NAG Library Function Document

nag_tsa_dickey_fuller_unit (g13awc)


    1  Purpose
    7  Accuracy


nag_tsa_dickey_fuller_unit (g13awc) returns the (augmented) Dickey–Fuller unit root test.


#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)


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: τ, τμ or ττ, 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, yt, for t=1,2,,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 τ constructed as
τ = β^1 σ11  
where  is the difference operator, with yt = yt- yt-1 , and where β^1 and σ11 are the least squares estimate and associated standard error for β1 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 τμ 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 ττ constructed as
ττ = β^1 σ11  
The distributions of the three test statistics; τ, τμ and ττ, are nonstandard. An associated probability can be obtained from nag_prob_dickey_fuller_unit (g01ewc).


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


1:     type Nag_TS_URTestTypeInput
On entry: the type of unit test for which the probability is required.
A unit root test will be performed and τ returned.
A unit root test with drift will be performed and τμ returned.
A unit root test with drift and deterministic time trend will be performed and ττ returned.
Constraint: type=Nag_UnitRoot, Nag_UnitRootWithDrift or Nag_UnitRootWithDriftAndTrend.
2:     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: p>0.
3:     n IntegerInput
On entry: n, the length of the time series.
  • if type=Nag_UnitRoot, n>2p;
  • if type=Nag_UnitRootWithDrift, n>2p+1;
  • if type=Nag_UnitRootWithDriftAndTrend, n>2p+2.
4:     y[n] const doubleInput
On entry: y, the time series.
5:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n>value.
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.
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.
On entry, p=value.
Constraint: p>0.
On entry, the design matrix used in the estimation of β1 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.
σ11=0, therefore depending on the sign of β^1, a large positive or negative value has been returned.



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 implementation-specific information.

Further Comments



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.

Program Text

Program Text (g13awce.c)

Program Data

Program Data (g13awce.d)

Program Results

Program Results (g13awce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017