Interpretation of results for unitroot test

Question

Let's say I have a pure random walk:

library(fUnitRoots)
library(urca)

set.seed(1130)
rndwlk1 <- filter(rnorm(1000,0,1),c(1),method="recursive",init=c(1))
plot(rndwlk1)

and I want perform a unitroot test with trend using ur.df, I would define my series as such:

$y_t = c + \delta t +\rho y_{t-1} + \epsilon_t $

summary(ur.df(rndwlk1,type=c("trend"),lags=0))

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

## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7054 -0.6353  0.0402  0.6363  2.9240 

## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -0.186081   0.077171  -2.411  0.01608 * 
## z.lag.1     -0.017740   0.005983  -2.965  0.00310 **
## tt          -0.000619   0.000233  -2.657  0.00802 **
## ---
## Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

## Residual standard error: 0.9726 on 996 degrees of freedom
## Multiple R-squared:  0.00875,    Adjusted R-squared:  0.00676 
## F-statistic: 4.396 on 2 and 996 DF,  p-value: 0.01256


## Value of test-statistic is: -2.9648 3.8345 4.3962 

## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2  6.09  4.68  4.03
## phi3  8.27  6.25  5.34

Can someone tell me what the test-statistics mean? As far as I understand the first one is the ADF-statistic for $\rho$ corresponds the -2.648 is the t-statiscics for $\rho$. What are the two other statistics?

The model with trend is not the right model, but I would like to test it against a model with drift. I have figured out that the F-statistics gives the results for the test of a unit root with drift. So:

H0 = $\rho$ = 1 , $\delta t$ = 0. But the p-value seems to be from a regular F distribution. How can get critical values for the F statistics?

Update

The p-value of the F-statistic is indeed from the regular F-distribution. pf(4.396,2,996,lower.tail=FALSE)

#0.01256662

Does anyone know where can we find the Dickey fuller critical values for the F-statistics on R?

Answer 1

First of all, the function ur.df will estimate the following equation:

$\Delta y_t = c+ (\rho-1)y_{t-1} + \delta t + \epsilon_t $

The value of the test-statistics:

## Value of test-statistic is: -2.9648 3.8345 4.3962 

représents the critical values for the following $H_0$.

|----------------+----------------------+--------------------|
|        -2.9648 |               3.8345 |             4.3962 |
|----------------+----------------------+--------------------|
| $(\rho-1) = 0$ | $\rho-1= c=\delta=0$ | $\rho-1= \delta=0$ |
|----------------+----------------------+--------------------|

• What are the two other statistics?

So the two other values are critical values for F-Test.

• How can get critical values for the F statistics?

The corresponding critical values are given in the table below the test statistics. the tau3, phi1 to phi3, are the notations DF used in their paper. I found the table and the explanation in Applied Economic Time Series, Walter Enders. Here is what the notations represent. Where $\tilde{\rho}$ = $\rho-1$

|--------+---------------------------------+-------------------------------------|
| <6>    | $H_0$                           | Interpretation                      |
|--------+---------------------------------+-------------------------------------|
| tau3   | $\tilde{\rho}= 0$               |                                     |
| phi1   | $\tilde{\rho}= 0 et c = 0$      | random walk without drift           |
| phi2   | $\tilde{\rho} = c = \delta = 0$ | random walk without drift and trend |
| phi3   | $\tilde{\rho} = \delta = 0$     | random walk without trend           |
|--------+---------------------------------+-------------------------------------|