# NAG FL Interfaceg13awf (uni_​dickey_​fuller_​unit)

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## 1Purpose

g13awf returns the (augmented) Dickey–Fuller unit root test.

## 2Specification

Fortran Interface
 Function g13awf ( type, p, n, y,
 Real (Kind=nag_wp) :: g13awf Integer, Intent (In) :: type, p, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: y(n)
#include <nag.h>
 double g13awf_ (const Integer *typ, const Integer *p, const Integer *n, const double y[], Integer *ifail)
The routine may be called by the names g13awf or nagf_tsa_uni_dickey_fuller_unit.

## 3Description

If the root of the characteristic equation for a time series is one then that series is said to have a unit root. Such series are nonstationary. g13awf returns one of three types of (augmented) Dickey–Fuller test statistic: $\tau$, ${\tau }_{\mu }$ or ${\tau }_{\tau }$, used to test for a unit root, a unit root with drift or a unit root with drift and a deterministic time trend, respectively.
To test whether a time series, ${y}_{t}$, for $\mathit{t}=1,2,\dots ,n$, has a unit root, the regression model
 $∇yt = β1 yt-1 + ∑ i=1 p-1 δi ∇ yt-i +εt$
is fitted and the test statistic $\tau$ constructed as
 $τ = β^1 σ11$
where $\nabla$ is the difference operator, with $\nabla {y}_{t}={y}_{t}-{y}_{t-1}$, and where ${\stackrel{^}{\beta }}_{1}$ and ${\sigma }_{11}$ are the least squares estimate and associated standard error for ${\beta }_{1}$ respectively.
To test for a unit root with drift the regression model
 $∇yt = β1 yt-1 + ∑ i=1 p-1 δi ∇ yt-i +α +εt$
is fit and the test statistic ${\tau }_{\mu }$ constructed as
 $τμ = β^1 σ11$
To test for a unit root with drift and deterministic time trend the regression model
 $∇yt = β1 yt-1 + ∑ i=1 p-1 δi ∇ yt-i +α +β2t +εt$
is fit and the test statistic ${\tau }_{\tau }$ constructed as
 $ττ = β^1 σ11$
The distributions of the three test statistics; $\tau$, ${\tau }_{\mu }$ and ${\tau }_{\tau }$, are nonstandard. An associated probability can be obtained from g01ewf.

## 4References

Dickey A D (1976) Estimation and hypothesis testing in nonstationary time series PhD Thesis Iowa State University, Ames, Iowa
Dickey A D and Fuller W A (1979) Distribution of the estimators for autoregressive time series with a unit root J. Am. Stat. Assoc. 74 366 427–431

## 5Arguments

1: $\mathbf{type}$Integer Input
On entry: the type of unit test for which the probability is required.
${\mathbf{type}}=1$
A unit root test will be performed and $\tau$ returned.
${\mathbf{type}}=2$
A unit root test with drift will be performed and ${\tau }_{\mu }$ returned.
${\mathbf{type}}=3$
A unit root test with drift and deterministic time trend will be performed and ${\tau }_{\tau }$ returned.
Constraint: ${\mathbf{type}}=1$, $2$ or $3$.
2: $\mathbf{p}$Integer Input
On entry: $p$, the degree of the autoregressive (AR) component of the Dickey–Fuller test statistic. When $p>1$ the test is usually referred to as the augmented Dickey–Fuller test.
Constraint: ${\mathbf{p}}>0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the length of the time series.
Constraints:
• if ${\mathbf{type}}=1$, ${\mathbf{n}}>2{\mathbf{p}}$;
• if ${\mathbf{type}}=2$, ${\mathbf{n}}>2{\mathbf{p}}+1$;
• if ${\mathbf{type}}=3$, ${\mathbf{n}}>2{\mathbf{p}}+2$.
4: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: $y$, the time series.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=11$
On entry, ${\mathbf{type}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{type}}=1$, $2$ or $3$.
${\mathbf{ifail}}=21$
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}>0$.
${\mathbf{ifail}}=31$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint:
• if ${\mathbf{type}}=1$, ${\mathbf{n}}>2{\mathbf{p}}$;
• if ${\mathbf{type}}=2$, ${\mathbf{n}}>2{\mathbf{p}}+1$;
• if ${\mathbf{type}}=3$, ${\mathbf{n}}>2{\mathbf{p}}+2$.
${\mathbf{ifail}}=41$
On entry, the design matrix used in the estimation of ${\beta }_{1}$ is not of full rank, this is usually due to all elements of the series being virtually identical. The returned statistic is, therefore, not unique and likely to be meaningless.
${\mathbf{ifail}}=42$
${\sigma }_{11}=0$, therefore, depending on the sign of ${\stackrel{^}{\beta }}_{1}$, a large positive or negative value has been returned.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g13awf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13awf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

In this example a Dickey–Fuller unit root test is applied to a time series related to the rate of the earth's rotation about its polar axis.

### 10.1Program Text

Program Text (g13awfe.f90)

### 10.2Program Data

Program Data (g13awfe.d)

### 10.3Program Results

Program Results (g13awfe.r)