J.F. Carriere Insurance: Mathematics and Economics 26 2000 193–202 195
2. A spline model
In this section, we present a spline model of the forward rates. Let 0 = κ κ
1
· · · κ
n− 1
κ
n
= 30 denote the fixed knots of the spline and let κκκ = κ
1
, . . . , κ
n− 1
′
denote the interior knots. Thus, the formula for a q
-polynomial spline is Splinem|q, κκκ =
q
X
i=
φ
i
m
i
+
n− 1
X
j = 1
ξ
j
m − κ
j q
1m κ
j
, 2.1
where 1e is an indicator function that is equal to 1 if the event e is true and 0 otherwise. This spline is a piecewise polynomial of degree q with q − 1 continuous derivatives at the interior knots. The parameter q = 1, 2, . . . is fixed
and it controls the smoothness. The parameter n = 1, 2, . . . controls the fit and it is also fixed. In this paper, we will let q = 2 and n = 12. The unknown parameters of this spline are φ
, . . . , φ
q
and ξ
1
, . . . , ξ
n− 1
. Consult Seber and Wild 1989 for more information about splines in regression analysis.
At this point, it will be convenient to introduce some matrix notation. Let p = q + n denote the total number of parameters in the spline model and let β
β β
denote a p × 1 parameter vector that is defined as β
β β =
[φ , . . . , φ
q
, ξ
1
, . . . , ξ
n− 1
]
′
. 2.2
Next, define x
x xm =
[1, m
1
, . . . , m
q
, m − κ
1 q
1m κ
1
, . . . , m − κ
n− 1
q
1m κ
n− 1
]
′
. 2.3
Thus, we can write Spline m|q, κκκ = x x
x
′
mβ β
β . We are now ready to specify the statistical model for the observed
data. Our model is F
k
= x x
x
′
¯ m
k
β β
β + ε
k
, 2.4
where ε
1
, ε
2
, . . . , ε
N
are random variables that are not necessarily independent and identically distributed. Let εεε = ε
1
, . . . , ε
N ′
, let F F
F = F
1
, . . . , F
N ′
and let
X X
X =
x x
x
′
¯ m
1
x x
x
′
¯ m
2
.. .
x x
x
′
¯ m
N
.
2.5
These definitions allow us to write our linear model in matrix notation. Specifically, F
F F = Xβ
Xβ Xβ + εεε.
2.6 In this paper, we will estimate β
β β
by a weighted least squares WLS method. Let W W
W denote an N × N diagonal
matrix where all the off-diagonal elements are equal to zero. Using our matrix notation, we can write the WLS loss function as
Lβ β
β = [F
F F − Xβ
Xβ Xβ
]
′
W W
W [F
F F − Xβ
Xβ Xβ
]. 2.7
Note that in this definition, W W
W is assumed to be fixed. Usually, we want to minimize Lβ
β β
subjected to a linear constraint of Aβ
Aβ Aβ = µ
µ µ
. An example of a desirable linear constraint is 1 1
1
′
Xβ Xβ
Xβ = 1
1 1
′
F F
F where 1
1 1 is a column vector of
ones. In this case A A
A = 1
1 1
′
X X
X and µ
µ µ =
1 1
1
′
F F
F . The constrained WLS estimator is that value that minimizes Lβ
β β
subject to the constraint. That value is equal to
ˆ β
β β = X
X X
′
WX WX
WX
−1
[X X
X
′
WF WF
WF + A A
A
′
λ λ
λ ],
2.8
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
