 ) 0 i(cid:17) 7 7 6 6 0 7 7 6 7 6 7 6 (cid:16) h i(cid:17) 7 6 7 6 6 6 6 6 v E e t ch (cid:17) (cid:16) b ( t 5 V y) t I h b fb " r P t t + (cid:21) 1 c g i(cid:17) 7 7 7 7 7 6 6 6 6 6 vech (cid:0) V 0 t (cid:2) "P t+1 (cid:3)(cid:1) 7 7 7 7 7 6 7 6 7 R t = 6 6 6 Et h (cid:17)( t 5y) If rt<c g i 7 7 7 6 6 6 0 7 7 7 : (20) 6 6 6 Et (cid:17) h( t 5 (cid:0) 10y) If rt (cid:21) c i g 7 7 7 (cid:0)6 6 6 0 7 7 7 6 7 6 7 6 6 6 Et h (cid:17)( t 5 (cid:0) 10y) If rt<c g i 7 7 7 6 6 6 0 7 7 7 6 6 6 Et h (cid:17)( t 5y) 2 If rt (cid:21) c g i 7 7 7 6 6 6 (cid:27)2 (cid:17) 7 7 7 6 (cid:20) (cid:21) 7 6 7 6 6 6 Et (cid:16) (cid:17)( t 5y) (cid:17)2 If rt<c g 7 7 7 6 6 6 (cid:27)2 (cid:17) 7 7 7 6 (cid:20) (cid:21) 7 6 7 6 6 6 Et (cid:17) (cid:16) t (5 (cid:0) 10y (cid:17) ) 2 If rt (cid:21) c g 7 7 7 6 6 6 (cid:27)2 (cid:17) 7 7 7 6 6 6 6 Et (cid:20) (cid:16) (cid:17) t (5 (cid:0) 10y) (cid:17)2 If rt<c g (cid:21) 7 7 7 7 6 6 6 4 (cid:27)2 (cid:17) 7 7 7 5 6 (cid:20) (cid:21) 7 (cid:16) (cid:17) 4 5 Here, we ignore the estimation uncertainty in the factors as accommodating this feature would make the estimation computationally infeasible.16 With more moment conditions than parameters, we obtain the GMM estimates for (cid:18) using numerical optimization of the 2 squared residuals in equation (20), where we upscale the weight assigned to the survey-based moments by 6 to account for the fact that surveys are only observed once every 6 months. For the B-SRM, we use a similar estimation procedure with regimes de(cid:133)ned based on the break point in the sample instead of the value of r . In the standard SRM and the B-SRM, t all short rate expectations are computed in closed form. In the R-SRM, the short rate expectations are computed by Monte Carlo simulation.17 16Our experience with other SRMs suggests that the e⁄ects of allowing for estimation uncertainty in the factors are generally very small when representing the yield curve by 25 points each period. 17We (cid:133)nd that a simulation using 500 simulated short rate paths generated using antithetic shocks is su¢ ciently accurate to allow estimation of the model. 42