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Surprises, sentiments, and the expectations hypothesis of the term structure of interest rates

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Abstract

This study shows that time-varying coefficients in the term structure of interest rates equation are correlated with the time-varying term premiums (TVTP) and expectation error (EE). Consistent with Froot (J Finance 44:283–305, 1989), TVTP and EE are the main factors that cause variations in the expectations hypothesis. Once the TVTP and the EE are appropriately incorporated into the model, the GARCH-M evidence fades away. This study documents that investors’ sentiment and macroeconomic surprises are the main driving forces behind the TVTP and EE. Evidence of significant sentiment and its interacting with macroeconomic surprises shed some light on the bias due to behavioral variations.

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Notes

  1. Some researchers prefer to use survey data (Froot 1989; Chiang and Chung 1993) to proxy for expectations, although it could have sample bias or forecasting bias problem.

  2. Bekaert et al. (2001) incorporate a peso problem into their model and find data consistent with the term structure of the US, the UK, and Germany. In a follow-up study of the same markets, Jardet (2008) finds a peso problem in the UK and Germany during the European exchange rate crisis in 1992. However, there is no rejection of the “no peso problem” hypothesis in the US.

  3. Some authors (Lesseig and Stock 1998; Chang, Lin and Yang 2015), however, demonstrate that that stochastic changes in interest rates will have impact on the capital structure of the firm.

  4. Other variables will be stated in due time. However, the full data explanation is given in “Appendix 1”.

  5. Wu and Zhang (1997) provide a procedure to test the unit-root of interest rate series.

  6. Empirical studies are further differentiated from each other by sample sizes/periods, frequency of observations, econometric methods, and continuous time versus discrete time setting, among others. In this study, we focus on the 6-month and 3-month interest rates (US Libor rates and US Treasury bill rates) to avoid the overlapping sample problem.

  7. For the case when the change point is known, one can form regular tests such as F, Wald, LM, or LR in a normal linear regression model. If we are interested in the case where the change point is unknown, we face the problem of testing for structural change with an unknown change point that does not fit into the standard "regular" testing framework. The sup F test statistic for this case is constructed to treat unknown change points as parameters (see Andrews 1993). Because the test statistics do not follow any regular distribution, Bai and Perron (1998) simulated a list of asymptotic critical values and reported them in their Table 1.

  8. This test assesses whether the overall minimum value of the sum of squared residuals in (l + 1) breaks is sufficiently smaller than that in l breaks to evaluate the significance of the marginal contribution by including an additional break.

  9. Engle and Ng (1993) find that adjusting the forward rate for the volatility-related forward premium can improve its performance as a predictor for the future spot rate.

  10. CME Eurodollar futures prices are determined by the market’s forecast of the 3-month USD Libor interest rate expected to prevail on the settlement date.

  11. To simplify the notations, we use \(\alpha_{t} {\text{and }}\beta_{t}\) to denote the intercept and slopes of the right-hand-side variables. In fact, in estimating Eq. (10), the transitional equation of \(\beta_{t}\) will be a vector for \(\{ \beta_{t}^{\left( 3 \right)} , \beta_{t}^{{\theta^{*} }} {\text{and}} \beta_{t}^{{\delta^{*} }} \}\).

  12. It should be noted that the purpose of using the Kalman filter procedure here (in Eqs. 11–13) is to trace out the time-varying coefficients, while in Eqs. (6) and (7), or in the Eqs. (12) and (13), the Kalman filter was used to generate the time-varying term premium, \(\theta_{t}\) or \(\theta_{t}^{*}\). By comparing Eqs. (1.3) and (11), it is clear that the constant coefficient model in Eq. (1.3) appears to be a special case of the time-varying coefficient model as represented by Eq. (11), since if we impose: \(\alpha_{1} = \alpha_{2} = \cdots = \alpha_{t - 1 } = \alpha_{t} = \alpha^{\left( 0 \right)}\) and \(\beta_{1} = \beta_{2} = \cdots = \beta_{t - 1} = \beta_{t} = \beta^{\left( 0 \right)}\), Eq. (11) reduces to (1.3), where \(X_{t} = FP_{t + 1,t} \,{\text{and}}\,\theta_{t} = 0,\,{\text{or}}\, \theta_{t}^{*} = \delta_{t + 1}^{ *} = 0.\)

  13. In Figs. 1 and 2, we plot the intercept and slope of the forward premium. The coefficients of the term premium and the expectation error are depicted in Fig. 3. Definitions of the variables are given in “Appendix 1”.

  14. Rational herding occurs when market participants react to information about the behavior of other economic agents or investors rather than the behavior of the market and fundamental factors (Devenow and Welch 1996).

  15. Table 7 in “Appendix 2” provides some supportive evidence.

  16. Data and measurement of sentiment and macroeconomic surprises are described in “Appendix 1”.

  17. “Appendix 2” reports the estimates of the economic determinants of time-varying term premium and expectation error. The evidence indicates that market irrationality driven by investors’ sentiment and macroeconomic surprises are important determinants.

  18. In Table 7 of “Appendix 2”, we regress \(\theta_{t}\) on sentiment, expectation error, and related macroeconomic variables, and find statistical significance.

  19. The evidence in Table 7 of “Appendix 2” indicates that \(\theta_{t}\) is significantly and positively correlated with the sentiment and expectation error; however, \(\theta_{t}^{ *}\) is not significantly correlated with these variables.

  20. If we compare the difference in slopes between model 2 and model 3 (\(\beta_{t}^{\left( 3 \right)} - \beta_{t}^{\left( 2 \right)} )\), only the macroeconomic surprises are significant, since both models include the information on the time-varying term premium, as shown in Eqs. (7) and (9), respectively.

  21. We estimated a similar equation by including both sentiment and U_PC in the equation. The estimated results (not reported) suggest that both coefficients are statistically significant. The estimated equation is available upon request.

  22. This is known as the GARCH-M model. We do not report the statistics of the variance equation to save space. However, we will supply them upon request. In general, we impose IGARCH (Franses 1995) in estimations due to the persistence variance resulting from the structural breaks reported in Table 2.

  23. We claim that the GARCH-M term may serve as a proxy for unspecified determinants, such as the term premium or the expectations error. In this sense, the model is consistent with the HJM framework, but with an additional parameter of the d second moment. As a result, the GARCH-M term may capture the term premium or the expectations error that could be one of the reasons for explaining why the GARCH-M term becomes less significant once we have incorporated the term premium or the expectation error into the model.

  24. An exception is Eq. (20)′, where an interacting term of (Sent*U_PC) is used as a regressor. The evidence shows that the GARCH-M term is statistically significant. We believe that this is due to the magnification effect during the crisis period presenting structural breaks. When we rerun the equation excluding the crisis period (drop the sample after mid-2007), the GARCH-M is insignificant.

  25. To deal with the possibility that sentiment proxies may contain rational assessments of the future interest rate, we need to extract the error beliefs in the sentiment variables. After we regress each sentiment variable on a set of rational predictors of the future interest rate, the residuals in the regression tend to be cleaner proxies for the belief errors in investor sentiment. The rational predictors of the future interest rate in the regression we use include: (1) monetary policy (i.e., measured by money growth or the M2 growth rate), (2) the growth rate of the industrial production index, (3) demand for liquidity (i.e., the difference between the 1-year USD swap interest rate and the Treasury yield), (4) term spread (i.e., the difference between the 5-year T-bond rate and the 6-month T-bill rate), and (5) the inflation rate (see Bekaert et al. 2001; Bartolini et al. 2002; Deuskar et al. 2008).

  26. Selection of these macroeconomic variables is motivated by the evidence in a number of research papers (see Bekaert et al. 2001; Bartolini et al. 2002, Benkert 2004; Deuskar et al. 2008; and Kuo and Chen 2011). These variables are measured as follows: the term spread, which is the slope of the yield curve (the difference between the 5-year T-bond rate and the 6-month T-bill rate), the credit spread (the difference between the 3-month U.S. interbank rate and the 3-month overnight indexed swap), the liquidity spread (the difference between the 3-month overnight indexed swap and the 3-month T-bill rate), volatility (the model-free implied volatility extracted from Eurodollar options markets), monetary policy (the M2 growth rate, which is the percentage change in the monthly M2), and the inflation rate (the percentage change in the CPI for each month).

  27. This surprise index, U_PC, constructed from the first principal component, explains 40 % of the total variance.

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Authors and Affiliations

  1. Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Cathy Yi-Hsuan Chen

  2. Department of Finance, Chung Hua University, 707, Sec. 2, WuFu Rd., Hsinchu, 300, Taiwan, ROC

    Cathy Yi-Hsuan Chen

  3. LeBow College of Business, Drexel University, Gerri LeBow Hall, 3220 Market Street, Philadelphia, PA, 19104, USA

    Thomas C. Chiang

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  1. Cathy Yi-Hsuan Chen
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Correspondence to Cathy Yi-Hsuan Chen.

Appendices

Appendix 1: Description of the data and data measurements

1.1 Interest rate data

Quarterly and non-overlapping 3-month Libor rate data are employed in this study. Using 3-month Eurodollar futures rates as the proxy for the market’s expectations for future 3-month interest rates enables us to measure the quantitative expectation error by comparing it with the realized 3-month Libor rate. The forecasting values from Eurodollar futures markets are more appealing because of their timely availability to the public and because they more sensitively reflect investor expectations in reacting to public information. Since trading Eurodollar futures contracts in CME start from 1998, we collect the 3-month Libor rate data ranging from the third quarter of 1998 to the fourth quarter of 2010. For the same sample period, we also use the 3-month and 6-month Treasury constant maturity rates. Those data were downloaded from the FRED economic database, Federal Reserve Bank of St. Louis.

1.2 Determinants

Following Baker and Wurgler (2006) and Ho and Hung (2009), we apply the principal component analysis (PCA) and construct a composite sentiment index comprising (1) the residuals in the bullish consensus for Eurodollar futures (Eurodollar Consensus); (2) the residuals in the net position in Eurodollar futures (CFTC); (3) the residuals in the AAII bearish percentage; and (4) the residuals in the Baker sentiment index.Footnote 25 We define Sent, an investor sentiment index, as the first principal component estimated by the PCA, capturing the common component of the four proxies.

Surprise is more likely to capture information associated with the peso bias and can be chosen as the residual terms from the VAR (Sims 1980) model for macroeconomic variables such as the term spread, the credit spread, the liquidity spread, interest rate market volatility, the money growth rate, and the inflation rate.Footnote 26 All of these variables are directly linked to interest rate movements. In addition, the unobserved regime changes or policy innovations can lead to surprises in macroeconomic fundamentals, subsequently resulting in the expectation error. It’s likely that the surprises extracted from various macroeconomic variables may share some common factors, and in turn, they may correlate with each other; that is, there should be some common factors that drive or produce surprises across macroeconomic variables. Likewise, we employ the PCA method again to construct a composite surprise index comprising these surprises and call it U_PC.Footnote 27 The expectation error, at least during certain sub-periods, might not always be white noise due to the existence of sentiment as well as surprise.

Appendix 2

See Table 7 and Figs. 4, 5.

Table 7 Estimates of the economic determinants of time-varying term premiums and expectation error
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Fig. 4
figure 4

The time variations of the term premium, expectation error, and sentiment. Time series plots of time-varying term premiums (\(\theta_{t} \,{\text{and}}\,\theta_{t}^{ *}\)), and expectation error (\(\delta_{t + 1}^{ *}\)), and sentiment index (Sent) are plotted against time. “Appendix 1” contains the illustration of deriving these variables. Structural breaks are in the region between the two vertical lines (2000:03-2002:09 and 2007:09–2009:06)

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Fig. 5
figure 5

The time variations of t statistics of time-varying coefficients. The t-statistics of time-varying alpha and beta coefficients are depicted in the upper and lower panels, respectively. Structural breaks are shown in the region between the two vertical lines (2000:03–2002:09 and 2007:09–2009:06). Using iterated extended Kalman filtering and importance sampling method (Durbin and Koopman 1997), we derive the conditional mean and conditional variance matrices of the state vector, as well as the approximated log-likelihood based on the Gaussian approximation to the state space model. The 5 % significance level corresponding to the t-statistic value, 1.96, is used to test the null hypothesis for \(\hat{\alpha }_{t} = 0\) or \(\hat{\beta }_{t} = 1\)

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Chen, C.YH., Chiang, T.C. Surprises, sentiments, and the expectations hypothesis of the term structure of interest rates. Rev Quant Finan Acc 49, 1–28 (2017). https://doi.org/10.1007/s11156-016-0584-y

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