2.1 Estimation of the multi-level factor model

The multi-level factor model has seen a number of applications in the literature. The examples include Gregory, Head, and Raynauld (1997), Kose, Otrok, and Whiteman (2003), and Moench, Ng, and Potter (2013) among others. In most applications, multi-level factors are estimated via Bayesian methods. We estimate the multi-level factor model in (2) via the procedure proposed in Choi et al. (2014), which is in line with the principal component method by Bai (2003) .

Rewrite the model in (2) in matrix notation.

where

Stacking Xmt′s vertically across t yields that

where

The estimation procedure proposed in Choi et al. (2014) is as follows.

Step 1: Select two sectors, say m = 1 and 2, and obtain an estimate of K_mt by the principal component method, say K^mt, which is consistent for some linear combinations of the global and sectoral factors, from the result of Bai (2003). Let Q^ab=T−1∑t=1TK^atK^bt′ (a,b=1,2) and λ^1≥⋯≥λ^s+r1 be the generalized eigenvalues in decreasing order, solving

The vector p^j is the eigenvector corresponding to λ^_j. Then, the initial estimate of the global factors is given by G^t(1)=[p^1′,…,p^s′]K^1t. The superscript (1) is employed to distinguish the current estimate from the final estimate of G_t obtained in Step 3. Given that sectoral factors from different sectors are uncorrelated, pre-multiplying the eigenvectors [p^1′,…,p^s′] has an effect to disentangle the global factors from the sectoral factors.

Step 2: Rewrite the model in (3) as

and, estimate Ψ_m and S_mt by the principal component method after projecting out G^t(1). These estimates are denoted as Ψ^m(1) and S^mt(1), respectively.

Step 3: Using Ψ^m(1)S^mt(1), rewrite the model in (3) as

stack them as

and estimate {Γ_m} and G_t by the principal component method. These estimates are written as {Γ^m(1)} and G^t(2). G^t(2) is the final estimate of the global factors.

Step 4: Using {Γ^m(1)} and G^t(2), rewrite the model in (3) as

and estimate Ψ_m and S_mt by the principal component method. Denote these estimates as Ψ^m(2) and S^mt(2), respectively. S^mt(2) is the final estimate of the sectoral factors. We refer to Choi et al. (2014) for further asymptotic theory such as consistency of these multi-level factor estimates.

One may think of splitting the panel at the sector level to extract sectoral factors. For example, Ang and Piazzesi (2003) extract a real factor from a group of real variables and an inflation factor from a group of price variables. Ludvigson and Ng (2011) also split the panel at the sectoral level and extract common factors separately from each sector. These factors are linear combinations of the global and sectoral factors from the perspective of the multi-level factor model and consequently have correlation with variables in other sectors due to the existence of global factors. Thus, even if one finds that common factors obtained from one specific sector have predictive power, one cannot tell if the source of predictive power is the time variation specific to that sector or the time variation common to all sectors. The estimation method employed in this paper avoids such an issue.

2.2 Selection of predictive regressors

Having estimated the global and sectoral factors, we determine which variable enters Z_t in (1). In a related study, Ludvigson and Ng (2009) estimate factors, without the multi-level structure we use, from a similar panel of macroeconomic variables and show that some of the factors possess significant predictive power. They also consider the squares and cubes of the factors as candidate predictors. A rationale for this is what enters Z_t is some unknown function of each factor which can be well approximated by the third order polynomial. We also set our candidate predictors as the global and sectoral factors themselves as well as their squares and cubes.

Two methods are considered to determine the composition of Z_t in (1). They are the best subset selection and the backward elimination.^[1] In principle, the best subset selection finds the combination of predictors that gives the smallest residual sum of squares for each subset size and chooses the best subset size by comparing the model fits penalized by the subset size. To save on computation time, we first find the preliminary best subset by excluding the squares and cubes from candidates. We then take the variables in the preliminary best subset as well as their squares and cubes as new candidates and find the final best subset. For the comparison across subset sizes, we use Bayes Information Criterion (BIC). The backward elimination starts from a model with all candidate predictors and sequentially deletes the predictor with the greatest p-value until the p-values of all predictors are under a certain threshold. The Newey-West robust standard errors which are obtained using 18 lags are used to compute p-values.^[2]

We prefer the best subset selection because the choice of a threshold in the backward elimination is difficult to justify while the final selection of predictors critically relies on the threshold. Furthermore, when the threshold is set at a common choice of 0.05, the resulting predictive regression yields an adjusted R² slightly lower than the predictive regression determined by the best subset selection.

4.1 In-sample prediction

The total number of factors is estimated as 8 by the information criteria proposed by Bai and Ng (2002), agreeing with other studies using the same data set. We assume that there are 2 global factors and 2 sectoral factors in each sector. As a matter of notation, we denote the global factors by G₁ and G₂, the real factors by S₁₁ and S₁₂, the financial factors by S₂₁ and S₂₂, and the inflation factors by S₃₁ and S₃₂. We estimate these factors as described earlier and now investigate with what variables in the panel each factor is correlated. Figure 1–Figure 4 report the marginal R² values from regressing each multi-level factor on each variable in the panel. Note that the marginal R² value is the square of the correlation coefficient and remains the same when the dependent variable and the regressor switch around. Figure 1 shows that both global factors are correlated with the variables in the real and financial sectors. In our definition, any factor that affects multiple sectors is regarded as a global one. Figure 2–Figure 4 respectively correspond to the real, financial, and inflation factors. These sectoral factors have correlation mostly in the sector of their name, as assumed by the model.

Figure 1:

Marginal R² for global factors.

Notes: The marginal R² values are obtained from regressing a global factor, G₁ or G₂, on each variable in the panel. The list of variables is provided in the appendix. The variables numbered from 1 to 70 are real variables, the ones from 71 to107 are financial variables, and the ones from 108 to 132 are inflation variables.

Figure 2:

Marginal R² for real factors.

Notes: The marginal R² values are obtained from regressing a real factor, S₁₁ or S₁₂, on each variable in the panel. The list of variables is provided in the appendix. The variables numbered from 1 to 70 are real variables, the ones from 71 to107 are financial variables, and the ones from 108 to 132 are inflation variables.

Figure 3:

Marginal R² for financial factors.

Notes: The marginal R² values are obtained from regressing a financial factor, S₂₁ or S₂₂, on each variable in the panel. The list of variables is provided in the appendix. The variables numbered from 1 to 70 are real variables, the ones from 71 to107 are financial variables, and the ones from 108 to 132 are inflation variables.

Figure 4:

Marginal R² for inflation factors.

Notes: The marginal R² values are obtained from regressing an inflation factor, S₃₁ or S₃₂, on each variable in the panel. The list of variables is provided in the appendix. The variables numbered from 1 to 70 are real variables, the ones from 71 to107 are financial variables, and the ones from 108 to 132 are inflation variables.

The best subset selection procedure is applied to the predictive regression where the predicted variable is the excess return on a 2-year discount bond. The selected variables are the two real factors (S₁₁ and S₁₂), the square of the first real factor (S112) and the second financial factor (S₂₂). These are the main predictors of ours. To check the robustness of our choice, we apply the best subset selection to subsamples in a recursive manner. The subsample first covers data up to 2005:2 and increases by a month until it includes the entire data. Table 1 shows the frequency that each factor is selected from the recursive exercise. The second and third rows correspond to the case where the predicted variable is the 2-year and 3-year bond excess returns respectively. The frequencies are similar across the two rows. The first real factor (S₁₁) and the second financial factor (S₂₂) are selected in all subsamples. The second global factor (G₂), the second real factor (S₁₂) and the first financial factor (S₂₁) are selected with a frequency less than a half, and the other factors are never selected.^[4] The first real factor (S₁₁) and the second financial factor (S₂₂) are the two most consistently selected predictors.

The predictive regression results using the main predictors are reported in Table 2(A). The upper and lower part of the table respectively correspond to the 2-year and 3-year bond excess returns. CP stands for the single forward rate factor that is shown to have predictive power by Cochrane and Piazzesi (2005). F4 is the projection image of excess bond returns averaged across maturities 2 and 3 years on the four main predictors. In other words, this is the linear combination of the main predictors that best explains the average excess bond return. In row (a), only CP is used as a predictor and it is significant at 5% significance level^[5] with the adjusted R² being close to 0.2. This confirms the predictive power of CP in our data. In row (c), the main predictors are added and the adjusted R² rises to 0.3. In row (b), CP is dropped and the resulting adjusted R² is slightly over 0.1, which is roughly the same as the increment in the adjusted R² from row (a) to row (c). This suggests that our main predictors possess some predictive power independent of CP. This result is in line with the findings in Ludvigson and Ng (2009) where unrestricted factors are used. In rows (d) and (e), the main predictors are replaced by the single predictor F4. Comparing them with rows (b) and (c), there is little change in the adjusted R² values. The single predictor F4 has as much predictive power as the set of four main predictors. While S₁₁ and S₂₂ are significant in all specifications, S₁₂ and S112 have less significance especially when the dependent variable is rxt+12(2). Hence, it can be said that S₁₁ and S₂₂ are more important predictors than S₁₂ and S112, which also matches the implication from Table 1.

In Table 2(B), we compare the predictive power between the real sector and the financial sector. In rows (f) and (g), we exclude the financial factor (S₂₂). The adjusted R² values do not drop much. They are slightly lower than 0.1 without CP and slightly lower than 0.3 with CP. In rows (h) and (i), we exclude the real factors (S₁₁, S₁₂ and S112). Now the adjusted R² values drop more substantially. Especially with CP, the adjusted R² values are around 0.2, which is only slightly higher than what is achievable with CP only (rows (a)).

To sum up, the main predictors seem to have predictive power irrespective of whether or not CP is controlled. Between the real factors and the financial factor, the real factors appear to have more predictive power. The financial factor might have some predictive power but its predictive power is less clear in the presence of CP.

We now investigate if the other factors not selected as the main predictors have any predictive power. 3 reports the predictive regression results with the global factors (G₁ and G₂), the first financial factor (S₂₁) and the inflation factors (S₃₁ and S₃₂). The model specification in each row is the same as in Table 2. In row (b), only these factors are used as predictors and the resulting adjusted R² is close to 0, suggesting that they have no predictive power. In row (c), CP is added and the adjusted R² is close to 0.2, what can be achieved with CP alone. (See Table 2 row (a).) Furthermore, none of these variables is significant. F5 is the single predictor constructed in the same way as F4, but based on G₁, G₂, S₂₁, S₃₁ and S₃₂. The results in rows (d) and (e) suggest that F5 does not have any meaningful predictive power. Hence, these factors are deemed to have no predictive power.

Next, we inspect if these factors can boost the predictive power of the main predictors, although they do not have independent predictive power. We augment the predictive regressions reported in Table 1 with G₁, G₂, S₂₁, S₃₁ and S₃₂. Table 4 reports the adjusted R² values of the augmented regressions. The column denoted by “None” has no added regressors and thus is the same as the adjusted R² values in Table 2(A). As we add the global factors, the first financial factor, the inflation factors or all of them, the adjusted R² values remain almost unchanged. Using variables other than the main predictors does not contribute to the predictability.

In the finance literature, a number of studies show that not only measures on real activities such as output gap, employment growth and Chicago Fed National Activity Index (CFNAI) but also measures on inflation are important predictors for excess bond returns. For example, see Ang and Piazzesi (2003), Cooper and Priestley (2011), and Joslin, Priebsch, and Singleton (2014). Our result indicates, however, that the time variation that exists exclusively in inflation measures does not help forecasting bond excess returns, at least for the data set we use. One possible reconciliation is that if an inflation variable with predictive power has a component that is correlated with real variables and if that component is not common among the other inflation variables, then that inflation variable can have some predictive power due to correlation with real variables while that correlation is not captured as a common factor among inflation variables.

4.2 Out-of-sample prediction

In Table 5(A), the 1 year ahead out-of-sample forecast performance of the main predictors is reported. The benchmark model uses only a constant or CP_t in addition to a constant, while the comparison model uses the main predictors in addition to the variables of the benchmark model. The model parameters and factors are recursively estimated in the estimation sample to generate 1 year ahead out-of-sample forecasts. For the forecast sample of 1996:8–2011:12, the initial estimation sample spans from 1976:6 to 1995:8, where the dependent variable spans from 1977:6 to 1995:8 and the predictors span from 1976:6 to 1994:8. The predictors are estimated from the panel spanning from 1976:6 to 1994:8. The forecast for the excess return at 1996:8 is obtained by combining the coefficient estimates with the value of predictors at 1995:8. Then the estimation sample expands by a month to include data from 1976:6 to 1995:9 and the forecast for 1996:9 is obtained in the same manner. This process continues until the forecast for 2011:12 is obtained.

The column labelled as “relative MSE” contains the ratio of the mean squared forecast error (MSE) of our model relative to that of the benchmark model. Under the presence of CP, the MSE of our model is smaller than that of the benchmark model for both maturities and for both forecast samples. However, the performance reverses in the absence of CP. The column labelled as “Test statistics” reports the values of the ENC-NEW test statistic proposed by Clark and McCracken (2001). The test rejects the null of equal predictive ability at the asymptotic 5% size in all cases, implying that the main predictors can improve the forecast of the benchmark model.

The superior performance of the main predictors is clear with CP, while the evidence is mixed without CP. This might be due to the fact that the main predictors are selected after CP is controlled. Having mentioned that, we cannot say that the above exercise is strictly out-of-sample, because the entire sample is used to select the main predictors.^[6] One might consider a forecast model which selects the predictors in each recursive estimation sample. Then, what can be obtained is the performance of a model that is made of a set of candidate predictors and a specific choice rule. This is different from what we are interested in, that is, the performance of our predictors. Instead, we consider an out-of-sample prediction exercise where only one of our main predictors is included, and report the results for the two most important predictors, the first real factor (S₁₁) and the second financial factor (S₂₂). This not only avoids the issues arising from the predictor selection but also is meant to compare the out-of-sample forecast ability between the two predictor factors.

Table 5(B) reports the results when only S₁₁ is used instead of the main predictors while Table 5(C) reports the results when only S₂₂ is used. The out-of-sample predictability is much clearer with S₁₁. With S₁₁, the relative MSE is smaller than one in all cases but one and the ENC-NEW test rejects in all cases. With S₂₂, the relative MSE is smaller than one in only two cases and the ENC-NEW test fails to reject in two cases.

4.3 Countercyclicality of excess bond returns

In Figure 5 and Figure 6, we plot each of S₁₁ and F4 against the growth of the industrial production (IP) as 12 month moving averages. In the figures, the shaded areas denote the recessions designated by the NBER. As can be seen, both S₁₁ and F4_t show clear countercyclicality, with their respective correlation coefficient with the IP growth being −0.82 and −0.55. Both S₁₁ and F4_t also have positive coefficients in the predictive regression across all maturities (See Table 2). This suggest that the excess bond returns have a strong countercyclical component.

Figure 5:

Countercyclicality of excess return, first real factor and IP growth.

Notes: S₁₁ stands for the 12 month moving average of the first real factor. IP growth stands for the 12 month moving average of the industrial production growth. The shaded areas denote the recessions designated by the NBER.

Figure 6:

Countercyclicality of excess return, single predictor factor and IP growth.

Notes: F4 stands for the 12 month moving average of the single predictor factor, constructed with the main predictors. IP growth stands for the 12 month moving average of the industrial production growth. The shaded areas denote the recessions designated by the NBER.

4.4 Choice of factor numbers

Our model, when collected across sectors, can be written as

which is a standard factor model when the zero restrictions in Λ′ are ignored.

If the numbers of global and sectoral factors are correctly specified, each unrestricted factor must be perfectly spanned by the multi-level factors, because the unrestricted factors are exact rotations of the multi-level factors, that is, FΨ = [G, S₁, …, S_M] for some invertible matrix Ψ. Naturally, one is tempted to choose the factor numbers by looking at how close to one the R² value from the regression of each unrestricted factor estimate on the multi-level factor estimates is.^[7] However, the interpretation of such R² values is challenging. Both the unrestricted and multi-level factors are obtained with estimation errors, and the cause of a low R² value can be on either side. Also, because each unrestricted factor estimate will be approximated with a varying degree of precision across possible combinations of factor numbers, one has to decide how much weight to put on each of the R² values to assess the adequacy of the model specification.

We choose the factor numbers so that the unrestricted factor estimates with predictive power are replicated as much as possible. This is motivated by the aim of our study to uncover the informational contents of Ludvigson and Ng type macro factor predictors. We first estimate eight unrestricted factors from our data set, say F = [F₁, …, F₈], via principal component analysis and obtain five predictors by the best subset selection. They are F₁, F₂, F₄, F₈ and F12. Table 6(A) reports the predictive regression results using these predictors. To choose the factor numbers, we obtain the projection images of the unrestricted factor estimates onto the space spanned by the multi-level factor estimates for each possible choice of factor numbers. Then, we look at how well the predictive regressions with the unrestricted factors in Table 6(A) are replicated when these projection images replace the unrestricted factor estimates. When there are two global factors and two sectoral factors in each of the three sectors, the predictive regressions with the unrestricted factor estimates are replicated remarkably well. Table 6(B) reports the results. As can be seen, the coefficient estimates and R²s are very similar across Table 6(A) and (B).

One may wonder if there is an one-to-one correspondence between the multi-level factors and the unrestricted factors. Figure 7 shows the marginal R² values from regressing an unrestricted predictor on each multi-level factor. F₁ is a linear combination of G₁ and S₁₁ where S₁₁ has greater importance than G₁. F₂ and F₄ are a linear combination of various multi-level factors. F₈ is not well represented as a linear combination of multi-level factors, but this has little impact on the predictive regression as evidenced in Table 6(B).

Figure 7:

Composition of unrestricted factors in terms of multi-level factors.

Notes: The marginal R² values are obtained from regressing an unrestricted factor, F₁, F₂, F₄ or F₈, on each multi-level factor.

4.5 Proxy macroeconomic variables

We search for macroeconomic variables that can be proxies for the main predictor factors by regressing each of the main predictors on each variable in the panel. The first real factor S₁₁ is highly correlated with variables representing employment growth. The log difference of “Employees on Nonfarm Payrolls: Total Private” gives the highest R² at 0.71, followed by similar employment variables. For the second real factor S₁₂, housing related variables show high correlation. The log of “Houses Authorized by Building Permits, West” yields the highest R² at 0.52, followed by a number of housing starts variables. The second financial factor S₂₂ is highly correlated with exchange rate variables. The log difference of the effective exchange rate (the multilateral exchange rate model index) gives the highest R² at 0.59, followed by the exchange rates with UK and Switzerland.

We replace in the predictive regression the main predictor factors with these variables. The regression results are reported in Table 7. The statistical significance of these proxy variables drops slightly, when compared with the main predictor factors in Table 2. This might be a natural consequence because these proxies are contaminated versions of the true factors. However, the adjusted R² values are almost as high as the ones observed in Table 2. P4_t is the single predictor factor constructed using these proxy variables.

Our finding is in agreement with Piazzesi and Swanson (2006), who show that excess returns on fed funds futures are well predicted by employment growth. Employment growth is one of the common choices for a macroeconomic predictor found in the literature. Our result justifies this common choice using the multi-level factor model. The macroeconomic variable that resembles the most the multi-level factor with the greatest predictive power is employment growth.