Palgrave Handbook of Econometrics: Applied Econometrics

858 Macroeconometric Modeling for Policy

Iffis a linear function ofyandxthenR=0 and so:

f(x,y)=a

(

y−y ̄+

δ

a

(x−x ̄)

)

=a(y−bx−c), (17.4)

in whichb=−δ/aandc=y ̄+(δ/a) ̄x.

17.2.2 Discretization

For a macroeconometric model, a discrete representation is usually practical, and it
can be worked out as follows. Lett 1 ,t 2 ,···,tk,···be a sequence of times spaced
hapart and lety 1 ,y 2 ,···,yk,···be the values of a continuous real variabley(t)
at these times. The backward-difference operator%is defined by the rule:

%yk=yk−yk− 1 , k≥1. (17.5)

By observing thatyk=( 1 −%)^0 ykandyk− 1 =( 1 −%)^1 yk, the value ofyat the
intermediate pointt=tk−sh( 0 <s< 1 )may be estimated by the interpolation
formula:

y(tk−sh)=yk−s=( 1 −%)syk, s∈[0, 1]. (17.6)

Whensis not an integer,( 1 −%)sshould be interpreted as the power series in the
backward-difference operator obtained from the binomial expansion of( 1 −x)s.
This is an infinite series of differences. Specifically:

( 1 −%)s= 1 −s%−

s( 1 −s)

2!

%^2 −

s( 1 −s)( 2 −s)

3!

%^3 −···. (17.7)

With this preliminary background, the differential equation:

dy

dt

=f(y,x), x=x(t), (17.8)

may be integrated over the time interval[tk,tk+ 1 ]to obtain:

y(tk+ 1 )−y(tk)=%yk+ 1 =

∫tk+ 1

tk

f(y(t),x(t))dt, (17.9)

in which the integral on the right-hand side of this equation is to be estimated by
using the backward-difference interpolation formula given in equation (17.7). The
substitutiont=tk+shis now used to change the variable of this integral from
t∈[tk,tk+ 1 ]tos∈[0, 1]. The details of this change of variable are:

∫tk+ 1

tk

f(y(t),x(t))dt=

∫ 1

0

f(y(tk+sh),x(tk+sh)) (hds)=h

∫ 1

0

fk+sds,