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I have a time series and I want to check whether it has a unit root or not. I want to use the Dickey-Fuller test. How should I choose the regression equation from the three options: regression without constant and trend, with constant (drift) and with constant and trend?

Is there a procedure that I should follow to select the regression? On what criteria is the choce of regression is based?

If the errors of the chosen regression are correlated then should I run the augmentd DF test with the same regression chosen in the first step?

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Including a trend and drift term when they are not necessary reduce the power of the test---that is, its ability to reject the null hypothesis of non-stationarity (i.e., the null of a unit root in the time series). Contrarily, the test is biased when these parameters are needed, but missing.

In economics, we typically don't worry about the trend term, which would imply a trend that was quadratic in time in our variable of interest. Drift implies a linear trend and is commonly incorporated.

You may plot a time series of your variable and look at the pattern to see if a trend is noticeable. A basic linear regression of the variable on a linear time trend may give you an idea of whether there is a linear trend as well (of course, you shouldn't pay attention to official hypothesis tests here because serial corrleation/non-stationarities could be biasing your results). Using a spline may also indicate whether there is a linear or quadratic trend in the variable. These visual cues are often good indicators of how you should conduct your Dickey-Fuller test.

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  • $\begingroup$ If I understand your answer correctly, we choose a Dickey-Fuller equation to match the form of the equation that appears to be underlying/"generating" our data, correct? For example, if my data appears to have drift and trend terms, I should use the DF equation with drift/trend terms equation #3 because that closely matches my underlying model, even though the equation on Wikipedia uses the first difference? $\endgroup$ – Ricardo Altamirano Nov 30 '12 at 1:09
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    $\begingroup$ The DF test always uses first differences as the outcome. $\endgroup$ – Charlie Nov 30 '12 at 1:34
  • $\begingroup$ So I need to choose a DF test based on the form of the first difference of the equation that (by assumption or other analyses) underlies/generates my time series, not the form of the underlying equation/model itself? $\endgroup$ – Ricardo Altamirano Nov 30 '12 at 1:36
  • $\begingroup$ Write down a model for $y_t$, which likely includes $\rho y_{t-1}$ as a predictor. Subtract $y_{t-1}$ from both sides, giving the first difference as your outcome and $(\rho - 1) y_{t-1}$ as a predictor. Use this model for your DF test of $\rho = 1$ or, more directly, of the coefficient$(\rho - 1) = 0$. $\endgroup$ – Charlie Nov 30 '12 at 14:40
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    $\begingroup$ @MichaelPerdue, the difference between two periods is basically the derivative with respect to time. If the derivative of a function is linear in $t$ (a linear time trend), then the function itself must be quadratic in $t$. In discrete time, you have $\Delta y_t = \beta[t^2 - (t-1)^2] = \beta[(t+t-1)(t-t+1)] \approx 2\beta t$. $\endgroup$ – Charlie Jul 19 '16 at 16:55
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Charlie's suggestion to use other information to help determine what deterministic components are included is good. I would add that theoretical considerations might suggest appropriate deterministic regressors.

Others have also suggested procedures for testing for a unit root that incorporate testing for the presence of deterministic regressors too. Enders "Applied Econometric Time Series" 2ed p213 has one such approach. I suspect there are others. Enders starts with a general formulation, tests for a unit root, if a unit root then tests for significance of the time trend, if time trend is not significant then tests for a unit root in a formulation without time trend and so on.

In any such procedure some caution is needed: 1 Critical values used depend on whether the test can assume a normal distribution or not. 2 Final results are effectively a result of a sequence of pretests, and each result is conditioned on the previous tests being correct. So true significance levels are difficult (impossible?) to work out. 3 Serial correlation should be addressed at each testing stage. Otherwise the test results generated at each stage may be misleading and ultimately give a misleading final result.

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There is a formal procedure to test for Unit Roots, when the true data-generating process is completely unknown. Enders mentions this in Appendix 4.2, where there is also a flowchart explaining the necessary steps. Alternatively, you could look at the underlying publication by Dolado, Jenkinson, and Sosvilla-Rivero (1990).

To summarize their approach, you would start at equation 3. If $\gamma=0$ is rejected, conclude there is no unit root. Otherwise, continue by estimating equation 2 and 3, until you find the true specification.

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