Augmented Dickey-Fuller Test with R

Last Update: December 22, 2020

First order trend stationary time series consist of random processes that have constant mean which don’t exhibit trend pattern.

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Augmented Dickey-Fuller test [1] consists of evaluating whether time series was first order trend stationary with null hypothesis that it had a unit root and was not stationary.

1. Formula notation.

1.1. Augmented Dickey-Fuller test formula notation.

\Delta y_{t}=c+\beta t+\gamma y_{t-1}+\sum_{i=1}^{p}\delta_{i}\Delta y_{t-i}+e_{t}

Where \Delta y_{t}=y_{t}-y_{t-1} = current period asset prices difference, c = regression constant term, \beta,\gamma,\delta_{i} = regression coefficients, t = linear trend variable, y_{t-1} = previous period asset price, \Delta y_{t-i} = previous periods asset prices differences, p = number of lags included within test, e_{t} = regression residuals or forecasting errors.

1.2. Augmented Dickey-Fuller test formula notation constant and linear trend variable assumptions options.

i)\;c=0\;and\;\beta=0

ii)\;c\neq0\;and\;\beta=0

iii)\;c\neq0\;and\;\beta\neq0

Where c = regression constant term, \beta = linear trend variable regression coefficient.

1.3. Augmented Dickey-Fuller test.

Augmented Dickey-Fuller individual test \gamma coefficient t-statistic approximated p-value:

  • If Augmented Dickey-Fuller individual test \gamma coefficient t-statistic approximated p-value<\alpha level of statistical significance then time series was first order trend stationary with (1-\alpha) level of statistical confidence.
  • If Augmented Dickey-Fuller individual test \gamma coefficient t-statistic approximated p-value>\alpha level of statistical significance then higher differentiation order needed for first order trend stationary time series with (1-\alpha) level of statistical confidence.

2. R script code example.

2.1. Load R packages [2].

library('quantmod')
library('tseries')

2.2. Augmented Dickey-Fuller test data reading, training and testing ranges delimiting.

  • Data: MSCI® Germany index replicating ETF (ticker symbol: EWG) daily adjusted close prices (2007-2016).
  • Training and testing ranges delimiting not fixed and only included for educational purposes.
data <- read.csv('Augmented-Dickey-Fuller-Test-Data.txt',header=T)
data <- xts(data[,2],order.by=as.Date(data[,1]))
colnames(data) <- 'ger'
tdata <- data['::2014-12-31']
fdata <- data['2015-01-02::']

2.3. Augmented Dickey-Fuller test prices chart.

  • Augmented Dickey-Fuller test prices chart within training range.
tger <- tdata
plot(tger,main='tger Prices Chart')

2.4. Augmented Dickey-Fuller test calculation and output.

  • Augmented Dickey-Fuller test calculation within training range.
  • Augmented Dickey-Fuller test function includes constant and linear trend variable by default.
  • Augmented Dickey-Fuller test function alternative hypothesis and lag order to calculate test statistic not fixed and only included for educational purposes.
In:
adf.test(tger,alternative='stationary',k=1)
Out:
	Augmented Dickey-Fuller Test

data:  tger
Dickey-Fuller = -1.785, Lag order = 1, p-value = 0.6694
alternative hypothesis: stationary
3. References.

[1] David A. Dickey and Wayne A. Fuller. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root”. Journal of the American Statistical Association. 1979

[2] Jeffrey A. Ryan and Joshua M. Ulrich. “quantmod: Quantitative Financial Modelling Framework”. R package version 0.4-17. 2020.

Adrian Trapletti and Kurt Hornik. “tseries: Time Series Analysis and Computational Finance”. R package version 0.10-47. 2019.