Abstract
This paper illustrates the concept of the natural yield curve and how to measure it. The natural yield curve extends the idea of the natural rate of interest defined at a single maturity to one defined for all maturities. If the actual real yield curve matches the natural yield curve, the output gap will converge to zero. An empirical analysis using data for Japan shows that past monetary easing programs expanded the gap between the actual real yield curve and the natural yield curve mainly for short and medium maturities and led to accommodative financial conditions. By contrast, the quantitative and qualitative monetary easing policy has expanded the gap for long maturities as well as short and medium maturities. The natural yield curve is expected to provide a useful benchmark in the conduct of both conventional monetary policy and unconventional monetary policy aiming to influence the entire yield curve.
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Notes
Justiniano and Primiceri (2010) argue that economic activity can be affected by not only the overnight interest rate gap at the current time but also future expectations of the interest rate gap.
Laubach and Williams’ (2003) model includes lagged terms of the output gap and the interest rate gap, which are omitted in this section for simplicity.
Another idea is to add the second curvature factor to the standard DNS model, as proposed by Christensen et al. (2009). Yet, we consider that the standard DNS model is simple but flexible enough to trace dynamics of Japan’s yield curve, which has only a single kink point at around a 7-year maturity during our sample period.
In their NYC model, Brzoza-Brzezina and Kotłowski (2014) assume a priori that \(\phi (\tau )\) follows a uniform distribution. They further assume that \(L_{t}^{*}\) is equal to \(L_{t}\), and that \(C_{t}^{*}\) is equal to the historical average of \(C_{t}\).
Given the latent variable, \(z_{t}^{*}= \left( y_{t}^{*}, L_{t}^{*}, S_{t}^{*}, C_{t}^{*}\right) '\), Eq. (11) can be rewritten as
$$\begin{aligned} z_{t}=z_{t}^{*}+ B\left( z_{t - 1}-z_{t}^{*}\right) + e_{t}, \end{aligned}$$where \(z_{t} =\left( y_{t}, L_{t}, S_{t}, C_{t}\right) '\); B and \(e_{t}\) are the coefficient matrix and the error term multiplied by the impact matrix, respectively. This equation yields, in the stationary,
$$\begin{aligned} z_{t}= Bz_{{t - 1}} +\left( 1 - B\right) z_{t}^{*}, \end{aligned}$$which means that \(z_{t}= z_{t}^{*}\).
Due to data limitations, we use nominal interest rates deflated by survey-based inflation expectations instead of directly observed market-based real interest rates. The estimates reported below, in particular the estimated levels of the natural yield curve and the actual real yield curve, should therefore be interpreted with some latitude. For example, the 10-year real interest rate of Japanese government inflation-indexed bonds (JGBi) was −0.5% at the end of 2014, while the corresponding rate plotted in Fig. 5 is −1.1%. In addition, adjustments for the direct effects of the consumption tax hikes on inflation expectations are made prior to the analysis. Specifically, we extrapolate 1-year inflation expectations based on a spline curve. The curve is fitted to inflation expectations for 2 and more years ahead, which are not affected by the consumption tax hikes.
Another alternative is to use the HP-filtered potential output and the corresponding output gap. When we use the HP-filtered series instead of the one provided by the Bank of Japan, there is only a slight difference in the results. Yet, we note that the HP-filtered series generally suffers from the end-of-sample issue: the end-of-sample estimates are likely to fluctuate when updating data. In this regard, the series provided by the Bank of Japan is relatively robust to such an issue.
From \(\mathop {\lim }\nolimits _{{\tau \rightarrow +\infty }}r_{{\tau , t}}=L_{t}\) and \(\mathop {\lim }\nolimits _{{\tau \rightarrow 0}}r_{{\tau ,t}}=L_{t}+ S_{t}, \mu _{L}\) and \(\mu _{L}+\mu _{S}\) correspond to the historical average of the long- and short-term interest rates, respectively.
In this section we focus on the relative size of \(\phi (\tau )\) for each \(\tau \). Although, more strictly speaking, the lagged effect of the output gap should also be taken into account when discussing \(\phi (\tau )\) we omit it here.
The mixture distribution depicted in Fig. 3c is the simple average of the distributions \( f_{1} \left( \alpha _{1}, \beta _{1}\right) \) and \(f_{2}\left( \alpha _{2}, \beta _{2}\right) \) for \(0<\alpha _{1}<1\) and \(1<\beta _{1}\le 10\) which are consistent with the estimated \(b_{s}/b\) and \(b_{c}/b\) under the conditions that \(\omega =0.5\) and \(\phi (\tau =1\,\hbox {year})-\phi (\tau> 1\,\hbox {year})>0.1\).
For the quantitative easing program, the policy commitment was to maintain the program until the consumer price index registers stably at zero percent or shows an increase year on year, while for the comprehensive monetary easing program it was to maintain the program until the Bank of Japan judges that the 1% goal is in sight.
In Fig. 10, the contributions of \(\varepsilon _{t}^{L},\varepsilon _{t}^{s}\) and \(\varepsilon _{t}^{C}\) to \(I_{t}\) are referred to as a downward shift, a steepening, and a bending of the yield curve, respectively.
References
Barsky R, Justiniano A, Melosi L (2014) The natural rate of interest and its usefulness for monetary policy. Am Econ Rev 104(5):37–43
Bomfim AN (1997) The equilibrium fed funds rate and the indicator properties of term-structure spreads. Econ Inquiry 35(4):830–846
Brzoza-Brzezina M (2003) Estimating the natural rate of interest: a SVAR approach. Working Paper 27, National Bank of Poland
Brzoza-Brzezina M, Kotłowski J (2014) Measuring the natural yield curve. Appl Econ 46(17):2052–2065
Christensen JHE, Diebold FX, Rudebusch GD (2009) An arbitrage-free generalized Nelson–Siegel term structure model. Econ J 12(3):33–64
Clark TE, Kozicki S (2005) Estimating equilibrium real interest rates in real time. N Am J Econ Financ 16(3):395–413
Cuaresma J, Gnan E, Ritzberger-Gruenwald D (2004) Searching for the natural rate of interest: a euro area perspective. Empirica 31(2–3):185–204
Diebold FX, Li C (2006) Forecasting the term structure of government bond yields. J Econ 130(2):337–364
Diebold FX, Rudebusch GD, Aruoba SB (2006) The macroeconomy and the yield curve: a dynamic latent factor approach. J Econ 131(1):309–338
Justiniano A, Primiceri GE (2010) Measuring the equilibrium real interest rate. Econ Perspect 34(1):14–27
Kamada K (2009) Japan’s equilibrium real interest rate. In: Fukao K (ed) Macroeconomy and industrial structures. Keio University Press (in Japanese), pp 387–427
King M, Low D (2014) Measuring the ‘world’ real interest rate. NBER Working Paper 19887
Laubach T, Williams JC (2003) Measuring the natural rate of interest. Rev Econ Stat 85(4):1063–1070
Laubach T, Williams JC (2016) Measuring the natural rate of interest redux. Bus Econ 51(2):57–67
Neiss KS, Nelson E (2003) The real-interest-rate gap as an inflation indicator. Macroecon Dyn 7(2):239–262
Nelson CR, Siegel AF (1987) Parsimonious modeling of yield curves. J Bus 60(4):473–489
Oda N, Muranaga J (2003) On the natural rate of interest: theory and estimates. Bank of Japan Working Paper Series, 03-J-5 (in Japanese)
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We are grateful for helpful comments from the Editor, an anonymous referee, and the staff of the Bank of Japan. The views expressed herein are those of the authors alone and do not necessarily reflect those of the Bank of Japan or Bank for International Settlements.
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Imakubo, K., Kojima, H. & Nakajima, J. The natural yield curve: its concept and measurement. Empir Econ 55, 551–572 (2018). https://doi.org/10.1007/s00181-017-1289-3
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DOI: https://doi.org/10.1007/s00181-017-1289-3