How we to estimate the parameters of models that involve
cointegrated variables?
We don’t need to worry about spurious regression results,
because the disturbances now contain no stochastic trend.
When error term contains a stochastic trend, OLS is
inconsistent. We use the Granger and Newbold method (by differencing \({{X}_{t}}\)
and \({{Y}_{t}}\) )
OLS is is consistent, even super consistent, when
explanators contain stochastic trends and error term do not, but the OLS not
asymptotically normal.
But what are we do about the continuing invalidity of our
usual test statistics?
More formally, if \({{Y}_{t}}\),\({{Z}_{1t}}\) and \({{Z}_{2t}}\)
are cointegrated variables, estimating the slopes in
\({{Y}_{t}}={{\beta
}_{0}}+{{\beta }_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{u}_{t}}\) (1)
is an instance of Case 2.
The explanators are stochastically
trending, but the disturbances is not.
OLS is consistent, but our usual test
statistics are no longer valid, except for one special case.
How are we to conduct inference when
variables in our model are cointegrated?
If we add a mean zero \(I\left( 0
\right)\) \({{X}_{t}}\) into this model,
as in
\({{Y}_{t}}={{\beta
}_{0}}+{{\beta }_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta
}_{3}}{{X}_{t}}+{{u}_{t}}\) (2)
In Eq(2) the variables \({{Y}_{t}}\),\({{Z}_{it}}\) and \({{Z}_{2t}}\)
contain stochastic trends, but \({{X}_{t}}\) and \({{u}_{t}}\) do not.
We can estimate all the coefficients consistently.
Estimating the \({{\beta }_{3}}\) is an instance of Case1; OLS is consistent but only our estimate of \({{\beta }_{3}}\) for the
associated t-statistics is
asymptotically normal.
Our standard procedure are valid for
estimating and testing the coefficients on variable such as \({{X}_{t}}\), but
not valid for variables that contain a stochastic trend (\({{Z}_{1t}}\) and \({{Z}_{2t}}\)).
But
the questions, “ How are we to draw inference about the coefficients on
variables such as the \(Z\)’s?”
Dynamic Ordinary
Least Squares Estimator (DOLS)
Stock and Watson (1993) proposed that we add seemingly
superfluous nontrending variable to the cointegrated regression of interest to
obtain a specification that falls into the exception to Case 2 – the
respecified model could be rewritten in a way that makes \({{\beta }_{1}}\) and
\({{\beta }_{2}}\) coefficients on a nontrending variable.
If the innovations of \({{Y}_{t}}\),\({{Z}_{1t}}\)
and \({{Z}_{2t}}\) are serially uncorrelated, it suffices to add to the
cointegrated regression the changes in the randomly walking explanators.
The OLS estimators of \({{\beta }_{1}}\)
and \({{\beta }_{2}}\) then asymptotically support our usual t-statistics
and F-statitsics based on t-distributions and F-distributions.
This strategy for estimating \({{\beta
}_{1}}\) and \({{\beta }_{2}}\) is called dynamic
OLS (DOLS).
Case:
no serial correlation;
We wish to estimate
\({{Y}_{t}}={{\beta }_{0}}+{{\beta
}_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta }_{3}}{{X}_{t}}+{{u}_{t}}\) (3)
Rewrite the model as;
\({{Y}_{t}}={{\beta }_{0}}+{{\beta
}_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta }_{3}}{{X}_{t}}+{{\beta
}_{4}}\Delta {{Z}_{1t}}+{{\beta }_{5}}\Delta {{Z}_{2t}}+{{u}_{t}}\) (4)
After controlling for \(\Delta
{{Z}_{it}}\) there is no stochastic trend in \({{Z}_{it}}\)
Case:
serial correlation;
Serial correlation in the innovations of \({{Z}_{1t}}\)
and \({{Z}_{2t}}\) complicate dynamic OLS somewhat, requiring that we add not
only the contemporaneous change in \({{Z}_{1t}}\) and \({{Z}_{2t}}\), but also
both past and future values (lags and leads) of those changes.
We wish to estimate
\({{Y}_{t}}={{\beta }_{0}}+{{\beta
}_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta }_{3}}{{X}_{t}}+{{u}_{t}}\) (5)
\({{Y}_{t}}\),\({{Z}_{1t}}\) and \({{Z}_{2t}}\) contain
stochastic trends.
Rewrite the model as;
\({{Y}_{t}}={{\beta }_{0}}+{{\beta
}_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta
}_{3}}{{X}_{t}}+{{u}_{t}}+{{\gamma }_{1}}\Delta {{Z}_{1,t+2}}+{{\gamma
}_{2}}\Delta {{Z}_{1,t+1}}+{{\gamma }_{3}}\Delta {{Z}_{1,t}}+{{\gamma
}_{4}}\Delta {{Z}_{1,t-1}}+{{\gamma }_{5}}\Delta {{Z}_{1,t-2}}+{{\tau
}_{1}}\Delta {{Z}_{2,t+2}}+{{\tau }_{2}}\Delta {{Z}_{2,t+1}}+{{\tau
}_{3}}\Delta {{Z}_{2,t}}+{{\tau }_{4}}\Delta {{Z}_{2,t-1}}+{{\tau }_{5}}\Delta
{{Z}_{2,t-2}}\) (6)
where
\(\Delta
{{Z}_{j,\left( t+k \right)}}={{Z}_{j,\left( t+k \right)}}-{{Z}_{j,\left( t+k-1
\right)}}\)
The OLS estimators for \({{\beta }_{1}}\)
and \({{\beta }_{2}}\) are consistent and efficient.
If there is also serial correlation in
, we should use the Newey-West serial
correlation –consistent estimated standard errors for those OLS estimators.
Little is known about how many leads and
lags of the changes in \({{Z}_{1t}}\) and \({{Z}_{2t}}\) to included.
In practice, adding two leads and lags is
standard.
Its is necessary to add equal numbers of
leads and lags for the respecified model to be an exception to Case 2.
DOLS is a simple and efficient approach
to estimating the coefficients of a cointegrating relationship.
To do the DOLS example, we use the data defisit1.
We model the long-term interest rate measure by 10-year treasury
bond rate (fygt10), as a function of the short-run
interest rate measured by the 1-year treasury bond rate (fygt1), inflation (infl), the U.S. government real deficit per capita (usdef)
and change in real per capita income (dy)
The data range for estimation is from the year 1953 to 1998.
Perform the unit root test (ADF test or PP test)
for the variables fygt10, fygt1, infl,
usdef and dy.
varsoc
fygt10
dfuller
fygt10,lags(1)
varsoc
D.fygt10
dfuller
D.fygt10,lags(0)
varsoc
fygt1
dfuller
fygt1,lags(3)
varsoc
D.fygt1
dfuller
D.fygt1,lags(2)
varsoc
usdef
dfuller
usdef,lags(1)
varsoc
D.usdef
dfuller
D.usdef,lags(0)
varsoc
dy
dfuller
dy,lags(0)
The unit root test show that the variables fygt10, fygt1 ,infl and usdef is non-stationary
in level but stationary in first difference, or \(I\left( 1 \right)\) while the variable dy is stationary in level form, or \(I\left( 0 \right)\).
OLS estimate of a model of long-term
interest rates for the year 1953 to 1998;
newey
fygt10 fygt1 infl usdef dy in 6/51, lag(3)
quietly
reg fygt10 fygt1 infl usdef dy in 6/51
estat
dwatson
If the variables from OLS estimation
above are cointegrated, the OLS coefficient estimates are consistent, but the
reported t-statistics
do not follow the
t-distribution.
To test the cointegration by
Engle-Granger test;
predict uhat,residual
dfuller uhat in
6/51,nocons
egranger fygt10 fygt1
infl usdef dy in 6/51
The DF test for disturbances show
that we reject the null of a stochastic trend in the disturbances.
The rejection of the null for DF
test or rejection of the null for the Engle-Granger test above indicate that
the variables fygt10, fygt1 ,infl usdef and dy is cointegrated.
Because the stochastically trending
variables in this model are cointegrated, the OLS estimates of their slopes are
consistent, but the associated t
-statistics
and
F-statistics
are not distributed according to t
-distribution
and F-distribution.
To obtain t
-statistics
that do follow the
t-distribution,
we re-estimate the model, augmenting it with simultaneous changes of the
stochastically trending explanators and two leads and two lags of these change;
newey
fygt10 fygt1 infl usdef dy L(-2/2)D.(fygt1 infl usdef) in 6/51,lag(3)
quietly
reg fygt10 fygt1 infl usdef dy L(-2/2)D.(fygt1 infl usdef)in 6/51
estat dwatson
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