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My project is about purchasing power parity (PPP). I am checking whether the real exchange rate of Canadian dollar(CAD)/US dollar(USD), Japanese Yen(JPY)/USD and Great Britain Pound(GBP)/USD has a unit root with Augmented Dickey Fuller (ADF).

  • H0 is - there is a unit root for the series and real ex. rate will follow the random walk and is non-stationary
  • H1 is - there is no unit root and the real exchange rate is stationary and PPP holds.

I have created a diagram of the real exchange rates over the sample period, unfortunately I can not attach photos, due to the site regulations (I don't have 10 reputation yet) Anyways my JPY ex rate is highly volatile, fluctuates between 1.7 and 2.3 over the period; GBP is giving similar line, only it fluctuates between -0.3 and 0.1; CAD is the one that seems to have stationarity, it is relatively flat, moves between -0.1 and 0.3.

All the critical values are the same for all three results, and only t-statistic and the p-values are different. 1% -3.442; 5% -2.871; 10% -2.570

In Cad/USD t-stat is -1.568 and p-value 0.4997; in JPY/USD t-stat -2.551 and p-value 0.1036; in GBP/USD t-stat is -3.410 and p-value 0.0106

It seems that with the rise of the t-stat result, p-value decreases; moreover my H0 is rejected at 5 and 10% for GBP and is not rejected at 1%. I am really puzzled at how to give the interpretation for that. (Maybe it's because it fluctuates under zero?)

My supervisor wants me to explain what is the connection between t-stat and the p-value. and if possible could you please explain in simple English what is the unit root, mathematical explanations were of no help. I have also read the article Intuitive explanation of unit root already, unfortunately I still could not get the main idea of the unit root

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    $\begingroup$ ELA, or economists love acronyms. PPP = purchasing power parity. ADF = augmented Dickey-Fuller. (I'm not an economist.) $\endgroup$ – Nick Cox Jun 25 '13 at 16:08
  • $\begingroup$ Isn't this question fully answered at stats.stackexchange.com/questions/31/…? $\endgroup$ – whuber Jun 25 '13 at 16:15
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    $\begingroup$ @whuber Nothing there on unit roots? $\endgroup$ – Nick Cox Jun 25 '13 at 16:31
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    $\begingroup$ @ whuber: not with respect to the unit root process $\endgroup$ – Metrics Jun 25 '13 at 16:39
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    $\begingroup$ @Metric Your contribution is appreciated, do not doubt that. But as far as SE policy goes, when you have a new answer to offer to an existing question, then the existing question is the place to post it. The rest of this question IMHO is explicitly one about $t$ vs. $p$, as is evident from the penultimate paragraph (why are $t$ and $p$ inversely related?). The only reason it hasn't been closed as a duplicate yet is that your interpretation indeed offers a different take on the situation. If it turns out to be an acceptable answer, that will be sufficient reason to keep this thread open. $\endgroup$ – whuber Jun 25 '13 at 20:03
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Answer part 1:

Example from using Stata available here but the interpretation is similar for the results obtained using other software.

a) Models including constant and trend: For example, using 1 lag in the chicken series, you will have the following result

dfuller  chic, regress trend lags(1)

Augmented Dickey-Fuller test for unit root         Number of obs   =        52

                               ---------- Interpolated Dickey-Fuller --------- 
                  Test         1% Critical       5% Critical      10% Critical 
               Statistic           Value             Value             Value 
------------------------------------------------------------------------------ 
 Z(t)             -1.998            -4.146            -3.498            -3.179 
------------------------------------------------------------------------------ 
* MacKinnon approximate p-value for Z(t) = 0.6030

------------------------------------------------------------------------------ 
D.chic   |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval] 
---------+-------------------------------------------------------------------- 
chic     | 
      L1 |  -.1820551   .0911164     -1.998   0.051       -.365257    .0011467 
      LD |  -.0861985   .1435294     -0.601   0.551      -.3747837    .2023867 
_trend   |  -315.6405   266.9686     -1.182   0.243      -852.4168    221.1358 
_cons    |   83287.07   42600.86      1.955   0.056      -2367.711    168941.8 
------------------------------------------------------------------------------ 

Stata runs the regression in first difference.

H0: The coefficient on lagged chic is 0 . In other words, there is a presence of unit root. 
Ha: The coefficient on lagged chic is less than 0. In other words, there is no presence of unit root. (Note that this is a one-tailed test.) 

Results: The augmented Dickey-Fuller statistic is -1.998 which is greater than the critical value (-3.179). Therefore, we cannot reject the presence of unit root which is confirmed by the MacKinnon approximate p-value for Z(t) = 0.6030.

Remarks: The usual p-value (next to t value) doesn't make any sense here. It is the value if we have t distribution. Under the unit root process, we have Tau distribution. So, we have to compare computed t value on lagged chic with the critical values (given above) which were computed using the Tau distribution (these are generally reported by software but they are available in standard text book on time series, e.g., Hamilton(1994)). The relevant p-value obtained from Tau distribution is MacKinnon approximate p-value provided at the bottom.

Answer part 2

In simple English (and using an example of economics), suppose there is a negative shock (say war), then if GDP is a unit root process (also called stochastic trend process), then the war will permanently lower GDP levels in the long run. There is a nice diagram here on this. Note that without understanding the basic mathematical derivation of unit root, it is impossible to interpret the results obtained from econometric software.

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  • $\begingroup$ thank you a lot for clarifying that usual p-value and MacKinnon approximate p-value are not the same. $\endgroup$ – Fazliddin Jun 26 '13 at 10:08
  • $\begingroup$ I have come across with the definition on the connection of t-stat and p-value "More extreme test-statistics will return lower p-values giving greater indication that the null hypothesis is false." So does it mean, that t-stat -1.568 and p-value 0.4997; t-stat -2.551 and p-value 0.1036; i-3.410 and p-value 0.0106 indicate that the H0 is (or is not) rejected with "p-value" of certainty? Or shall I interpret it as "probability of observing at least the same t-stat assuming the H0 is true?" and will that mean that if the series has a unit root, the % of getting the same t-stat is equal p-value? $\endgroup$ – Fazliddin Jun 26 '13 at 10:23
  • $\begingroup$ By the way, I am grateful for the explanation of the unit root process too, I have come across with the definition that "Having a unit root in a series mean that there is more than one trend in the series." (Torres-Reyna, (2012) Time Series v. 1.5 Princeton University, Data and Statistical Services, Available at dss.princeton.edu/training/TS101.pdf) It makes a perfect sense to me, can you please comment on that too? $\endgroup$ – Fazliddin Jun 26 '13 at 10:26
  • $\begingroup$ @Fazliddin: Please re-read the answer before asking the next (but the same original) question. If p<0.10, null is rejected at 10% which means there is no unit root(remember null is unit root).I assume that p you are using is based on MacKinnon critical value. Your second comment should appear as the new question. $\endgroup$ – Metrics Jun 26 '13 at 13:34
  • $\begingroup$ thank you for your answer, I do appreciate your effort. Yes I am using MacKinnon approximate p-value provided by ADF itself. Could you please tell me what I should do when my test is giving t-stat -3.410, which is less then critical values 5 and 10% (-2.871 and -2.570), but more than 1% value -3.442. Shall I reject the null at 5 and 10%, and not reject in 1%? Will unit root still exist in this example, where the MacKinnon approximate p-value is 0.0106? $\endgroup$ – Fazliddin Jun 26 '13 at 13:58

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