Palgrave Handbook of Econometrics: Applied Econometrics

Gunnar Bårdsen and Ragnar Nymoen 859

wherefk+s=f(y(tk+sh),x(tk+sh)). The value of this latter integral is now computed
using the interpolation formula based on (17.7). Thus:
∫ 1

0

fk+sds=

∫ 1

0

( 1 −%)−sfkds

=

∫ 1

0

(

fk+s%fk+

s( 1 +s)

2!

%^2 fk+

s( 1 +s)( 2 +s)

3!

%^3 fk+···

)

ds

=fk+

1

2

%fk+

5

12

%^2 fk+

3

8

%^3 fk+···.

The final form for the backward-difference approximation to the solution of this
differential equation is therefore

%yk+ 1 =hfk+

h

2

%fk+

5 h

12

%^2 fk+

3 h

8

%^3 fk+···. (17.10)

17.2.3 Equilibrium correction representations and cointegration

The discretization scheme (17.10) applied to the linearized model (17.3), withk=
t−1 andh=1, gives the equilibrium correction model, EqCM, representation

%yt=a(y−bx−c)t− 1 +Rt− 1 +

1

2

a(%yt− 1 −b%xt− 1 )+

1

2

%Rt− 1

+

5

12

a(%^2 yt− 1 −b%^2 xt− 1 )+

5

12

%^2 Rt− 1 +···.

At this point two comments are in place. The first is that an econometric specifi-
cation will mean a truncation of the polynomial both in terms of powers and lags.
Diagnostic testing is therefore imperative to ensure a valid local approximation,
and indeed to test that the statistical model is valid (see Hendry, 1995a, p. 15.1;
Spanos, 2008). The second is that the framework allows for flexibility regarding
the form of the steady state. The standard approach in DSGE modeling has been
to filter the data, typically using the so-called Hodrick–Prescott filter, to remove
trends, hopefully achieving stationary series with constant means, and then work
with the filtered series. Another approach, popular at present, is to impose the
theoretical balanced growth path of the model on the data, expressing all series
in terms of growth corrected values. However, an alternative approach is to esti-
mate the balanced growth paths in terms of finding the number of common trends
and identifying and estimating cointegrating relationships. The present approach
allows for all of these interpretations.
To illustrate the approach in terms of cointegration, consider real wages to be
influenced by productivity, as in many theories.^5 Assume that the logs of the real
wagerwtand productivityztare each integrated of order one, but found to be
cointegrated, so:

rwt∼I( 1 ),rwt∼I( 0 ) (17.11)

zt∼I( (^1) ),zt∼I( (^0) ) (17.12)
(rw−βz)t∼I( 0 ). (17.13)