Pesaran, Shin and Smith (PSS)(2001) developed a
new approach to cointegration testing which is applicable irrespective of
whether the regressor variables are \(I\left( 0 \right)\),\(I\left( 1 \right)\)
or mutually cointegrated.
The statistic underlying PSS procedure is the
familiar Wald or F-statistic in a generalized Dickey-Fuller type
regression used to test the significance of lagged of the variables under
consideration in an unrestricted error correction regression.
The test show that the asymptotic distribution
of both statistics are non-standard under the null hypothesis that there exist
no relationship between the level of the included variables; irrespective of
whether the regressor are \(I\left( 0 \right)\),\(I\left( 1 \right)\) or
mutually cointegrated.
PSS provide two sets of asymptotic critical
values for the two polar case: one which assumes that all the regressor are , \(I\left(
1 \right)\) and the other assuming that they are \(I\left( 0 \right)\) .
Since these two sets of critical values provide
critical value bounds for all classifications
of the regressors into \(I\left( 1 \right)\) and/or \(I\left( 0 \right)\) PSS
propose a bound testing procedure.
If the computed Wald or F-statistic falls outside the critical value bounds, a conclusive
inference can be drawn without needing to know whether the underlying regressor
are \(I\left( 1 \right)\): cointegrated amongst themselves or individually
\(I\left( 0 \right)\) .
The bound test methodology has a number
features that many researcher feel give it some advantages over conventional
cointegration testing;
1. It can be used
with mixture of \(I\left( 0 \right)\) and \(I\left( 1 \right)\) data.
2. It involves
just a single-equation set-up, making it simple to implement and interpret.
3. Different
variables can be assigned different lag-lengths as they enter the model.
There are the basic steps that we are going to
follow to perform the bound test and the basic model form of ARDL model is;
\({{Y}_{t}}={{\beta
}_{0}}+\sum\limits_{i=1}^{p}{{{\beta
}_{1}}{{Y}_{t-i}}+\sum\limits_{i=0}^{{{q}_{1}}}{{{a}_{1}}{{X}_{1,t-i}}\sum\limits_{i=0}^{{{q}_{2}}}{{{a}_{2}}{{X}_{2,t-i}}+}}}\sum\limits_{i=0}^{{{q}_{3}}}{{{a}_{3}}{{X}_{3,t-i}}+}\sum\limits_{i=0}^{q4}{{{a}_{4}}{{X}_{4,t-i}}+}{{\varepsilon
}_{t}}\) (1)
Step 1
Make sure that none of the variables are \(I\left(
2 \right)\), as such data will invalidate the methodology.
Step 2
Formulate the following model;
\({{Y}_{t}}={{\beta
}_{0}}+\sum\limits_{i=1}^{p}{{{\beta
}_{1}}{{Y}_{t-i}}+\sum\limits_{i=0}^{{{q}_{1}}}{{{a}_{1}}{{X}_{1,t-i}}\sum\limits_{i=0}^{{{q}_{2}}}{{{a}_{2}}{{X}_{2,t-i}}+}}}\sum\limits_{i=0}^{{{q}_{3}}}{{{a}_{3}}{{X}_{3,t-i}}+}\sum\limits_{i=0}^{q4}{{{a}_{4}}{{X}_{4,t-i}}}+{{\theta
}_{0}}{{Y}_{t-1}}+{{\theta }_{1}}{{X}_{1,t-1}}+{{\theta
}_{2}}{{X}_{2,t-1}}+{{\theta }_{3}}{{X}_{3,t-1}}+{{\theta }_{4}}{{X}_{4,t-1}}+{{\varepsilon
}_{t}}\)
(2)
Step 3
The range of summation in various term
in Eq (2) are from 1 to p ,and 0 to q1, 0 to q2, 0 to q3 and 0 to q4
respectively. We need to select the appropriate values for maximum lags, p, q1,
q2, q3 and q4.
Use the information criterion (AIC,SIC
or Bayesian information) to select appropriate
lags. The smaller the value of an information criterion the better the
results.
Step 4
Perform the bound testing. Again;
\({{Y}_{t}}={{\beta
}_{0}}+\sum\limits_{i=1}^{p}{{{\beta
}_{1}}{{Y}_{t-i}}+\sum\limits_{i=0}^{{{q}_{1}}}{{{a}_{1}}{{X}_{1,t-i}}\sum\limits_{i=0}^{{{q}_{2}}}{{{a}_{2}}{{X}_{2,t-i}}+}}}\sum\limits_{i=0}^{{{q}_{3}}}{{{a}_{3}}{{X}_{3,t-i}}+}\sum\limits_{i=0}^{q4}{{{a}_{4}}{{X}_{4,t-i}}}+{{\theta
}_{0}}{{Y}_{t-1}}+{{\theta }_{1}}{{X}_{1,t-1}}+{{\theta
}_{2}}{{X}_{2,t-1}}+{{\theta }_{3}}{{X}_{3,t-1}}+{{\theta }_{4}}{{X}_{4,t-1}}+{{\varepsilon
}_{t}}\) (2=3)
Perform an “F-test” of the hypothesis \({{H}_{0}}:{{\theta
}_{0}}={{\theta }_{1}}=...={{\theta }_{4}}=0\) against
the alternative that \({{H}_{0}}\) is
not true. The rejection implies that we have a long-run relationship.
Because the
distribution of F-test for Eq(4) is non-standard, Pesaran et.al (2001) supply
bounds on the critical values for the asymptotic distribution of the
F-statistics.
In each case, the lower
bounds is based on the assumption that all the variables are I(0), and the
upper bound is based on the assumption that all the variables are I(1).
Step 5
If the bound test leads to conclusion of cointgration, we
can meaningfully estimate the long-run
equilibrium relationship between the variables:
\({{Y}_{t}}={{\beta
}_{0}}+{{a}_{1}}{{X}_{1,t}}+{{a}_{2}}{{X}_{2,t}}+{{a}_{3}}{{X}_{3,t}}+{{a}_{4}}{{X}_{4,t}}+{{\varepsilon
}_{t}}\) (4)
as well as the usual ECM:
\(\Delta {{Y}_{t}}={{\beta
}_{0}}+\sum\limits_{i=1}^{p}{{{\beta }_{1}}\Delta
{{Y}_{t-i}}+\sum\limits_{i=0}^{{{q}_{1}}}{{{a}_{1}}\Delta
{{X}_{1,t-i}}\sum\limits_{i=0}^{{{q}_{2}}}{{{a}_{2}}\Delta
{{X}_{2,t-i}}+}}}\sum\limits_{i=0}^{{{q}_{3}}}{{{a}_{3}}\Delta
{{X}_{3,t-i}}+}\sum\limits_{i=0}^{q4}{{{a}_{4}}\Delta {{X}_{4,t-i}}}+\delta {{Z}_{t-1}}+{{\varepsilon
}_{t}}\) (5)
where \({{Z}_{t-1}}=\left( {{Y}_{t-1}}-{{a}_{0}}-{{a}_{1}}{{X}_{1,t-1}}-{{a}_{2}}{{X}_{2,t-1}}-{{a}_{3}}{{X}_{3,t-1}}-{{a}_{4}}{{X}_{4,t-1}}
\right)\)
We use the data Macro.
Perform the unit root test (ADF test or PP
test) and conform that none the variables
is \(I\left( 2 \right)\).
To test the bound test with an ardl method;
We want perform cointegration test between the variables
gdp, pdi, and pce with the bound
test.
* Case 1 - no intercept and no determistic
trend
ardl pce pdi gdp,
ec1 aic noconstant
ardl,noctable btest
* Case 2 - restricted intercepts and no
deterministic trends
ardl pce pdi gdp,
ec1 aic constant restricted
ardl,noctable btest
* Case 3 - unrestricted intercepts and no
deterministic trends
ardl pce pdi gdp,
ec1 aic constant
ardl,noctable btest
* Case 4 - unrestricted intercepts restricted
deterministic trend
ardl pce pdi gdp,
ec1 aic constant trendvar(year) restricted
ardl,noctable btest
* Case 5 - unrestricted intercepts and trends
ardl pce pdi gdp,
ec1 aic trendvar(year)
ardl,noctable btest
The ardl results show that the optimal lags for
our model ardl is ARDL (3,0,1).
The results for Case 3 show that the value of \({{F}_{s}}=4.259\)
which is greater than
the critical value of upper bound \({{F}_{ub}}=4.14\) at 10% level which is null for no cointegration can
be rejected.
It means that there is cointegration between the
variables pce, pdi and gdp.
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