Luciano Vereda1 Eduardo Bevilaqua1

Ana Luiza Abrão1

Adapting the Diebold and Li Methodology to Deal with Heteroskedasticity in Factor Dynamics

1 IAPUC (Institute for Financial and Actuarial Risk Manegement at PUC-Rio)

IntroductionIntroduction

Why modeling the term structure of interest rates? Forecasting the returns of traditional fixed

income securities of various maturities, which are in the core of the portfolio of any pension insurance company;

Controlling the risks associated to investing in such assets (specially interest rate and reinvestment risk);

IntroductionIntroduction

Why modeling the term structure of interest rates? Calculating the discount factors that are necessary

to mark to market assets and liabilities (procedure that is in the heart of solvency analysis);

Important input of any ALM system. These objectives naturally call for realistic

econometric models, which should take into account all information available.

Introduction This paper builds on three main ideas…

Expectations Hypothesis

Long rates are risk-adjusted averages of futureexpected short rates → the former can help predict the later.

11

11

k

t t t l tl

y E y Lk

Risk premium

Average of expected future one period rates

The yield curve as a whole conveys valuable information.

1

Introduction This paper builds on three main ideas…

Diebold and Li (2006): trade-off between complexity (which increases the ability to describe observed dynamics) and simplicity (which usually improves forecasting performance).

Reccomendation: summarize the information content of the term structure by modeling its driving forces.

The Nelson and Siegel framework is suitable for this goal because the yield curve is represented by means of only three factors (level, slope and curvature).

2

Introduction This paper builds on three main ideas…

Longstaff and Schwartz (1992), Christiansen and Lund (2002), Pérignon and Smith (2004), etc...

3

Financial time series behave in such a way that some periods are more volatile than others.

Level effects, GARCH effects and regime shifts are required to adequately model interest rate volatility.

These properties are transmitted to the factors that “explain” yield curve dynamics.

Introduction Question: is it worth adding these attributes to

models which are aimed at forecasting? It is hard to say a priori…

SimplicityExplanatory Power

Perhaps the answer depends on the economy at hand. Emerging economies usually experience greater volatility levels → Probably these attributes are more important for them!

The main purpose of this (working) paper is answering these questions.

OutlineOutline1) Discussing the U. S. yield curve → show that interest rate

volatilities vary over time;2) Describing the Nelson and Siegel representation of the yield

curve;3) Discussing the factors → show that their volatilities vary over

time;4) Describing the Diebold and Li methodology;5) Describing our first attempt to adapt the Diebold and Li

methodology; 6) Analyzing the forecasting performance (for three different

forecasting horizons) of our variant, comparing it with the performance achieved by the Diebold and Li proposal;

7) Some preliminary conclusions;8) Future research.

The U.S. Term StructureThe U.S. Term Structure Yields of zero-coupon bonds of several maturities (1, 3 and 6

months; 1, 2, 3, 5, 7, 10, 20 and 30 years); Source → Federal Reserve Economic Data (FRED), St. Louis

FED. The yield curve is upward sloping on average; Rates are highly autocorrelated to their past values and to

current and past values of other rates; The first order autocorrelation of squared rates is highly

significant; Cross-section autocorrelation decreases with the distance

between maturities; Their distribution cannot be considered normal (long right tails

and excess kurtosis for short rates); Their variance decreases with maturity;

The U.S. Term StructureThe U.S. Term Structure

0

2

4

6

8

10

12

14

1985 1990 1995 2000 2005

GS1MGS3MGS6MGS1

GS2GS3GS5GS7

GS10GS20

The U.S. Term StructureThe U.S. Term Structure Mean Std. Dev. Skewness Kurtosis Jarque-Bera AC(1) AC2(1) gs1m 5.0725 2.2492 0.0361 2.4571 3.6867 0.993 0.990 gs3m 5.2667 2.3314 0.0516 2.5218 2.9413 0.992 0.989 gs6m 5.4751 2.4089 0.1008 2.6020 2.4469 0.988 0.982 gs1 5.6789 2.4569 0.2076 2.6846 3.3426 0.985 0.977 gs2 6.1048 2.5072 0.3440 2.7868 6.3769 0.983 0.975 gs3 6.3164 2.4710 0.4502 2.8225 10.3517 0.982 0.975 gs5 6.6359 2.3949 0.6270 2.9119 19.4268 0.981 0.976 gs7 6.8726 2.3518 0.6933 2.8993 23.7607 0.982 0.977 gs10 7.0013 2.2987 0.7496 2.8902 27.7773 0.982 0.978 gs20 7.3410 2.1274 0.8822 3.1179 38.4384 0.982 0.980 gs30 7.4117 2.0227 0.9336 3.1995 43.3403 0.980 0.979

Evidence of heteroskedasticity

The U.S. Term StructureThe U.S. Term Structure

gs1m gs3m gs6m gs1 gs2 gs3 gs5 gs7 gs10 gs20 gs30 gs1m 1 0.997 0.989 0.976 0.954 0.936 0.900 0.878 0.851 0.819 0.753 gs3m 0.997 1 0.997 0.990 0.971 0.955 0.921 0.899 0.874 0.843 0.778 gs6m 0.989 0.997 1 0.997 0.982 0.968 0.937 0.915 0.891 0.860 0.797 gs1 0.976 0.990 0.997 1 0.993 0.983 0.957 0.938 0.916 0.888 0.829 gs2 0.954 0.971 0.982 0.993 1 0.997 0.983 0.969 0.953 0.930 0.879 gs3 0.936 0.955 0.968 0.983 0.997 1 0.993 0.984 0.971 0.952 0.907 gs5 0.900 0.921 0.937 0.957 0.983 0.993 1 0.998 0.992 0.980 0.947 gs7 0.878 0.899 0.915 0.938 0.969 0.984 0.998 1 0.998 0.990 0.963 gs10 0.851 0.874 0.891 0.916 0.953 0.971 0.992 0.998 1 0.996 0.977 gs20 0.819 0.843 0.860 0.888 0.930 0.952 0.980 0.990 0.996 1 0.988 gs30 0.753 0.778 0.797 0.829 0.879 0.907 0.947 0.963 0.977 0.988 1

The U.S. Term StructureThe U.S. Term Structure

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1985 1990 1995 2000 2005

RESID_GS1M

0.0

0.2

0.4

0.6

0.8

1.0

1985 1990 1995 2000 2005

RESID_GS1

0.0

0.2

0.4

0.6

0.8

1.0

1985 1990 1995 2000 2005

RESID_GS10

Evidence of volatility clustering

ttt eyy ,1110 )1()1( ttt eyy ,12110 )12()12( ttt eyy ,120110 )120()120(

te ,1

te ,1te ,12 te ,120

The Nelson and Siegel FrameworkThe Nelson and Siegel Framework

1, 2, 3,

1 11t t tt t

e ey e

1st factor 2nd factor 3rd factor

y y y

y

+ +

How Factors Look Like?How Factors Look Like?

-8

-4

0

4

8

12

16

1985 1990 1995 2000 2005

LEVEL SLOPE CURVATURE

How Factors Look Like?How Factors Look Like?

Level Slope CurvatureLevel 1.0000 -0.2312 0.5611Slope -0.2312 1.0000 0.3514Curvature 0.5611 0.3514 1.0000

Mean Std. Dev. Skewness Kurtosis Jarque-Bera AC(1) AC2(1)Level 7.4688 1.9891 0.9251 3.2837 42.4841 0.985 0.986Slope -2.3970 1.6054 -0.0674 1.9585 13.3733 0.977 0.977Curvature -0.2970 2.2226 -0.6336 3.7963 27.1573 0.956 0.930

Evidence of heteroskedasticity

Cross correlations can be important...

How Factors Look Like?How Factors Look Like?

0.0

0.2

0.4

0.6

0.8

1.0

1985 1990 1995 2000 2005

RESID_LEVEL

0

1

2

3

4

5

6

1985 1990 1995 2000 2005

RESID_SLOPE

0.0

0.4

0.8

1.2

1.6

2.0

2.4

1985 1990 1995 2000 2005

RESID_CURV

Evidence of volatility clustering

tLtt eLL ,110 tStt eSS ,110 tCtt eCC ,110

tLe ,tSe , tCe ,

The Diebold and Li ProposalThe Diebold and Li Proposal

1, 2, 3,

1 11t t tt t

e ey e

titiitiiiti c ,2,2,1,1,,

Our Variant of the Diebold and Li Our Variant of the Diebold and Li ProposalProposal

1, 2, 3,

1 11t t tt t

e ey e

titiitiiiti c ,2,2,1,1,,

21,

21,

2, tiitiiiti

Mean zero, variance obbeys...

Empirical ProcedureEmpirical Procedure Observations were taken in a monthly frequency. The complete sample is comprised by data coming from September 1982

(after the so called “monetary experience”) until March 2007. Our procedure for examining out-of-sample forecasts is very conventional:

Estimate the models using observations from September 1982 until and including October 2003;

Calculate h-month-ahead forecasts (h = 1, 6 and 12 months) of all yields. Repeat the first step using observations from September 1982 until and including

November 2003.

Stop when the estimation sample comprises data from July 1999 until March 2006. Observations that are not used during the estimation process are put apart in order to evaluate forecast errors.

Estimations were made by applying the OLS technique. The criterion that we use to judge forecasting performance is the mean

squared deviation between actual and forecasted rates.

ResultsResultsh = 1

AR(1) AR(2) AR(1) AR(2) AR(1) AR(2)NS NS NS GARCH NS GARCH

gs1m 0,03975 0,03258 0,01630 0,02772 0,01563 0,02741gs5 0,04438 0,04455 0,04785 0,05104 0,04744 0,05066gs10 0,03494 0,03440 0,09252 0,09481 0,08949 0,09389

h = 6

AR(1) AR(2) AR(1) AR(2) AR(1) AR(2)NS NS NS GARCH NS GARCH

gs1m 0,79807 0,56817 0,41212 0,37563 0,40500 0,34579gs5 0,15603 0,16088 0,13584 0,18185 0,11380 0,16075gs10 0,11212 0,14411 0,19963 0,27017 0,17356 0,24964

h = 12

AR(1) AR(2) AR(1) AR(2) AR(1) AR(2)NS NS NS GARCH NS GARCH

gs1m 2,95808 2,15175 1,65472 1,42796 1,70036 1,36765gs5 0,27703 0,19290 0,26112 0,32956 0,20887 0,28179gs10 0,16626 0,24072 0,35629 0,50798 0,30148 0,45828

Final Remarks

Results suggest that our very simple way of adding sthocastic volatility to the Diebold and Li proposal enhances forecasting performance for the set of horizons that was considered.

This result is valid for our three proxies of short, medium and long term rates.

This result holds true even for the U.S. economy, which is a very stable one.

Final Remarks

It is possible to improve performance further... Refine the strategy adopted to model sthocastic

volatility... GARCH-M. Volatility as a function of the magnitude of the factors. Volatility as a function of macroeconomic variables.

Testing if sthocastic volatility depends on the characteristics of the economy (emerging vs. developed countries).

Final Remarks

GARCH-M

1, 2, 3,

1 11t t tt t

e ey e

titiitiitiiiti c ,,2,2,1,1,,

The values assumed by the factors may depend on prevailing volatility levels.

Final Remarks

21,

21,

2, tiitiiiti

GARCH with level effects

1, 2, 3,

1 11t t tt t

e ey e

titiitiiiti c ,2,2,1,1,,

titititi ei,,,, white noise

iti

titi

1,

,,

Final Remarks

Volatility depending on the state of the economy.

1, 2, 3,

1 11t t tt t

e ey e

titLitiitiiiti c ,,2,2,1,1,,

...,,, etcYf tttL