’0.5

0 12 24 36 48 60 72 84 96 108 120

Term to maturity (in months)

2f non-id. 2f ident. 2f ident.2

(b)

Figure 3.10 Term premium curves: (a) 1986“1998 and (b) 1972“1998

112 Applied Quantitative Methods for Trading and Investment

9.0

8.5

8.0

7.5

%

7.0

6.5

6.0

5.5

5.0

0 12 24 36 48 60 72 84 96 108 120

Time to settlement (in months)

2f non-id. 3f non-id. 2f ident. 2f ident.2

(a)

9.0

8.5

8.0

7.5

%

7.0

6.5

6.0

5.5

5.0

0 12 24 36 48 60 72 84 96 108 120

Time to settlement (in months)

2f non-id. 2f ident. 2f ident.2

(b)

Figure 3.11 Average forward curves: (a) 1986“1998 and (b) 1972“1998

Modelling the Term Structure of Interest Rates 113

monotonically to a positive value, around 4% in maturities higher than 20 years.45 In the

non-identi¬ed models, the term premium rises rapidly up to 1.5 in the 5-year maturity,

converging to around 1.6 and 1.25 in the 10-year maturity, respectively using the two-

and the three-factor model.46

Accordingly, the expected short-term interest rates exhibit differences between the iden-

ti¬ed and the non-identi¬ed models, as well as between the samples (Figure 3.12a and

b). The most remarkable feature is the positive slope in the identi¬ed models and using

the shorter sample, namely with the model in (3.56). In the longer sample, the short-term

interest rate is roughly constant in all models, as the shape of the one-period forward and

the term premium curves are similar.

Regarding the time-series results, Figure 3.13a and b show the observed and the esti-

mated yields for several maturities considered, while in Figure 3.14a and b the observed

and the estimated in¬‚ation are shown (according to the models in (3.55) and (3.56)).

It can be concluded that all estimates reproduce very closely both the yields across the

maturity spectrum and the in¬‚ation.47 However, the ¬t for the yields is poorer in two

subperiods: the ¬rst in Spring 1994, where predicted short-term rates are above actual

rates and predicted long-term rates are below actual rates;48 the second in 1998, when

long bond yields fell as a consequence of the Russian and Asian crises, whilst short rates

remained stable. Therefore, the estimated long-term interest rates remained higher than

the actual rates.

The identi¬ed model in (3.56) provides more accurate estimates for the in¬‚ation, as it

uses a speci¬c factor for this variable, instead of the in¬‚ation being explained only by

the second factor of the term structure.

Focusing on the parameter estimates (Table 3.4), the factors are very persistent, as

the estimated •™s are close to one and exhibit low volatility.49 It is the second factor

that contributes positively to the term premium (σ2 »2 ) and exhibits higher persistency.

The market price of risk of this factor becomes higher when it is identi¬ed with in¬‚a-

tion. Conversely, its volatility decreases. All variables are statistically signi¬cant, as the

parameters evidence very low standard deviations (Table 3.5). This result con¬rms the

general assertion about the high sensitivity of the results to the parameter estimates.50

As was seen in the previous charts, the results obtained with the three-factor model and

using the shorter sample exhibit some differences to those obtained with the two-factor

45

Notice that the identi¬ed model in (3.56) provides lower estimates for the risk premium as the market price

of risk of the ¬rst factor is higher, giving a lower contribution for the risk premium.

46

The term premium estimate for the 10-year maturity using the two-factor non-identi¬ed model is in line with

other estimates obtained for different term structures. See, e.g., De Jong (1997), where an estimate of 1.65 for

the 10-year term premium in the US term structure is presented.

47

The quality of the ¬t for the yields contrasts sharply with the results found for the US by Gong and Remolona

(1997c). In fact, in the latter it was concluded that at least two different two-factor heteroskedastic models are

needed to model the whole US term structures of yields and volatilities: one to ¬t the medium/short end of the

curves and another to ¬t the medium/long terms. The lowest time-series correlation coef¬cients between the

observed and the estimated ¬gures is 0.87, for the 10-year maturity, and the cross-section correlation coef¬cients

are, in most days of the sample, above 0.9.

48

The surprising behaviour of bond yields during 1994 is discussed in detail in Campbell (1995) with reference

to the US market.

49

Nevertheless, given that the standard deviations are low, the unit root hypothesis is rejected and, consequently,

the factors are stationary. The comments of Jerome Henry on this issue are acknowledged.

50

Very low standard deviations for the parameters have usually been obtained in former Kalman ¬lter estimates

of term structure models, as in Babbs and Nowman (1998), Gong and Remolona (1997a), Remolona et al.

(1998), and Geyer and Pichler (1996).

114 Applied Quantitative Methods for Trading and Investment

7.0

6.8

6.6

6.4

6.2

%

6.0

5.8

5.6

5.4

5.2

5.0

0 12 24 36 48 60 72 84 96 108 120

Horizon (in months)

2f non-id. 3f non-id. 2f ident. 2f ident.2

(a)

7.0

6.8

6.6

6.4

6.2

%

6.0

5.8

5.6

5.4

5.2

5.0

0 12 24 36 48 60 72 84 96 108 120

Horizon (in months)

2f non-id. 2f ident. 2f ident.

(b)

Figure 3.12 Average expected short-term interest rates: (a) 1986“1998 and (b) 1972“1998

Modelling the Term Structure of Interest Rates 115

(i) 1-month (ii) 1-year

14

14

12

12

10

10

8

8

6 6

4 4

2 2

0 0

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed 2f non-id. Observed 2f non-id.

3f non-id. 2f ident. 3f non-id. 2f ident.

2f ident.2 2f ident.2

(iv) 5-year

(iii) 3-year

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0

0

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed 2f non-id. Observed 2f non-id.

3f non-id. 2f ident. 3f non-id. 2f ident.

2f ident.2 2f ident.2

(v) 7-year (vi) 10-year

14

14

12

12

10

10

8

8

6 6

4 4

2 2

0 0

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed 2f non-id. Observed 2f non-id.

3f non-id. 2f ident. 3f non-id. 2f ident.

2f ident.2 2f ident.2

(a)

Figure 3.13 Time-series yield estimation results: (a) 1986“1998 and (b) 1972“1998

model. In order to opt between the two- and the three-factor model, a decision must be

made based on the estimation accuracy of the models. Consequently, a hypothesis test is

performed, considering that both models are nested and the two-factor model corresponds

to the three-factor version imposing •3 = σ3 = 0.

This test consists of computing the usual likelihood ratio test l = ’2(ln v ’ ln v — ) ∼

χ 2 (q), v and v* being the sum of the likelihood functions respectively of the two- and

116 Applied Quantitative Methods for Trading and Investment

(i) 1-year (ii) 3-year

14

14

12

12

10

10

8 8

6 6

4 4

2 2

0 0

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed Observed

2f non-id. 2f non-id.

2f ident. 2f ident.

2f ident.2 2f ident.2

(iv) 7-year

(iii) 5-year

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0

0

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed Observed

2f non-id. 2f non-id.

2f ident. 2f ident.

2f ident.2 2f ident.2

(v) 8-year (vi) 10-year

14

14

12

12

10

10

8

8

6 6

4 4

2 2

0 0

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

Observed Observed

2f non-id. 2f non-id.

2f ident. 2f ident.

2f ident.2 2f ident.2

(b)

Figure 3.13 (continued )

the three-factor models and q the number of restrictions (in this case two restrictions are

imposed). The values for ln v and ln v* were ’306.6472 and ’304.5599, implying that

l = 4.1746. As χ 2 (2)0.95 = 5.991, the null hypothesis is not rejected and the two-factor

model is chosen. This conclusion is also suggested by the results already presented,

due to the fact that the three-factor model does not seem to increase signi¬cantly the

Modelling the Term Structure of Interest Rates 117

6

5

4

3

2

%

1

0

’1

’2

’3

’4

Jan-86

Jan-87

Jan-88

Jan-89

Jan-90

Jan-91

Jan-92

Jan-93

Jan-94

Jan-95

Jan-96

Jan-97

Jan-98

Observed Estimated Estimated 2

(a)

5

4

3

2

1

%

0

’1

’2

’3

’4

’5

Sep-72

Sep-74

Sep-76

Sep-78

Sep-80

Sep-82

Sep-84

Sep-86

Sep-88

Sep-90

Sep-92

Sep-94

Sep-96

Sep-98

Observed Estimated Estimated 2

(b)

Figure 3.14 Observed and estimated yearly in¬‚ation: (a) 1986“1998 and (b) 1972“1998

Table 3.4 Parameter estimates

δ σ1 •1 »1 σ1 σ2 •2 »2 σ2

1986“1998

2f non-id. 0.00489 0.00078 0.95076 0.22192 0.00173 0.98610 ’0.09978

3f non-id. 0.00494 0.00114 0.97101 1.13983 0.00076 0.97543 ’1.71811

2f ident. 0.00498 0.00156 0.98163 3.09798 0.00109 0.98216 ’4.44754

2f ident.2 0.00566 0.00154 0.98169 3.20068 0.00111 0.98225 ’4.45554

1972“1998

2f non-id. 0.00508 0.00127 0.96283 0.10340 0.00165 0.99145 ’0.10560

2f ident. 0.00547 0.00159 0.99225 2.30797 0.00070 0.99259 ’5.29858

2f ident.2 0.00526 0.00141 0.95572 0.09198 0.00160 0.99298 ’0.09869

σ1π •1π »1π σ1π »2π σ1π B1π B2π σ3 •3 »3 σ3

1986“1998

2f non-id. “ “ “ “ “ “ “ “ “

3f non-id. “ “ “ “ “ “ 0.001317 0.989882 0.006502

2f ident. “ “ “ “ “ “ “ “ “

2f ident.2 6.89340 0.99773 1.72529 1.55172 0.00004 “ “ “

’0.05438

1972“1998

2f non-id. “ “ “ “ “ “ “ “ “

2f ident. “ “ “ “ “ “ “ “ “

2f ident.2 0.00931 0.99748 0.08405 0.25683 0.06685 “ “ “

’2.65631

Modelling the Term Structure of Interest Rates 119

Table 3.5 Standard deviation estimates

δ σ1 •1 »1 σ1 σ2 •2 »2 σ2

1986“1998

2f non-id. 0.00009 0.00001 0.00142 0.00648 0.00003 0.00100 0.00084

2f ident. 0.00001 0.00001 0.00000 0.00000 0.00071 0.00100 0.00005

2f ident.2 0.00001 0.00001 0.00000 0.00000 0.00071 0.00100 0.00005

1972“1998

2f non-id. 0.00001 0.00001 0.00000 0.00000 0.00249 0.00084 0.00013

2f ident. 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

2f ident.2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

σ1π •1π »1π σ1π »2π σ1π B1π B2π σ3 •3 »3 σ3

1986“1998

2f non-id. “ “ “ “ “ “ “ “ “

2f ident. “ “ “ “ “ “ “ “ “

2f ident.2 0.12343 0.00154 0.00653 0.00000 0.00000 0.00004 “ “ “

1972“1998

2f non-id. “ “ “ “ “ “ “ “ “

2f ident. “ “ “ “ “ “ “ “ “

2f ident.2 0.00000 0.00000 0.00000 0.00000 0.00864 0.00025 “ “ “

¬tting quality. Consequently, the factor analysis and the identi¬cation are focused on the

two-factor model.

In what concerns to the factor loadings (Figure 3.15a and b), the ¬rst factor “ less

persistent and volatile “ is relatively more important for the short end of the curve, while

the second assumes that role in the long end of the curve.51

The results already presented illustrate two ways of assessing the identi¬cation problem.

A third one, as previously mentioned, is to analyse the correlation between the factors

and the variables with which they are supposed to be related “ the real interest rate and

the expected in¬‚ation “ including the analysis of the leading indicator properties of the

second factor concerning the in¬‚ation rate.

The comparison between the unobservable factors, on the one hand, and a proxy for the

real interest rate and the in¬‚ation rate, on the other hand, shows (in Figure 3.16a“d) that

the correlation between the factors and the corresponding economic variables is high. In

fact, using ex-post real interest rates as proxies for the ex-ante real rates, the correlation

coef¬cients between one-month and three-month real rates, on the one side, and the ¬rst

factor, on the other side, are around 0.75 and 0.6 respectively in the shorter and in the

longer sample.52 Using the identi¬ed models, that correlation coef¬cient decreases to 0.6

and 0.7 in the shorter sample, respectively using the models in (3.55) and (3.56), while

51

Excluding the identi¬ed two-factor models for the shorter sample.

52

This ¬nding contrasts with the results in Gerlach (1995) for Germany, between January 1967 and January

1995. However, it is in line with the conclusion of Mishkin (1990b) that US short rates have information

content regarding real interest rates.

120 Applied Quantitative Methods for Trading and Investment

1.2

1.0

0.8

0.6

%

0.4

0.2

0.0

0 12 24 36 48 60 72 84 96 108 120

Term to maturity (in months)

F1 non-id. F2 non-id. F1 ident.

F2 ident. F1 ident.2 F2 ident.2

(a)

1.2

1.0

0.8

0.6

%

0.4

0.2

0.0

0 12 24 36 48 60 72 84 96 108 120

Term to maturity (in months)

F1 non-id. F2 non-id. F1 ident.

F2 ident. F1 ident.2 F2 ident.2

(b)

Figure 3.15 Factor loadings in the two-factor models: (a) 1986“1998 and (b) 1972“1998

in the larger sample it increases to 0.75 using the model in (3.55) and slightly decreases

to 0.56 with the model in (3.56).

The correlation coef¬cients between the second factor and the in¬‚ation are close to 0.6

and 0.7 in the same samples. When the identi¬ed model in (3.55) is used, the second factor

basically reproduces the in¬‚ation behaviour in the shorter sample, with the correlation

Modelling the Term Structure of Interest Rates 121

0.15

6

0.10

4

Ident. and real rate (%)

0.05

2

Non-ident.

0.00

0

’0.05

’2

’0.10

’4

’0.15

’6

Jan-86

Jan-87

Jan-88

Jan-89

Jan-90

Jan-91

Jan-92

Jan-93

Jan-94

Jan-95

Jan-96

Jan-97

Jan-98

Ident. Ident.2 1m.Real Rate 3m.Real Rate Non-ident.

(a)

6 0.6

0.4

4

Ident. and real rate (%)

Non-ident. and ident.2

0.2

2

0.0

0

’0.2

’2

’0.4

’4

’0.6

’6

Sep-72

Sep-74

Sep-76

Sep-78

Sep-80

Sep-82

Sep-84

Sep-86

Sep-88

Sep-90

Sep-92

Sep-94

Sep-96

Sep-98

Ident. 1m. Real Rate 3m. Real Rate Non-ident. Ident.2

(b)

Figure 3.16 Time-series evolution: (a) ¬rst factor, 1986“1998; (b) ¬rst factor, 1972“1998;

(c) second factor, 1986“1998; (d) second factor, 1972“1998; (e) in¬‚ation factor, 1986“1998;

(f) in¬‚ation factor, 1972“1998

coef¬cient achieving ¬gures above 0.98 in both samples. As the model in (3.56) includes

a speci¬c factor, the in¬‚ation, the correlation between the second factor and that variable

decreases to around 0.5 in the shorter sample and remains close to 0.7 in the larger one.

In Figure 3.16e and f, the behaviour of the speci¬c in¬‚ation factor vis-` -vis the observed

a

in¬‚ation is presented. This factor in both samples is highly correlated to the in¬‚ation,

122 Applied Quantitative Methods for Trading and Investment

4

3

2

1

%

0

’1

’2

’3

’4

Jan-86

Jan-87

Jan-88

Jan-89

Jan-90

Jan-91

Jan-92

Jan-93

Jan-94

Jan-95

Jan-96

Jan-97

Jan-98

Non-ident. Ident. Ident.2 Yearly CPI

(c)

6

4

2

%

0

’2

’4

’6

Sep-72

Sep-74

Sep-76

Sep-78

Sep-80

Sep-82

Sep-84

Sep-86

Sep-88

Sep-90

Sep-92

Sep-94

Sep-96

Sep-98

Non-ident. Ident. Ident.2 Yearly CPI

(d)

Figure 3.16 (continued )

which implies that the correlation between the second factor and the in¬‚ation rate becomes

lower.

One of the most striking results obtained is that the second factor appears to be a

leading indicator of in¬‚ation, according to Figure 3.16c and d. This is in line with the

pioneer ¬ndings of Fama (1975) regarding the forecasting ability of the term structure

Modelling the Term Structure of Interest Rates 123

6

100 000

80 000

5

60 000

4

40 000

Inflation (%)

20 000

3

Factor

0

2

’20 000

’40 000 1

’60 000

0

’80 000

’1

’100 000

Jan-86

Jan-87

Jan-88

Jan-89

Jan-90

Jan-91

Jan-92

Jan-93

Jan-94

Jan-95

Jan-96

Jan-97

Jan-98

Factor Inflation

(e)

20 8

7

15

6

10

5

5

Inflation (%)

4

Factor

0

3

’5

2

’10

1

’15 0

’20 ’1

Sep-72

Sep-74

Sep-76

Sep-78

Sep-80

Sep-82

Sep-84

Sep-86

Sep-88

Sep-90

Sep-92

Sep-94

Sep-96

Sep-98

Factor Inflation

(f)

Figure 3.16 (continued )

on future in¬‚ation. It is also consistent with the idea that in¬‚ation expectations drive

long-term interest rates, given that the second factor is relatively more important for the

dynamics of interest rates at the long end of the curve, as previously stated.

If z2t is a leading indicator for πt , then the highest (positive) correlation should occur

between lead values of πt and z2t . Table 3.6 shows cross-correlation between πt and z2t

124 Applied Quantitative Methods for Trading and Investment

in the larger sample. The shaded ¬gures are the correlation coef¬cients between z2t and

lags of πt . The ¬rst ¬gure in each row is k and the next corresponds to the correlation

between πt’k and z2t (negative k means lead). The next to the right is the correlation

between πt’k’1 and z2t , etc.

According to Table 3.6, the highest correlation (in bold) is at the fourth lead of in¬‚ation.

Additionally, that correlation increases with leads of in¬‚ation up to four and steadily

declines for lags of in¬‚ation.

Table 3.7 presents the Granger causality test and Figure 3.17 the impulse-response func-

tions. The results strongly support the conjecture that z2,t has leading indicator properties

for in¬‚ation. At the 5% level of con¬dence one can reject that z2,t does not Granger cause

in¬‚ation. This is con¬rmed by the impulse-response analysis.

A positive shock to the innovation process of z2t is followed by a statistically signi¬cant

increase in in¬‚ation, as illustrated by the ¬rst panel in Figure 3.17, where the con¬dence

interval of the impulse-response function does not include zero. However a positive shock

to the innovation process of in¬‚ation does not seem to be followed by a statistically

signi¬cant increase in z2t , according to the last panel in Figure 3.17, where the con¬dence

interval includes zero. These results suggest that an innovation to the in¬‚ation process

does not contain “news” for the process of expectation formation. This is in line with

the forward-looking interpretation of shocks to z2t as re¬‚ecting “news” about the future

course of in¬‚ation.

Table 3.6 Cross-correlations of series πt and z2t (monthly data from 1972:09 to 1998:12)

k

’25: 0.3974759 0.4274928 0.4570631 0.4861307 0.5129488 0.5394065

’19: 0.5639012 0.5882278 0.6106181 0.6310282 0.6501370 0.6670820

’13: 0.6829368 0.6979626 0.7119833 0.7235875 0.7324913 0.7385423

’7: 0.7432024 0.7453345 0.7457082 0.7458129 0.7443789 0.7411556

’1: 0.7368274 0.7311008 0.7144727 0.6954511 0.6761649 0.6561413

5: 0.6360691 0.6159540 0.5949472 0.5726003 0.5478967 0.5229193

11: 0.4960254 0.4706518 0.4427914 0.4156929 0.3870690 0.3590592

17: 0.3311699 0.3011714 0.2720773 0.2425888 0.2126804 0.1829918

23: 0.1527293 0.1226024 0.0927202 0.0636053 0.0362558 0.0085952

’0.0167756 ’0.0421135 ’0.0673895 ’0.0911283 ’0.1139021 ’0.1356261

29:

Note: Correlation between πt’k and z2t (i.e., negative k means lead).