# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

# High-frequency movements of the term structure of US interest rates: the role of oil market uncertainty

Elie Bouri, Rangan Gupta, Clement Kyei and Sowmya Subramaniam

####
Need to know

- The authors study the impact of oil volatility on the latent factors of the US Treasury term structure.
- No evidence of causality is found when using a linear model.
- A quantile approach is applied that considers the entire conditional distribution.
- Oil volatility holds predictive power.

####
Abstract

Using daily data from January 3, 2001 to July 17, 2020 we analyze the impact of oil market uncertainty (computed based on the realized volatility of five-minute intraday oil returns) on the level, slope and curvature factors derived from the term structure of US interest rates covering maturities from 1 to 30 years. The results of the linear Granger causality tests show no evidence of the predictive ability of oil uncertainty for the three latent factors. However, evidence of nonlinearity and structural breaks indicates misspecification of the linear model. Accordingly, we use a data-driven approach: the nonparametric causality-in-quantiles test, which is robust to misspecification due to nonlinearity and regime change. Notably, this test allows us to model the entire conditional distribution of the level, slope and curvature factors, and hence accommodate, via the lower quantiles, the zero lower bound situation observed in our sample period. Using this robust test, we find overwhelming evidence of causality from oil uncertainty for the entire conditional distribution of the three factors, suggesting the predictability of the entire US term structure based on information contained in oil market volatility. Our results have important implications for academics, investors and policy makers.

####
Introduction

## 1 Introduction

Existing theories of investment under uncertainty and real options predict that uncertainty, such as oil price uncertainty, induces firms to postpone their investment decisions (Phan et al 2019), thereby leading to a decline in aggregate output (see, for example, Bernanke 1983; Dixit and Pindyck 1994; Bloom 2009). Empirical evidence in favor of this line of reasoning (ie, the recessionary impact of oil price uncertainty) can be found in the works of Elder and Serletis (2010, 2011) for the US economy. Oil market uncertainty has also been shown to drive overall macroeconomic uncertainty (Hailemariam et al 2019) in equity markets (Alsalman 2016; Demirer et al 2020a,b).

With the well-known role of US Treasury securities as a traditional safe haven (Kopyl and Lee 2016; Habib and Stracca 2015; Hager 2017) due to their ability to offer portfolio diversification and hedging benefits during periods of heightened uncertainty that negatively impacts the equity market (Chuliá et al 2017; Gupta et al 2020), a pertinent research question we aim to address in this paper is whether the oil price uncertainty is predictive for the term structure of US Treasury securities. Understandably, the accurate predictability of movement in Treasury securities is important for both central bankers and bond investors. For central bankers, understanding the evolution of interest rates helps in the fine-tuning of monetary policies. For bond market investors, correct prediction of interest rates will likely result in better bond return performance, especially given the US bond market capitalization stands at USD40.7 trillion (compared with a corresponding value of USD30 trillion associated with the stock market) and represents nearly two-thirds of the value of the global bond market.^{1}^{1} 1 Securities Industry and Financial Markets Association data. URL: http://www.sifma.org/.

Despite of the importance of the research question, studies on the predictive ability of oil market uncertainty have been restricted to the context of equity markets (see, for example, Demirer et al 2020a). Very few studies have considered the predictive power of oil price uncertainty for movements of US Treasury securities, with the exception of Balcilar et al (2020) and Nazlioglu et al (2020), who use post-World War II data. Some recent studies look into the impact of oil prices or returns and structural oil shocks on the first moment of the US government bond market (see, for example, Kang et al 2014; Wan and Kao 2015; Ioannidis and Ka 2018; Demirer et al 2020b; Gupta et al 2021; Nguyen et al 2020). Specifically, Balcilar et al (2020) analyze causality between oil market uncertainty and bond premiums of US Treasury securities, based on a $k$th-order nonparametric causality-in-quantiles framework to account for misspecification due to uncaptured nonlinearity and structural breaks. They find that oil uncertainty predicts the first and second moments of monthly US bond premiums associated with maturities of between two and five years better than those with one-year maturities. However, Nazlioglu et al (2020), using daily data for 10-year government bond returns and accounting for structural shifts as a smooth process, find no evidence of volatility spillover from the oil markets to the bond markets (but they do find it in the opposite direction). Unlike Nazlioglu et al (2020), Coronado et al (2021) detect time-varying evidence of bidirectional spillovers between oil and 10-year government bond (and high-yield corporate bond) returns and volatility, using historical monthly data over the period from October 1859 to March 2019. In this regard, it should be noted that Gormus et al (2018) also detect significant Granger causality from the oil market to the high-yield bond market in terms of volatility (and price).

Given this sparse background on the predictive ability of oil market volatility for the US bond market, we aim to extend the academic literature by examining the effects of crude oil uncertainty on the term structure of US interest rates. In light of the suggestion of McAleer and Medeiros (2008) that the rich information available from intraday data could produce more accurate estimates of daily realized volatility (RV) than estimates derived from generalized autoregressive conditional heteroscedasticity (GARCH) models, we use a measure of oil market uncertainty based on five-minute subsamples of oil returns (though we also check for the robustness of our results using the Crude Oil Exchange-Traded Fund Volatility Index (OVX) of the Chicago Board Options Exchange (CBOE)). We relate these metrics of uncertainty to the term structure of interest rates using the framework of Nelson and Siegel (1987) (henceforth NS), which is well established in the finance literature. The NS model summarizes the entire term structure into three latent yield factors (level, slope and curvature), which are considered the only relevant factors that characterize the yield curve (Litterman and Scheinkman 1991). The combination of the factor model of the interest rate term structure for US Treasury securities with maturities from 1 to 30 years and the uncertainties associated with oil price movements enables us to characterize the responses of the yield curve to oil market uncertainty and to calculate the entire yield curve movement in the wake of these second-moment oil market effects.

Specifically, we rely on daily estimates of oil RV, obtained from five-minute intraday data for the period from January 3, 2001 to July 17, 2020 (and May 10, 2007 to July 17, 2020 for the OVX), and we relate oil uncertainty to the corresponding daily movements of the level, slope and curvature of the yield curve using the causality-in-quantiles framework of Jeong et al (2012). This nonparametric causality-in-quantiles framework allows us to test for predictability emanating from oil uncertainty over the entire conditional distribution of the level, slope and curvature of the yield curve by controlling for misspecification due to uncaptured nonlinearity and regime change (both of which we show to exist in a formal statistical fashion in Section 3). Given that the period of study involves the zero lower bound (ZLB) situation of US interest rates in the wake of the global financial crisis, the use of a quantile-based framework makes perfect sense, since different quantiles can (without having to specify an explicit number of regimes as in a Markov-switching model) accurately capture the various phases of the three latent factors, with the lower, median and upper quantiles corresponding to low, normal and high interest rates, respectively. Naturally, high-frequency prediction of the term structure of interest rates would aid investors in the timely design of optimal portfolios involving US government bonds. Further, using the predictability of daily information, policy makers can gauge where the low-frequency real and nominal variables in the economy are headed by feeding the information into mixed-frequency models (Caldeira et al 2019) given that the entire yield curve is considered a predictor of economic activity (Hillebrand et al 2018). This allows policy makers to make appropriate monetary decisions in a timely manner and thereby avoid probable recessions well ahead of time (ie, even before low-frequency macroeconomic data becomes available).

To the best of our knowledge, our paper is the first to study the predictive ability of oil market uncertainty at a daily frequency for the entire conditional distribution of the level, slope and curvature factors characterizing the complete term structure of US interest rates. It extends the work of Balcilar et al (2020) by using high-frequency (ie, daily) data based on more reliable and accurate estimates of oil price uncertainty, which are derived from intraday data instead of a GARCH model. Further, unlike Balcilar et al (2020), who studied US Treasury maturities of 1 to 5 years, we study the entire term structure, associated with maturities of 1 to 30 years, as summarized by the three latent factors of level, slope and curvature. As outlined above, our analysis at a daily frequency for the entire term structure is of tremendous value not only to bond investors but also to policy makers.

## 2 Data and econometric methodologies

In this section we present the data and the basics of the two methodologies used for our empirical analyses.

### 2.1 Data

We collect daily zero-coupon yields of Treasury securities with maturities from 1 to 30 years to estimate the yield curve factors for the United States. The zero-coupon bond yields are based on the work of Gürkaynak et al (2007) and are retrieved from DataStream, maintained by Thomson Reuters. The daily data of Gürkaynak et al (2007) provides researchers and practitioners with a long history of high-frequency yield curve estimates of the Federal Reserve Board. They use a well-known and simple smoothing method that is shown to fit the data very well, with the resulting estimates used to compute yields for any horizon.

The data for the RV of oil returns, as a measure of oil market uncertainty, is obtained directly from Risk Lab, maintained by Dacheng Xiu of the Booth School of Business, University of Chicago.^{2}^{2} 2 URL: https://dachxiu.chicagobooth.edu/#risklab. Risk Lab collects trades at their highest frequencies available and cleans them using the prevalent national best bid and offer available on a second-by-second basis. The estimation procedure for RV follows Xiu (2010) and is based on quasi-maximum likelihood estimates of volatility built on moving-average models $\mathrm{MA}(q)$, using nonzero returns of transaction prices sampled up to the highest frequency available, for days with at least 12 observations. In this paper we use RV estimates based on five-minute subsampled returns of New York Mercantile Exchange light crude oil futures, the only publicly available source of robust estimates of RV associated with the oil market. Our main analysis covers the period January 3, 2001 to July 17, 2020, with the start and end dates determined by the availability of data on RV and zero-coupon, respectively. As a robustness check, we also use the CBOE’s OVX as an alternative measure of oil-related uncertainty to the RV. The OVX data is derived from the Federal Reserve Economic Data database of the Federal Reserve Bank of St. Louis,^{3}^{3} 3 URL: https://fred.stlouisfed.org/series/OVXCLS. and, based on data availability, the corresponding period of coverage is May 10, 2007 to July 17, 2020.

### 2.2 Methodology

#### 2.2.1 Extraction of the yield curve factors

The dynamic three-factor NS model (DNS) of Diebold and Li (2006) is applied in this study to fit the yield curve of zero-coupon US Treasury securities. The yield curve is decomposed into three latent factors using the NS representation in a dynamic form. The DNS model with time-varying parameters is represented as

$${r}_{t}(\tau )={L}_{t}+{S}_{t}\left(\frac{1-{\mathrm{e}}^{-\lambda \tau}}{\lambda \tau}\right)+{C}_{t}\left(\frac{1-{\mathrm{e}}^{-\lambda \tau}}{\lambda \tau}-{\mathrm{e}}^{-\lambda \tau}\right),$$ | (2.1) |

where ${r}_{t}$ represents the yield rate at time $t$ and $\tau $ is the time to maturity. The factor loading of ${L}_{t}$ is 1 and loads equally for all maturities. A change in ${L}_{t}$ can change all yields equally, and hence it is the level factor that represents the movements of long-term yields. The loading of ${S}_{t}$ starts at 1 and monotonically decays to 0. ${S}_{t}$ changes the slope of the yield curve, and hence it is the slope factor that represents the movements of short-term yields. The loading for ${C}_{t}$ starts at 1 and decays to 0, with a hump in the middle. An increase in ${C}_{t}$ leads to an increase in the yield curve curvature, and hence it is the curvature factor that represents medium-term yield movements. In this regard it is important to note that $\lambda $ is a constant controlling the decay rate. The DNS model follows a vector autoregressive (VAR) process and is modeled in state-space form using the Kalman filter. The measurement equation relating the yields and latent factors is

$\left(\begin{array}{c}\hfill {r}_{t}({\tau}_{1})\hfill \\ \hfill {r}_{t}({\tau}_{2})\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {r}_{t}({\tau}_{n})\hfill \end{array}\right)={\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{-{\tau}_{1}\lambda}}{{\tau}_{1}\lambda}}\right)\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{-{\tau}_{1}\lambda}}{{\tau}_{1}\lambda}}-{\mathrm{e}}^{-{\tau}_{1}\lambda}\right)\hfill \\ \hfill 1\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{{\tau}_{2}\lambda}}{{\tau}_{2}\lambda}}\right)\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{-{\tau}_{2}\lambda}}{{\tau}_{2}\lambda}}-{\mathrm{e}}^{-{\tau}_{2}\lambda}\right)\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill 1\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{-{\tau}_{n}\lambda}}{{\tau}_{n}\lambda}}\right)\hfill & \hfill \left({\displaystyle \frac{1-{\mathrm{e}}^{-{\tau}_{n}\lambda}}{{\tau}_{n}\lambda}}-{\mathrm{e}}^{-{\tau}_{n}\lambda}\right)\hfill \end{array}\right)}^{\prime}{f}_{t}+\left(\begin{array}{c}\hfill {u}_{t}({\tau}_{1})\hfill \\ \hfill {u}_{t}({\tau}_{2})\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {u}_{t}({\tau}_{n})\hfill \end{array}\right),$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{u}_{t}\sim N(0,R).$ | (2.2) |

The transition equation relating the dynamics of the latent factors is

$${\stackrel{~}{f}}_{t}=\mathrm{\Gamma}{\stackrel{~}{f}}_{t-1}+{\eta}_{t},{\eta}_{t}\sim N(0,G),$$ | (2.3) |

where ${r}_{t}(\tau )$ and ${u}_{t}$ are $m\times 1$-dimensional vectors for yield rates with given maturities (in our case 1 to 30 years) and the error terms, respectively. The coefficient matrix in the measurement equation follows the structure introduced by Nelson and Siegel (1987): ${f}_{t}=[{L}_{t},{S}_{t},{C}_{t}]$ is a $3\times 1$-dimensional vector and comprises the yield rate shape parameters that vary over time. In the transition equation, ${\stackrel{~}{f}}_{t}={f}_{t}-\overline{f}$ is the demeaned time-varying shape parameter matrix, $\mathrm{\Gamma}$ illustrates the dynamic relationship across shape parameters, ${\eta}_{t}$ is a $3\times 1$-dimensional error vector that is assumed to be independent of ${u}_{t}$, $G$ is an $m\times m$-dimensional diagonal matrix and $R$ is a $3\times 3$-dimensional variance–covariance matrix that allows the latent factors to be correlated. Details of the estimation procedure are beyond the scope of this study, and interested readers are referred to Diebold and Li (2006). Further details on the parameter estimates of the model are available from the corresponding author on request.

#### 2.2.2 Causality-in-quantiles model

We now describe the nonparametric causality-in-quantiles approach of Jeong et al (2012).

Let ${y}_{t}=\{{L}_{t},{S}_{t},{C}_{t}\}$ and let ${x}_{t}$ correspond to ${\mathrm{RV}}_{t}$. Further, let ${Y}_{t-1}\equiv ({y}_{t-1},\mathrm{\dots},{y}_{t-p})$, ${X}_{t-1}\equiv ({x}_{t-1},\mathrm{\dots},{x}_{t-p})$ and ${Z}_{t}=({X}_{t},{Y}_{t})$, and denote by ${F}_{{y}_{t}\mid j}({y}_{t}\mid j)$ the conditional distribution of ${y}_{t}$ given $j$. Defining ${Q}_{\theta}({Z}_{t-1})\equiv {Q}_{\theta}({y}_{t}\mid {Z}_{t-1})$ and ${Q}_{\theta}({Y}_{t-1})\equiv {Q}_{\theta}({y}_{t}\mid {Y}_{t-1})$, we have ${F}_{{y}_{t}\mid {Z}_{t-1}}\{{Q}_{\theta}({Z}_{t-1})\mid {Z}_{t-1}\}=\theta $, with probability 1. The hypotheses to be tested regarding (non)causality in the $\theta $th quantile are

${\mathrm{H}}_{0}:$ | $P\{{F}_{{y}_{t}\mid {Z}_{t-1}}\{{Q}_{\theta}({Y}_{t-1})\mid {Z}_{t-1}\}=\theta \}$ | $=1,$ | (2.4) | |||

${\mathrm{H}}_{1}:$ | $P\{{F}_{{y}_{t}\mid {Z}_{t-1}}\{{Q}_{\theta}({Y}_{t-1})\mid {Z}_{t-1}\}=\theta \}$ | $$ | (2.5) |

Jeong et al (2012) show that the feasible kernel-based test statistic has the following form:

$${\widehat{J}}_{T}=\frac{1}{T(T-1){h}^{2p}}\sum _{t=p+1}^{T}\sum _{\begin{array}{c}s=p+1,\\ s\ne t\end{array}}^{T}K\left(\frac{{Z}_{t-1}-{Z}_{s-1}}{h}\right){\widehat{\epsilon}}_{t}{\widehat{\epsilon}}_{s},$$ | (2.6) |

where $K(\cdot )$ is the kernel function with bandwidth $h$, $T$ is the sample size, $p$ is the lag order and ${\widehat{\epsilon}}_{t}\mathrm{\U0001d7cf}\{{y}_{t}\le {\widehat{Q}}_{\theta}({Y}_{t-1})\}-\theta $ is the regression error, where ${\widehat{Q}}_{\theta}({Y}_{t-1})$ is an estimate of the $\theta $th conditional quantile and $\mathrm{\U0001d7cf}\{\cdot \}$ is the indicator function. The Nadarya–Watson kernel estimator of ${\widehat{Q}}_{\theta}({Y}_{t-1})$ is given by

$${\widehat{Q}}_{\theta}({Y}_{t-1})=\frac{{\sum}_{s=p+1,s\ne t}^{T}L(({Y}_{t-1}-{Y}_{s-1})/h)\mathrm{\U0001d7cf}\{{y}_{s}\le {y}_{t}\}}{{\sum}_{s=p+1,s\ne t}^{T}L(({Y}_{t-1}-{Y}_{s-1})/h)},$$ | (2.7) |

with $L(\cdot )$ denoting the kernel function.

The empirical implementation of Granger causality testing via quantiles entails specifying three key parameters: the bandwidth ($h$), the lag order ($p$) and the kernel types for $K(\cdot )$ and $L(\cdot )$. We use a lag order of 4 based on the Schwarz information criterion (SIC), which is known to provide a parsimonious selection of the lag order and, importantly, prevents the overparameterization encountered in nonparametric estimations (see the detailed discussion in this regard in Balcilar et al (2018)). The optimal lag length results are presented in Table A1 in the online appendix, which also shows that the four lags ensure that there is no evidence of autocorrelation up to five lags in the residuals. We determine $h$ by the leave-one-out least-squares cross-validation method. Finally, for $K(\cdot )$ and $L(\cdot )$ we use Gaussian kernels.

Although the predictive inference using the causality-in-quantiles test is robust, it is also interesting to estimate the effect of the sign of oil uncertainty on the level, slope and curvature at various quantiles. However, in a nonparametric framework, this is not straightforward, as we need to employ the first-order partial derivatives. Estimation of the partial derivatives for nonparametric models can be complicated, because nonparametric methods exhibit slow convergence rates, due to the dimensionality and smoothness of the underlying conditional expectation function. However, we can look at a statistic that summarizes the effect on the global curvature (ie, the global sign and magnitude) rather than the entire derivative curve. In this regard, a natural measure of the global curvature is the average derivative (AD) using the conditional pivotal quantile based on approximation or the coupling approach of Belloni et al (2019). The latter allows us to estimate the partial ADs. The pivotal coupling approach can also approximate the distribution of ADs using Monte Carlo simulation.

The details of the AD estimation are as follows. We define ${x}_{t}$ as the key variable for which we want to evaluate the derivative with respect to ${y}_{t}$, and we define ${R}_{t}=({x}_{t},{v}_{t})$, where ${v}_{t}$ is a vector of other covariates, which in our case includes lagged values. Following Belloni et al (2019), we can model the $\theta $th quantile of ${y}_{t}$ conditional on ${R}_{t}$ using the partially linear quantile model:

$${Q}_{\theta \mid {R}_{t}}({y}_{t}\mid {R}_{t})=f({x}_{t},\theta )+{v}_{t}^{\prime}\gamma (\theta ).$$ | (2.8) |

Belloni et al (2019) develop a series approximation to ${Q}_{\theta \mid {R}_{t}}({y}_{t}\mid {R}_{t})$, which we can represent as follows:

$$\begin{array}{cc}\hfill {Q}_{\theta \mid {R}_{t}}({y}_{t}\mid {R}_{t})& \approx W{({R}_{t})}^{\prime}\beta (\theta ),\hfill \\ \hfill \beta (\theta )& ={(\alpha {(\theta )}^{\prime},\gamma (\theta ))}^{\prime},\hfill \\ \hfill W({R}_{t})& ={(W({x}_{t}),{v}_{t}^{\prime})}^{\prime}.\hfill \end{array}\}$$ | (2.9) |

In (2.9) we approximate the unknown function $f({x}_{t},\theta )$ by linear combinations of the series terms $W({x}_{t})\alpha {(\theta )}^{\prime}$. Ideally, $W({x}_{t})$ would include transformations of ${x}_{t}$ that possess good approximation properties. The transformations $W({x}_{t})$ may include polynomials, B-splines and trigonometric terms. Once we have defined the transformations $W({x}_{t})$, we can generate the first-order derivative with respect to ${x}_{t}$ as follows:

$$h({x}_{t},\theta )=\frac{\partial {Q}_{\theta \mid {R}_{t}}({y}_{t}\mid {R}_{t})}{\partial {x}_{t}}=\frac{\partial f({x}_{t},\theta )}{\partial {x}_{t}}=\frac{W({x}_{t})\alpha {(\theta )}^{\prime}}{\partial {x}_{t}}.$$ | (2.10) |

Based on the first-order derivative estimates in (2.10), we can derive the first-order AD with respect to ${x}_{t}$ as follows:

$$\overline{h}(\theta )=\int \frac{\partial f({x}_{t},\theta )}{\partial {x}_{t}}d\mu ({x}_{t}),$$ | (2.11) |

where $\mu ({x}_{t})$ is the distribution function of ${x}_{t}$.

## 3 Empirical results

### 3.1 Preliminary analyses

Dependent | Nonlinear | |
---|---|---|

variable | Linear | (Diks–Panchenko) |

Level | 0.469 | 1.466${}^{*}$ |

(0.759) | (0.071) | |

Slope | 1.172 | 3.339${}^{***}$ |

(0.321) | (0.000) | |

Curvature | 0.201 | 2.831${}^{***}$ |

(0.938) | (0.002) |

The data for the three level, slope and curvature yield curve factors are summarized together with the RV of the oil market in Table A2 and Figure A1 in the online appendix. Looking at the dependent variables, we see that the average value of the slope factor is negative, indicating that, on average, yields increase with maturities. The curvature, associated with medium-term maturities, has a higher average value than the level factor, which corresponds to long-term yields. This result, which is in line with Kim and Park (2013), who also use daily US bond yields, is indicative of liquidity issues for bonds with very long maturities. The most volatile of the three factors is the curvature factor, followed by the level factor and then the slope factor. As is evident from the rejection of the null hypothesis of normality under the Jarque–Bera test, level, slope and oil uncertainty are strongly nonnormal, while curvature is weakly nonnormal. This result, particularly for ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, provided our preliminary motivation to take a quantile-based approach to analyzing the influence of RV on these variables.

Before we discuss the findings from the causality-in-quantiles tests, for completeness and comparability we conduct a standard linear Granger causality test, with a lag length of 4, as determined by the SIC. The resulting ${\chi}^{2}(4)$ statistics (with $p$-values in parentheses) associated with the causality running from ${\mathrm{RV}}_{t}$ to ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ are shown in the second column of Table 1. Given these results, the null hypothesis (ie, oil uncertainty does not Granger-cause the three latent factors of the yield curve considered in turn in a bivariate setup) cannot be rejected at the conventional 5% level of significance, or even at the weak 10% level of significance. Therefore, based on the standard linear test, we conclude that there are no significant effects related to oil uncertainty on the level, slope or curvature of the US yield curve.

Given the nonsignificant results obtained from the linear causality test, we examine the presence of nonlinearity and structural breaks in the relationship between the three latent factors of the term structure and the RV of oil statistically. Nonlinearity and regime change, if present, would motivate the use of the nonparametric causality-in-quantiles approach, as the quantile-based test would formally address nonlinearity and structural breaks in the relationship between the variables under investigation in a bivariate setup. For this purpose, we apply the Brock–Dechert–Scheinkman (BDS) test of nonlinearity (Brock et al 1996) to the residuals from the ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ equations involving four lags of the three factors and ${\mathrm{RV}}_{t}$.

The results of the BDS test are presented in Table A3 in the online appendix presents. These show strong evidence, at the highest level of significance, for the rejection of the null hypothesis of independent and identically distributed residuals at various embedded dimensions, $m$. This, in turn, is indicative of nonlinearity in the relationship between the factors and oil uncertainty.

To further motivate the causality-in-quantiles approach, we next use the powerful unweighted double maximum (UDmax) and weighted double maximum (WDmax) tests of Bai and Perron (2003) to detect $1,\mathrm{\dots},M$ structural breaks in the relationship between ${L}_{t}$, ${S}_{t}$, ${C}_{t}$ and ${\mathrm{RV}}_{t}$, allowing for heterogeneous error distributions across the breaks. When we apply these tests to the ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ equations involving four lags of the three factors and ${\mathrm{RV}}_{t}$ in a bivariate structure, we detect two breaks in each of the three cases:

- •
February 11, 2004 and August 29, 2006 (level);

- •
March 2, 2005 and April 19, 2007 (slope); and

- •
April 7, 2004 and August 29, 2006 (curvature).

The break dates are in line with sharp oil price increases and associated volatility between 2004 and 2007. The issue of instability is further confirmed by applying the Andrews (1993) and Andrews and Ploberger (1994) tests of parameter stability on the ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ equations involving four lags of the three factors and ${\mathrm{RV}}_{t}$ in a bivariate structure. As can be seen from the results in Table A4 in the online appendix, the null hypothesis of stability is overwhelmingly rejected by the maximum likelihood ratio (LR) and Wald $F$-statistic, by the exponential (Exp) LR and Wald $F$-statistic and by the average (Ave) LR and Wald $F$-statistic.

### 3.2 Causality-in-quantiles test: main results and robustness check

Given the strong evidence of nonlinearity and structural breaks in the relationship between the latent factors and oil uncertainty, we now turn our attention to the causality-in-quantiles test, which is robust to misspecification because of its nonparametric (ie, data-driven) approach. However, before that, given the presence of nonlinearity and instability in the relationship of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ with ${\mathrm{RV}}_{t}$, we adopt the suggestion of an anonymous referee and apply the conditional-mean-based nonlinear and nonparametric Granger causality test of Diks and Panchenko (2006), which alleviates the risk of bias in rejecting the null hypothesis of no Granger causality in the popular nonlinear Granger causality test of Hiemstra and Jones (1994). As can be seen from the third column in Table 1, the test statistics for causality running from ${\mathrm{RV}}_{t}$ to ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ provide evidence of predictability from ${\mathrm{RV}}_{t}$ for ${S}_{t}$ and ${C}_{t}$, while the corresponding effect is weak for ${L}_{t}$. Clearly, accounting for nonlinearity matters in obtaining evidence of causality, but this test, being conditional-mean based, is unable to analyze predictability over the entire conditional distribution of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, which we turn to next. As outlined in Section 1, accounting for the entire distribution is important, since it allows our sample period to include the ZLB scenario, which will be captured by the lower quantiles of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, without us having to perform subsample (ie, pre- and post-ZLB) analysis.

Figure 1 reports the results of the causality-in-quantiles test for the quantile range 0.05–0.95; unlike the complete lack of causality observed under the linear framework, it can be seen that the null hypothesis (ie, ${\mathrm{RV}}_{t}$ does not Granger-cause ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$) is overwhelmingly rejected over virtually the entire conditional distribution at the 5% level of significance (given the critical value of 1.96), with the exception of the lowest quantile of ${L}_{t}$ and the highest quantile of ${C}_{t}$, as causality in both holds at the 10% level (ie, for a critical value of 1.645). In fact, the null hypothesis is rejected at the 1% level of significance (given the critical value of 2.575) over the quantile range 0.10–0.90 in all cases. In other words, when we account for nonlinearity and structural breaks using a nonparametric approach, we find strong evidence of predictability effects originating from oil market uncertainty, as captured by ${\mathrm{RV}}_{t}$, on the three factors characterizing the term structure of US interest rates, with the highest impact at the 0.65, 0.45 and 0.70 quantiles for ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, respectively. In particular, oil market uncertainty can predict the yield curve factors, irrespective of the magnitude of these factors as captured by the various quantiles of the conditional distribution of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$.

Our findings on the importance of oil uncertainty are in line with those of Balcilar et al (2020), but we show in addition that these shocks actually affect the entire yield curve over all its phases rather than just the bonds with maturities from one to five years, with the effect being strongest for long-term maturities, as captured by the level factor (in 11 of the 19 quantiles considered), followed by the medium-term US Treasury securities, as captured by the curvature factor (in the remaining 8 quantiles).

To delve deeper into why it may be that ${\mathrm{RV}}_{t}$ causes the entire conditional distribution of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, we use the dynamic conditional correlation multivariate-GARCH (DCC-MGARCH) time-varying tests for Granger causality of Lu et al (2014). These tests allows us to investigate whether and to what extent the nature of the information spillover from ${\mathrm{RV}}_{t}$ to ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$ changes across time. The key appeal of the DCC-MGARCH tests is that causality can be analyzed at each point in time, allowing us to establish evidence of any instantaneous causality. To remove the volatility-of-volatility effect, we fit the heterogeneous autoregressive RV (HAR-RV) model of Corsi (2009) to the RV of crude oil and then recover the residuals from the model and square them, then use these squared residuals in the DCC-MGARCH model to test for causality.

As can be seen from Figure 2, consistent with evidence of quantile-causality over the entire conditional distribution of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, we find that ${\mathrm{RV}}_{t}$ tends to consistently predict (in the standard causal sense as well as instantaneously) the three factors at virtually all points in our sample period.

As a robustness check, we replace RV with OVX, and rerun the causality-in-quantiles test to analyze the impact on the three latent factors, as shown in Figure 3. The linear Granger causality test again fails to pick up any evidence of predictability effects of OVX on the three factors even at the 10% level of significance, which is unsurprising given the strong evidence of nonlinearity detected using the BDS test as well as the regime changes picked up in June 2009 for the level and curvature factors and October 2009 for the slope factor. (Further details and results are available from the corresponding author on request.)

There are some subtle differences between the two metrics of oil market uncertainty in terms of the pattern of causality: the highest impact is now observed at the 0.70 and 0.35 quantiles for ${S}_{t}$ and ${C}_{t}$, respectively, while the effect is strongest for medium-term maturities (in 12 of the 19 quantiles considered) followed by the long- and then short-term US Treasury securities (in 4 and 3 quantiles, respectively). However, our main message remains the same. That is, we again find strong evidence of predictability from the OVX over the entire conditional distribution of ${L}_{t}$, ${S}_{t}$ and ${C}_{t}$, at least at the 5% level of significance, with stronger predictability (ie, at the 1% level) holding over the 0.10–0.90 quantile range. In other words, our result showing oil uncertainty impacting the entire conditional distribution of the complete term structure of US interest rates is robust across measures of oil market uncertainty.

Having derived robust predictive inference based on the causality-in-quantiles test, we now investigate the effect of the direction of oil uncertainty on the level, slope and curvature at various quantiles. Figure 4 reports the partial ADs, ie, the change in sign of the impacts of oil uncertainty on the three latent factors. It can be seen that oil uncertainty reduces short-term yields, as it negatively impacts the slope factor at all quantiles (barring the 0.30 quantile), while long-term yields primarily go down at lower conditional quantiles of the level factor (and at the 0.50 and 0.65 quantiles). Figure 4 also plots the change in sign of the oil uncertainty’s impact on the curvature factor, which corresponds to medium-term maturities of US Treasury securities, and shows that, in general, it has an intermittent negative impact (barring the 0.25–0.40 and 0.50–0.55 quantiles) up until the 0.80 quantile, and thereafter it has a positive effect. These results suggest that, in the wake of heightened oil uncertainty, agents prefer to invest mainly in short- and medium-term government bonds. Bonds with long-term maturities primarily attract investment when they produce high returns corresponding to the lower conditional quantiles of their yields.^{4}^{4} 4 A qualitatively similar pattern emerges when we look at the effect of the quantile-specific sign of the OVX on the three factors. Further details are available from the corresponding author on request. In other words, when there is an increase in oil return uncertainty, the flight-to-safety channel associated with the safe haven of government bonds is specific to maturities and quantiles, ie, the initial values of the yields. The consistent negative impact on the slope factor is unsurprising when we bear in mind that it captures bonds with short-term maturities and reflects monetary policy decisions (Ioannidis and Ka 2018), in this case an expansionary policy given increases in oil market uncertainty and the associated recessionary impact on the real economy. Interestingly, at the upper conditional quantiles of medium- and long-term government bonds, oil price uncertainty impacts the corresponding yields positively, suggesting that greater uncertainty causes agents to look beyond bonds with low returns and possibly to invest in other types of safe haven, such as commodities (eg, gold) and currencies (eg, Swiss francs).

## 4 Conclusion

In light of the sparse literature on the impact of oil uncertainty on the US government bond market, we analyze the impact of the daily RV of crude oil prices on the level, slope and curvature factors derived from the term structure of US interest rates covering maturities of 1 to 30 years. Using daily data covering the period from January 3, 2001 to July 17, 2020 we find that standard linear tests of Granger causality fail to detect any evidence of predictability effects of RV on the three yield curve factors. However, we show that the linear model is misspecified due to nonlinearity and structural breaks. Given this, we use a nonparametric causality-in-quantiles framework to reconsider the impact of RV on the three factors, with this econometric model allowing us to test for predictability over the entire conditional distribution of level, slope and curvature. Being a data-driven approach, the framework is robust to the misspecification due to nonlinearity and regime change associated with the linear model. Note that, with our sample period including the ZLB situation of US interest rates, the lower quantiles of the level, slope and curvature allow us to capture this situation without carrying out a subsample analysis involving pre- and post-global-financial-crisis data.

Using the causality-in-quantiles test, we find overwhelming evidence of predictability deriving from RV throughout the entire conditional distribution of the three factors of the US term structure. In other words, our results highlight the importance of controlling for model misspecification in order to make correct inferences when analyzing the impact of oil RV on the US term structure. Our findings evidence that oil market uncertainty is an important driver of the entire yield curve, irrespective of its different phases. They are confirmed by a time-varying test of causality. Moreover, our results continue to hold when we use an alternative metric of oil market uncertainty, based on its implied volatility, ie, the OVX. From the perspective of gauging the safe-haven property of US Treasury securities, we find that, in the wake of heightened oil uncertainty, investors prefer to invest mainly in short- and medium-term government bonds. Investment in bonds with long-term maturities primarily occurs when yields are low.

Our findings using high-frequency (ie, daily) data naturally have implications in several areas. First, the observation that oil uncertainty contains predictive information on the evolution of future interest rates in a nonparametric setup can help policy makers fine-tune their monetary policy models, given that oil volatility affects the slope factor of the yield curve, which captures movements of short-term interest rates. Second, bond investors can improve their investment strategies by exploiting the role of oil uncertainty in their interest rate prediction models, while risk managers can develop asset allocation decisions conditional on the level of the volatility of the oil market. Third, researchers may use our findings to explain deviations from asset-pricing models by embedding oil uncertainty in their pricing kernels (which should, however, be nonlinear).

While this study concentrates on US Treasury securities, given their global dominance in the sovereign bond market, as future research it would be interesting to extend our analysis to the term structure factors associated with the government bond markets of other developed countries as well as emerging countries, and also possibly to distinguish between oil exporters and importers.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

We thank the editor, Farid AitSahlia, and an anonymous referee for many helpful comments. Any remaining errors are solely ours.

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