# Connection of t-statistic and p-value in augmented Dickey-Fuller test

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

• ELA, or economists love acronyms. PPP = purchasing power parity. ADF = augmented Dickey-Fuller. (I'm not an economist.) – Nick Cox Jun 25 '13 at 16:08
• Isn't this question fully answered at stats.stackexchange.com/questions/31/…? – whuber Jun 25 '13 at 16:15
• @whuber Nothing there on unit roots? – Nick Cox Jun 25 '13 at 16:31
• @ whuber: not with respect to the unit root process – Metrics Jun 25 '13 at 16:39
• @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. – whuber Jun 25 '13 at 20:03

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.