Question

Describe a process to determine the number of lagged differences to use in an Augmented DF test.

Answer 1

Named for American statisticians David Dickey and Wayne Fuller, who developed the test in 1979, the Dickey-Fuller test is used to determine whether a unit root (a feature that can cause issues in statistical inference) is present in an autoregressive model. The formula is appropriate for trending time series like asset prices. It is the simplest approach to test for a unit root, but most economic and financial times series have a more complicated and dynamic structure than what can be captured by a simple autoregressive model, which is where the augmented Dickey-Fuller test comes into play.

Development
With a basic understanding of that underlying concept of the Dickey-Fuller test, it is not difficult to jump to the conclusion that an augmented Dickey-Fuller test (ADF) is just that: an augmented version of the original Dickey-Fuller test. In 1984, the very same statisticians expanded their basic autoregressive unit root test (the Dickey-Fuller test) to accommodate more complex models with unknown orders (the augmented Dickey-Fuller test).

Similar to the original Dickey-Fuller test, the augmented Dickey-Fuller test is one that tests for a unit root in a time series sample. The test is used in statistical research and econometrics, or the application of mathematics, statistics, and computer science to economic data.

The primary differentiator between the two tests is that the ADF is utilized for a larger and more complicated set of time series models. The augmented Dickey-Fuller statistic used in the ADF test is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root. Of course, this is only at some level of confidence. That is to say that if the ADF test statistic is positive, one can automatically decide not to reject the null hypothesis of a unit root. In one example, with three lags, a value of -3.17 constituted rejection at the p-value of .10.

A. Test on white noise
Let’s start by doing the test on data that we know are stationary, white noise. We will use an Augmented Dickey-Fuller test where we use the default number of lags (amount of time-dependency) in our test. For a time-series of 100, this is 4.

TT <- 100
wn <- rnorm(TT) # white noise
tseries::adf.test(wn)
Warning in tseries::adf.test(wn): p-value smaller than printed p-value

Augmented Dickey-Fuller Test

data: wn
Dickey-Fuller = -4.8309, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary
The null hypothesis is rejected.

Try a Dickey-Fuller test. This is testing with a null hypothesis of AR(1) stationarity versus a null hypothesis with AR(4) stationarity when we used the default k.

tseries::adf.test(wn, k = 0)
Warning in tseries::adf.test(wn, k = 0): p-value smaller than printed p-value

Augmented Dickey-Fuller Test

data: wn
Dickey-Fuller = -10.122, Lag order = 0, p-value = 0.01
alternative hypothesis: stationary
Notice that the test-statistic is smaller. This is a more restrictive test and we can reject the null with a higher significance level.