In
statistics, the Dickey–Fuller test 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 an
autoregressive
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
time series model. The
alternative hypothesis is different depending on which version of the test is used, but is usually
stationarity or
trend-stationarity. The test is named after the
statisticians
David Dickey and
Wayne Fuller
Wayne Arthur Fuller (born June 15, 1931) is an American statistician who has specialised in econometrics, survey sampling and time series analysis. He was on the staff of Iowa State University from 1959, becoming a Distinguished Professor in 1983 ...
, who developed it in 1979.
Explanation
A simple
AR(1) model is
:
where
is the variable of interest,
is the time index,
is a coefficient, and
is the
error
An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'.
In statistics ...
term (assumed to be
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
). A unit root is present if
. The model would be non-stationary in this case.
The regression model can be written as
:
where
is the
first difference operator and
. This model can be estimated and testing for a unit root is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to testing
. Since the test is done over the residual term rather than raw data, it is not possible to use standard
t-distribution to provide critical values. Therefore, this
statistic has a specific
distribution simply known as the
Dickey–Fuller table.
There are three main versions of the test:
1. Test for a unit root:
::
2. Test for a unit root with constant:
::
3. Test for a unit root with constant and deterministic time trend:
::
Each version of the test has its own critical value which depends on the size of the sample. In each case, 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 ...
is that there is a unit root,
. The tests have low
statistical power in that they often cannot distinguish between true unit-root processes (
) and near unit-root processes (
is close to zero). This is called the "near observation equivalence" problem.
The intuition behind the test is as follows. If the series
is
stationary (or
trend-stationary), then it has a tendency to return to a constant (or deterministically trending) mean. Therefore, large values will tend to be followed by smaller values (negative changes), and small values by larger values (positive changes). Accordingly, the level of the series will be a significant predictor of next period's change, and will have a negative coefficient. If, on the other hand, the series is integrated, then positive changes and negative changes will occur with probabilities that do not depend on the current level of the series; in 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 ...
, where you are now does not affect which way you will go next.
It is notable that
::
may be rewritten as
::
with a deterministic trend coming from
and a stochastic intercept term coming from
, resulting in what is referred to as a ''stochastic trend''.
There is also an extension of the Dickey–Fuller (DF) test called the
augmented Dickey–Fuller test (ADF), which removes all the structural effects (autocorrelation) in the time series and then tests using the same procedure.
Dealing with uncertainty about including the intercept and deterministic time trend terms
Which of the three main versions of the test should be used is not a minor issue. The decision is important for the size of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is one) and the power of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is not one). Inappropriate exclusion of the intercept or deterministic time trend term leads to
bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
in the coefficient estimate for ''δ'', leading to the actual size for the unit root test not matching the reported one. If the time trend term is inappropriately excluded with the
term estimated, then the power of the unit root test can be substantially reduced as a trend may be captured through the
random walk with drift model. On the other hand, inappropriate inclusion of the intercept or time trend term reduces the power of the unit root test, and sometimes that reduced power can be substantial.
Use of prior knowledge about whether the intercept and deterministic time trend should be included is of course ideal but not always possible. When such prior knowledge is unavailable, various testing strategies (series of ordered tests) have been suggested, e.g. by Dolado, Jenkinson, and Sosvilla-Rivero (1990) and by Enders (2004), often with the ADF extension to remove autocorrelation. Elder and Kennedy (2001) present a simple testing strategy that avoids double and triple testing for the unit root that can occur with other testing strategies, and discusses how to use prior knowledge about the existence or not of long-run growth (or shrinkage) in ''y''. Hacker and Hatemi-J (2010) provide
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the s ...
results on these matters, including simulations covering the Enders (2004) and Elder and Kennedy (2001) unit-root testing strategies. Simulation results are presented in Hacker (2010) which indicate that using an
information criterion such as the
Schwarz information criterion may be useful in determining unit root and trend status within a Dickey–Fuller framework.
See also
*
KPSS test In econometrics, Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests are used for testing a null hypothesis that an observable time series is stationary around a deterministic trend (i.e. trend-stationary) against the alternative of a unit root.
...
*
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 ...
References
Further reading
*
*
External links
Statistical tables for unit-root tests– Dickey–Fuller table
How to do a Dickey-Fuller Test Using Excel
{{DEFAULTSORT:Dickey-Fuller test
Time series statistical tests