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  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.
Where = current period asset prices difference, = regression constant term, = regression coefficients, = linear trend variable, = previous period asset price, = previous periods asset prices differences, = number of lags included within test, = regression residuals or forecasting errors.
1.2. Augmented Dickey-Fuller test formula notation constant and linear trend variable assumptions options.
Where = regression constant term, = linear trend variable regression coefficient.
1.3. Augmented Dickey-Fuller test.
Augmented Dickey-Fuller individual test coefficient t-statistic approximated :
- If Augmented Dickey-Fuller individual test coefficient t-statistic approximated level of statistical significance then time series was first order trend stationary with level of statistical confidence.
- If Augmented Dickey-Fuller individual test coefficient t-statistic approximated level of statistical significance then higher differentiation order needed for first order trend stationary time series with level of statistical confidence.
2. R script code example.
2.1. Load R packages .
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.
Out: Augmented Dickey-Fuller Test data: tger Dickey-Fuller = -1.785, Lag order = 1, p-value = 0.6694 alternative hypothesis: stationary
 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
 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.