1. Macro Risks and the Term Structure of Interest Rates
Geert Bekaert1 Eric Engstrom2 Andrey Ermolov3
2017 The views expressed herein do not necessarily reflect those of the Federal Reserve System, its Board of Governors, or staff.
1Columbia University and NBER
2Federal Reserve Board of Governors
2Gabelli School of Business, Fordham University
Geert Bekaert, Eric Engstrom, Andrey Ermolov

2. I. Introduction Big Picture
Significant variation in bond risk premiums:
Macro level factors help predict bond returns (Ludvigson and Ng, 2009)
Implied risk premiums are counter-cyclical
Inflation risk premiums (See Bekaert and Wang; 2010, survey):
High in stagflations (70s)
Low in recent Great Recession
Economic intuition: inflation and bond risk premiums should be higher (lower) in “aggregate supply (AS)“ (“aggregate demand (AD)”) environments.
This paper: links AS/AD “Macro Risks” to the term structure.
Geert Bekaert, Eric Engstrom, Andrey Ermolov

3. II. Modeling Macro Risks Main Idea
Consider shocks to real growth and inflation:
Model them as functions of supply/demand shocks (Blanchard, 1989):
(*)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

4. II. Modeling Macro Risks Main Idea
If supply/demand shocks are heteroskedastic
Demand shock environment nominal bonds hedge
real risk
Supply shock environment nominal bonds
exacerbate real risk
Idea goes back to Fama (1981)
𝐶𝑜 𝑣 𝑡−1 𝑢 𝑡 𝑔 , 𝑢 𝑡 𝜋 >0
𝐶𝑜 𝑣 𝑡−1 𝑢 𝑡 𝑔 , 𝑢 𝑡 𝜋 <0
Geert Bekaert, Eric Engstrom, Andrey Ermolov

5. II. Modeling Macro Risks Identification
The (*) – system is not identified in a Gaussian framework
Use higher order moments
Coskewness and identification:
𝐸 𝑢 𝑡 𝑔 ( 𝑢 𝑡 𝜋 ) 2 = 𝜎 𝑔𝑑 𝜎 𝜋𝑑 2 𝐸 ( 𝑢 𝑡 𝑑 ) 3 + 𝜎 𝑔𝑠 𝜎 𝜋𝑠 2 𝐸 ( 𝑢 𝑡 𝑠 ) 3 ,
𝐸 (𝑢 𝑡 𝑔 ) 2 𝑢 𝑡 𝜋 = 𝜎 𝑔𝑑 2 𝜎 𝜋𝑑 𝐸 ( 𝑢 𝑡 𝑑 ) 3 − 𝜎 𝑔𝑠 2 𝜎 𝜋𝑠 𝐸 𝑢 𝑡 𝑠
Imagine: 𝐸 ( 𝑢 𝑡 𝑠 ) 3 ≈0; 𝐸 𝑢 𝑡 𝑑 3 <0
Co-Skewness Moments admit identification of 𝜎 𝜋𝑑 , 𝜎 𝑔𝑑
If 𝐸 𝑢 𝑡 𝑔 ( 𝑢 𝑡 𝜋 ) 2 <𝐸 ( 𝑢 𝑡 𝑔 ) 2 𝑢 𝑡 𝜋
⇒ 𝜎 𝜋𝑑 > 𝜎 𝑔𝑑
(all else equal!)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

6. II. Modeling Macro Risks Modelling the Shocks
Demand (and supply) shocks are “BEGE”-distributed (BEGE= Bad Environment – Good Environment)
and follow de-meaned gamma distributions
time-varying shape parameters 𝑝 𝑡−1 𝑑 , 𝑛 𝑡 −1 𝑑
unit scale parameters
Geert Bekaert, Eric Engstrom, Andrey Ermolov

7. II. Modeling Macro Risks Digression on the Gamma Distribution
ωp,t nt
Variancet pt
Skewnesst
Excess Kurtosist
Geert Bekaert, Eric Engstrom, Andrey Ermolov

8. II. Modeling Macro Risks BEGE Distributions
“Large” and equal pt and nt: Gaussian limit
Geert Bekaert, Eric Engstrom, Andrey Ermolov

9. II. Modeling Macro Risks BEGE Distributions
“Small” but still equal pt and nt: excess kurtosis
Geert Bekaert, Eric Engstrom, Andrey Ermolov

10. II. Modeling Macro Risks BEGE Distributions
Relatively large nt: negative skewness: “Bad Environment”
Geert Bekaert, Eric Engstrom, Andrey Ermolov

11. II. Modeling Macro Risks BEGE Distributions
Relatively large pt: positive skewness “Good Environment”
Geert Bekaert, Eric Engstrom, Andrey Ermolov

12. II. Modeling Macro Risks BEGE Distributions
The BEGE distribution has some advantages…
Realistic
Fits some financial and macro economic data well:
Bekaert and Engstrom (2017, JPE, Consumption Growth and the VIX)
Bekaert, Engstrom and Ermolov (2015, JEc; Stock Returns)
Ermolov (2017, Stock & Bond Return Correlations)
Bekaert, Engstrom, Xu (2017, Time-varying risk appetite model)
Tractable
Fits in the affine class of asset pricing models
Also: B(ekaert) E(ngstrom)
G(eert) E(ric)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

13. II. Modeling Macro Risks BEGE Distributions
…. but we have no affiliation with the Bee Gees
… and some disadvantages
We have no affiliation with the Bee Gees!
Geert Bekaert, Eric Engstrom, Andrey Ermolov

14. II. Modeling Macro Risks The Time-Variation in Macro Risks
Implications for conditional distribution supply shocks, for example:
𝐸 𝑡−1 𝑢 𝑡 𝑠 =0,
𝐸 𝑡−1 [ (𝑢 𝑡 𝑠 ) 2 ]= (𝜎 𝑝 𝑠 ) 2 𝑝 𝑡 𝑠 + (𝜎 𝑛 𝑠 ) 2 𝑛 𝑡 𝑠 ,
𝐸 𝑡−1 [ (𝑢 𝑡 𝑠 ) 3 ]= 2(𝜎 𝑝 𝑠 ) 3 𝑝 𝑡 𝑠 − 2 (𝜎 𝑛 𝑠 ) 3 𝑛 𝑡 𝑠 ,
𝐸 𝑡−1 [ (𝑢 𝑡 𝑠 ) 4 ]− 3(𝐸 𝑡−1 (𝑢 𝑡 𝑠 2 ]) 2 = 6(𝜎 𝑝 𝑠 ) 4 𝑝 𝑡 𝑠 + 6 (𝜎 𝑛 𝑠 ) 4 𝑛 𝑡 𝑠 .
Macro risks vary through time in autoregressive fashion (Gourieroux and Jasiak (2006)):
𝑝 t d , 𝑝 t s : “Good”(Positive Skew)Uncertainty
𝑛 𝑡 𝑑 , 𝑛 𝑡 𝑠 : “Bad”(Negative Skew)Uncertainty
Geert Bekaert, Eric Engstrom, Andrey Ermolov

15. III. Identifying Macro Risks Methodological Steps
Three steps applied to macro data only [1962:Q2 to 2016:Q4 (219 quarters]:
Identify conditional means versus shocks in real activity (also includes unemployment gap) and inflation (also core inflation) data - projections :
Core inflation: key focus of the Fed
Ajello, Benzoni, and Chyruk (2012)
Recover supply and demand shocks – unconditional GMM
Estimate BEGE processes (Bates, 2006) → 𝑝 𝑡 𝑑 , 𝑛 𝑡 𝑑 , 𝑝 𝑡 𝑠 , 𝑛 𝑡 𝑠
Geert Bekaert, Eric Engstrom, Andrey Ermolov

16. III. Identifying Macro Risks Recover Supply/Demand Shocks (2)
Shock Structure:
𝑢 𝑡 𝑥 = 𝑢 𝑡 𝑠 𝑢 𝑡 𝑑 Ω 𝑣 𝑡 𝑥
𝐶𝑜𝑣 𝑢 𝑡 𝑑 , 𝑢 𝑡 𝑠 =0,𝑉𝑎𝑟 𝑢 𝑡 𝑑 =𝑉𝑎𝑟 𝑢 𝑡 𝑠 =1
𝑢 𝑡 𝑑 , 𝑢 𝑡 𝑠 have skewness, kurtosis
𝑣 𝑡 𝑥 :𝑖𝑖𝑑; unit variance; zero skewness and excess kurtosis
4x x2 4x4
with x representing the 4 state variables
measurement error
Geert Bekaert, Eric Engstrom, Andrey Ermolov

17. III. Identifying Macro Risks Recover Supply/Demand Shocks (2)
Σ(8);Ω(4); skewness, kurtosis: 16 parameters
Identification, using GMM, and unconditional shocks from
covariance matrix (10)
(co)skewness (16)
(co)kurtosis (22)
Can infer structural shocks:
𝑢 𝑡 𝑠 𝑢 𝑡 𝑑 = Σ ′ (Σ Σ ′ +Ω Ω ′ ) −1 𝑢 𝑡 𝑥
Geert Bekaert, Eric Engstrom, Andrey Ermolov

18. III. Identifying Macro Risks Recover Supply/Demand Shocks (2)
volatility
𝑢 𝑡 𝝅
𝑢 𝑡 𝒈
𝑢 𝑡 𝝅 𝒄𝒐𝒓𝒆
𝑢 𝑡 𝒖
Data
0.5655***
0.7078***
0.3252***
0.2658***
Standard Error
(0.0867)
(0.0781)
(0.0531)
(0.0228)
Fitted
0.5655
0.7078
0.3252
0.2658
Skewness
0.4956
0.1144
0.3745***
(1.0067)
(0.3714)
(0.3808)
(0.1879)
0.2308
Excess kurtosis **
2.5052**
2.0640**
1.0528***
(5.7197)
(1.0656)
(0.8233)
(0.4056)
1.9051
1.1046
0.9798
1.0160
Geert Bekaert, Eric Engstrom, Andrey Ermolov
Geert Bekaert, Eric Engstrom, Andrey Ermolov 18

19. III. Identifying Macro Risks Recover Supply/Demand Shocks (2)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

20. III. Identifying Macro Risks Recover Supply/Demand Shocks (2)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

21. III. Identifying Macro Risks Recover Macro Risks (3)
Dynamics of the Macro Risk Factors (Bates, 2006, Approximate MLE):
𝑝 𝑡 𝑑 = − 𝑝 𝑡−1 𝑑 𝜔 𝑝,𝑡 𝑑 ,
𝑝 𝑡 𝑠 = − 𝑝 𝑡−1 𝑠 𝜔 𝑝,𝑡 𝑠 ,
𝑛 𝑡 𝑑 = − 𝑛 𝑡−1 𝑑 𝜔 𝑛,𝑡 𝑑 ,
𝑛 𝑡 𝑠 = − 𝑛 𝑡−1 𝑠 𝜔 𝑛,𝑡 𝑠 .
Geert Bekaert, Eric Engstrom, Andrey Ermolov

22. IV. Macro Results Demand/Supply Shocks
Unconditional moments of supply and demand shocks:
Some macro facts:
70-recessions feature mostly large negative supply shocks
The Great Recession is mostly but not purely demand driven
See also Ireland (2011), Mulligan (2012) versus Bils, Klenow, and Malin (2012); Mian and Sufi (2014)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

23. IV. Macro Results Demand Shocks
Geert Bekaert, Eric Engstrom, Andrey Ermolov

24. IV. Macro Results Supply Shocks
Geert Bekaert, Eric Engstrom, Andrey Ermolov

25. IV. Macro Results Structural Variances
Demand Variances
Geert Bekaert, Eric Engstrom, Andrey Ermolov

26. IV. Macro Results Structural Variances
Supply Variances
Geert Bekaert, Eric Engstrom, Andrey Ermolov

27. IV. Macro Results The Great Moderation
Great Moderation: Secular decline in the variability of:
Inflation: since 1990Q1 (Baele et al., 2015)
Real GDP growth: since 1984Q1
(McConnell and Perez-Quiros, 2000; Stock et al., 2002)
Questions:
What is the source of the decline?
Mostly “good demand variance”
Is it over?
No.
(see also Gadea, Gomez Loscos and Perez-Quiros, 2015)
Geert Bekaert, Eric Engstrom, Andrey Ermolov

28. IV. Macro Results The Great Moderation
Aggregate Inflation
Real GDP Growth
Data till 2000
Data till 2016
Aggregate variance
*** ** ** **
(0.0450)
(0.0457)
(0.0686)
(0.0688)
Supply variance
***
***
***
***
(0.0033)
(0.0035)
(0.0158)
(0.0147)
Good supply Variance
***
***
***
***
(0.0016)
(0.0021)
(0.0103)
(0.0102)
Bad Supply Variance
0.0017
0.0013
(0.0023)
(0.0029)
(0.0078)
(0.0107)
Demand Variance
*** * *
(0.0431)
(0.0437)
(0.0591)
(0.0604)
Good demand variance
***
*** * **
(0.0388)
(0.0381)
(0.0532)
(0.0524)
Bad demand variance
0.0254*
0.0187
(0.0066)
(0.0140)
(0.0142)
(0.0186)
Geert Bekaert, Eric Engstrom, Andrey Ermolov
Geert Bekaert, Eric Engstrom, Andrey Ermolov 28

29. IV. Macro Results Real Skewness
Geert Bekaert, Eric Engstrom, Andrey Ermolov

30. IV. Macro Results Nominal Skewness
Geert Bekaert, Eric Engstrom, Andrey Ermolov

31. IV. Macro Results Real-Nominal Covariance
We can recover the implied correlation between real growth and inflation:
Geert Bekaert, Eric Engstrom, Andrey Ermolov

32. V. Macro Risks and the Term Structure Yields
Adjusted R² of Macro factors for Yields:
Level
Slope
Curvature
Macro level factors
0.7146
0.5713
0.2808
Macro level factors + macro risks
0.7902***
0.5975*
0.4072***
Geert Bekaert, Eric Engstrom, Andrey Ermolov

33. V. Macro Risks and the Term Structure Bond Return Predictability
Macro (level) factors have additional explanatory power over financial factors (Ludvigson and Ng, 2009; Joslin, Priebsch and Singleton, 2014)…. But evidence weaker under Bauer-Hamilton (2017) bootstrap!
Explanatory Power (Adjusted R²) of Macro Risk Factors for Quarterly Excess Bond Returns:
1 year bond
5 year bond
3 financial factors
6.66%
7.08%
3 financial factors + macro level factors
9.62%*
7.74%
3 financial factors + macro risks
13.38%***
11.01%**
3 financial factors + macro level factors + macro risks
14.29%**
10.65%*
Geert Bekaert, Eric Engstrom, Andrey Ermolov
Geert Bekaert, Eric Engstrom, Andrey Ermolov 33

34. V. Macro Risks and the Term Structure Bond Risk Premiums
Regression of Returns on macro factors:
Implied risk premiums on NBER dummy; demand-supply variance ratio (and interaction):
Counter-cyclicality: insignificant
1 year bond
5 year bond
𝑝 𝑡 𝑑
- 0.87***
- 3.15***
𝑛 𝑡 𝑑
- 0.23***
- 1.66***
𝑝 𝑡 𝑠 , 𝑛 𝑡 𝑠
positive, mostly insignificant coefficients
1 year bond
5 year bond
Demand-supply ratio
-0.50***
-2.25***
Geert Bekaert, Eric Engstrom, Andrey Ermolov

35. V. Macro Risks and the Term Structure Bond Return Variances
Future realized bond return variances on term structure and macro factors (Adjusted 𝑅 2 s):
Most significant factor: bad demand variance
3 financial factors:
13.90%
Macro level factors:
18.90%
Macro Risks:
34.73%
3 financial factors + macro risks:
42.67%***
Macro level factors + macro risks:
42.00%***
3 financial factors + macro level factors + macro risks:
44.08%***
Geert Bekaert, Eric Engstrom, Andrey Ermolov

36. Conclusions Contributions of this paper
Macroeconomics
New framework to model real activity and inflation with AS/AD interpretation, including a novel identification scheme.
Evidence for non-Gaussian shocks with time-varying distributions
⟹“Macro Risk” instead of “Macro Level” factors
⟹ “Bad” and “Good” Risks
AS/AD interpretation of recessions; re-interpret the Great
Moderation.
Asset Pricing
Macro variables, incl. new macro risk variables, drive 80% of variation in yields
Non-Gaussian macro risk factors explain variation in bond risk premiums and bond return variances
Bond Risk and Term premiums decrease in “AD” environments.
Geert Bekaert, Eric Engstrom, Andrey Ermolov

37. Conclusions On the Agenda
A new TS model:
Macro factors with intuitive macro-economic interpretation (AS/AD macro risks)
Accommodates non-Gaussianities, but yields are affine in the state variables
Intuitive decomposition of (time variation in) inflation risk premiums
Geert Bekaert, Eric Engstrom, Andrey Ermolov

38. Conclusions On the Agenda
Affine models
Latent variables
Macro variables (Ang and Piazzesi (2003); Joslin, Priebsch and Singleton (2014); Chernov and Mueller (2012), …)
Less focus on economics; “Fits” data
DSGE models
Optimizing agents
Complex equations
Lots of economics; tightly parameterized
Many models are still (conditionally) Gaussian
Geert Bekaert, Eric Engstrom, Andrey Ermolov

39. Appendix Alternative Models
Geert Bekaert, Eric Engstrom, Andrey Ermolov

40. IV. Macro Risks and the Term Structure Term Premiums
5 Year Bond
10 Year Bond
𝐸 𝑡 𝜋 𝑡+1 𝑐𝑜𝑟𝑒
6.7811***
8.0065***
(0.9978)
(0.9994)
𝐸 𝑡 𝜋 𝑡+1
***
***
(0.0026)
(0.0008)
𝐸 𝑡 𝑔 𝑡+1
0.8876*
1.0378*
(0.9720)
(0.9608)
𝑢𝑔𝑎𝑝 𝑡
0.0769
0.1164
(0.5100)
(0.6018)
𝑝 𝑡 𝑑 * **
(0.0412)
(0.0206)
𝑛 𝑡 𝑑 *
(0.2678)
(0.0318)
𝑝 𝑡 𝑠
0.5720***
0.6415***
(0.9998)
(0.9996)
𝑛 𝑡 𝑠
(0.2614)
(0.2928)
Adjusted 𝑅 2 without macro risks
0.6513
0.6543
Adjusted 𝑅 2 with macro risks
0.6914*
0.6941*
Explanatory Power (Adjusted 𝑹 𝟐 ) of Macro factors for Term Premiums
Geert Bekaert, Eric Engstrom, Andrey Ermolov

41. Appendix Alternative Models
Geert Bekaert, Eric Engstrom, Andrey Ermolov