# 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.

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

Where $\Delta&space;y_{t}=y_{t}-y_{t-1}$ = 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, $\Delta&space;y_{t-i}$ = 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.

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:

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.

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