In
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, an augmented Dickey–Fuller test (ADF) tests the
null hypothesis
The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
that a
unit root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...
is present in a
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
sample. The
alternative hypothesis
In statistical hypothesis testing, the alternative hypothesis is one of the proposed propositions in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting ...
depends on which version of the test is used, but is usually
stationarity or
trend-stationarity. It is an augmented version of the
Dickey–Fuller test for a larger and more complicated set of time series models.
The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence.
Testing procedure
The procedure for the ADF test is the same as for the
Dickey–Fuller test but it is applied to the model
:
where
is a constant,
the coefficient on a time trend and
the lag order of the autoregressive process. Imposing the constraints
and
corresponds to modelling a
random walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a rand ...
and using the constraint
corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous those of the
Dickey–Fuller test. (See that article for a discussion on dealing with uncertainty about including the intercept and deterministic time trend terms in the test equation.)
By including lags of the order ''p'', the ADF formulation allows for higher-order autoregressive processes. This means that the lag length ''p'' must be determined in order to use the test. One approach to doing this is to test down from high orders and examine the
''t''-values on coefficients. An alternative approach is to examine information criteria such as the
Akaike information criterion,
Bayesian information criterion or the
Hannan–Quinn information criterion.
The unit root test is then carried out under the null hypothesis
against the alternative hypothesis of
Once a value for the test statistic
:
is computed, it can be compared to the relevant critical value for the Dickey–Fuller test. As this test is asymmetric, we are only concerned with negative values of our test statistic
. If the calculated test statistic is less (more negative) than the critical value, then the null hypothesis of
is rejected and no unit root is present.
Intuition
The intuition behind the test is that if the series is characterised by a unit root process, then the lagged level of the series (
) will provide no relevant information in predicting the change in
besides the one obtained in the lagged changes (
). In this case, the
and null hypothesis is not rejected. In contrast, when the process has no unit root, it is stationary and hence exhibits reversion to the mean - so the lagged level will provide relevant information in predicting the change of the series and the null hypothesis of a unit root will be rejected.
Examples
A model that includes a constant and a time trend is estimated using sample of 50 observations and yields the
statistic of −4.57. This is more negative than the tabulated critical value of −3.50, so at the 95% level, the null hypothesis of a unit root will be rejected.
Alternatives
There are alternative
unit root test
In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either Stationary process, s ...
s such as the
Phillips–Perron test
In statistics, the Phillips–Perron test (named after Peter C. B. Phillips and Pierre Perron) is a unit root test. That is, it is used in time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) ...
(PP) or the
ADF-GLS test procedure (ERS) developed by Elliott, Rothenberg and Stock (1996).
Software implementations
*
R:
** package
forecast
function
ndiffs
handles multiple popular unit root tests
** package
tseries
function
adf.test
** package
fUnitRoots
function
adfTest
** package
urca
*
Gretl
gretl is an open-source statistical package, mainly for econometrics. The name is an acronym for ''G''nu ''R''egression, ''E''conometrics and ''T''ime-series ''L''ibrary.
It has both a graphical user interface (GUI) and a command-line interf ...
*
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
** the Econometrics Toolbox function
adfTest
** the Spatial Econometrics toolbox (free)
*
SAS PROC ARIMA
*
Stata command
dfuller
*
EViews the
Unit Root Test
*
Python
** package
statsmodels
function
adfuller
** package
ARCH
*
Java
Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
project
SuanShu
package
com.numericalmethod.suanshu.stats.test.timeseries.adf
class
AugmentedDickeyFuller
*
Julia package
HypothesisTests
''
'' function
ADFTest
See also
*
Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test
References
Further reading
*
*
{{DEFAULTSORT:Augmented Dickey-Fuller Test
Time series statistical tests