Robust Bond Risk Premia Predictability Test in the Quantiles

Robust Bond Risk Premia Predictability Test in the Quantiles

\def\spacingset#1{ {#1}} \spacingset{1}

{\it Keywords:} Macro-spanning Hypothesis, Highly Persistent Predictors, Size Control, Predictive Quantile Regression, Bond Risk Premia. \spacingset{1.5}

Introduction

Treasury bonds are a crucial component in numerous investment portfolios. Therefore, understanding the risk and return dynamics of this asset class is of fundamental importance from an economic perspective. One focal topic in the study of U.S. Treasury bond returns is the famous macro-spanning hypothesis/puzzle on whether the yield curve contains all available information about future bond risk premia. \footnote{See the beginning of Section (ref) for the mathematical representation of the hypothesis. } Numerous studies have explored the predictive power of macro variables on bond risk premiums while controlling for the CP factor CochranePiazzesi2005 representing the information of the yield curve. These studies include CooperPriestley2008, ludvigson2009macro, Bansal2013, Joslin2014, GreenwoodVayanos2014, CieslakPovala2015, Ghysels2018, Bauer2018, ZhaoZhouguofu2021. The literature mentioned above employs mean regression to assess the macro-spanning hypothesis, which suggests that there is no predictability of bond returns beyond the information provided by the yield curve. However, it is uncertain whether these results are consistent across the entire distribution or mainly pertain to the mean return. This paper studies the predictability of bond risk premia in a predictive quantile regression framework. First, the quantile model explores the heterogeneous bond return predictability, which enrichs the study of macro-spanning hypothesis. In the literature, AndreasenEngsted2021 and Borup2024 have examined the heterogeneity of bond risk premia predictability, taking into account the business cycle and economic uncertainty. However, the heterogeneous predictability of bond risk premia at various quantile levels remains unexplored. Second, quantile model helps predict bond risk premia in different scenarios, e.g., investors often pay more attention to the tail risk of bonds. Third, the estimation and inference in mean regression are significantly affected by outliers, whereas the quantile regression is robust to the outliers. There are notable challenges to conduct the test of spanning hypothesis in a quantile model. First, Bauer2018 noted that the predictors commonly employed in the literature exhibit high persistence, leading to size distortion in predictive quantile regression test statistics and potentially causing spurious findings. Second, the conventional test statistics suffer size distortion at tail quantiles due to inaccurate density estimator of quantile regression errors. Third, when investigating the macro-spanning hypothesis, it is common for previous empirical studies to utilize four or more predictors. These predictors typically include the CP factor, two LN factors ludvigson2009macro, and at least one additional macro predictor. \footnote{See the details of the variables description in Section (ref).} In summary, testing the spanning hypothesis at the quantiles needs a robust marginal inference approach (meaning testing the predictive power of one variable while controlling for others, as opposed to conducting a joint test) in a predictive quantile regression framework that includes a large number of highly persistent predictors. Unfortunately, existing inference methods suffer size distortion in marginal tests with many highly persistent predictors. In the literature, Lee2016 extended the IVX method PhillipsMagdalinos2009 in mean regression to the quantile regression framework (IVX-QR) to test the predictability at various quantile levels, with a focus on joint tests but suffers severe size distortion at tail quantiles due to inaccuracy of nonparametric density estimator. To avoid size distortions at tail quantiles, FanLee2019 combined the moving block bootstrap technique and IVX-QR approach for the joint test, which circumvents estimating this density. LiuYangetal2023 developed a unified predictability test for univariate predictive quantile regression with highly persistent predictors. All three methods above are unsuitable for the marginal tests in multivariate models. CaiChenLiao2023 (hereafter referred to as CCL2023) constructed the instrumental variable (IV) estimator based on a double-weighted method for predictive QR (DW-QR) to conduct the joint and marginal tests in multivariate models. However, the size performance of CCL2023 is still not very satisfactory at tail quantiles and with many predictors.

The Testing Procedure and Contributions

Our new inference procedure fits the task of testing the macro-spanning hypothesis for bond risk premia. First, we construct a consistent IV estimator under both the null and the alternative hypotheses by a two-step quantile regression. Nonetheless, the test statistics constructed by the IV estimator above continue to experience size distortions with many highly persistent predictors, which is induced by two higher-order terms detailed in Section (ref). Second, we improve the size and power performance of the test. To address the size distortion issue, we adopt the sample splitting method proposed by liao2024robust to eliminate one higher-order term. We select a conservative tuning parameter in constructing instrumental variables to reduce the size distortion induced by another higher-order term. To amend the power loss due to the conservative tuning parameter choice, we enhance the power of the aforementioned test by adding a modified test statistic from the conventional test, which converges to zero at the rate $\sqrt{T}$ under the null hypothesis. This way, we fully utilize the good power performance of the traditional test without the bundling of its size distortion effect. Third, when constructing the final test statistic, we introduce a new and accurate density estimator of the error term, which utilizes a simulation-based estimator instead of the inefficient nonparametric estimator. \footnote{See CCL2023 for more details about the procedure using an irrelevant (but suitably picked) auxiliary variable, such that the resulting test do not depend on regressor persistence under the null. } Next, we apply the above procedure for the macro-spanning hypothesis by testing the predictive power of the LN factors, TDH, and TI at various quantiles while controlling for the CP factor. The main empirical findings are summarized as follows. \footnote{In our empirical study, the variable has predictive power if its rejection rate is less than 1%.} On the one hand, the CP factor has significant predictive power, while all macro variables, including the LN factors, have negligible predictive power at center quantiles, which strongly supports the macro-spanning hypothesis. On the other hand, strong empirical evidence against the macro-spanning hypothesis is found in tail quantiles. In particular, TDH has predictive power for bonds of all maturities at right tail quantiles, while it only has predictive power for 2- and 3-year bonds at left tail quantiles. Moreover, for bonds of all maturities, TI has significant predictive power at all quantiles except for center quantiles. Potential economic explanations for the findings are discussed in detail in Section (ref). Our contributions are threefold.

The rest of this paper is organized as follows. Section (ref) describes the data characteristics of bond risk premia and its predictors. Section (ref) introduces our inference approach in the predictive quantile regression with highly persistent predictors. Section (ref) presents the heterogeneous predictability of bond risk premia at various quantiles. Section (ref) shows the excellent in-sample and out-of-sample performance compared with CCL2023. Section (ref) concludes the paper. The online appendix includes an algorithm for the inference procedure, additional theoretical and numerical results, and an application on the left and the right tail risk indicators. Throughout this paper, the standard notations $\Rightarrow$, $\xrightarrow{d}$, $\xrightarrow{p}$ and $\overset{d}{=}$ are used to represent weak convergence, convergence in distribution and in probability, and equivalence in distribution, respectively. All limits are for $T\rightarrow \infty$ in all limit theories, and $O_p (1)$ is asymptotically bounded while $o_p(1)$ is asymptotically negligible.

Data Characteristics

In this section, we illustrate the variables that are used to test the macro-spanning hypothesis. Following CochranePiazzesi2005, we refer to the difference of the yield-to-maturity (YTM) of the n-year and 1-year maturity of U.S. discount bond as the dependent variable, {\bf{bond risk premia}} rx(n), where $n=2,3,4,5$. The five {\bf{predictors}} are CP factor, LN factors (LN1 and LN2), and TI, all from previous literature, and trend in debt holding (TDH), which is new in the literature. The n-year U.S discount bond data rx(n) is available on Center for Research in Security Prices (CRSP) and the macroeconomic factors are from the paper of ludvigson2009macro. We compute the CP factor following CochranePiazzesi2005. The consumer price index (CPI) and the debt holding data are from the CEIC database. \footnote{See more details in https://fiscaldata.treasury.gov/americas-finance-guide/national-debt/.} The sample used in this study consists of monthly data from January 1980 to December 2022. We refer to quantile levels close to 0.5 as center quantiles and quantile levels much less (greater) than 0.5 as left (right) tail quantiles. \footnote{We give specific definitions of the left (right) tail quantiles in the illustrative Figure (ref).} Next, we explain the predictors in detail. First, the CP factor CochranePiazzesi2005 represents the information of the yield curve, which is the linear combination of the forward rate $F_{i,t-1}$, $i=1,2,3,4,5$. Second, the LN factors, LN1 and LN2 constructed by ludvigson2009macro are fitted values of $\frac{1}{4} \sum_{n=2}^5 r x(n)_t$ regressing on macroeconomic factors which are the first eight principal components from a large dateset of 132 macroeconomic indicators. Third, the trend inflation (TI) is proposed by CieslakPovala2015 to test the predictive power of highly persistent expected inflation dynamics on bond excess returns. Specifically, TI at period (t-1) is equal to $(1-\check{w}) \sum_{i=1}^{t-1} \check{w}^i \check{\pi}_{t-i}$, where $\check{\pi}_{t-i}$ is inflation rate in the core CPI and $\check{w}$ is 0.9 in this paper, $\check{w}^i$ is the $i$th power of $\check{w}$. \footnote{We also set $\check{w}$ to other values such as 0.88 and 0.92, and obtain similar simulation and empirical results. To save space, these results are not shown here but are available upon request. } Besides the above popular predictors in existing literature, we introduce the trend in debt holding (TDH), defined as the trend in U.S. federal debt held by the public and by various government agencies (inter-governmental holdings). TDH represents the rate at which the federal government's borrowed funds increase in order to address the outstanding balance of expenses over time. It contains the information of the demand shift for the national bonds such as treasury bonds, treasury inflation-protected securities (TIPS) and discount bonds. Specifically, we construct TDH at period (t-1) in the same manner as TI, which is equal to $(1-\check{w}) \sum_{i=1}^{t-1} \check{w}^i \check{d}_{t-i}$ and $\check{d}_{t-i}$ is the increment of debt holding at time $t-i$. Figure (ref) shows that the demand shift for U.S. treasury bonds represented by TDH frequently precedes fluctuations in bond returns, which indicates the potential predictability. The statistical and economic significance of the TDH predictor is further explained in later sections.

We first demonstrate the correlation coefficients between bond risk premia rx(n) and one-period lagged predictors in Figure (ref). The size of the colored square (maximum=1, colors indicate the values of positive/negative correlations) means the absolute value of the correlation coefficient. It shows that the one-period lagged CP factor, which only contains the information of yield curve, has the strongest correlations with bond risk premia while the other macroeconomic predictors have much weaker correlations with bond risk premia. The correlation coefficients only reveal linear relationship at the center of the distribution. Next, we demonstrate the relationship between bond risk premia and one-period lagged predictors, specifically TDH and TI, at different quantiles in their time series plot in Figure (ref). For convenience of discussion, we first define the left tail risk period in Figure (ref) as the period in which one of bond risk premia rx(n) is less than or equal to its unconditional quantile at 0.05 level, \footnote{This criterion of 0.05 is not essential for our test in the following sections.} for n=2,3,4,5. E.g., Jul. 1980--Sep. 1981 is the left tail risk period by above definition. During periods of left tail risk, the difference between YTM of 1-year and that of n-year bonds (for n=2,3,4,5) is significantly smaller compared to normal conditions. In the extreme instance of the left tail risk period, such as Aug. 2022--Dec. 2022, the inverted yield curve (yields on short-term bonds are higher than those on long-term bonds) occurs. In the left tail quantiles of risk premia for bonds with maturities of two to five years, the relatively high YTM of 1-year bonds suggests that investors are concerned about short-term risks in the bond market. This perception likely stems from current economic indicators or market volatility, which typically leads investors to demand higher yields for assuming additional risk over the near term. Consequently, in such market conditions, investors tend to favor bonds with longer maturities (two to five years) over 1-year bonds. They perceive these longer-term bonds as offering a more attractive balance of risk and return, particularly when short-term forecasts appear unstable. This preference is especially pronounced at the left tail quantiles, where risk sensitivity is heightened. Likewise, we define the right tail risk period as the period in which one of bond risk premia rx(n) is greater than or equal to its unconditional quantile at 0.95 level, for n=2,3,4,5. E.g., Jun. 1982--Dec. 1982, is the right tail risk period. Correspondingly, YTM of n-year bonds (for n=2,3,4,5) is much bigger than that of 1-year bonds in the right tail risk period. In the right tail quantiles of YTM of n-year bonds (for n=2,3,4,5), there is an indication that market risks are expected to emerge over the medium term rather than the immediate future. This higher YTM reflects a risk premium that investors demand due to anticipated economic uncertainties or rising interest rates affecting these longer maturities. Consequently, in these scenarios, investors tend to prefer 1-year bonds over those with longer durations (two to five years). This preference is driven by the perceived safety and greater liquidity of shorter-term bonds, especially when facing potential medium-term market fluctuations. Some interesting patterns are observable in Figure (ref). First, in the left tail risk periods, e.g., Jul. 1980--Sept. 1981, Dec. 1981--Mar. 1982 and Aug. 2022--Dec. 2022, one-period lagged TI and TDH have an opposite trend of n-year bond risk premia, for n=2,3,4,5. Second, in the right tail risk periods, e.g., Dec. 2001--Sept. 2002 and Oct. 2008--Dec. 2008, one-period lagged TI and TDH have the same ascending trend of n-year bond risk premia, for n=2,3,4,5. This suggests that TDH and TI may have predictive power at tail quantiles. Granted, the correlations of risk premia with predictors shown in Figures (ref) and (ref) are insufficient to infer the macro-spanning hypothesis, it motivates us to test the predictability of bond returns using CP factor, LN1, LN2, TDH and TI, not only at center quantiles but also at tail quantiles.

Table (ref) shows that the AR(1) coefficients of CP factor, LN1, LN2, TDH and TI are close to 1, indicating that these predictors are highly persistent. To obtain a reliable test about the macro-spanning hypothesis, which requires marginal test in a multivariate model with many highly persistent predictors, we present the new procedure in the following section.

Statistical Model

Predictive Quantile Regression Model

Denote the bond risk premia at period t as $y_t$ and its $\tau$th conditional quantile is $Q_{y_{t}}\left(\tau \mid \mathcal{F}_{t-1}\right)$ such that $\operatorname{Pr}\left[y_{t} \leq Q_{y_{t}}\left(\tau \mid \mathcal{F}_{t-1}\right) \mid \mathcal{F}_{t-1}\right]=\tau \in(0,1)$, where $\mathcal{F}_{t-1}$ is the information set available at time $t-1$. For simplicity, a linear conditional quantile of $y_t$ is imposed as follows.

where $x_{t-1}=\left(x_{1,t-1},x_{2,t-1} \cdots,x_{K,t-1}\right)^\top$ is the vector of predictors ($K=5$ in this study) including CP factor, LN factors (LN1 and LN2), TDH and TI, representing $\mathcal{F}_{t-1}$. And $\beta_{\tau} = (\beta_{1\tau},\beta_{2\tau},\cdots,\beta_{K\tau})^\top$ is a $K$-dimensional coefficient vector given any $\tau$. The macro-spanning hypothesis essentially means that only the $\beta_{1\tau}$ of the CP factor is nonzero, and the $\beta_{i\tau}$'s of all other macro variables, for $i=2,3,4,5$, are zeroes at all quantile levels $\tau$. We define the quantile measurement error $u_{t\tau}\equiv y_t-Q_{y_t}(\tau|\mathcal{F}_{t-1})$ and the quantile score function $\psi_{\tau} (u_{t\tau})\equiv\tau-1(u_{t\tau}<0)$. {Since the AR(1) coefficients of predictors in Table (ref) are close to one (highly persistent), it is common in the literature to model these predictors as follows.}

where $x_{i,0}=o_p(\sqrt{T})$, $\rho_i=1+c_i/T^\alpha$, $t=1,\cdots,T$, $i=1,\cdots,K$, $v_t=(v_{1,t},v_{2,t},\cdots,v_{K,t})^\top$, $\alpha=0$ or 1, $-2<c_i<0$ and $c=\operatorname{diag}(c_1,c_2,\cdots,c_K)$. We assume no cointegration relationship exist among predictors $x_{t-1}$. Two types of persistency with different values of $c_i$ and $\alpha$ are considered for theoretical purpose: (1) Strong Dependence [SD]: $\alpha=1$ and $c_i$ is a constant; (2) Weak dependence [WD]: $\alpha=0$ and $|1+c_i|<1$. \footnote{ We also permit $x_{i,t-1}$ with $i=1,2,\cdots,K$ to have mixed types of persistence, that is to say, $\alpha$ could be a function of $i$. In this setting, the theoretical results for the test statistics are almost the same.}

Following Lee2016 and CCL2023, we impose the following general weakly dependent structure for innovation $\{v_t\}$ in ((ref)).

Lee2016 and CCL2023 state the functional central limit theorem (FCLT) for $\{\psi_\tau(u_{t\tau}), v_t\}$ as follows.

where $[B_{\psi_\tau}(r), B_v(r)]^\top$ is a vector of Brownian motions. Additionally, SD predictors $x_{\lfloor rT \rfloor}/{\sqrt{T}} \Rightarrow J_x^c(r)$ for $0\le r\le 1$, where $J_x^c(r)=\int_0^r e^{(r-s)c}d B_v(s)$ is Ornstein-Uhlenbeck process Phillips1987. Next, we impose some regularity assumptions on the conditional density of $u_{t\tau}$, which are also adopted by Xiao2009 and CCL2023.

The conventional test statistics tend to find spurious predictability with SD predictors and non-zero contemporary correlation between the error term of bond risk premia and predictors. This occurs because their asymptotic distribution contains nuisance parameters that cannot be estimated consistently, which is shown in Lee2016 and CCL2023.

The Inference Procedure and Theoretical Results

We improve IVX-QR Lee2016 to reduce the size distortions arising from the following sources: 1, the inconsistency of IV estimator under alternative hypothesis; 2, density function estimation for the measurement error; 3, the higher-order terms in the test statistic causing the bias. In the following subsection (ref), we develop the consistent estimator. In subsection (ref), we address the other two issues of size distortions.

A Two-step Regression for IV Estimator Construction

To offer an IV estimator that is consistent under both the null and the alternative hypothesis, we extend the two-step mean regression to quantile regression. We follow Lee2016 and define the instrumental variable $z_t=(z_{1,t},z_{2,t}\cdots,z_{K,t})^\top$ as follows,

where $z_{i,0}= x_{i,0} - x_{i,-1}$, $\Delta x_{i,t} = x_{i,t} - x_{i,t-1}$, $\rho_z = 1+ c_z/T^\delta $, $c_z<0$ and $1/2<\delta<1$ (here we set $\delta=0.95$). Next, the new two-step regression is conducted as follows.

The intuition of the above two-step IV estimator is the following. First, by the equations $x_{t-1}= \tilde{x}_{t-1}+\tilde{v}_{t-1} $ and ((ref)), it follows that

{It is clear that both $\hat{\beta}_\tau$ and $\hat{\gamma}_\tau$ in ((ref)) are consistent estimators of $\beta_\tau$ under both the null and the alternative hypothesis, since they are the estimates of the same true $\beta_{\tau}$ in (ref)}. Second, the OLS property guarantees that $ \tilde{v}_{t-1}$ is orthogonal to $\tilde{x}_{t-1}$ and the intercept term. The distribution of $\hat{\beta}_\tau$ dependents on $\tilde{x}_{t-1}$ rather than $\tilde{v}_{t-1}$, and thus, it only dependents on $z_{t-1}$ since $\tilde{x}_{t-1}= \hat{\mu}_x+\hat{\theta} z_{t-1}$. Since $z_{t-1}$ is mildly integrated with SD predictors, the asymptotic mixture normal distribution of $\hat{\beta}_\tau$ is guaranteed. To obtain the asymptotic distribution of $\hat{\beta}_\tau$, we first establish the Bahadur representation in the following theorem.

Since the first step decomposes predictors $x_{t-1}$ to orthogonal components, i.e., $\sum_{t=1}^T \tilde{x}_{t-1}\tilde{v}_{t-1}^\top =0$ and $\sum_{t=1}^T \tilde{v}_{t-1} =0$, the following theorem holds by Proposition (ref) and the asymptotic property of instrumental variable $z_{t-1}$ Kostakisetal2015.

Equation ((ref)) in Theorem (ref) reveals that $\hat{\beta}_\tau$ constructed by the new two-step regression is the instrumental variable estimator in the quantile regression. It is straightforward to construct the Wald type test statistic $Q_{ivx-qr}$ by self-normalization for the null hypothesis $H_0:R\beta_\tau =r_\tau$ where $R$ is a $J\times K$ predetermined matrix with rank $J$, $r_\tau$ is a predetermined vector with dimension $K$.

where $\operatorname{\widehat{Avar}}( \hat{\beta}_\tau)=\frac{1}{\tilde{f}_{u_\tau}(0)^2 } \hat{\Omega}_{zx}^{-1}\hat{\Omega}_{zz}\left( \hat{\Omega}_{zx}^{-1}\right)^\top$, $\hat{\Omega}_{zx}=\sum_{t=1}^T \bar{z}_{t-1} x_{t-1}^\top$ and $\hat{\Omega}_{zz}=\tau(1-\tau) \sum_{t=1}^T \bar{z}_{t-1} \\ \bar{z}_{t-1}^\top$. $\tilde{f}_{u_\tau}(0)$ is some consistent estimator of $f_{u_\tau}(0)$, e.g., the one obtained by the nonparametric method of Lee2016. Moreover, we define the t-test statistic for the one-sided marginal test with $J=1$ such as $H_0:\beta_{i\tau}=0$, for $i=1,2,...K$.

Although the asymptotic distributions of $Q_{ivx-qr}$ and $\check{Q}_{ivx-qr}$ are known, they still suffer from size distortions in finite sample due to the following two reasons. First, both the size distortions of $Q_{ivx-qr}$ and $\check{Q}_{ivx-qr}$ arise due to two higher-order terms $B_T$ and $C_T$ similar to those in mean regression models. \footnote{For more details on the higher-order terms $B_T$ and $C_T$, refer to Proposition (ref) in the subsequent section and the mean regression model counterpart discussed in liao2024robust.} Second, the nonparametric estimator $\tilde{f}_{u_\tau}(0)$ is inaccurate at tail quantiles and when the number of macro variables is large. This inaccuracy causes both $Q_{ivx-qr}$ and $\check{Q}_{ivx-qr}$ to suffer from size distortion in these cases.

Reduce Size Distortions

We now address the size distortion issues of the preliminary test statistic $Q_{ivx-qr}$ and $\check{Q}_{ivx-qr}$ in the following aspects. First, we employ a new simulation-based estimator for the density function of quantile regression error $f_{u_\tau}(0)$ to reduce the size distortion at tail quantiles and with a large number of predictors. Second, we apply a sample splitting procedure introduced by liao2024robust to eliminate one of the higher-order terms {$C_T$}. Third, we choose a conservative value for the tuning parameter $c_z$ to reduce the size distortion effect of another higher-order term $B_T$ and point out its negative effect on its power performance. The details are shown below.

A New Simulation-based Estimator for $f_{u_\tau}(0)$

The simulation-based approach to estimate $f_{u_\tau} (0)$ is as follows. We repeat the following two steps for $i=1,2,\cdots,M_1$, where $M_1 \in \mathbb{N}^{+}$ is a predetermined positive integer.

Since $\xi_{t-1}^{(i)}$ is randomly generated, it is independent of the main sample, then the following asymptotic distribution of $\xi_{t-1}^{(i)}$ holds. As the sample size $T\rightarrow\infty$,

Equation ((ref)) and the fact that $\overrightarrow{\xi_{t-1}}\equiv \big(\xi_{1,t-1}^{(1)},\xi_{2,t-1}^{(1)}, \cdots,\xi_{M_2,t-1}^{(1)},\xi_{1,t-1}^{(2)},\xi_{2,t-1}^{(2)}, \cdots,\xi_{M_2,t-1}^{(2)},\\ \cdots\cdots, \xi_{1,t-1}^{(M_1)},\xi_{2,t-1}^{(M_1)}, \cdots,\xi_{M_2,t-1}^{(M_1)}\big)^\top$ follows i.i.d. $\operatorname{N}(0_{M},I_M)$ imply that

as $M_1\rightarrow \infty$ and $T\rightarrow \infty$ and $M_2$ is constant. Therefore, the simulation-based estimator of $f_{u_\tau} (0)$ is defined as follows

as $M_1\rightarrow \infty$ and $T\rightarrow \infty$ and $M_2$ is constant. Note that ((ref)) holds by the continuous mapping theorem and ((ref)). In practice, setting $M_2$ to a relatively large number, such as $M_2=50$, is advisable as it allows for a smaller number of loops ($M_1$) to construct $\hat{f}_{u_\tau}(0)$. For example, we set $M_1=100$, which guarantees speedy simulations to accommodate computation efficiency. {Equations ((ref)) and ((ref)) imply that, unlike the nonparametric estimators, the accuracy of the estimator $\hat{f}_{u_\tau}(0)$ is not affected by the number of predictors and the quantile level $\tau$, since $\hat{f}_{u_\tau}(0)$ does not require the estimator of $u_{t\tau}$.} Additionally, $\operatorname{Var}\Big\{\frac{1}{M} \sum_{i=1}^{M_1}\sum_{j=1}^{M_2} \big[\sqrt{T}\hat{l}_{j,\tau}^{(i)}\big]^2 \Big| \mathcal{F}_T \Big\} = \frac{2}{M} [\tau(1-\tau)]^2 f_{u_\tau}(0)^{-4} +o_p(1)$ shown in equation ((ref)) of the appendix imply that the accuracy of $\hat{f}_{u_\tau}(0)$ could be improved by enlarging M. Therefore, applying $\hat{f}_{u_\tau}(0)$ in constructing the test statistics could improve the size performance at tail quantiles and with a large number of predictors.

Higher-order Terms

Another source of size distortion of $Q_{ivx-qr}$ and $\check{Q}_{ivx-qr}$ is the two higher-order terms, $C_T$ and $B_T$, shown in the following Proposition (ref). To demonstrate the concept of higher-order terms, we present them using the univariate case of $\check{Q}_{ivx-qr}$. \footnote{Higher-order terms in the multivariate model closely resemble Proposition 1 by liao2024robust, and thus their detailed derivations are omitted for simplicity.} A similar conclusion can be drawn for $Q_{ivx-qr}$ since $Q_{ivx-qr}=\check{Q}_{ivx-qr}^2$ with $J=1$.

The slow convergence of $\frac{1}{T}\sum_{t=1}^T z_{t-1}$ in $\check{Q}_{ivx-qr}$ induces $C_T$, while the correlation between the numerator and the denominator of $\check{Q}_{ivx-qr}$ leads to $B_T$. To eliminate $C_T$, we follow liao2024robust to define a new instrumental variable $\tilde{z}_{t-1}$ based on IV $z_{t-1}$ in ((ref)). Specifically, define $\tilde{z}_{t-1} = (\operatorname{I_K}-S_a)z_{t-1}$ when $1\leq t\leq T_0=\lambda\, T$ \footnote{Following liao2024robust, we set $\lambda=0.5$ to evenly split the full sample in half.} and $\tilde{z}_{t-1} = (\operatorname{I_K}-S_b)z_{t-1}$ when $T_0+1 \leq t \leq T$, where $S_a = \frac{1}{T}\sum\nolimits_{t=1}^{T} z_{t-1}\big(\frac{1}{T_0}\sum\nolimits_{t=1}^{T_0} z_{t-1}^\top \big) \big(\frac{1}{T_0}\sum\nolimits_{t=1}^{T_0} z_{t-1}^\top \frac{1}{T_0} \sum\nolimits_{t=1}^{T_0} z_{t-1}\big)^{-1}$ and $S_b = \frac{1}{T}\sum\nolimits_{t=1}^{T} z_{t-1}\big(\frac{1}{T-T_0}\sum\nolimits_{t=T_0+1}^{T} z_{t-1}^\top \big) \big(\frac{1}{T-T_0}\sum\nolimits_{t=T_0+1}^{T} z_{t-1}^\top \frac{1}{T-T_0}\sum\nolimits_{t=T_0+1}^{T} z_{t-1}\big)^{-1}$. Since the mean of the IV $\tilde{z}_{t-1}$ is exactly zero, the source of $C_T$ is eliminated. Hereafter, we apply the IV $\tilde{z}_{t-1}$ to run the two-step regression introduced in ((ref)) and ((ref)) to obtain the consistent IV estimator of $\beta_\tau$ under both the null and the alternative hypothesis. Specially, we run regression $ (\hat{\mu}_x^l,\hat{\theta}^l) = \arg \; \min_{\mu_x,\theta} \sum_{t= 1}^T \left(x_{t-1}- \mu_x - \theta \tilde{z}_{t-1} \right)^\top \left(x_{t-1}- \mu_x - \theta \tilde{z}_{t-1} \right)$ to obtain the fitted value $\tilde{x}_{t-1}^l = \hat{\mu}_x^l+\hat{\theta}^l \tilde{z}_{t-1}$ and the residual $\tilde{v}_{t-1}^l = x_{t-1}- \tilde{x}_{t-1}^l$ in the first step, and $ \big[\hat\mu_\tau^l,(\hat\beta_\tau^l)^\top,(\hat\gamma_\tau^l)^\top\big]^\top =\arg \min_{\mu_\tau, \beta_\tau,\gamma_\tau} \sum_{t= 1}^T \rho_{\tau}\left(y_{t}-\mu_\tau-\beta_\tau^\top \tilde{x}_{t-1}^l -\gamma_\tau^\top \tilde{v}_{t-1}^l\right)$ in the second step. Notice $\hat{\beta}_\tau^l$ uses the full sample and will not lose finite sample efficiency. The following theorem gives the asymptotic distribution of $\hat{\beta}_\tau^l$.

Then the test statistic ${Q_{l-qr}}$ is constructed for the null hypothesis $H_0:R\beta_\tau=r_\tau$,

where $\operatorname{\widehat{Avar}}(\hat{\beta}_\tau^l )= \hat{f}_{u_\tau}(0)^{-2} ( \sum_{t=1}^T \tilde{z}_{t-1} x_{t-1}^\top )^{-1}[ \tau(1-\tau) \sum_{t=1}^T \tilde{z}_{t-1} \tilde{z}_{t-1}^\top ] ( \sum_{t=1}^T x_{t-1} \tilde{z}_{t-1}^\top )^{-1}$. Moreover, we construct the t-test statistic ${\check{Q}_{l-qr}}$ when $J=1$ for right-sided test $H_0:\beta_i=0$ vs $H_a^r:\beta_i>0$ and left-sided test $H_0:\beta_i=0$ vs $H_a^l:\beta_i<0$.

The following theorem gives the asymptotic distribution of $\check{Q}_{l-qr}$.

Next, the local power of the test statistics ${\check{Q}_{l-qr}}$ and ${Q_{l-qr}}$ are shown as follows.

By the definition $\Omega_{zz}= \tau(1-\tau) \Omega_{vv}/(-2c_z)$ and Theorem (ref), it follows that $\operatorname{Avar}(\hat{\beta}_\tau^l)=-c_z \frac{\tau(1-\tau)}{2f_{u_\tau}(0)^2} (\Sigma_{zx}^*)^{-1}\Sigma_{zz}^* \left[(\Sigma_{zx}^*)^{-1}\right]^\top$ is proportional to $-c_z$ with SD predictors, where $\Sigma_{zz}^* = \lambda(\operatorname{I_K} - \tilde{S}_a)\Omega_{vv}(\operatorname{I_K} - \tilde{S}_a)^\top +(1-\lambda)(\operatorname{I_K}-\tilde{S}_b)\Omega_{vv} (\operatorname{I_K} - \tilde{S}_b)^\top$ and $\Sigma_{zx}^* = (\operatorname{I_K} -\tilde{S}_a) \int_0^{\lambda} dJ_x^c(r)\, J_x^c(r)^\top +(\operatorname{I_K}-\tilde{S}_b) \int_{\lambda}^{1} dJ_x^c(r) \, J_x^c(r)^\top + [ \operatorname{I_K}- \lambda\tilde{S}_a -(1-\lambda)\tilde{S}_b ]\operatorname{E}(v_tv_t^\top)$. Thus, the larger the magnitude of $|c_z|$, the smaller the absolute values of the test statistics $\check{Q}_{l-qr}$ and $Q_{l-qr}$ under the local alternative hypothesis $H_{a}$, resulting in reduced power for both $\check{Q}_{l-qr}$ and $Q_{l-qr}$. On the other hand, Proposition (ref) in the appendix shows that while the higher-order term $C_T$ vanishes in both $\check{Q}_{l-qr}$ and $Q_{l-qr}$, the higher order term $B_T^l$, as defined in Proposition (ref) of the appendix persists for the same reasons as $B_T$. Moreover, the smaller the magnitude of $|c_z|$, the more pronounced the size distortions induced by $B_T^l$ becomes. Thus, we mitigate the size distortion induced by the higher-order term $B_T$ by selecting a conservative tuning parameter $c_z=-8-2K$. However, Proposition (ref) suggests that this conservative setting of $c_z$ improves the size performance at the expense of reduced power.

A Linear Combination Test Statistic to Enhance the Power

To amend the loss of power due to the conservative choice of turning parameter $c_z$, we construct the following linear combination test statistic $\check{Q}_{m-qr}$ when $J=1$ for one-sided marginal test.

where $\check{q}_{0.999}$ is 99.9% quantile of the standard normal distribution and $1/(1-\delta)=20$ here. And $\check{Q}_{o-qr}$ is the conventional test statistic defined as $\check{Q}_{o-qr} = (R\hat{\beta}_\tau^o - r_\tau) \big[ \tau(1-\tau) R\big(\sum_{t=1}^T \bar{x}_{t-1} \bar{x}_{t-1}^\top\big)^{-1} R^\top \big]^{-1/2} \hat{f}_{u_\tau}(0)$, where $\bar{x}_{t-1}=x_{t-1}-\frac{1}{T}\sum_{t=1}^T x_{t-1}$ and $\big[\hat{\mu}_\tau^o,(\hat{\beta}_\tau^o)^\top \big]^\top =\arg \,\min_{\mu_\tau, \beta_\tau} \sum_{t=1}^T \rho_{\tau}\left(y_{t}-\mu_\tau-\beta_\tau^\top x_{t-1} \right)$. {We put $\check{q}_{0.999}$ here to ensure that, under the null hypothesis, most realizations of $|\check{Q}_{o-qr}|$ remain below one. Consequently, this facilitates the second term $\frac{1}{\sqrt{T}} \left(\frac{ \check{Q}_{o-qr}}{\check{q}_{0.999}}\right)^{1/(1-\delta)}\operatorname{sign}(\check{Q}_{o-qr})$ in maintaining the good size performance of $\check{Q}_{m-qr}$ in finite sample without exacerbating it.} Note that

where $ \check{Q}_{o-qr}^{*} \overset{d}{=} \beta_\tau^* \big[ \tau(1-\tau) R \big(\int_0^1 \bar{J}_x^c(r)\bar{J}_x^c(r)^\top \big)^{-1} R^\top \big]^{-1/2} $, $\beta_\tau^* \overset{d}{=} \big[\int_0^1 \bar{J}_x^c(r)\bar{J}_x^c(r)^\top \big]^{-1} \int_0^1 \bar{J}_x^c(r)\\ dB_{\psi_\tau}(r)$, and $ \check{Q}_{o-qr}^{**} \overset{d}{=} b_\tau \big[ \tau(1-\tau) R \big(\int_0^1 \bar{J}_x^c(r)\bar{J}_x^c(r)^\top \big)^{-1} R^\top \big]^{-1/2}$ with SD predictors; $ \check{Q}_{o-qr}^{*} \overset{d}{=} \operatorname{N} \big[ 0,1\big]$ and $ \check{Q}_{o-qr}^{**} \overset{d}{=} b_\tau \big[ \tau(1-\tau) R \operatorname{Var}(x_{t-1})^{-1} R^\top \big]^{-1/2} $ for WD predictors. And $\bar{J}_x^c(r)=J_x^c(r)-\int_0^1 J_x^c(r)dr$ with $0\leq r \leq 1$. Then we have the following theorem on the local power of $\check{Q}_{m-qr}$.

Next, we construct a linear combination test statistic $Q_{m-qr}$ for two-sided tests using the similar procedure to that used for constructing $\check{Q}_{m-qr}$.

where ${q}_{0.999}$ is 99.9% quantile of the $\chi_J^2$ distribution and ${Q}_{o-qr}$ is the conventional test statistic defined as ${Q}_{o-qr} \equiv (R\hat{\beta}_\tau^o - r_\tau) \big[ \tau(1-\tau) R\big(\sum_{t=1}^T \bar{x}_{t-1} \bar{x}_{t-1}^\top\big)^{-1} R^\top \big]^{-1} (R\hat{\beta}_\tau^o - r_\tau)^\top \hat{f}_{u_\tau}(0)^{-2}$. Note that

where ${Q}_{o-qr}^{*} \overset{d}{=} \beta_\tau^* \big[ \tau(1-\tau) R \big(\int_0^1 \bar{J}_x^c(r)\bar{J}_x^c(r)^\top \big)^{-1} R^\top \big]^{-1}\big( \beta_\tau^*\big)^\top$, ${Q}_{o-qr}^{**} \overset{d}{=} b_\tau \big[ \tau(1-\tau) R \big(\int_0^1 \bar{J}_x^c(r)$ \quad $ \bar{J}_x^c(r)^\top \big)^{-1} R^\top \big]^{-1} b_\tau ^\top$ and ${Q}_{o-qr}^{***} \overset{d}{=}\beta_\tau^* \big[ \tau(1-\tau) R \big(\int_0^1 \bar{J}_x^c(r)\bar{J}_x^c(r)^\top \big)^{-1} R^\top \big]^{-1} b_\tau^\top$ with SD predictors; ${Q}_{o-qr}^{*} \overset{d}{=} \chi_J^2$, ${Q}_{o-qr}^{**} \overset{d}{=} b_\tau \big[ \tau(1-\tau) R \operatorname{Var}(x_{t-1})^{-1} R^\top \big]^{-1}b_\tau ^\top$ and ${Q}_{o-qr}^{***} \overset{d}{=} \beta_\tau^* \big[ \tau(1-\tau) R \operatorname{Var}(x_{t-1})^{-1} R^\top \big]^{-1} b_\tau^\top$ for WD predictors. We can then establish the following theorem, which is analogous to Theorem (ref).

As indicated in Remark (ref), Theorem (ref) shows that ${Q}_{m-qr}$ not only retains the favorable size performance of ${Q}_{l-qr}$ but also adopts the good power performance of ${Q}_{o-qr}$.

Robust Inference for Unspanned Predictability

In this section we test the macro-spanning hypothesis using the inference approach outlined in Section (ref). We apply the proposed test statistic $Q_{m-qr}$ to test the predictive power of macro variables for n-year bond risk premia rn(x), with n= 2,3,4,5, controlling for CP factor at quantile level $\tau_j\in (0.01,0.02,\cdots,0.99)$, where $j =1,2,\cdots,J$ and $J=99$. \footnote{The same choice of $\tau_j$ and $J$ is seen in many other empirical studies such as RiskyOil2024.} We report the testing results in Section (ref) and give the potential economic explanation for our test results in Section (ref). We compare our results with those of CCL2023, the only feasible method in existing literature that is designed to do marginal test in the quantile predictive model with many highly persistent predictors.

Inference Results

The results for the two-sided marginal test are reported in Figure (ref). \footnote{See appendix of subsection (ref) for results of the one-sided test statistic $\check{Q}_{m-qr}$.} The horizontal axis is $\tau_j$ and the vertical axis is $\ln{[\max(0.001,P^*)]}$, where $P^*$ is p-values of $Q_{m-qr}$ and CCL2023. We take the logarithm of the p-value instead of the original p-value to show the statistical significance more clearly in Figure (ref). The dashed horizontal line at $-4.605$ corresponds to the significant level, which is set at $0.01$. In this way, the conclusion of predictability is more reliable with small type I error, and the corresponding out-of-sample performance is shown to be good. Additionally, we take the maximum value of p-value and 0.001 to avoid $\ln{(P^*)}$ being too small to disrupt the scale of that figure. The macro variable exhibits significant predictive power when it is within the gray region of Figure (ref).

The following conclusions could be drawn from the empirical results. First, at center quantiles, the CP factor has positive predictive power while all macro variables have negligible predictive power. This observation supports the macro-spanning hypothesis and aligns with the empirical results reported in studies by Ghysels2018 and Bauer2018 using mean regressions. Secondly, the CP factor consistently exhibits positive predictive power across all quantiles, while LN1 and LN2 show no predictive power at any quantile, strongly supporting the spanning hypothesis. Third, TDH demonstrates predictive power at the right tail quantiles for bonds of all maturities and at the left tail quantiles for 2- and 3-year bonds. Fourth, TI exhibits predictive power for bonds of all maturities across all quantiles, except for the median quantiles. The size distortion due to the highly persistence of macro variable is significantly mitigated by the proposed two-sided marginal ($Q_{m-qr}$) test. This study effectively eliminates the spurious predictive power of LN1 and LN2 across all quantiles, and of TDH and TI at the center quantiles. {Although CCL2023 makes the same conclusion for most cases}, our findings diverge from those of CCL2023 regarding the predictive power of TDH and TI at tail quantiles. First, we find that TDH has significant predictive power at right tail quantiles for bonds of all maturities, a finding not reported by CCL2023. This capability of TDH is particularly valuable for predicting tail risks associated with bond risk premia. Second, we find that TI has predictive power at both tails quantiles for bonds of all maturities, while CCL2023 fails to find the predictive power of TI at right tail quantiles for 4- and 5-year bonds. Third, we find the significant predictive power of TDH at left tail quantiles for 2- and 3-year bonds, which CCL2023 does not find. Considering the simulations presented in the appendix, which demonstrate that our size and power performance surpass those of CCL2023 across all quantiles, the most plausible explanation for the discrepancies between our findings and those of CCL2023 is that CCL2023 likely commits type II errors (failing to detect true predictive power) at tail quantiles, whereas our analysis does not. The predictability at tail quantiles (high-risk region) is practically beneficial, e.g., for constructing the tail risk indicator in the appendix Section (ref).

Economic Explanations of Testing Results

The macro-spanning hypothesis states that the yield curve contains all available information about future bond risk premia, which provides economic interpretation about the empirical findings at the center quantiles. The CP factor contains the information of the yield curve so it has significant predictive power while LN1 and LN2 have negligible predictive power at all quantiles. Moreover, the spanning hypothesis also explains the test results that all other variables, including TDH and TI, have no predictive power at center quantiles. The most interesting empirical finding of this paper is the significant predictive power of TDH at right tail quantiles and the significant predictive power of TI at both tails quantiles for bonds of all maturities. The predictive power of TDH at left tail quantiles for 2- and 3-year bonds are also found. Subsequently, we offer potential economic explanations for these findings. Since investors prefer 1-year bonds over n-year bonds (n=2,3,4,5) at right tail quantiles, \footnote{See more discussions of this point in Section (ref).} this preference results in a more pronounced increase in demand for 1-year bonds compared to n-year bonds when the TDH rises. Consequently, as TDH increases, the price of 1-year bonds experiences a sharper increase relative to the prices of n-year bonds (n=2,3,4,5). This sharper price increase leads to a more substantial decrease in the YTM of 1-year bonds compared to that of n-year bonds. Given that the risk premium of n-year bonds (n=2, 3, 4, 5) is defined as the difference between the YTM of n-year and 1-year bonds, it follows that the risk premia for n-year bonds will increase if TDH rises at the right tail quantiles. In other words, TDH has positive predictive power for bonds of all maturities at right tail quantiles. On the other hand, when investors exhibit a preference for bonds with maturities of two and three years over 1-year bonds at the left tail quantiles, the demand for these 2- and 3-year bonds significantly increases. This heightened demand occurs particularly if the TDH rises. Consequently, the prices of 2- and 3-year bonds rise more sharply compared to 1-year bonds under the same conditions. This increase in price leads to a more pronounced decrease in the YTM of 2- and 3-year bonds than that observed in 1-year bonds. As a result, the bond risk premia, which represent the difference between the returns of 2- and 3-year bonds and 1-year bonds, decrease if the TDH increases at the left tail quantiles. This effect highlights the sensitivity of bond risk premia to shifts in investor preference and market conditions at the left tail quantiles. In other words, TDH has negative predictive power for both 2- and 3-year bonds at left tail quantiles. Besides that, the demand for n-year (n=2,3,4,5) and 1-year bonds may rise comparably if TDH increases within the center quantiles. Additionally, at the left tail quantiles of the risk premia for bonds with longer maturities (n=4,5 years), the risk associated with 1-year bonds is high, while bonds with maturities of four and five years continue to face elevated risks due to their longer durations. Consequently, if the TDH increases at the left tail quantiles, the demand for both 1-year bonds and longer-term bonds (n=4,5 years) may rise comparably. As a result, the prices and YTM for both 1-year and longer-term bonds (n=4,5 years) may move in the same direction and by approximately the same magnitude, leading to relatively stable bond risk premia. That is, TDH has no predictive power at center quantiles for bonds of all maturities and left tail quantiles for 4- and 5-year bonds. Next, we discuss the predictive power of TI found at all quantiles except center quantiles. If TI increases, the real return of investors decrease, leading to a reduced demand for all 1- to 5-year bonds. However, the demand for bonds with maturities of two to five years decreases more sharply than that for 1-year bonds when TI increases since investors prefer 1-year bonds at right tail quantiles. \footnote{See more discussions of this point in Section (ref).} As a result, the prices of n-year (n=2,3,4,5) bonds decrease more sharply than that of a 1-year bond when TI increases. Thus, the YTM of these longer-term bonds rise more significantly than those of 1-year bonds. Given that the risk premia for bonds with maturities of two to five years are defined as the difference between their YTM and that of 1-year bonds, the risk premia for these longer-term bonds (n=2, 3, 4, 5) increase as TI rises. That is, TI has positive predictive power for all bonds at right tail quantiles. Following the same logic, TI has negative predictive power for all bonds at left tail quantiles.

In-sample and Out-of-sample Performance Evaluation

In this section, we use the test results of $Q_{m-qr}$ in Section (ref) to obtain the in-sample and out-of-sample predictions and compare the performance with those using the test of CCL2023. We also use the out-of-sample predictive power to introduce a new tail risk indicator of the U.S. bond market in the appendix Section (ref). The quantile-weighted continuous ranked probability score (qw-CRPS) criterion introduced by Gneiting2011 is useful to evaluate the in-sample and out-of sample prediction performance of our method at left tail quantiles, center quantiles, right tail quantiles and both tails quantiles. Specifically, the qw-CRPS is a scoring rule that takes into account information from the entire predictive density, but it also permits researchers to focus more on certain areas of interest. For example, evaluating how effectively a model predicts downside risks could be of independent interest, for this task qw-CRPS puts more emphasis on the area in the right tail quantiles. {The in-sample prediction of condition quantile of bond risk premia $\hat{y}_{t\tau_j}$ is obtained as follows.} First, using the two-sided test results of $Q_{m-qr}$ in Section (ref), we select predictors $\dot{x}_{t-1}^j=(x_{i_{j1},t-1},x_{i_{j2},t-1},\cdots,x_{i_{jI},t-1},)^\top$ in CP factor, LN factors, TI and TDH if the p-values of $x_{i_{j1},t-1},x_{i_{j2},t-1},\cdots,x_{i_{jI},t-1}$ are less than 1%, where $1\leq i_{j1} < i_{j2} <\cdots<i_{jI} \leq K$. That is, $\dot{x}_{t-1}^j$ has predictive power at quantile $\tau_j$. Second, we run the following quantile regressions $\big(\hat{\dot\mu}_{\tau_j}, \hat{\dot\beta}_{\tau_j}^\top \big)^\top =\arg \,\min_{\mu_{\tau_j}, \beta_{\tau_j}} \sum_{t=1}^T \rho_{\tau_j} \, (y_{t}-\mu_{\tau_j}-\dot\beta_{\tau_j}^\top \dot{x}_{t-1}^j )$ at quantile $\tau_j$, where $j=1,2,\cdots,J${, which are defined in the beginning of Section (ref)}. Finally, we obtain in-sample prediction value $\hat{y}_{t\tau_j}=\hat{\dot\mu}_{\tau_j}+\hat{\dot\beta}_{\tau_j}^\top \dot{x}_{t-1}^j$ and the in-sample error estimator $\hat{u}_{t\tau_j}=y_t-\hat{y}_{t\tau_j}$. Then the qw-CRPS, which is calculated by applying weights to $\hat{y}_{t\tau}$ for $J-1$ different quantiles, is computed as follows

where $W\left(\tau_j\right)$ is a deterministic weighting function. The considered weighting functions are: $W\left(\tau_j\right)=\tau_j\left(1-\tau_j\right)$, which gives more weight to the center of the predictive distribution; $W\left(\tau_j\right)=\left(1-\tau_j\right)^2$ (and $W\left(\tau_j\right)=\tau_j^2$), which assigns greater weight to the left (right) tail to emphasize downside (upside) risks; and $W\left(\tau_j\right)=\left(2 \tau_j-1\right)^2$ which places emphasis on the prediction performance at both tails. Using the qw-CRPS criteria defined in ((ref)), we obtain the following in-sample performance of our proposed approach and CCL2023 in Table (ref). \footnote{{In-sample performance of CCL2023 is obtained by the same way of $Q_{m-qr}$.}} It shows that the in-sample performance of the proposed test $Q_{m-qr}$ is much better than CCL2023 at center quantiles, left tail quantiles, right tail quantiles and both tails quantiles, since all of its $\operatorname{qw}_c$ values are much less than those of CCL2023. The excellent in-sample performance of $Q_{m-qr}$ arises from its success of finding the significant predictive power of TDH and TI at right tail quantiles. This result is also shown in Table (ref), that the difference between $\operatorname{qw}_c$ of $Q_{m-qr}$ and CCL2023 when the weight $W(\tau_j)$ emphasizing in right tail quantiles is much larger than those in left tail quantiles and center quantiles. {8pt} \captionsetup{ width=0.8\textwidth, }

Next, we present the out-of-sample prediction results. To evaluate the out-of-sample prediction performance (in the out-of-sample period from $T_m$ to $T$, and we set $T_m$ to be 1992/01 and $T$ to be 2022/12), we undertake the following steps. To predict the conditional quantile of bond risk premia at period $\tilde{T}$ and quantile level $\tau_j$, where $j=1,2,\cdots,J$ and $T_m \le \tilde{T}\le T$, we first use the sample during period $1-(\tilde{T}-1)$ to run the following quantile regression to obtain the real time coefficient estimator $ \big[\hat{\dot\mu}_{\tau_j}(\tilde{T}), \hat{\dot\beta}_{\tau_j}(\tilde{T})^\top \big]^\top =\arg \,\min_{\mu_{\tau_j}, \beta_{\tau_j}} \sum_{t=1}^{\tilde{T}-1} \rho_{\tau_j} (y_{t}-\mu_{\tau_j}-\dot\beta_{\tau_j}^\top \dot{x}_{t-1}^j )$, where $\dot{x}_{t-1}^j$ is the one used in ((ref)). Then the out-of-sample predicted value of bond risk premia at quantiles $\tau_j$ is $\hat{y}_{\tau_j}^o(\tilde{T})=\hat{\dot\mu}_{\tau_j}(\tilde{T})+ \hat{\dot\beta}_{\tau_j}(\tilde{T})^\top \dot{x}_{\tilde{T}-1}^j$ and the out-of-sample quantile regression error is $\hat{u}_{\tau_j}^o (\tilde{T})=y_{\tilde{T}}-\hat{y}_{\tau_j}^o (\tilde{T})$. The out-of-sample performance in the period from $T_0$ to T is evaluated by qw-CRPS. Specifically, the out-of-sample evaluation criteria are $\operatorname{qw}-{\operatorname{CRPS}_{\tilde{T}}}=\frac{2}{J-1} \sum_{j=1}^{J-1} W\left(\tau_j\right) \rho_{\tau_j}[\hat{u}_{\tau_j}^o (\tilde{T})]$, and $ \operatorname{qw}_c(T_m) = \frac{1}{T-T_m+1} \sum_{\tilde{T}=T_m}^T \operatorname{qw}-{\operatorname{CRPS}_{\tilde{T}}}$. By this criteria, we obtain the out-of-sample performance in the period 1992--2022, which is shown in Table (ref). {8pt} \captionsetup{ width=0.8\textwidth, }

The out-of-sample performance of $Q_{m-qr}$ is better than those of CCL2023, as $\operatorname{qw}_c(T_0)$ of $Q_{m-qr}$ is much less than those of CCL2023 in all quantiles except for rx(2) at center quantiles and left tail quantiles where they are equivalent. The excellent out-of-sample performance of $Q_{m-qr}$ also arises from its success of finding the significant predictive power of TDH and TI at right tail quantiles. The difference between $\operatorname{qw}_c(T_0)$ of $Q_{m-qr}$ and CCL2023 is much larger at right tail quantiles than those at left tail quantiles, center quantiles and both tails quantiles.

Conclusion

We examine the predictability of bond risk premia using predictive quantile regression with many highly persistent predictors, employing a reliable inference procedure tailored for this analysis. Our results lend support to the macro-spanning hypothesis at center quantiles, while evidence at tail quantiles contradicts this hypothesis. This dichotomy helps clarify inconsistencies observed in previous studies that utilized mean regressions. Specifically, all predictors, with the exception of the CP factor, exhibit negligible predictive power at the center quantile. Additionally, TDH demonstrates predictive power at the right tail quantiles, and TI shows predictive power at both left and right tail quantiles. Furthermore, we demonstrate that both in-sample and out-of-sample prediction performance using our proposed inference method significantly outperforms existing methods. \setcounter{page}{1} \counterwithin{figure}{section} \counterwithin{table}{section} \counterwithin{thm}{section} \counterwithin{prop}{section} \counterwithin{equation}{section} { \bf

} In this supplementary material, we provide the following parts: Appendix (ref) gives the details of computation algorithm to construct our test statistics used in the main text. Appendix (ref) collects all the results for Monte Carlo simulations. The numerical results show that it is necessary to use the new method introduced in the main text Section (ref) for the macro-spanning hypothesis test. Appendix (ref) includes additional asymptotic properties for the higher-order terms that appear in the test statistics construction, whose results we applied in the main text Section (ref). Appendix (ref) provides additional empirical results. A new tail risk indicator is also introduced there, which further showcases the usefulness of our test. Appendix (ref) presents selected proofs for main theorems.