NAG Library Function Document
nag_tsa_dickey_fuller_unit (g13awc)
1
Purpose
nag_tsa_dickey_fuller_unit (g13awc) returns the (augmented) Dickey–Fuller unit root test.
2
Specification
#include <nag.h> 
#include <nagg13.h> 
double 
nag_tsa_dickey_fuller_unit (Nag_TS_URTestType type,
Integer p,
Integer n,
const double y[],
NagError *fail) 

3
Description
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. nag_tsa_dickey_fuller_unit (g13awc) 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
is fitted and the test statistic
$\tau $ constructed as
where
$\nabla $ is the difference operator, with
$\nabla {y}_{t}={y}_{t}{y}_{t1}$, and where
${\hat{\beta}}_{1}$ 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
is fit and the test statistic
${\tau}_{\mu}$ constructed as
To test for a unit root with drift and deterministic time trend the regression model
is fit and the test statistic
${\tau}_{\tau}$ constructed as
The distributions of the three test statistics;
$\tau $,
${\tau}_{\mu}$ and
${\tau}_{\tau}$, are nonstandard. An associated probability can be obtained from
nag_prob_dickey_fuller_unit (g01ewc).
4
References
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
5
Arguments
 1:
$\mathbf{type}$ – Nag_TS_URTestTypeInput

On entry: the type of unit test for which the probability is required.
 ${\mathbf{type}}=\mathrm{Nag\_UnitRoot}$
 A unit root test will be performed and $\tau $ returned.
 ${\mathbf{type}}=\mathrm{Nag\_UnitRootWithDrift}$
 A unit root test with drift will be performed and ${\tau}_{\mu}$ returned.
 ${\mathbf{type}}=\mathrm{Nag\_UnitRootWithDriftAndTrend}$
 A unit root test with drift and deterministic time trend will be performed and ${\tau}_{\tau}$ returned.
Constraint:
${\mathbf{type}}=\mathrm{Nag\_UnitRoot}$, $\mathrm{Nag\_UnitRootWithDrift}$ or $\mathrm{Nag\_UnitRootWithDriftAndTrend}$.
 2:
$\mathbf{p}$ – IntegerInput

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:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the length of the time series.
Constraints:
 if ${\mathbf{type}}=\mathrm{Nag\_UnitRoot}$, ${\mathbf{n}}>2{\mathbf{p}}$;
 if ${\mathbf{type}}=\mathrm{Nag\_UnitRootWithDrift}$, ${\mathbf{n}}>2{\mathbf{p}}+1$;
 if ${\mathbf{type}}=\mathrm{Nag\_UnitRootWithDriftAndTrend}$, ${\mathbf{n}}>2{\mathbf{p}}+2$.
 4:
$\mathbf{y}\left[{\mathbf{n}}\right]$ – const doubleInput

On entry: $y$, the time series.
 5:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>\u2329\mathit{\text{value}}\u232a$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_ORDERS_ARIMA

On entry, ${\mathbf{p}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{p}}>0$.
 NW_SOLN_NOT_UNIQUE

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.
 NW_TRUNCATED

${\sigma}_{11}=0$, therefore depending on the sign of ${\hat{\beta}}_{1}$, a large positive or negative value has been returned.
7
Accuracy
None.
8
Parallelism and Performance
nag_tsa_dickey_fuller_unit (g13awc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_dickey_fuller_unit (g13awc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
None.
10
Example
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.
10.1
Program Text
Program Text (g13awce.c)
10.2
Program Data
Program Data (g13awce.d)
10.3
Program Results
Program Results (g13awce.r)