In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, an augmented Dickey–Fuller test (ADF) tests the
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
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 1 is ...
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. Exa ...
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
. The
alternative hypothesis
In statistical hypothesis testing, the alternative hypothesis is one of the proposed proposition in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting ...
is different depending 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
In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationa ...
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 testing procedure for the ADF test is the same as for the
Dickey–Fuller test
In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationa ...
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 is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
and using the constraint
corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous to the ones discussed on
Dickey–Fuller test
In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationa ...
(see that page 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'' has to be determined when applying the test. One possible approach 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
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to e ...
,
Bayesian information criterion
In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on ...
or the
Hannan–Quinn information criterion In statistics, the Hannan–Quinn information criterion (HQC) is a criterion for model selection. It is an alternative to Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is given as
: \mathrm = -2 L_ + 2 k \ln(\ln ...
.
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 asymmetrical, 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 percent 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 analysis to test the null hypothesis that a time series is integrated of order 1. It builds ...
(PP) or the
ADF-GLS test
In statistics and econometrics, the ADF-GLS test (or DF-GLS test) is a test for a unit root in an economic 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 ...
procedure (ERS) developed by Elliott, Rothenberg and Stock (1996).
Implementations in statistics packages
* In
R, there are various packages supplying implementations of the test. The ''forecast'' package includes a ''ndiffs'' function (which handles multiple popular unit root tests), the ''tseries'' package includes an ''adf.test'' function and the ''fUnitRoots'' package includes an ''adfTest'' function. A further implementation is supplied by the "urca" package.
*
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 interfa ...
includes the Augmented Dickey–Fuller test.
* In
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, implementation ...
, the ''adfTest'' function is part of the Econometrics Toolbox, and a free version is available as part of the 'Spatial Econometrics' toolbox
* In
SAS, ''PROC ARIMA'' can perform ADF tests.
* In
Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
, the ''dfuller'' command is used for ADF tests.
* In
EViews
EViews is a statistical package for Microsoft Windows, Windows, used mainly for time-series oriented econometrics, econometric analysis. It is developed by Quantitative Micro Software (QMS), now a part of IHS Inc., IHS. Version 1.0 was released ...
, the ''Augmented Dickey-Fuller'' is available under "Unit Root Test."
* In
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
, the ''adfuller'' function is available in the
Statsmodels
Statsmodels is a Python package that allows users to explore data, estimate statistical models, and perform statistical tests. An extensive list of descriptive statistics, statistical tests, plotting functions, and result statistics are available f ...
package and the ARCH package also provides an Augmented Dickey–Fuller test.
* In
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
, the ''AugmentedDickeyFuller'' class is included in ''SuanShu'' available under the ''com.numericalmethod.suanshu.stats.test.timeseries.adf'' package.
* In
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g. ...
, the ''ADFTest'' function is available in the ''HypothesisTests'' package.
See also
*
Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test
References
Further reading
*
*
{{DEFAULTSORT:Augmented Dickey-Fuller Test
Time series statistical tests