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Since we have observations of y, we can compute the left-hand side of (
5.152
)
and use the results to build a linear model of the right-hand side of the equation,
cf. Sect.
5.2.2
.
The purpose of this project is to introduce a third idea. Using this idea, we elim-
inate the use of (
5.150
) and thus avoid the inaccuracies introduced by numerical
differentiation. We do, however, introduce errors due to numerical integration. The
essence of the third idea is to observe that
y
0
.t /
y.t/
:
d
dt
ln y.t/
D
(5.153)
In this project we will assume that we have data given by .t
n
;y.t
n
//.
(a) Use (
5.153
) to show that
Z
t
nC1
y
0
.t /
y.t/
dt
D
ln y.t
nC1
/
lny.t
n
/:
(5.154)
t
n
(b) Consider (
5.148
)andshowthat
ln y.t
nC1
/
ln y.t
n
/
t
nC1
t
n
˛
D
:
(5.155)
(c) Define
ln y.t
nC1
/
ln y.t
n
/
t
nC1
t
n
˛
n
D
(5.156)
and compute ˛
0
, ˛
1
, ˛
3
,and˛
4
using the data in Table
5.2
on page
167
.Note
that t
0
D
0 corresponds to 1950, t
1
D
1 corresponds to 1951, and so on.
(d) Verify that
4
X
1
5
˛
D
˛
n
0:01688
(5.157)
nD0
and compare your result with the result given in (
5.82
)onpage
168
.
(e) Define
p.t/
D
2:555 e
0:01688 t
(5.158)
and graph both p givenby(
5.158
)andp given by (
5.83
)fort ranging from
0 to 50.