The role of uncertainty in the term structure of interest rates: A GARCH-ATSM approach

Abstract

This article examines the roles of uncertainties regarding various macro-variables in determining risk premiums of bond yields. We develop a multivariate GARCH-VAR to quantify uncertainties regarding inflation, real activities and monetary policy as time-varying conditional variances. We jointly estimate the multivariate GARCH and no-arbitrage bond pricing equations using a maximum likelihood method. The results indicate that the inflation uncertainty is the largest contributor to the dynamics of long-term yields since the 1980s, while the monetary policy uncertainty also plays noticeable roles.

Keywords:

• GARCH
• estimation
• term structure of interest rates
• financial

JEL Classification:

• C13
• C32
• E43
• E44
• E52

Acknowledgements

We are grateful to Ken Singleton, Tack Yun, Kosuke Aoki, Tatsuyoshi Okimoto and seminar participants at International Symposium in Computational Economics and Finance 2014 for their useful comments and discussions. We thank Yoshifumi Hisata and Masaki Higurashi for the role they played at an early stage of our work.

Notes

¹ Haubrich et al. (2012) propose an attempt to incorporate time-varying volatility into bond pricing using a different framework from both Heston and Nandi’s (2003) model and our model.

² In a related context, Favero and Mosca (2001) report empirical evidence that the volatility of shocks to the Federal Funds rate decreased over time.

³ It could be argued that Equation 1 indicates reduced form dynamics of the short-term interest rate rather than the structural ‘Taylor rule’, depending on how the structural rule is defined. In this context, we specify Equation 1 as a Taylor rule following prior studies, such as Ang and Piazzesi (2003) and others.

⁴ See Bollerslev et al. (1988) for the same simplification. Allowing interdependence among hⁱ does not drastically change the main results while making the estimation results less stable. See Section III for more on this issue.

⁵ A number of recent empirical studies, such as Cogley and Sargent (2005), Primiceri (2006), and Sargent et al. (2006), support the view that little evidence is found pointing to drastic changes in coefficients of the Fed’s monetary policy function. For robustness check, we estimated the same model for a shorter sample period from 1988 to 2012 which basically covers ex-Chairman Greenspan’s term. We confirmed that the estimated results remain unchanged qualitatively. The result using the short sample period is available upon request.

⁶ On the choice of the lag length of ${\mathbf{f}}^o$ (the VAR of inflation and real activity measures with the lagged short rate), we rely on the Schwarz criteria (BIC). In choosing lag length for $h^i$, we rely on Ljung-Box Q-statistics, which also point to one as the best choice.

⁷ The estimated GARCH processes are stationary because the absolute values of the corresponding polynomial roots are all greater than one. The estimated parameters representing the dynamics of inflation and the output gap $\left({\delta }_0,{\delta }_1,\mathbf{\Phi }\right)$ and those included in the Taylor rule $\left({\mu }_0,{\mu }_1,{\mu }_2\right)$ have the right signs as predicted by the economic interpretation. The SEs are calculated by computing the Hessian matrix using numerical derivatives.

⁸ For a more detailed description, see for example, Goodfriend and King (2005).

⁹ This result is consistent with Joslin et al. (2014) suggestion that macro-factors are ‘unspanned’. However, their arguments are not readily applicable to this article’s GARCH-ATSM because the yield equation explicitly includes unobservable conditional variance terms.

Additional information

Funding

Junko Koeda acknowledges funding from the Nomura Foundation for Academic Promotion and the Ministry of Education, Culture, Sports, Science, and Technology of the Japanese government [grant number 26870124]. Views expressed in this article are those of the authors and do not necessarily reflect the official views of the Bank of Japan. All remaining errors are our own.