Interpretation of ur.df - "p-value"?

Question

When applying the "urca" package function ur.df, like

summary(ur.df(data$col1, type = c("none"), lags = 12, selectlags = c("AIC")))

I get following result:

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

Test regression trend 


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

Residuals:
      Min        1Q    Median        3Q       Max 
-12928366  -2888728   1284718   4218373   7179531 

Coefficients:
                 Estimate    Std. Error  t value  Pr(>|t|)   
(Intercept)  5.391984e+07  1.638362e+07  3.29108 0.0043123 **
z.lag.1     -2.438154e+00  7.557134e-01 -3.22629 0.0049588 **
tt           6.579260e+05  2.730453e+05  2.40959 0.0275861 * 
z.diff.lag1  1.712004e+00  6.595980e-01  2.59553 0.0188537 * 
z.diff.lag2  1.402824e+00  6.379412e-01  2.19899 0.0420083 * 
z.diff.lag3  1.321555e+00  5.294537e-01  2.49607 0.0231329 * 
z.diff.lag4  1.099430e+00  4.720412e-01  2.32910 0.0324428 * 
z.diff.lag5  8.132753e-01  4.181477e-01  1.94495 0.0685140 . 
z.diff.lag6  1.797331e-01  3.654326e-01  0.49184 0.6291254   
z.diff.lag7  5.890640e-01  2.939590e-01  2.00390 0.0612825 . 
z.diff.lag8  3.919041e-01  2.794371e-01  1.40248 0.1787705   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6708593 on 17 degrees of freedom
Multiple R-squared:  0.7237276, Adjusted R-squared:  0.5613144 
F-statistic: 4.253547 on 10 and 17 DF,  p-value: 0.003348755


Value of test-statistic is: -3.2263 3.9622 5.2635 

Critical values for test statistics: 
      1pct  5pct 10pct
tau3 -4.15 -3.50 -3.18
phi2  7.02  5.13  4.31
phi3  9.31  6.73  5.61

Now the question:

1. I do understand that "-3.2263" is the critical value (t-value)
2. There is a unit root with trend since -3.2263 > -3.18 (tau3@10pct) This means the time-series is non-stationary at a 10% significance level.
3. But, what is the meaning of "p-value: 0.003348755"? Should I list this value in a table summarizing my unit root test results or rather mark the 0.1 significance level (*10%)?

The documentation says that critical values are based on Hamilton (1994) and Dickey and Fuller (1981)".

Answer 1

1. -3.2263 is a test statistic tau3, not a critical value. The corresponding critical values are listed under "Critical values" (no wonder) in the row beginning with tau3.
2. At 95% confidence level you cannot reject the null hypothesis of presence of unit root -- since the test statistic is greater than the respective critical value, -3.2263 > -3.50.
   Meanwhile, at 90% confidence level you can reject the null hypothesis of presence of unit root -- since the test statistic is lesser than the respective critical value, -3.2263 < -3.18.
3. p-value: 0.003348755 is associated with the overall significance of the regression model used for testing for the presence of the unit root. It is not associated with the statistical significance of the tau3 statistic which is the relevant one in the unit root test. I am not sure if reporting it would be useful.