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I tested for trend in a time series using the Augmented Dickey-Fuller (ADF) test in R. I am having trouble interpreting the output below:

trend <- summary(ur.df(air_xt, lag = 0, type = 'trend'))
trend 

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

Test regression trend 

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

Residuals:
     Min       1Q   Median       3Q      Max 
-163.429   -0.018    0.021    0.293    8.521 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  5.857e-02  1.867e-03  31.376   <2e-16 ***
z.lag.1     -3.544e-04  9.636e-06 -36.778   <2e-16 ***
tt          -3.503e-10  2.206e-10  -1.588    0.112    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.342 on 7635596 degrees of freedom
Multiple R-squared:  0.0001771, Adjusted R-squared:  0.0001769 
F-statistic: 676.3 on 2 and 7635596 DF,  p-value: < 2.2e-16

Value of test-statistic is: -36.7776 450.8666 676.2999 

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

I am wondering if anyone can help me decide the acceptance of the $H_0$ hypothesis (i.e., no trend) or rejection of the $H_1$ hypothesis (i.e., trend).

Based on the following post: Interpreting R's ur.df (Dickey-Fuller unit root test) results, I understand that $H_0$ is rejected as the test-statistic (i.e., 450.8666) is outside the critical threshold values (phi2). But the $p$-value of tt is 0.112 (i.e., $p > 0.05$). The significance level indicates strong support for rejecting the alternative hypothesis.

Questions

(1) Does this contradict the results in my output? If not, can anyone help me understand it properly?

(2) Could someone please clear up my confusion as to why the $t$-value in the coefficients table for tt is -1.588 instead of 450.8666?

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  • $\begingroup$ t-values are not really important here. t-value is just the usual t-value reported in regression. However here the regression coefficients do not follow t-distribution under the null hypothesis, unlike standard linear regression. The critical values for this hypothesis are separately reported (the ones in the end). See this for more details $\endgroup$
    – Dayne
    Commented Dec 24, 2020 at 9:24
  • $\begingroup$ Hi, @Dayne, thanks a lot for your answer. I understand that t-values are not important, but what about the test statistic compared with the critical values? Why the significance test and test statistics results are contradictory? Any feedback is really appreciated. $\endgroup$
    – NewLearner
    Commented Dec 25, 2020 at 21:28
  • $\begingroup$ which significance test and test statistic are you referring to? The test statistic are reported towards the end and critical values for the three statistic. It appears all the test statistics are in the rejecting $H_0$ region, indicating that the series is stationary. Where is the contradiction? $\endgroup$
    – Dayne
    Commented Dec 28, 2020 at 9:41
  • $\begingroup$ Hi, $H_0$ is rejected as the test-statistic (i.e., 450.8666) is outside the critical threshold values (phi2). But the p-value of tt is 0.112 (i.e., p>0.05). This significance level indicates strong support for rejecting the alternative hypothesis according to the link I have given above. $\endgroup$
    – NewLearner
    Commented Dec 29, 2020 at 10:24
  • $\begingroup$ I am talking about the rejection of $H_0$, which refers to the unit root with drift and trend. If $H_0$ = phi2: $a_0$=$\gamma$=$a_2$=0, is rejected then one, two, OR all three of these terms was NOT zero. tau3 and phi3 are in the rejection region when comparing the test statistic with critical values and p-values found in the Coefficients table. $\endgroup$
    – NewLearner
    Commented Dec 29, 2020 at 10:24

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