Dynamic Ordinary Least Squares Estimator (DOLS) with Stata (Time Series)

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OLS is consistent, but our usual test statistics are no longer valid, except for one special case.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)

${{Y}_{t}}$,${{Z}_{1t}}$ and ${{Z}_{2t}}$ contain stochastic trends.

Rewrite the model as;

${{Y}_{t}}={{\beta }_{0}}+{{\beta }_{1}}{{Z}_{1t}}+{{\beta }_{2}}{{Z}_{2t}}+{{\beta }_{3}}{{X}_{t}}+{{\beta }_{4}}\Delta {{Z}_{1t}}+{{\beta }_{5}}\Delta {{Z}_{2t}}+{{u}_{t}}$       (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