196 J.F. Carriere Insurance: Mathematics and Economics 26 2000 193–202
where λ
λ λ =
[A A
AX X
X
′
WX WX
WX
−1
A A
A
′
]
−1
[µ µ
µ − A A
AX X
X
′
WX WX
WX
−1
X X
X
′
WF WF
WF ].
2.9 Using this estimator, we find that we can estimate the forward rates with ˆ
f
m
= x x
x
′
m ˆ β
β β
. We found a very good-fitting curve by using the constraint 1
1 1
′
Xβ Xβ
Xβ = 1
1 1
′
F F
F and a weighting matrix equal to the identity matrix. The fixed parameters
were q = 2, n = 12 and κ
j
= 30jn for j = 1, 2, . . . , n − 1. The residuals are defined as e
k
= F
k
− ˆ f
¯ m
k
. 2.10
These residuals were unusual because they exhibited some heteroscedasticity and correlation.
3. Estimation of the variance structure
Traditionally, statisticians will assume that ε
1
, ε
2
, . . . , ε
N
are independent and identically distributed random variables. However, the residuals in our case do not exhibit that behaviour. We will assume that Varε
k
= σ
2 gk
. In this definition, gk is a map of {1, 2, . . . , N } onto {1, 2, . . . , n}, that is defined as
gk =
n
X
j = 1
j 1κ
j − 1
¯ m
k
≤ κ
j
. 3.1
Thus, the variance is σ
2 j
on the interval κ
j − 1
, κ
j
] and we get an estimable heteroscedastic error structure. It will be useful to define the standard deviation matrix S
S S
. This is a diagonal matrix where the diagonal elements are equal to σ
gk
and all the off-diagonal elements are zero. Now let us describe how S S
S can be estimated. The number of
observations in the interval κ
j − 1
, κ
j
] is N
j
=
N
X
k= 1
1κ
j − 1
¯ m
k
≤ κ
j
. Let e
1
, e
2
, . . . , e
N
denote the residuals and let e
k
be associated with the maturity of ¯ m
k
. A standard non-parametric estimator of σ
2 j
is ˆ
σ
2 j
= 1
N
j N
X
k= 1
e
2 k
1κ
j − 1
¯ m
k
≤ κ
j
− 1
N
j N
X
k= 1
e
k
1κ
j − 1
¯ m
k
≤ κ
j 2
. 3.2
Using ˆ σ
j
we can construct an estimator of S S
S , denoted as ˆ
S S
S . In Section 5, we showed how a WLS estimator of β
β β
can be constructed for a fixed weighting matrix W
W W
, without discussing what W W
W should be. Ideally, we would let
W W
W = [S
S S S
S S
]
−1
. 3.3
However, S S
S is not known. Thus, we recommend that β
β β
and S S
S be estimated by iterated weighted least squares. This
method will generate a sequence of estimators, denoted as ˆ β
β β
1
, ˆ β
β β
2
, ˆ β
β β
3
, . . . and ˆ
S S
S
1
, ˆ S
S S
2
, ˆ S
S S
3
, . . . . As a starting value,
we suggest that ˆ S
S S
1
be set to the identity matrix. Let ˆ β
β β
1
be the estimator in this case. Using ˆ β
β β
1
, we can calculate the residuals and estimate ˆ
S S
S
2
. In turn, ˆ β
β β
2
is found by letting W W
W = [ ˆ
S S
S
2
ˆ S
S S
2
]
−1
. The process is repeated until we have convergence in successive estimates of the parameters. Table 1 shows ˆ
σ
j
for j = 1, 2, . . . , n = 12. Note that the standard deviations have stabilized after five iterations. Also note that the standard deviations in the first iteration
are very close to those in the last iteration.
J.F. Carriere Insurance: Mathematics and Economics 26 2000 193–202 197
Table 1 Estimates of the standard deviations
Parameter Iteration
1 2
3 4
5 ˆ
σ
1
1.00 0.00280
0.00279 0.00280
0.00280 ˆ
σ
2
1.00 0.00413
0.00390 0.00385
0.00384 ˆ
σ
3
1.00 0.01006
0.01009 0.01009
0.01009 ˆ
σ
4
1.00 0.00889
0.00898 0.00900
0.00900 ˆ
σ
5
1.00 0.00598
0.00548 0.00544
0.00543 ˆ
σ
7
1.00 0.00982
0.00979 0.00978
0.00978 ˆ
σ
6
1.00 0.00598
0.00548 0.00544
0.00543 ˆ
σ
7
1.00 0.01237
0.01273 0.01285
0.01287 ˆ
σ
8
1.00 0.06292
0.06299 0.06301
0.06301 ˆ
σ
9
1.00 0.00587
0.00492 0.00482
0.00480 ˆ
σ
10
1.00 0.02288
0.02282 0.02281
0.02280 ˆ
σ
11
1.00 0.03222
0.03263 0.03275
0.03277 ˆ
σ
12
1.00 0.04663
0.04663 0.04664
0.04664
Fig. 2. Spline estimates of the forward rates. This is a plot of F
k
, ˆ f
1 ¯
m
k
and ˆ f
4 ¯
m
k
versus the maturity ¯ m
k
.
198 J.F. Carriere Insurance: Mathematics and Economics 26 2000 193–202
Let ˆ f
i ¯
m
k
= x x
x
′
¯ m
k
ˆ β
β β
i
denote the estimated forward rates at iteration i = 1, 2, . . . . We suggest that the iterations should stop whenever the average relative error ARE is less than 0.001, i.e.,
ARE
i+ 1
= 1
N
N
X
k= 1
ˆ f
i+ 1
¯ m
k
ˆ f
i ¯
m
k
− 1 0.001,
3.4 where N = 121. Note that the ARE of 0.001 could be smaller but the effect on the estimated values would be
minimal. In our application we found that ARE
2
= 0.0274, ARE
3
= 0.0047, ARE
4
= 0.0007. Thus, we stopped after the fourth iteration. Fig. 2 is a plot of the observed rates F
k
against the estimated rates from the first and fourth iterations.
Next, let ˆ σ
gk
denote the estimated standard deviations and let ˆ S
S S
denote the associated matrix from the last iteration. Also, let eee = e
1
, e
2
, . . . , e
N ′
denote the residuals from the last iteration. Our statistical model suggests that the standardized errors
˜eee = ˆ S
S S
−1
eee 3.5
should not exhibit any heteroscedasticity. If ˜e
k
is the kth element of ˜eee, then ˜e
k
= e
k
ˆ σ
gk
. Fig. 3 is a plot of ˜e
k
versus ¯ m
k
. Note the lack of heteroscedasticity.
Fig. 3. This is a plot of the standardized errors ˜e
k
versus the maturity ¯ m
k
for k = 1, 2, . . . , N = 121.
J.F. Carriere Insurance: Mathematics and Economics 26 2000 193–202 199
4. Estimation of the correlation structure
