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. 6
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’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).

<<

. 6
( 19)



>>