libopenturns-dev 1.7-3 / usr / include / openturns / swig / DickeyFullerTest_doc.i

%feature("docstring") OT::DickeyFullerTest
"The Dickey-Fuller stationarity test.

Notes
-----
The Dickey-Fuller test checks the stationarity of a scalar time series using one time series. It assumes that the :math:`X: \\\\Omega \\\\times \\\\cD \\\\rightarrow \\\\Rset` process with :math:`\\\\cD \\\\in \\\\Rset`, discretized on the time grid :math:`(t_0, \\\\dots, t_{N-1})` writes:

.. math::
    :label: DFmodel

    X_t = a + bt + \\\\rho X_{t-1} + \\\\varepsilon_{t}

where :math:`\\\\rho > 0` and where :math:`a` or :math:`b` or both :math:`(a,b)` can be assumed to be equal to 0.

The Dickey-Fuller test checks whether the random perturbation at time :math:`t` vanishes with time.

When :math:`a \\\\neq 0` and :math:`b=0`, the model :eq:`DFmodel` is said to have a *drift*. When :math:`a = 0` and :math:`b \\\\neq 0`, the model :eq:`DFmodel` is said to have a *linear trend*.

In the model :eq:`DFmodel`, the only way to have stochastic non stationarity is to have :math:`\\\\rho = 1` (if :math:`\\\\rho > 1`, then the process diverges with time which is readily seen in the data). In the general case, the Dickey-Fuller test is a unit root test to detect whether :math:`\\\\rho=1` against :math:`\\\\rho < 1`:

The test statistics and its limit distribution depend on the a priori knowledge we have on :math:`a` and :math:`b`. In case of absence of a priori knowledge on the structure of the model, several authors have proposed a global strategy to cover all the subcases of the model :eq:`DFmodel`, depending on the possible values on :math:`a` and :math:`b`. 

The strategy implemented in OpenTURNS, is recommended by Enders (*Applied Econometric Times Series*, Enders, W., second edition, John Wiley \\\\& sons editions, 2004.).



We note :math:`(X_1, \\\\hdots, X_n)` the data, by :math:`W(r)` the Wiener process, and :math:`W^{a}(r) = W(r) - \\\\int_{0}^{1} W(r) dr`, :math:`W^{b}(r) = W^{a}(r) - 12 \\\\left(r - \\\\frac{1}{2} \\\\right) \\\\int_{0}^{1} \\\\left(s - \\\\frac{1}{2} \\\\right) W(s) ds`.


**1.** We assume the model :eq:`Model1`:

.. math::
    :label: Model1

    \\\\boldsymbol{X_t = a + bt + \\\\rho X_{t-1} + \\\\varepsilon_{t}}

The coefficients :math:`(a,b,\\\\rho)` are estimated by :math:`(\\\\Hat{a}_n, \\\\Hat{b}_n, \\\\Hat{\\\\rho}_n)` using ordinary least-squares fitting, which leads to:

.. math::
    :label: Model1Estim

    \\\\underbrace{\\\\left(
       \\\\begin{array}{lll}
         \\\\displaystyle n-1 &\\\\sum_{i=1}^n t_{i} &\\\\sum_{i=2}^n y_{i-1}\\\\\\\\
         \\\\displaystyle \\\\sum_{i=1}^n t_{i} &\\\\sum_{i=1}^n t_{i}^2 &\\\\sum_{i=2}^n t_{i} y_{i-1}\\\\\\\\
         \\\\displaystyle \\\\sum_{i=2}^n y_{i-1}& \\\\sum_{i=2}^n t_{i}y_{i-1} &\\\\sum_{i=2}^n y_{i-1}^2
       \\\\end{array}
       \\\\right)}_{\\\\mat{M}}
     \\\\left(
       \\\\begin{array}{c}
        \\\\hat{a}_n\\\\\\\\
        \\\\hat{b}_n\\\\\\\\
        \\\\hat{\\\\rho}_n
       \\\\end{array}
     \\\\right)=
     \\\\left(
     \\\\begin{array}{l}
       \\\\displaystyle \\\\sum_{i=1}^n y_{i} \\\\\\\\
       \\\\displaystyle \\\\sum_{i=1}^n t_{i} y_{i}\\\\\\\\
       \\\\displaystyle \\\\sum_{i=2}^n y_{i-1} y_{i}
     \\\\end{array}
     \\\\right)


We first test:

.. math::
    :label: TestModel1

    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\cH_0: & \\\\rho = 1 \\\\\\\\
      \\\\cH_1: & \\\\rho < 1
    \\\\end{array}
    \\\\right.

thanks to the Student statistics:

.. math::

    t_{\\\\rho=1} = \\\\frac{\\\\rho_n-1}{\\\\hat{\\\\sigma}_{\\\\rho_n}}

where :math:`\\\\sigma_{\\\\rho_n}` is the least square estimate of the standard deviation of :math:`\\\\Hat{\\\\rho}_n`, given by:

.. math::

    \\\\sigma_{\\\\rho_n}=\\\\mat{M}^{-1}_{33}\\\\sqrt{\\\\frac{1}{n-1}\\\\sum_{i=2}^n\\\\left(y_{i}-(\\\\hat{a}_n+\\\\hat{b}_nt_i+\\\\hat{\\\\rho}_ny_{i-1})\\\\right)^2}


which converges in distribution to the Dickey-Fuller distribution associated to the model with drift and trend:

.. math::

    t_{\\\\rho = 1} \\\\stackrel{\\\\mathcal{L}}{\\\\longrightarrow} \\\\frac{\\\\int_{0}^{1}W^{b}(r) dW(r)}{\\\\int_{1}^{0} W^{b}(r)^2 dr}

The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel1` is accepted when :math:`t_{\\\\rho=1} > C_{\\\\alpha}` where :math:`C_{\\\\alpha}` is the test threshold of level :math:`\\\\alpha`.

The quantiles of the Dickey-Fuller statistics for the model with drift and linear trend are:

.. math::
    
    \\\\left\\\\{
    \\\\begin{array}{ll}
        \\\\alpha = 0.01, & C_{\\\\alpha} = -3.96 \\\\\\\\
        \\\\alpha = 0.05, & C_{\\\\alpha} = -3.41 \\\\\\\\
        \\\\alpha = 0.10, & C_{\\\\alpha} = -3.13
    \\\\end{array}
    \\\\right.


**1.1. Case 1:** The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel1` is rejected

We test whether :math:`b=0`:

.. math::
    :label: TestSousModele1_1

    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\cH_0: & b = 0 \\\\\\\\
      \\\\cH_1: & b \\\\neq 0
    \\\\end{array}
    \\\\right.

where the statistics :math:`t_n = \\\\frac{|\\\\hat{b}_n|}{\\\\sigma_{b_n}}` converges in distribution to the Student distribution :class:`~openturns.Student` with :math:`\\\\nu=n-4`, where :math:`\\\\sigma_{b_n}` is the least square estimate of the standard deviation of :math:`\\\\Hat{b}_n`, given by:

.. math::

    \\\\sigma_{b_n}=\\\\mat{M}^{-1}_{22}\\\\sqrt{\\\\frac{1}{n-1}\\\\sum_{i=2}^n\\\\left(y_{i}-(\\\\hat{a}_n+\\\\hat{b}_nt_i+\\\\hat{\\\\rho}_ny_{i-1})\\\\right)^2}

The decision to be taken is:
    - If :math:`\\\\cH_0` from :eq:`TestSousModele1_1` is rejected, then the model 1 :eq:`Model1` is confirmed. And the test :eq:`TestModel1` proved that the unit root is rejected : :math:`\\\\rho < 1`. We then conclude that the final model is : :math:`\\\\boldsymbol{X_t = a + bt + \\\\rho X_{t-1} + \\\\varepsilon_{t}}` whith :math:`\\\\boldsymbol{\\\\rho < 1}` which is a **trend stationary model**.

    - If :math:`\\\\cH_0` from :eq:`TestSousModele1_1` is accepted, then the model 1 :eq:`Model1` is not confirmed, since the trend presence is rejected and the test :eq:`TestModel1` is not conclusive (since based on a wrong model). **We then have to test the second model** :eq:`Model2`.


**1.2. Case 2:** The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel1` is accepted

We test whether :math:`(\\\\rho, b) = (1,0)`:

.. math::
    :label: TestSousModele1_2

    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\cH_0: & (\\\\rho, b) = (1,0) \\\\\\\\
      \\\\cH_1: & (\\\\rho, b) \\\\neq (1,0)
    \\\\end{array}
    \\\\right.

with the Fisher statistics:

.. math::

    \\\\displaystyle \\\\hat{F}_1 = \\\\frac{(S_{1,0} - S_{1,b})/2}{S_{1,b}/(n-3)}

where :math:`S_{1,0}=\\\\sum_{i=2}^n\\\\left(y_i-(\\\\hat{a}_n+y_{i-1})\\\\right)^2` is the sum of the square errors of the model 1 :eq:`Model1` assuming :math:`\\\\cH_0` from :eq:`TestSousModele1_2` and :math:`S_{1,b}=\\\\sum_{i=2}^n\\\\left(y_i-(\\\\hat{a}_n+\\\\hat{b}_nt_i+\\\\hat{\\\\rho}_ny_{i-1})\\\\right)^2` is the same sum when we make no assumption on :math:`\\\\rho` and :math:`b`.

The statistics :math:`\\\\hat{F}_1` converges in distribution to the Fisher-Snedecor distribution :class:'~openturns.FisherSnedecor` with :math:`d_1=2, d_2=n-3`. The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel1` is accepted when :math:`\\\\hat{F}_1 < \\\\Phi_{\\\\alpha}` where :math:`\\\\Phi_{\\\\alpha}` is the test threshold of level :math:`\\\\alpha`.

The decision to be taken is:
    - If :math:`\\\\cH_0` from :eq:`TestSousModele1_2` is rejected, then the model 1 :eq:`Model1` is confirmed since the presence of linear trend is confirmed. And the test :eq:`TestModel1` proved that the unit root is accepted: :math:`\\\\rho = 1`. We then conclude that the model is: :math:`\\\\boldsymbol{X_t = a + bt + X_{t-1} + \\\\varepsilon_{t}}` which is a **non stationary model**.
    
    - If :math:`\\\\cH_0` from :eq:`TestSousModele1_2` is accepted, then the model 1 :eq:`Model1` is not confirmed, since the presence of the linear trend is rejected and the test :eq:`TestModel1` is not conclusive (since based on a wrong model). **We then have to test the second model** :eq:`Model2`.


**2.** We assume the model :eq:`Model2`:

.. math::
    :label: Model2

    \\\\boldsymbol{X_t = a + \\\\rho X_{t-1} + \\\\varepsilon_{t}}


The coefficients :math:`(a,\\\\rho)` are estimated as follows:

.. math::
    :label: Model2Estim

    \\\\underbrace{\\\\left(\\\\begin{array}{lll}
       \\\\displaystyle n-1 &\\\\sum_{i=2}^n y_{i-1}\\\\\\\\
       \\\\displaystyle \\\\sum_{i=2}^n y_{i-1} &\\\\sum_{i=2}^n y_{i-1}^2
                      \\\\end{array}
     \\\\right)}_{\\\\mat{N}}
     \\\\left(
      \\\\begin{array}{c}
        \\\\hat{a}_n\\\\\\\\
        \\\\hat{\\\\rho}_n
      \\\\end{array}
     \\\\right)=
     \\\\left(
      \\\\begin{array}{l}
        \\\\displaystyle \\\\sum_{i=1}^n y_{i} \\\\\\\\
        \\\\displaystyle \\\\sum_{i=2}^n y_{i-1} y_{i}
       \\\\end{array}
     \\\\right)
   

We first test:

.. math::
    :label: TestModel2

    \\\\left\\\\{
     \\\\begin{array}{lr}
       \\\\mathcal{H}_0: & \\\\rho = 1 \\\\\\\\
       \\\\mathcal{H}_1: & \\\\rho < 1
     \\\\end{array}
     \\\\right.

thanks to the Student statistics:

.. math::

    t_{\\\\rho=1} = \\\\frac{\\\\rho_n-1}{\\\\sigma_{\\\\rho_n}}

where :math:`\\\\sigma_{\\\\rho_n}` is the least square estimate of the standard deviation of :math:`\\\\Hat{\\\\rho}_n`, given by:

.. math::

    \\\\sigma_{\\\\rho_n}=\\\\mat{N}^{-1}_{22}\\\\sqrt{\\\\frac{1}{n-1}\\\\sum_{i=2}^n\\\\left(y_{i}-(\\\\hat{a}_n+\\\\hat{\\\\rho}_ny_{i-1})\\\\right)^2}

which converges in distribution to the Dickey-Fuller distribution associated to the model with drift and no linear trend:

.. math::

    t_{\\\\rho = 1} \\\\stackrel{\\\\mathcal{L}}{\\\\longrightarrow} \\\\frac{\\\\int_{0}^{1}W^{a}(r) dW(r)}{\\\\int_{1}^{0} W^{a}(r)^2 dr}

The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel2` is accepted when :math:`t_{\\\\rho=1} > C_{\\\\alpha}` where :math:`C_{\\\\alpha}` is the test threshold of level :math:`\\\\alpha`.

The quantiles of the Dickey-Fuller statistics for the model with drift are:

.. math::
    
    \\\\left\\\\{
    \\\\begin{array}{ll}
        \\\\alpha = 0.01, & C_{\\\\alpha} = -3.43 \\\\\\\\
        \\\\alpha = 0.05, & C_{\\\\alpha} = -2.86 \\\\\\\\
        \\\\alpha = 0.10, & C_{\\\\alpha} = -2.57
    \\\\end{array}
    \\\\right.


**2.1. Case 1:** The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel2` is rejected

We test whether :math:`a=0`:

.. math::
    :label: TestSousModele2_1
    
    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\mathcal{H}_0: & a = 0 \\\\\\\\
      \\\\mathcal{H}_1: & a \\\\neq 0
    \\\\end{array}
    \\\\right.

where the statistics :math:`t_n = \\\\frac{|\\\\hat{a}_n|}{\\\\sigma_{a_n}}` converges in distribution to the Student distribution :class:`~openturns.Student` with :math:`\\\\nu=n-3`, where :math:`\\\\sigma_{a_n}` is the least square estimate of the standard deviation of :math:`\\\\Hat{a}_n`, given by:

.. math::

    \\\\sigma_{a_n}=\\\\mat{N}^{-1}_{11}\\\\sqrt{\\\\frac{1}{n-1}\\\\sum_{i=2}^n\\\\left(y_{i}-(\\\\hat{a}_n+\\\\hat{\\\\rho}_ny_{i-1})\\\\right)^2}

The decision to be taken is:
    - If :math:`\\\\cH_0` from :eq:`TestSousModele2_1` is rejected, then the model 2 :eq:`Model2` is confirmed. And the test :eq:`TestModel2` proved that the unit root is rejected: :math:`\\\\rho < 1`. We then conclude that the final model is: :math:`\\\\boldsymbol{X_t = a + \\\\rho X_{t-1} + \\\\varepsilon_{t}}` whith :math:`\\\\boldsymbol{\\\\rho < 1}` which is a **stationary model**.

    - If :math:`\\\\cH_0` from :eq:`TestSousModele2_1` is accepted, then the model 2 :eq:`Model2` is not confirmed, since the drift presence is rejected and the test :eq:`TestModel1` is not conclusive (since based on a wrong model). **We then have to test the third model** :eq:`Model3`.


**2.2. Case 2:** The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel2` is accepted

We test whether :math:`(\\\\rho, a) = (1,0)`:

.. math::
    :label: TestSousModele2_2

    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\mathcal{H}_0: & (\\\\rho, a) = (1,0) \\\\\\\\
      \\\\mathcal{H}_1: & (\\\\rho, a) \\\\neq (1,0)
    \\\\end{array}
    \\\\right.

with a Fisher test. The statistics is:

.. math::

    \\\\displaystyle \\\\hat{F}_2 = \\\\frac{(SCR_{2,c} - SCR_{2})/2}{SCR_{2}/(n-2)}

where :math:`SCR_{2,c}` is the sum of the square errors of the model 2 :eq:`Model2` assuming :math:`\\\\cH_0` from :eq:`TestSousModele2_2` and :math:`SCR_{2}` is the same sum when we make no assumption on :math:`\\\\rho` and :math:`a`.

The statistics :math:`\\\\hat{F}_2` converges in distribution to the Fisher-Snedecor distribution :class:`~openturns.FisherSnedecor` with :math:`d_1=2, d_2=n-2`. The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel1` is accepted if when :math:`\\\\hat{F}_2 < \\\\Phi_{\\\\alpha}` where :math:`\\\\Phi_{\\\\alpha}` is the test threshold of level :math:`\\\\alpha`.

The decision to be taken is:
    - If :math:`\\\\cH_0` from :eq:`TestSousModele2_2` is rejected, then the model 2 :eq:`Model2` is confirmed since the presence of the drift is confirmed. And the test :eq:`TestModel2` proved that the unit root is accepted: :math:`\\\\rho =1`. We then conclude that the model is: :math:`\\\\boldsymbol{X_t = a + X_{t-1} + \\\\varepsilon_{t}}` which is a **non stationary model**.

    - If :math:`\\\\cH_0` from :eq:`TestSousModele2_2` is accepted, then the model 2 :eq:`Model2` is not confirmed, since the drift presence is rejected and the test :eq:`TestModel2` is not conclusive (since based on a wrong model). **We then have to test the third model** :eq:`Model3`.



**3.** We assume the model :eq:`Model3`:

.. math::
    :label: Model3

    \\\\boldsymbol{X_t = \\\\rho X_{t-1} + \\\\varepsilon_{t}}

The coefficients :math:`\\\\rho` are estimated as follows:

.. math::
    :label: Model3Estim

    \\\\hat{\\\\rho}_n=\\\\frac{\\\\sum_{i=2}^ny_{i-1}y_i}{\\\\sum_{i=2}^ny_{i-1}^2}

We first test:

.. math::
    :label: TestModel3
  
    \\\\left\\\\{
    \\\\begin{array}{lr}
      \\\\mathcal{H}_0: & \\\\rho = 1 \\\\\\\\
      \\\\mathcal{H}_1: & \\\\rho < 1
    \\\\end{array}
    \\\\right.

thanks to the Student statistics:

.. math::

    t_{\\\\rho=1} = \\\\frac{\\\\hat{\\\\rho}_n-1}{\\\\sigma_{\\\\rho_n}}

where :math:`\\\\sigma_{\\\\rho_n}` is the least square estimate of the standard deviation of :math:`\\\\Hat{\\\\rho}_n`, given by:

.. math::

    \\\\sigma_{\\\\rho_n}=\\\\sqrt{\\\\frac{1}{n-1}\\\\sum_{i=2}^n\\\\left(y_{i}-\\\\hat{\\\\rho}_ny_{i-1}\\\\right)^2}/\\\\sqrt{\\\\sum_{i=2}^ny_{i-1}^2}

which converges in distribution to the Dickey-Fuller distribution associated to the random walk model:

.. math::

    t_{\\\\rho = 1} \\\\stackrel{\\\\mathcal{L}}{\\\\longrightarrow} \\\\frac{\\\\int_{0}^{1}W(r) dW(r)}{\\\\int_{1}^{0} W(r)^2 dr}

The null hypothesis :math:`\\\\cH_0` from :eq:`TestModel3` is accepted when :math:`t_{\\\\rho=1} > C_{\\\\alpha}` where :math:`C_{\\\\alpha}` is the test threshold of level :math:`\\\\alpha`.

The quantiles of the Dickey-Fuller statistics for the random walk model are:

.. math::
    
    \\\\left\\\\{
    \\\\begin{array}{ll}
        \\\\alpha = 0.01, & C_{\\\\alpha} = -2.57 \\\\\\\\
        \\\\alpha = 0.05, & C_{\\\\alpha} = -1.94 \\\\\\\\
        \\\\alpha = 0.10, & C_{\\\\alpha} = -1.62
    \\\\end{array}
    \\\\right.

The decision to be taken is:
    - If :math:`\\\\cH_0` from :eq:`TestModel3` is rejected, we then conclude that the model is : :math:`\\\\boldsymbol{X_t = \\\\rho X_{t-1} + \\\\varepsilon_{t}}` where :math:`\\\\rho < 1` which is a **stationary model**.

    - If :math:`\\\\cH_0` from :eq:`TestModel3` is accepted, we then conclude that the model is: :math:`\\\\boldsymbol{X_t = X_{t-1} + \\\\varepsilon_{t}}` which is a **non stationary model**.


Examples
--------
Create an ARMA process and generate a time series:

>>> import openturns as ot
>>> arcoefficients = ot.ARMACoefficients([0.3])
>>> macoefficients = ot.ARMACoefficients(0)
>>> timeGrid = ot.RegularGrid(0.0, 0.1, 100)
>>> whiteNoise = ot.WhiteNoise(ot.Normal(), timeGrid)
>>> myARMA = ot.ARMA(arcoefficients, macoefficients, whiteNoise)

>>> realization = ot.TimeSeries(myARMA.getRealization())
>>> test = ot.DickeyFullerTest(realization)

Test the stationarity of the data without any asumption on the model:

>>> globalRes = test.runStrategy()

Test the stationarity knowing you have a drift and linear trend model:

>>> res1 = test.testUnitRootInDriftAndLinearTrendModel(0.95)

Test the stationarity knowing you have a drift model:

>>> res2 = test.testUnitRootInDriftModel(0.95)
 
Test the stationarity knowing you have an AR1 model:

>>> res3 = test.testUnitRootInAR1Model(0.95)

"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testUnitRootInDriftAndLinearTrendModel
"Test for unit root in model with drift and trend.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestModel1`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testUnitRootInDriftModel
"Test for unit root in model with drift.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestModel2`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testUnitRootInAR1Model
"Test for unit root in AR1 model.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestModel3`.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::DickeyFullerTest::testUnitRootAndNoLinearTrendInDriftAndLinearTrendModel
"Test for linear trend in model with unit root.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestSousModele1_2`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testNoUnitRootAndNoLinearTrendInDriftAndLinearTrendModel
"Test for trend in model without unit root.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestSousModele1_1`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testUnitRootAndNoDriftInDriftModel
"Test for null drift in model with unit root.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestSousModele2_2`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::testNoUnitRootAndNoDriftInDriftModel
"Test for null drift in model without unit root.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the test detailed in :eq:`TestSousModele2_1`.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::DickeyFullerTest::runStrategy
"Test the stationarity without any assumption on the model.

Parameters
----------
alpha : float, :math:`0 < \\\\alpha < 1`
    The first order error of the test.

    By default, :math:`\\\\alpha=0.95`.

Returns
-------
testResult : :class:`~openturns.TestResult`
    Results container of the tests. The strategy if the one described above.
"