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| In [[statistics]], the '''Dickey–Fuller test''' tests whether a [[unit root]] is present in an [[autoregressive]] model. It is named after the [[statistician]]s [[D. A. Dickey]] and [[Wayne Arthur Fuller|W. A. Fuller]], who developed the test in 1979.<ref>{{cite journal |last=Dickey |first=D. A. |first2=W. A. |last2=Fuller |year=1979 |jstor=2286348 |title=Distribution of the Estimators for Autoregressive Time Series with a Unit Root |journal=[[Journal of the American Statistical Association]] |volume=74 |issue=366 |pages=427–431 }}</ref>
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| ==Explanation==
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| A simple [[Autoregressive|AR]](1) model is
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| : <math>y_{t}=\rho y_{t-1}+u_{t}\,</math>
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| where ''y''<sub>''t''</sup> is the variable of interest, ''t'' is the time index, ''ρ'' is a coefficient, and ''u''<sub>''t''</sub> is the [[errors and residuals in statistics|error]] term. A unit root is present if ''ρ'' = 1. The model would be non-stationary in this case.
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| The regression model can be written as
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| : <math> \nabla y_{t}=(\rho-1)y_{t-1}+u_{t}=\delta y_{t-1}+u_{t}\,</math>
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| where ∇ is the [[Finite difference|first difference operator]]. This model can be estimated and testing for a unit root is [[Equivalence relation|equivalent]] to testing ''δ'' = 0 (where ''δ'' = ''ρ'' − 1). Since the test is done over the residual term rather than raw data, it is not possible to use standard [[Student's t-distribution|t-distribution]] to provide critical values. Therefore this [[statistic]] ''τ'' has a specific [[Probability distribution|distribution]] simply known as the [[Dickey–Fuller distribution|Dickey–Fuller table]].
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| There are three main versions of the test:
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| 1. Test for a unit root:
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| :: <math>\nabla y_t =\delta y_{t-1}+u_t \,</math>
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| 2. Test for a unit root with drift:
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| :: <math>\nabla y_t =a_0+\delta y_{t-1}+u_t \,</math>
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| 3. Test for a unit root with drift and deterministic time trend:
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| :: <math>\nabla y_t = a_0+a_1t+\delta y_{t-1}+u_t \,</math>
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| Each version of the test has its own critical value which depends on the size of the sample. In each case, the [[null hypothesis]] is that there is a unit root, ''δ'' = 0. The tests have low [[statistical power]] in that they often cannot distinguish between true unit-root processes (''δ'' = 0)and near unit-root processes (''δ'' is close to zero). This is called the "near observation equivalence" problem.
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| The intuition behind the test is as follows. If the series ''y'' is [[stationary process|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]], where you are now does not affect which way you will go next.
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| It is notable that
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| :: <math>\nabla y_t =a_0 + u_t \,</math>
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| may be rewritten as
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| :: <math>y_t = y_0 + \sum_{i=1}^t u_i + a_0t </math>
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| with a deterministic trend coming from <math>a_0t</math> and a stochastic intercept term coming from <math>y_0 + \sum_{i=1}^t u_i</math>, resulting in what is referred to as a ''stochastic trend''.<ref>{{cite book |last=Enders |first=W. |year=2004 |title=Applied Econometric Time Series |edition=Second |publisher=John Wiley & Sons |location=Hoboken |isbn=0471230650 }}</ref>
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| 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.
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| ==Dealing with uncertainty about including the intercept and deterministic time trend terms==
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| 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 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 <math>a_0 </math> 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.<ref>{{cite journal |last=Campbell |first=J. Y. |last2=Perron |first2=P. |year=1991 |title=Pitfalls and Opportunities: What Macroeconomists Should Know about Unit Roots |journal=NBER Macroeconomics Annual |volume=6 |issue=1 |pages=141–201 |jstor=3585053 }}</ref> 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.
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| 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)<ref>{{cite journal |last=Dolado |first=J. J. |last2=Jenkinson |first2=T. |last3=Sosvilla-Rivero |first3=S. |year=1990 |title=Cointegration and Unit Roots |journal=[[Journal of Economic Surveys]] |volume=4 |issue=3 |pages=249–273 |doi=10.1111/j.1467-6419.1990.tb00088.x }}</ref> 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''.<ref>{{cite journal |last=Elder |first=J. |last2=Kennedy |first2=P. E. |year=2001 |title=Testing for Unit Roots: What Should Students Be Taught? |journal=[[Journal of Economic Education]] |volume=32 |issue=2 |pages=137–146 |doi=10.1080/00220480109595179 }}</ref> Hacker and Hatemi-J (2010) provide simulation results on these matters,<ref>{{cite paper |last=Hacker |first=R. S. |last2=Hatemi-J |first2=A. |year=2010 |url=http://ideas.repec.org/p/hhs/cesisp/0214.html |title=The Properties of Procedures Dealing with Uncertainty about Intercept and Deterministic Trend in Unit Root Testing |work=CESIS Electronic Working Paper Series, Paper No. 214 |publisher=Centre of Excellence for Science and Innovation Studies, The Royal Institute of Technology, Stockholm, Sweden }}</ref> 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.<ref>{{cite paper |last=Hacker |first=R. S. |year=2010 |url=http://cesis.abe.kth.se/documents/CESISWP213.pdf |title=The Effectiveness of Information Criteria in Determining Unit Root and Trend Status |work=CESIS Electronic Working Paper Series, Paper No. 213 |publisher=Centre of Excellence for Science and Innovation Studies, The Royal Institute of Technology, Stockholm, Sweden }}</ref>
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| ==See also==
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| * [[Augmented Dickey–Fuller test]]
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| * [[Phillips–Perron test]]
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| * [[Unit root]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.economics.utoronto.ca/jfloyd/stats/statabs.ps Statistical tables for unit-root tests] – Dickey–Fuller table
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| * [http://www.scribd.com/doc/80877200/How-to-do-a-Dickey-Fuller-Test-using-Excel How to do a Dickey-Fuller Test Using Excel]
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| {{DEFAULTSORT:Dickey-Fuller test}}
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| [[Category:Time series analysis]]
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| [[Category:Statistical tests]]
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Nice to satisfy you, my title is Refugia. My working day occupation is a meter reader. To do aerobics is a factor that I'm totally addicted to. South Dakota is where I've usually been living.
Also visit my webpage ... at home std testing