Return to Answer

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following model of GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

If we subtract $y_{t-1}$ from both sides we get that

$\Delta y_{t}=\alpha+(\beta-1) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$. The test depends on whether we account for the constant ($\alpha$) and the linear trend ($\gamma$) or not. If the true process has a linear trend but $\beta<1$ then you will not detect a unit root when including a linear trend in the test. On the other hand if you estimate the model without a linear trend and conduct the unit root test without an exogenous linear trend then the unit root test should not reject the null since the estimate for $\beta$ will move towards 1 to try and capture the linear trend. For this same reason if you have large structural breaks in the data the estimate of $\beta$ will move towards 1 and suggest evidence of a unit root.

Long story short if you include the linear trend in the test even if you exclude it from the estimated regression model then you are unlikely to find evidence of a unit root unless the data were even more explosive than it appears (i.e. closer to I(2)).

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following model of GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

If we subtract $y_{t-1}$ from both sides we get that

$\Delta y_{t}=\alpha+(\beta-1) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$. The test depends on whether we account for the constant ($\alpha$) and the linear trend ($\gamma$) or not. If the true process has a linear trend but $\beta<1$ then you will not detect a unit root when including a linear trend in the test. On the other hand if you estimate the model without a linear trend and conduct the unit root test without an exogenous linear trend then the unit root test should not reject the null since the estimate for $\beta$ will move towards 1 to try and capture the linear trend. For this same reason if you have large structural breaks in the data the estimate of $\beta$ will move towards 1 and suggest evidence of a unit root (regardless of an inclusion or a linear trend or not).

Long story short if you include the linear trend in the test even if you exclude it from the estimated regression model then you are unlikely to find evidence of a unit root in an empirical process where a linear trend dominates unless the data were even more non-stationary than it appears.

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following process for GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

Now if we take the difference of this series we get that

$\Delta y_{t}=\alpha+(1-\beta) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$ conditional on the existence of the trend. To see what impact the linear trend has we can difference the series in another way as

$\Delta y_{t}=\beta \Delta y_{t-1}+\gamma + \Delta e_{t}$

So now the trend term is gone and $\beta$ captures the impact of the lagged change in the series. Now if $\beta=1$ then that implies that the series has a unit root in addition to the linear trend. From the picture it is fairly clear that $\beta \neq 1$ and so once you account for the linear trend the process is probably fairly stationary with the exception of a structural break in 2008/2009.

I hope that makes sense.

Note that you claim to have removed the linear trend in your test but the results that you posted clearly indicate that the linear trend is included in the test with a p-value of 1%.

If this is not the case, then it might be useful to share more or your results and or your data since there may be something happening in terms of outliers in the estimation which gives you these results.

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following model of GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

If we subtract $y_{t-1}$ from both sides we get that

$\Delta y_{t}=\alpha+(\beta-1) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$. The test depends on whether we account for the constant ($\alpha$) and the linear trend ($\gamma$) or not. If the true process has a linear trend but $\beta<1$ then you will not detect a unit root when including a linear trend in the test. On the other hand if you estimate the model without a linear trend and conduct the unit root test without an exogenous linear trend then the unit root test should not reject the null since the estimate for $\beta$ will move towards 1 to try and capture the linear trend. For this same reason if you have large structural breaks in the data the estimate of $\beta$ will move towards 1 and suggest evidence of a unit root.

Long story short if you include the linear trend in the test even if you exclude it from the estimated regression model then you are unlikely to find evidence of a unit root unless the data were even more explosive than it appears (i.e. closer to I(2)).

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following process for GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

Now if we take the difference of this series we get that

$\Delta y_{t}=\alpha+(1-\beta) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$ conditional on the existence of the trend. To see what impact the linear trend has we can difference the series in another way as

$\Delta y_{t}=\beta \Delta y_{t-1}+\gamma + \Delta e_{t}$

So now the trend term is gone and $\beta$ captures the impact of the lagged change in the series. Now if $\beta=1$ then that implies that the series has a unit root in addition to the linear trend. From the picture it is fairly clear that $\beta \neq 1$ and so once you account for the linear trend the process is probably fairly stationary with the exception of a structural break in 2008/2009.

I hope that makes sense.

Note that you claim to have removed the linear trend in your test but the results that you posted clearly indicate that the linear trend is included in the test with a p-value of 1%.

I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following process for GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

Now if we take the difference of this series we get that

$\Delta y_{t}=\alpha+(1-\beta) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$ conditional on the existence of the trend. To see what impact the linear trend has we can difference the series in another way as

$\Delta y_{t}=\beta \Delta y_{t-1}+\gamma + \Delta e_{t}$

So now the trend term is gone and $\beta$ captures the impact of the lagged change in the series. Now if $\beta=1$ then that implies that the series has a unit root in addition to the linear trend. From the picture it is fairly clear that $\beta \neq 1$ and so once you account for the linear trend the process is probably fairly stationary with the exception of a structural break in 2008/2009.

I hope that makes sense.

Note that you claim to have removed the linear trend in your test but the results that you posted clearly indicate that the linear trend is included in the test with a p-value of 1%.

If this is not the case, then it might be useful to share more or your results and or your data since there may be something happening in terms of outliers in the estimation which gives you these results.