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_uni_dickey_fuller_unit (g13aw) 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 fit 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_stat_prob_dickey_fuller_unit (g01ew).

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

Parameters

Compulsory Input Parameters

1: $type$ – int64int32nag_int scalar

The type of unit test for which the probability is required.

$type = 1$: A unit root test will be performed and $τ$ returned.
$type = 2$: A unit root test with drift will be performed and $τ_{μ}$ returned.
$type = 3$: A unit root test with drift and deterministic time trend will be performed and $τ_{τ}$ returned.

Constraint:

type = 1

2

3

2: $p$ – int64int32nag_int scalar

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: $y (n)$ – double array

y

, the time series.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar

Default: the dimension of the array y.

n

, the length of the time series.

Constraints:

if $type = 1$ , $n > 2 p$ ;
if $type = 2$ , $n > 2 p + 1$ ;
if $type = 3$ , $n > 2 p + 2$ .

Output Parameters

1: $ts$ – double scalar
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 11$: Constraint: $type = 1$ , $2$ or $3$ .

$ifail = 21$: Constraint: $p > 0$ .

$ifail = 31$

Constraint:

if $type = 1$ , $n > 2 p$ ;
if $type = 2$ , $n > 2 p + 1$ ;
if $type = 3$ , $n > 2 p + 2$ .

$ifail = 41$: 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.

$ifail = 42$: $σ_{11} = 0$ , therefore depending on the sign of ${\hat{β}}_{1}$ , a large positive or negative value has been returned.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

None.

Further Comments

None.

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.

Open in the MATLAB editor: g13aw_example

function g13aw_example


fprintf('g13aw example results\n\n');

% Test type
type = int64(1);

% Order of the AR process
p = int64(1);

% Time series
y = [ -217; -177; -166; -136; -110;  -95;  -64;  -37;  -14;  -25;
       -51;  -62;  -73;  -88; -113; -120;  -83;  -33;  -19;   21;
        17;   44;   44;   78;   88;  122;  126;  114;   85;   64];

% Calculate the Dickey-Fuller test statistic
[ts,ifail] = g13aw(type,p,y);

% The p-value routine can issue a warning, but still return
% sensible results, so save current warning state and turn warnings on
warn_state = nag_issue_warnings();
nag_issue_warnings(true);

% Get the associated p-value
n = int64(size(y,1));
[pvalue,~,ifail] = g01ew(type,n,ts);

% Reset the warning state to its initial value
nag_issue_warnings(warn_state);

% Print the results
fprintf('Dickey-Fuller test statistic     = %6.3f\n', ts);
fprintf('associated p-value               = %6.3f\n', pvalue);

g13aw example results

Dickey-Fuller test statistic     = -2.540
associated p-value               =  0.013

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox