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

τ_{μ}

τ_{τ}

, 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

t = 1, 2, \dots, n

, has a unit root, the regression model

\nabla y_{t} = β_{1} y_{t - 1} + \sum_{i = 1}^{p - 1} δ_{i} \nabla y_{t - i} + ε_{t}

is fitted and the test statistic

τ

constructed as

τ = \frac{{\hat{β}}_{1}}{σ_{11}}

where

\nabla

is the difference operator, with

\nabla y_{t} = y_{t} - y_{t - 1}

, and where

{\hat{β}}_{1}

and

σ_{11}

are the least squares estimate and associated standard error for

β_{1}

respectively.

To test for a unit root with drift the regression model

\nabla y_{t} = β_{1} y_{t - 1} + \sum_{i = 1}^{p - 1} δ_{i} \nabla y_{t - i} + α + ε_{t}

is fit and the test statistic

τ_{μ}

constructed as

τ_{μ} = \frac{{\hat{β}}_{1}}{σ_{11}}

To test for a unit root with drift and deterministic time trend the regression model

\nabla y_{t} = β_{1} y_{t - 1} + \sum_{i = 1}^{p - 1} δ_{i} \nabla y_{t - i} + α + β_{2} t + ε_{t}

is fit and the test statistic

τ_{τ}

constructed as

τ_{τ} = \frac{{\hat{β}}_{1}}{σ_{11}}

The distributions of the three test statistics;

τ

τ_{μ}

and

τ_{τ}

, 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: $type$ – Nag_TS_URTestTypeInput

On entry: the type of unit test for which the probability is required.

$type = Nag_UnitRoot$: A unit root test will be performed and $τ$ returned.
$type = Nag_UnitRootWithDrift$: A unit root test with drift will be performed and $τ_{μ}$ returned.
$type = Nag_UnitRootWithDriftAndTrend$: A unit root test with drift and deterministic time trend will be performed and $τ_{τ}$ returned.

Constraint:

type = Nag_UnitRoot

Nag_UnitRootWithDrift

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.

Constraints:

if $type = Nag_UnitRoot$ , $n > 2 p$ ;
if $type = Nag_UnitRootWithDrift$ , $n > 2 p + 1$ ;
if $type = Nag_UnitRootWithDriftAndTrend$ , $n > 2 p + 2$ .

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).

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 $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 〈value〉$ .
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, $p = 〈value〉$ .
Constraint: $p > 0$ .
NW_SOLN_NOT_UNIQUE: On entry, the design matrix used in the estimation of $β_{1}$ 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: $σ_{11} = 0$ , therefore depending on the sign of ${\hat{β}}_{1}$ , 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 implementation-specific information.

9

Further Comments

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.

NAG Library Function Document

nag_tsa_dickey_fuller_unit (g13awc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results