COINTEGRATION - IASRI

Cointegration
Let, denote the realized value of the likelihood ratio statistic Q. If λ r+1 =…= λ n =0, the test
statistic should be small (close to zero), which delivers the critical value under the null. The
test is based on non-standard asymptotic distribution. Asymptotic critical values are provided
in Doornik (1998). The asymptotic distributions depend on whether there is a constant and/or
a trend in the model. In many economic applications, the size of the sample is often quite
small and the tabulated asymptotic distributions can be rather poor approximations as has
been demonstrated in many papers. Another reason of concern is that when using correct
small sample distributions for the trace test, the size of the test is correct, but the power can
be low, sometimes even of the same magnitude as the size. In such cases, a 5% test procedure
will reject a unit root incorrectly 5% of the time, but accept a unit root incorrectly 95% of the
time.
If dummy variables enter the deterministic part of the multivariate model, then these critical
values are only indicative. Similarly, if there is only a small sample of observations on y t ,
then there are likely to be problems with the power and size properties of the above test when
using asymptotic critical values. Many studies has shown that in such a situation the Johansen
procedure over-rejects when the null is true. Thus he suggests taking account of the number
of parameters to be estimated in the models and making an adjustment for degrees of
freedom.
5. SAS syntax for cointegration test
Consider the quarterly data for the years 1954 to 1987 for the US econlomy (Lütkepohl 1993,
Table E.3.). The program is taken from SAS help and documentation.
title 'Analysis of U.S. Economic Variables';
data us_money;
date=intnx( 'qtr', '01jan54'd, _n_-1 );
format date yyq. ;
input y1 y2 y3 y4 @@;
y1=log(y1);
y2=log(y2);
label y1='log(real money stock M1)'
y2='log(GNP in bil. of 1982 dollars)'
y3='Discount rate on 91-day T-bills'
y4='Yield on 20-year Treasury bonds';
datalines;
450.9 1406.8 0.010800000 0.026133333
453.0 1401.2 0.0081333333 0.025233333
... more lines ...
run;
proc varmax data=us_money;
id date interval=qtr;
model y1-y4 / p=2 lagmax=6 dftest
print=(iarr(3) estimates diagnose)
cointtest=(johansen=(iorder=2))
ecm=(rank=1 normalize=y1);
cointeg rank=1 normalize=y1 exogeneity;
run;
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(3.4)