ECON 3350/7350 Cointegration
Eric Eisenstat
The University of Queensland
Tutorial 6
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 6
1 / 7
Spurious Regression vs Cointegration
What are the implications for empirical economic research of having I(1) variables?
Spurious Regressions or Cointegration
It is generally true that any combination of two I(1) variables will also be I(1).
Spurious Regression
Conclude there is a significant relationship when there is none.
Cointegration
Linear combinations of I(1) variables are I(0).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6
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Spurious Regression (cont.)
Indications of a spurious regression:
Signicant t-values; respectable (sometimes high) R2; low Durbin-Watson (DW) statistics.
The signicant t-values occur because the random walks tend to wander, and this wandering looks like a trend.
If they wander in the same direction for a while (say for the time of the observed sample), there appears to be a relationship.
In:
yt = α + βxt + εt, εt ∼ I(1) so the regression is meaningless. This explains why DW is low.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6
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Cointegration and Equilibrium
The economic interpretation and signicance of cointegration We may regard the cointegrating relation
zt = xt − ayt as a stable equilibrium relation.
Although xt and yt are themselves unstable as they are I(1), they are attracted to a stable relationship that exists between them—i.e.,
zt ∼ I(0).
For example, there is strong evidence that interest rates are I(1). But the spread between two rates of different maturities, within the same market, appear to be I(0).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 4 / 7
Cointegration Order
If wt = (w1,t, w2,t, …, wn,t)′ ∼ I(1) but wt′β ∼ I(0)
where
β′wt = w1,tβ1 + w2,tβ2 + …wn,tβn w1,t
′ w2,t =(β1,β2,…,βn) . .
wn,t
Then we say that components of the vector wt are cointegrated of
order (1, 1), denoted CI(1, 1).
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 6
5 / 7
Testing for Cointegration
Consider the case of three variables: xt,yt, and zt
Estimate: xt = αt + β1yt + β2zt + et (by OLS)
Test the residual, et, for a unit root. If et ∼ I(0), then xt, yt, and zt cointegrate.
There are a number of ways we could perform this test. We will look at using the Dickey-Fuller test statistic (Engle-Granger) and the Durbin-Watson statistic.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 6 / 7
Testing for Cointegration (cont.)
Because the residual et comes from a potential cointegrating relation, the test statistics will not have the usual distributions so we cannot use the same critical values.
The Augmented Dickey-Fuller test to test for cointegration. We proceed as usual but use critical values from Table C in Enders.
We estimate the ADF equation: ∆et = γet−1 + νt (νt is WN) and test H0 : γ = 0.
The Durbin-Watson test to test for cointegration (CRDW). We proceed as usual but use critical values from Table 9.3 in Verbeek. CRDW is not widely used as it is applicable only for AR(1) processes.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 7 / 7