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After performing an ADF test with the function ur.df I get the result below.

My question is, why do I get two $t$-statistics?

I thought this test checks two $H_0$ hypotheses: The first one that there is a unit root and the second one the joint hypothesis that there is a unit root and there is no drift. I assume for the first one we need only one $t$-statistic, while for the latter an $F$-statistic as it is a joint hypothesis.

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
    Min      1Q  Median      3Q     Max 
-42.894  -5.809   0.153   6.279  72.989 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.221812   2.027465   1.096    0.273
z.lag.1     -0.002348   0.002314  -1.015    0.310
z.diff.lag  -0.008356   0.027735  -0.301    0.763

Residual standard error: 10.9 on 1301 degrees of freedom
Multiple R-squared:  0.0008888, Adjusted R-squared:  -0.0006471 
F-statistic: 0.5787 on 2 and 1301 DF,  p-value: 0.5608


Value of test-statistic is: -1.0147 0.7078 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78
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  • $\begingroup$ Your statements make sense, but as you see the output is not what you would like it to be. I checked the help file for the ur.df function -- no luck. I skimmed through a few time series textbooks (in some of them R examples are used) -- again, no luck. Perhaps I just missed it, or perhaps I looked at wrong sources. But so far I think you will have to do with individual hypotheses rather than a joint hypothesis... $\endgroup$ Commented May 18, 2015 at 19:54

1 Answer 1

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The different statistics are for different tests. With or without trend and with or without drift and unit root. I think if you write in R (using the ur.df function from the "urca" package),

t.d.u.MODEL <- ur.df(DEPVAR, lags=XX, type="trend")

you test both for trend, drift and unit root. I think the notation in the follwing command follows Enders "Applied Econometric Time Series" textbook.

t.d.u.MODEL@teststat   # gives you t-stats with labels of coefficents

t.d.u.MODEL@cval       # gives you critical values

EDIT: Here is an example on how you should interpret the two t-statistics (notice that you have to scroll in the "code-window").

   > d.u.pce <- ur.df(pce, lags=3, type="drift")
   > summary(d.u.pce)

   ############################################### 
   # Augmented Dickey-Fuller Test Unit Root Test # 
   ############################################### 

   Test regression drift 


   Call:
   lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

   Residuals:
       Min      1Q  Median      3Q     Max 
   -65.551  -9.446   1.567  12.744  34.394 

   Coefficients:
                Estimate Std. Error t value Pr(>|t|)  
   (Intercept) 17.173227  11.551492   1.487   0.1411  
   z.lag.1     -0.003154   0.004391  -0.718   0.4747  
   z.diff.lag1  0.115378   0.108963   1.059   0.2929  
   z.diff.lag2  0.125322   0.108807   1.152   0.2529  
   z.diff.lag3  0.233230   0.108960   2.141   0.0354 *
   ---
   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

   Residual standard error: 17.81 on 79 degrees of freedom
   Multiple R-squared:  0.1129, Adjusted R-squared:  0.06799 
   F-statistic: 2.514 on 4 and 79 DF,  p-value: 0.04809

   Value of test-statistic is: -0.7182 4.1235 

   Critical values for test statistics: 
   1pct  5pct 10pct
   tau2 -3.51 -2.89 -2.58
   phi1  6.70  4.71  3.86

   #gives the same statistics as above but with "paramamter names"
   > d.u.pce@teststat
            tau2     phi1
   statistic -0.7182212 4.123487

   > d.u.pce@cval
         1pct  5pct 10pct
   tau2 -3.51 -2.89 -2.58
   phi1  6.70  4.71  3.86

   #INTERPRETATION
   ## Reject the null-hypothesis of a unit root with drift if:
   #  Tau_2-stat (t) < Tau_2 (critical value)
   #  In our case: We  CANNOT reject the hypothesis that 
   #  there is a unit root with drift since -0.718 > -2.89

   ## Reject the null-hypothesis of a unit root with no drift if (i.e. single unit root test):
   #  Phi_1-stat (F) > Phi_1 (critical value)
   #  In our case: We CANNOT rejcect the null since 4.12 < 4.71
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  • $\begingroup$ How would you reconcile your answer with the fact that there are two $t$-values in the output shown in the original post? ur.df(..., type="trend") does not yield an $F$-statistic (with a non-standard null distribution) for a joint test for the null hypothesis {zero trend, no drift, a unit root} but rather yields two $t$-statistics. $\endgroup$ Commented May 19, 2015 at 17:39
  • 1
    $\begingroup$ I think @efi is not testing for a trend but is using the command, ur.df(...,type="drift"), hence he only gets two t-statistics. $\endgroup$
    – Cederlöf
    Commented May 20, 2015 at 20:04
  • $\begingroup$ That may be right. $\endgroup$ Commented May 20, 2015 at 20:24
  • $\begingroup$ @RichardHardy Do you know why the answer rejects the first hypothesis if tau2 < critical value, but reject the second if phi1 > critical value? $\endgroup$
    – Nox
    Commented Apr 27, 2017 at 14:48
  • 1
    $\begingroup$ @Nox, you normally reject something that is more extreme than the critical value. If you look at the progression of critical values for 10%, then 5% and then 1% you will realize that for tau2, the extemity is going to the left on the real line, while for phi1 it is going to the right. $\endgroup$ Commented Apr 27, 2017 at 15:21

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