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The regression equation of dickey fuller test is defined as:

$$\Delta y_t = py_{t-1}+e_t$$

Trends and constants may also be added.The test statistic can be derived in the following way:

$$y_t=\beta y_{t-1}+ u_t \space \space \space \space\space | -y_{t-1}$$ $$\Delta y_t=(\beta-1)y_{t-1}+u_t$$

Where the $(\beta-1)$ is the p in the first equation.

Just one question, if you actually run the two different regressions. The coefficient of the first equation won't be the same as the coefficient of the 2nd+1. I am not sure why this is as the equations are exactly the same. Any thoughts? Perhaps the coefficients would be the same if that was actually the true population model. Given this, are there some shortcomings in the test when applying to empirical data?

(Same applies to Johansen test, Engle Granger, and others with the same principle).

Dole
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  • One issue you seem to be facing repeatedly (across different posts) is that algebraic equivalence between models (equivalence of different representations) does not necessarily yield equivalence of the estimated models. When you think about it, it is not too surprising. Some representations fit the assumptions needed for the estimation method to yield "good" results while some equivalent representations do not. – Richard Hardy Apr 25 '16 at 21:23
  • @RichardHardy Indeed! It's a bit annoying that the text books et cetera, never explain in what sort of way, if any, the algebraic relationship is supposed to hold. In this case it doesn't hold at least on empirical data (as I tested it). – Dole Apr 25 '16 at 21:23
  • Take $$y_i=\beta x_i+\varepsilon_i,$$ a model that can be estimated using OLS (take on any assumptions you need for that). Then exponentiate it to obtain $$exp(y_i)=exp(\beta x_i+\varepsilon_i),$$ and you will get something nasty where you suddenly need optimization routines (OLS matrix algebra is no longer applicable). Algebraic manipulations are often done to obtain a representation that is suitable for estimation with some nice method. – Richard Hardy Apr 25 '16 at 21:27
  • @RichardHardy Indeed, but unfortunately the effects of the algebraic manipulations are often not very far dwelled into. This case is a bit strange given there is no distortion of the error term, and yet still the results are completely different. – Dole Apr 25 '16 at 21:41
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    The assumption being violated here (the distortion) is not about the error term but about finite moments of the $X'X$ matrix. I had a [similar question](http://stats.stackexchange.com/questions/145864/estimation-of-unit-root-ar1-model-with-ols) before, check it out. – Richard Hardy Apr 25 '16 at 21:57

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