## Abstract

Based on the classic Gaussian dynamic term structure model \( {\mathbb{A}}_{0} \left( 3 \right) \), we rotate the model to a special representation, the so called “Companion Form Realization”, in which the state variables comprise the short rate and its related expectations. This unique feature makes the representation very useful in analyzing the response of the yield curve to the shocks in the short rate and its related expectations, and monitoring market expectations. Using the estimated model, we quantify a variety of yield responses to the changes in these important state variables; and also give an “unsurprising” pattern in which changes in state variables have little impact on the long end of the yield curve. Estimated state variables have strong explanatory power for expected inflation. Three case studies of the unconventional monetary policies are presented.

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## Notes

- 1.
The expected inflation mentioned here is in the sense Haubrich et al. (2012) and Potter (2012). The data is publicly available at http://www.clevelandfed.org/research/data/inflation_expectations/index.cfm.

- 2.
\( mV \) is chosen to be \( \left[ {\begin{array}{*{20}c} {3\,{\text{month}}} & {2\,{\text{year}}} & {10\,{\text{year}}} \\ \end{array} } \right]^{{ \intercal }} \) in the empirical analysis.

- 3.
Here is the link: https://www.federalreserve.gov/releases/h15/data.htm.

- 4.
There are some noticeable exceptions after 2009. These discrepancies might be due to the distortion at the short end of the yield curve which has been artificially set to “zero” since 2009. This distorted nominal short rate can barely reflect the real economic fluctuation. By fitting to the whole yield curve, the model implied short end of the yield curve could be more consistent with the real economy, therefore, gives a better description of the actually nominal short rate.

- 5.
Here we assume the easing is conducted via a conventional monetary policy, e.g., setting the Fed funds rate’s targets.

- 6.
The downward impact on the short end would be bounded by the zero lower bound.

- 7.
Since the data used to estimate the expected inflation provided by the Federal Reserve Bank of Cleveland includes the TIPS data, and the 10-year TIPS are most liquid, the 10-year expected inflation is relatively free of estimation errors.

- 8.
Notice that \( \Delta X_{2,t}^{1} \) and \( \Delta X_{3,t}^{1} \) are not directly included in the regressions, instead residual \( \xi_{t} \) is used in the regressions to avoid the multicollinearity issue and have meaningful interpretations for coefficients.

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## Acknowledgements

We thank the participants at WB/BIS Joint Fourth Public Investors’ Conference for their helpful comments. All errors are ours.

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## Appendices

### Appendix 1: State Variables as the Forward Curve Characteristics

This section shows that the state variables \( Z_{t} \) and \( X_{t}^{h} \) can precisely be interpreted as some characteristics of the forward curve.

In what follows, \( \Delta t \) denotes the limit of a time interval which approaches to zero infinitely. For convenience, the notation of \( lim \) is dropped.

Given the parameter estimates, we have:

Comparing to the magnitudes of \( Z_{2,t} \) and \( Z_{3,t} \), these error terms are negligible. Therefore it is clear that \( Z_{2,t} \) and \( Z_{3,t} \) represent the slope and curvature of the forward curve \( r\left( {t,x} \right) \) at \( x = 0 \), respectively.

As shown in Fig. 17, \( \frac{{{\varvec{\Theta}}^{*} \left( 0 \right) - {\varvec{\Theta}}^{*} \left( h \right)}}{h} \) (evaluated at the parameter estimates) is no more than five bps even when \( h = 10\,{\text{year}} \); for \( h < 2\,{\text{year}} \), this error is less than one bp. So, \( \frac{{{\varvec{\Theta}}^{*} \left( 0 \right) - {\varvec{\Theta}}^{*} \left( h \right)}}{h} \) is negligible too.

Therefore, \( X_{2,t}^{h} \) represents the slope between \( r\left( {t,h} \right) \) and \( r\left( {t;0} \right) \) and \( X_{3,t}^{h} \) represents the difference between slopes at \( r\left( {t,h} \right) \) and \( r\left( {t,0} \right) \).

### Appendix 2: Expected Inflation and Short Rate Expectations

Although the strong linkage between the expected inflation and the slope of the yield curve or forward curve or term spreads has been mentioned in many of previous studies, see, e.g., Ang et al. (2008) and Söderlind and Svensson (1997); in this appendix, we again empirically verify the strong relationship between the long term expected inflation and short rate expectations to support the points about the expected inflation made in this article. The 10-year expected inflation data used here are estimated by the Federal Reserve Bank of Cleveland according to a model proposed in Haubrich et al. (2012) and Potter (2012),^{Footnote 7} and publicly available at https://www.clevelandfed.org/our-research/indicators-and-data/inflation-expectations.aspx In what follows, the expected inflation is denoted by \( \pi^{e} \).

Form Table 3, we can see \( \Delta X_{2,t}^{1} \)(\( \Delta X_{3,t}^{1} \)) alone can explain about 50% (26%) of the variation in \( \Delta \pi^{e} \); \( \Delta X_{2,t}^{1} \) and \( \Delta X_{3,t}^{1} \) together can explain about 60% of the variation.^{Footnote 8} and as expected \( \Delta \pi^{e} \) is positively related to \( \Delta X_{2,t}^{1} \), i.e., the slop of the forward curve (the coefficient in front of \( \Delta X_{2,t}^{1} \) is 0.23 with 5% significance). We can also see that the change of the short rate (\( \Delta {\text{Z}}_{1,t} \)) can barely explain any variation in \( \Delta \pi^{e} \), as the \( R^{2} \) of the regression is only 0.2% and the coefficient is insignificant. However, interestingly, it is noted that the short rate (\( Z_{1,t} \)) per se is able to explain the level of the expected inflation (\( \pi^{e} \)) up to 66%. These observations tell us that the change of the expected inflation from one period to another is primarily driven by the changes of the market expectations, and in the long run the level of the expected inflation is mainly affected by the short rate. These observations are also presented in Fig. 18.

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Luo, J., Ye, X. (2018). Term Structure, Market Expectations of the Short Rate, and Expected Inflation. In: Mili, M., Samaniego Medina, R., di Pietro, F. (eds) New Methods in Fixed Income Modeling. Contributions to Management Science. Springer, Cham. https://doi.org/10.1007/978-3-319-95285-7_1

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