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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.
 
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