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I want to prove that my data is non-stationary at level, but stationary after first differencing. I am trying to do this with the ur.df() function in R, but I am a little confused about the results

Since the goal is to do multiple linear regression with several independent variables (and no lags), I did lag = 0

summary(ur.df(stresstest$U.S._Mortgage_rate_Base, lag = 0))

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

Test regression none 


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

Residuals:
     Min       1Q   Median       3Q      Max 
-0.73167 -0.22969  0.06761  0.17445  0.76329 

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
z.lag.1 0.007198   0.008796   0.818    0.417

Residual standard error: 0.279 on 47 degrees of freedom
Multiple R-squared:  0.01405,   Adjusted R-squared:  -0.006932 
F-statistic: 0.6696 on 1 and 47 DF,  p-value: 0.4173


Value of test-statistic is: 0.8183 

Critical values for test statistics: 
      1pct  5pct 10pct
tau1 -2.62 -1.95 -1.61

Do I need to look at the p-value here, or the test statistic? I interpreted this as the data has a unit root, because the p-value is greater than 0.05.

I then redid the analysis with the first differences:

summary(ur.df(diff(stresstest$U.S._Mortgage_rate_Base), lag = 0))

In this case, the p-value was less than 0.05 so the data is stationary. Therefore, the data is I(1) stationary

Is this a correct interpretation?

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The lag = 0 is the Dickey-Fueller test when the lag > 0 it's called Augmented DF.

Why we do have DF or ADF ? It's because if you have autocorrelated errors in zt = zt-1 + ut then you can't trust the result of your DF so to correct autocorrelation you do an augmentation of the lag until the ut are not correlated.

Now not considering this aspect but only your results, you did interpret well, the golden rule is to know your H0, here unit root presence and to always reject it when p < 0.05 for alpha = 0.05.

So here you had indeed an I(1) series because the yt - yt-1 series is now I(0).

It's always important to verify that the differenced series is I(0) because the ADF or DF test always state first that the series is AT LEAST I(1) but can be I(2), I(3), etc... but that is fairly rare.

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