Real Time Macro Factors in Bond Risk Premium

Dashan HuangSingapore Management University

Fuwei JiangCentral University of Finance and Economics

Guoshi TongRenmin University of China

September 20, 2018

Tsinghua PBCSF Workshop

This Paper

This paper contributes to two broad literature

Empirical macro finance

Could we forecast time-varying bond risk premium with real timemacro variables?

Answer: Yes, but need a better econometric method

Econometric methodology

Could we incorporate forecasting target info. into the PCA factors?

Answer: Yes, we develop a Scaled PCA (sPCA) method

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Bond Return Predictability

Fama and Bliss (1987): Expectation hypothesis

Bond returns are predicted by yield curve: level, slope, and curvature

Joslin, Priebsch, and Singleton (2014): Spanning hypothesis

1 Bond returns are predicted by yield curve and macro variables (e.g.,inflation).

2 Expected bond returns are countercyclical.

Ludvigson and Ng (LN, 2009) extract PCA factors from 130 macrovariables and find forecasting power for the first 8 components.

1 The first factor is the most powerful.2 Bond risk premium forecasted by macro factors is countercyclical.

Google citations: Joslin, et al. (2014), 328; LN (2009), 693

It seems that macro data do have incremental forecasting power.

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But...

I. Macro data are released with lags:

1 Macro data of month t − 1 are released on the first Friday of month t.

2 On June 30, to predict returns in July, we cannot use the data inJune, but in May, which is released on June 1.

Time stamp is important for return predictability with macro data.

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But...

II. Macro data are revised after release:

April’s nonfarm payrolls is 164k on May 31, but should be used as159k on June 30.

When using macro data for predictability, we should consider revisions overtime.

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Snapshot of Vintages of “Total Nonfarm Payrolls”

Data PAYEMS PAYEMS PAYEMS PAYEMS ... PAYEMSDate 20170203 20170310 20170407 20170505 ... 20171208

2016-07 144,457 144,457 144,457 144,457 ... 144,4572016-08 144,633 144,633 144,633 144,633 ... 144,6332016-09 144,882 144,882 144,882 144,882 ... 144,8822016-10 145,006 145,006 145,006 145,006 ... 145,0062016-11 145,170 145,170 145,170 145,170 ... 145,1702016-12 145,327 145,325 145,325 145,325 ... 145,3252017-01 145,554 145,563 145,541 145,541 ... 145,5412017-02 145,798 145,760 145,773 ... 145,7732017-03 145,858 145,852 ... 145,8232017-04 146,063 ... 146,030

Joslin et al. (2014) and LN (2009) use the last column data (i.e.,final revised).

Ghysels et al. (2017) use real time data.

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Ghysels, Horan, and Moench (2017)

Forecasting bond returns without considering release lags andrevisions would lead to a look-ahead bias.

Using real time macro data, Ghysels et al. (2017) show that

1 the forecasting power of macro variables is substantially weakened andeven insignificant.

2 bond risk premium forecasted by macro factors is less countercyclical.

3 The forecasting power of macro data, found in Joslin et al. and LN, ismainly from data revision.

Real time macro data’s forecasting power is limited, if there is any.

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This Paper:Can Real Time Macro Data Predict Bond Returns?

We agree with Ghysels et al. (2017) that real time data are morenoisy than the final macro data.

And argue that more efficient econometric approaches can separatethe wheat from the chaff:

1 Bond returns can be predicted by real time macro variables.

2 Predicted bond returns are countercyclical.

3 Our real time macro factors capture the current and future macroconditions better.

To some extent, we reconcile Joslin et al. (2014) and LN (2009) andGhysels et al. (2017).

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Why Do Econometric Approaches Matter?

Because of measurement errors and structural breaks, jointly usingmultiple variables is preferred.

Since macro data are highly dimensional, dimension-reduction isimportant to avoid over-fitting.

However, the conventional PCA used in LN (2009) and Ghysels et al.(2017) is not necessarily efficient for predictability.

1 It does not incorporate the target—forecasting bond returns—whenextracting factors.

2 The PCA treats all macro variables equally important for predictability.

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The Forecasting Power Varies across Macro Variables

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The PCA Factor

Suppose (real time or final) macro data

X =

x1 x2 . . . xN

x11 x12 . . . x1Nx21 x22 . . . x2N

......

. . ....

xT1 xT2 . . . xTN

The first PCA factor is a linear combination of (x1, x2, · · · , xN):

F = w1x1 + w2x2 + · · ·+ wNxN ,

where w = (w1, · · · ,wN)′ is the solution to

maxwVar(Xw), s.t. w ′w = 1.

F does not use any information about bond returns.11 / 25

Our Approach: the Scaled PCA (sPCA) Factor

1 Let βi be the slope of rxt+1 = α + βixti + εt+1 for i = 1, · · · ,N.

2 Scale each xi by βi , i.e.,

β · X =

β1x1 β2x2 . . . βNxN

β1x11 β2x12 . . . βNx1Nβ1x21 β2x22 . . . βNx2N

......

. . ....

β1xT1 β2xT2 . . . βNxTN

3 The sPCA factor is a linear combination of (β1x1, β2x2, · · · , βNxN):

f = w1β1x1 + w2β2x2 + · · ·+ wNβNxN

= w̃1x1 + w̃2x2 + · · ·+ w̃NxN

where w is the solution to maxwVar(β · Xw), s.t. w ′w = 1.

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sPCA vs. PCA

1 The sPCA extracts factors from the space spanned by expected bondreturns, since βixti is the expected bond return predicted by xi .

(β1x1, β2x2, · · · , βNxN)

In contrast, the PCA extracts factors from the space spanned byexpected bond returns and the space spanned by others.

(x1, x2, · · · , xN)

2 The sPCA puts more weights on those variables with higher predictivepower.

The PCA treats all macro variables equally.

The sPCA factor should outperform the PCA factor.

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Two Alternative Machine Learning Approaches1

1 Target PCA (tPCA) (Bai and Ng, 2008)

Use LASSO to choose a subset of variables, say (x1, x2, x3), from(x1, x2, · · · , xN), and extract PCA factors from (x1, x2, x3).

2 Partial least squares (PLS, Kelly and Pruitt, 2013, 2015)

f = w1x1 + w2x2 + · · ·+ wNxN ,

where w is the solution to

maxwCov2(R,Xw), s.t. w ′w = 1.

Our sPCA is complementary to both tPCA and PLS.

1Gu, Kelly, and Xiu (2018) “Empirical Asset Pricing via Machine Learning”.14 / 25

Data

1 Dependent variable:

2- to 5-year Treasury bond excess returns, rx(n)t+1, (Fama-Bliss data).

2 Independent variables:

Real time data from the Archival Federal Reserve Economic Data(ALFRED)

60 macro variables used in Ghysels et al. (2017)

Vintages over 1982:03–2016:12

Out-of-sample is over 1990:01–2016:12

3 Recursively estimate real time macro factors for prediction.

F1,t|T : the first PCA factor extracted from final dataF1,t−1|t : the first PCA factor extracted from real time dataf1,t−1|t : the first sPCA factor extracted from real time data

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Performance Evaluation

Statistically

1 In-sample R2 and regression slope

2 Out-of-sample R2OS (Campbell and Thompson, 2008)

R2OS = 1−

∑T−12t=m (rx

(n)m+12 − r̂x

(n)m+12)2∑T−12

t=m (rx(n)m+12 − rx

(n)m+12)2

.

Economically (out-of-sample): mean-variance portfolio allocation

Utility gain = utility with predictability − utility without predictability

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Macro Factors (i.e., risk premiums)

1985 1990 1995 2000 2005 2010 2015−5

−4

−3

−2

−1

0

1

2

3

4

f1,t−1|tF1,t−1|t

F1,t|T

f1,t−1|t : real time sPCA factor; F1,t−1|t : real time PCA factor; F1,t|T : final

revised PCA factor).

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In-Sample Forecasting Results

Dependent variable: 5-year bond return

F1,t|T −0.91∗

[−1.78]

F1,t−1|t −0.58 1.34[−1.16] [1.00]

f1,t−1|t −0.82∗∗ −2.07∗

[−2.05] [−1.95]

f +1,t−1|t −0.76∗∗

[−1.98]

Adj-R2 0.03 0.01 0.03 0.03 0.02

f1,t−1|t : real time sPCA factor; F1,t−1|t : real time PCA factor; F1,t|T : finalrevised PCA factor; f ∗1,t−1|t : residual of regressing f1,t−1|t on F1,t−1|t .

The real time sPCA factor has comparable forecast power as the final macro PCA

factor, and outperforms the real time PCA factor, which is not significant.

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Out-of-Sample Forecasting Results

R2OS (%) MSPE p-value R2

OS (%) MSPE p-value

2-year bond return: rx(2)t+12 3-year bond return: rx

(3)t+12

F1,t|T 13.54∗∗∗ 2.60 0.00 14.68∗∗∗ 2.68 0.00F1,t−1|t 9.80∗∗∗ 2.27 0.01 11.49∗∗∗ 2.42 0.01f1,t−1|t 11.16∗∗ 1.97 0.02 17.04∗∗∗ 2.31 0.01

4-year bond return: rx(4)t+12 5-year bond return: rx

(5)t+12

F1,t|T 13.95∗∗∗ 2.66 0.00 12.78∗∗∗ 2.62 0.00F1,t−1|t 11.42∗∗∗ 2.44 0.01 10.85∗∗∗ 2.42 0.01f1,t−1|t 19.03∗∗∗ 2.50 0.01 19.54∗∗∗ 2.61 0.00

f1,t−1|t : real time sPCA factor; F1,t−1|t : real time PCA factor; F1,t|T : final

revised PCA factor.

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Out-of-Sample Forecasting with tPCA and PLS Factors

R2OS (%) MSPE p-value R2

OS (%) MSPE p-value

2-year bond return 5-year bond return

F1,t−1|t 9.80∗∗∗ 2.27 0.01 10.85∗∗∗ 2.42 0.01

tPCAH1 14.50∗∗ 2.06 0.02 16.02∗∗∗ 2.53 0.01

tPCAH2 12.21∗∗ 1.76 0.04 21.13∗∗∗ 2.36 0.01

tPCAH3 −38.87 1.61 0.05 −2.13 2.12 0.02

tPCAS1 10.09∗∗∗ 2.25 0.01 12.38∗∗∗ 2.58 0.00

tPCAS2 3.77∗∗ 2.03 0.02 11.93∗∗∗ 2.56 0.01

tPCAS3 −7.73 1.93 0.03 12.19∗∗∗ 2.56 0.01

PLS 5.81∗∗ 1.98 0.02 15.32∗∗ 2.53 0.01

Finding I: Bond returns can be significantly predicted by real time macrofactors, in- and out-of-sample.

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Term Premium is Countercyclical

tp(n)t =

1

n[Et (rx

(n)t+1) + Et (rx

(n−1)t+2 )+, ...,+Et (rx

(2)t+n−1)],

1985 1990 1995 2000 2005 2010 2015−4

−2

0

2

corr=−0.22**

A: Term premium implied by f1,t− 1|t

Term premiumCFNAI

1985 1990 1995 2000 2005 2010 2015−4

−2

0

2

corr=−0.10**

B: Term premium implied by F1,t− 1|t

Term premiumCFNAI

1985 1990 1995 2000 2005 2010 2015−4

−2

0

2

corr=−0.37***

C: Term premium difference between f1,t− 1|t and F1,t− 1|t

Term premium differenceCFNAI

Term premium is implied by a VAR model that includes a macro factor as the predictor. IP

growth: monthly growth in industrial production.21 / 25

Correlation Between Expected Bond Return and Macro Condition

corr(ERt ,Yt)

ERt Yt

Output Cay Inflation Consumption Recessiongap uncertainty uncertainty probability

E [rx(5)t+12|F1,t−1|t ] −0.39 −0.22 0.61 0.60 0.78

E [rx(5)t+12|f1,t−1|t ] −0.55 −0.33 0.70 0.67 0.82

∆ERt −0.57 −0.36 0.55 0.49 0.50

∆ER = E [rx(5)t+12|f1,t−1|t ]− E [rx

(5)t+12|F1,t−1|t ]

Finding II: The predicted bond returns (i.e., term premium) arecountercyclical.

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Forecasting Macro Data Revision

Dependent variable: revisiont|T = F1,t|T − F1,t−1|t

f +1,t−1|t 0.49∗∗∗ 0.49∗∗∗

[3.98] [3.60]

F1,t−1|t −0.12 −0.12[−1.05] [−1.20]

Adj-R2 0.06 0.03 0.09

f +1,t−1|t is the residual of regressing the real time sPCA factor f1,t−1|t on the real

time PCA factor F1,t−1|t .

Finding IIIa: The sPCA factor outperforms the PCA factor because itbetter captures current macro condition.

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Forecasting Next Year Macro Conditions

∆Gt+12 = α + βf +1,t−1|t + εt+12

CFNAI Consumption Employment Macro Yielduncertainty spread

β 0.20∗ 0.14∗∗ 0.23∗∗ −0.23∗∗ −0.10∗∗

t-stat 1.87 2.40 2.05 −2.31 −2.49

adj-R2 0.02 0.03 0.03 0.06 0.04

f +1,t−1|t is the residual of regressing the real time sPCA factor f1,t−1|t on the real

time PCA factor F1,t−1|t .

Finding IIIb: The sPCA factor predicts future bond returns because itsignificantly predicts future macro condition.

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Conclusion

Econometric approaches matter for extracting real time macro factors.

1 Bond returns can be significantly predicted by real time macro factors.

2 The predicted bond risk premium is countercyclical.

3 Our sPCA factor outperforms the PCA factor because it bettercaptures current and future macro conditions.

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