To be regarded as the main driver of risk premia, the (1-year) RFF should also summarize variations in expected bond returns considering holding periods different from 1 year. In this section we show that it does not. The 1-year RFF outperforms the first three PC only in 1-year excess return regressions such as those in . In regressions of excess returns for holding period longer than a year, the first three PC provide a better fit of excess returns. And for sufficiently long holding periods, the PC have three times the explanatory power of the RFF.

We first regress the average 1-year excess return ${\overline{rx}}_{t\to t+1}$ on the forward rates **f**_{t} using both the FB and GSW datasets. We call this the “1-year-ahead” RFF because it maximizes the forecasting performance in a regression of the average 1-year excess return. Likewise, we label the “m-year-ahead” RFF as the fitted value of a regression of the average m-year excess return ${\overline{rx}}_{t\to t+m}$ on the forward rates **f**_{t}. Let $RF{F}_{t}^{(m)}$ denote the m-year-ahead return forecasting factor computed using this procedure (below we consider $m=1,2,3,4$).

Next, we run predictive regressions of the expected h-year excess return averaged across available maturities ${\overline{rx}}_{t\to t+h}$, on the m-year return forecasting factor $RF{F}_{t}^{(m)}$, and on the first three principal components $P{C}_{1t}$, $P{C}_{2t}$, and $P{C}_{3t}$ extracted from the same datasets,^{5}

$${\overline{rx}}_{t\to t+h}={\alpha}_{0}^{(h)}+{\alpha}_{1}^{(h)}RF{F}_{t}^{(m)}+{\u03f5}_{t+h}$$(3)

$${\overline{rx}}_{t\to t+h}={\delta}_{0}^{(h)}+{\delta}_{1}^{(h)}P{C}_{1t}+{\delta}_{2}^{(h)}P{C}_{2t}+{\delta}_{3}^{(h)}P{C}_{3t}+{\eta}_{t+h}.$$(4)

Figure 1 displays the *R*^{2} of the regressions of the excess holding returns on the return forecasting factors using the Fama-Bliss dataset. For example, the top left panel of the figure reports the *R*^{2} of the regressing the 1- through 4-year ahead excess returns on the 1-year-ahead return forecasting factor (dotted-circled line) and on the first three principal components (dotted-squared line) as a function of the holding period $h=1,2,3,4$. While the RFF has more predictive ability than the PC at the 1-year holding horizon, the relation is reversed at longer holding periods: the RFF explains less than 10 percent of the variability of the 4-year excess return while the PC explain about 25 percent of the variance.

Figure 1: *R*^{2} of the average h-year excess holding return regressions using Fama-Bliss yields. The figure shows the *R*^{2}s of a regression of the average h-period excess holding returns (h = 1, 2, 3, 4) on the return forecasting factor and on the first three principal components $P{C}_{1t}$, $P{C}_{2t}$, and $P{C}_{3t}$ together.

In the upper left panel the return forecasting factor is computed using a regression of 1-year excess returns on the current forwards; in the upper right panel the return return forecasting factor is computed with a regression of 2-year excess returns on the current forwards, and so forth.

In a similar fashion, the upper right panel of Figure 1 shows the *R*^{2} of regressions of the average *h*-period excess holding returns ($h=1,2,3,4$) on the 2-year ahead return forecasting factor, $RF{F}^{(2)}$, and the PC, and likewise for the lower two panels of the figure but using $RF{F}_{t}^{(3)}\mathit{\text{and}}RF{F}_{t}^{(4)}$. Again, while the *h*-year-ahead RFF outperforms the PC at the *h*-year holding horizon, the PC tend to do better than the RFF at other holding horizons.

This pattern is reinforced when we consider the GSW database. Figure 2 is identical to Figure 1 but using a larger set of holding periods *h*. For example, the upper left panel of the figure shows the *R*^{2} of regressions of the average excess holding returns ${\overline{rx}}_{t\to t+h}$ for $h=1/12,2/12,...,108/12$ on the 1-year-ahead return forecasting factor $RF{F}_{t}^{(1)}$ and the first three principal components, as a function of the holding period *h*. Note that for longer holding horizons, the predictive power of the principal components is more than three times larger than that of the 1-year-ahead return forecasting factor and twice as large as that of the 2-year-ahead return forecasting factor. Finally, the other two return forecasting factors, $RF{F}_{t}^{(3)}$ and $RF{F}_{t}^{(4)}$, do not provide any predictive improvement over the principal components.^{6}

Figure 2: *R*^{2} of h-year excess holding return regressions using GSW data.

The figure shows *R*^{2}s of regressions of the average *h*-year excess holding returns on the 1- though 4-year ahead RFF and on the first three PC. In the upper upper left panel the RFF is computed using a regression of 1-year excess returns on 1- through 10-year forwards; in the upper right panel the RFF is computed with a regression of 2-year excess returns on 1- through 10-year forwards, and so forth.

The observation that the 1-year-ahead return forecasting factor performs poorly when attempting to explain excess holding returns for other horizons can be rationalized in several ways. The 1-year expected excess return obtained from the regressions can be thought as a model-free version of the expected excess return obtained from a structural model or an affine model of bond prices. For example, Bansal and Shaliastovich (2013) propose an equilibrium long-run risk model with Epstein-Zin preferences in which bond risk premia depend on expected economic growth and inflation. This model is able to reproduce 1-year excess returns remarkably well. Yet, since in the model investors prefer early resolution of uncertainty, risks that affect returns in the short run can be very different from those that affect long-run returns.

Then we can understand the previous results in two alternative ways. The first is that the determinants of short and long run risks are different. Therefore, trying to fit short and long period excess returns using a single 1-year return forecasting factor is most likely deemed to fail. Alternatively, since the Bansal and Shaliastovich (2013) model predicts a positive relation between bond risk premia and the preference for early resolution of uncertainty, estimating the model using data for different holding horizons will most likely generate different measures of risk premia. The empirical counterpart of this observation is that the h-year return forecasting factor will be a good predictor of the h-year excess return but not necessarily for other holding horizons. Trying to fit a 4-year excess return using the 1-year return forecasting factor is akin to wrongly calibrating the model.

A possible exception to the previous argument is when there exists a single factor structure in all excess returns, a property that is not supported by the data as we show in Section 2.4. But before doing that we evaluate the out-of-sample forecast properties of the empirical models discussed above.

We now analyze the relative merits of the 1-year-ahead RFF, the first three PCs, and a constant in terms of their ability to forecast excess returns.^{7} We compute the out-of-sample forecasts based on a series of recursive forecasts beginning in 2003M12 and extending through 2014M12 for a forecast horizon of 1 month and for holding periods of 12, 24, 36, and 48 months.^{8}

reports the results of this exercise. The forecasting ability of the two empirical models (using the RFF and the PCs) is poor, and both are outperformed by a constant for all holding periods. Which of the two models (RFF or PCs) forecast returns better depend on which holding horizon is forecast. Perhaps surprisingly, the principal components outperform the 1-year-ahead RFF only for a holding period of 1 year. Figure 3 displays the out-of-sample forecasts of the excess returns for the four holding periods. Overall, these results show the poor forecasting performance of both models.

Table 2: Out-of-sample forecast accuracy.

Figure 3: Out-of-sample performance of the empirical models.

One-month ahead out of sample forecasts of h-period excess returns of the 1-year-ahead RFF, the principal components, and a constant.

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