# Unit roots and order of differencing

I'm studying the stationarity with unit root tests and the order of integration in time series $\ln(x)$ and $\ln(y)$ found here. I'm using Dickey-Fuller test with constant but no trend.

From what I understand, the null hypothesis for ADF test is that there is a unit root present (non-stationary, random walk) and $I(d)$ process is stationary after differenced $d$ times. I tried the test for my data:

    df <- read.table(file="ts.txt", header=TRUE, sep="\t")
x <- as.ts(log(df$x)) #ln(x) y <- as.ts(log(df$y)) #ln(y)

testx <- ur.df(x,type="drift",lags=1) #drift should add constant but no trend right?
summary(textx)

testy <- ur.df(y,type="drift",lags=1)
summary(texty)


What I get for $\ln(x)$

    Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.040945   0.018828   2.175   0.0300 *
z.lag.1     -0.008988   0.004265  -2.107   0.0355 *
z.diff.lag   0.281566   0.037438   7.521  1.8e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.06311 on 656 degrees of freedom
Multiple R-squared:  0.08326,   Adjusted R-squared:  0.08047
F-statistic: 29.79 on 2 and 656 DF,  p-value: 4.133e-13

Value of test-statistic is: -2.1074 2.434

Critical values for test statistics:
1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


And for $\ln(y)$

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.032859   0.016654   1.973   0.0489 *
z.lag.1     -0.007538   0.003989  -1.890   0.0592 .
z.diff.lag   0.288379   0.037367   7.717 4.44e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.06192 on 656 degrees of freedom
Multiple R-squared:  0.08604,   Adjusted R-squared:  0.08325
F-statistic: 30.88 on 2 and 656 DF,  p-value: 1.53e-13

Value of test-statistic is: -1.8899 2.0388

Critical values for test statistics:
1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


What values should I be looking when rejecting/accepting $H_0$? And how can I find the order of integration in this case?

• So $ln(y)$ process IS stationary because $-1.8899<-2.86$? And for that one I cannot find order of difference (the order is 0 basically). – ELEC Apr 1 '15 at 19:19