ADF: Reject or keep null hypothesis (difference p-value & test statistic)

Question

I'm working at my thesis and wanted to test my time series for Stationarity by using the Augmented Dickey Fuller Test with the program R. When I received the output, the p-value was close to zero, so that I had to reject the null-hypothesis of non-stationarity. But with a look at the test statistic, which was higher than the critical value of tau, I couldn't reject it. Now I am very confused, which result gives me the correct one (or if I made a mistake). I would assume that the p-value is the crucial one, because it is not least derived from the test statistic.

That is the extract of my output:

Residual standard error: 1.607 on 1585 degrees of freedom
Multiple R-squared:  0.06058,   Adjusted R-squared:  0.05347 
F-statistic: 8.518 on 12 and 1585 DF,  p-value: 7.398e-16


Value of test-statistic is: 0.0957 

Critical values for test statistics: 
  1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62

I would be really glad, if someone could explain this difference to me. Thanks in advance!

Answer 1

You need to examine the rest of the output to see the source of the test statistic.

The Augmented Dickey Fuller Test runs an regression of the first difference of the time series against a lag of the level values of the time series plus lagged first differences. The test statistic is based on the significance of the lagged level values, not the significance of the overall regression via the F-statistic.

The test statistic is the t-value of the lag of the level values of the time series. A rough guide to significance would be its associated p-value. However, the reported critical values at the end of the output are more appropriate. These are the adjusted critical values as calculated by MacKinnon for the ADF.

It looks like you are using the urca package of R. Here's is the extended output on a simulated monthly time series with annual autocorrelation and a unit root. I set the (max) lags to twice the frequency and let ur.df (correctly) select the number of lags using the AIC.

Note the test-statistic corresponds to the t-value.

> arima.sim(list(ar=c(rep(0,11),0.8),order=c(12,1,0)),1000)->x
> summary(ur.df(x,type="none",lags=24,selectlags="AIC"))

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

    Test regression none


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

    Residuals:
        Min      1Q  Median      3Q     Max
    -3.1882 -0.6450  0.0481  0.7121  3.4243

    Coefficients:
                   Estimate Std. Error t value Pr(>|t|)
    z.lag.1      -0.0004062  0.0006635  -0.612   0.5405  <-- test statistic is this t-value
    z.diff.lag1  -0.0147038  0.0202857  -0.725   0.4687
    z.diff.lag2  -0.0057840  0.0202965  -0.285   0.7757
    z.diff.lag3  -0.0284008  0.0203133  -1.398   0.1624
    z.diff.lag4  -0.0468366  0.0203374  -2.303   0.0215 *
    z.diff.lag5   0.0085345  0.0203880   0.419   0.6756
    z.diff.lag6   0.0153530  0.0203456   0.755   0.4507
    z.diff.lag7  -0.0340281  0.0203676  -1.671   0.0951 .
    z.diff.lag8  -0.0190015  0.0203950  -0.932   0.3517
    z.diff.lag9  -0.0012032  0.0203491  -0.059   0.9529
    z.diff.lag10 -0.0128488  0.0203783  -0.631   0.5285
    z.diff.lag11 -0.0080163  0.0203859  -0.393   0.6942
    z.diff.lag12  0.7818341  0.0203577  38.405   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

    Residual standard error: 1.025 on 963 degrees of freedom
    Multiple R-squared:  0.6271,    Adjusted R-squared:  0.622
    F-statistic: 124.6 on 13 and 963 DF,  p-value: < 2.2e-16

Test-statistic is from the first t-value above. Critical values are from MacKinnon.

    Value of test-statistic is: -0.6123

    Critical values for test statistics:
          1pct  5pct 10pct
    tau1 -2.58 - 1.95 - 1.62

The overall regression is highly significant as it picks up the AR(12) nature of the time series. However, the lagged level variable is not significant. Therefore, we would not reject the null hypothesis of a unit root under this test.