naginterfaces.library.tsa.uni_​dickey_​fuller_​unit

naginterfaces.library.tsa.uni_dickey_fuller_unit(ut_type, p, y)

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

For full information please refer to the NAG Library document for g13aw

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13awf.html

Parameters

ut_typeint

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

$ut\_type=1$

A unit root test will be performed and $\tau$ returned.

$ut\_type=2$

A unit root test with drift will be performed and ${\tau }_{\mu }$ returned.

$ut\_type=3$

A unit root test with drift and deterministic time trend will be performed and ${\tau }_{\tau }$ returned.

pint

$p$, 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.

yfloat, array-like, shape $\left(n\right)$

$y$, the time series.

Returns

tsfloat

The (augmented) Dickey–Fuller test statistic: $\tau$ when $ut\_type=1$; ${\tau }_{\mu }$ when $ut\_type=2$ or ${\tau }_{\tau }$ when $ut\_type=3$.

Raises

NagValueError

(errno $11$)

On entry, $ut\_type=⟨value⟩$.

Constraint: $ut\_type=1$, $2$ or $3$.

(errno $21$)

On entry, $p=⟨value⟩$.

Constraint: $p>0$.

(errno $31$)

On entry, $n=⟨value⟩$.

Constraint: if $ut\_type=1$, $n>2p$ if $ut\_type=2$, $n>2p+1$ if $ut\_type=3$, $n>2p+2$

Warns

NagAlgorithmicWarning

(errno $41$)

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.

(errno $42$)

standard error of the parameter estimate is zero, therefore, depending on the sign of the parameter estimate, a large positive or negative value has been returned.

Notes

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. uni_dickey_fuller_unit 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 $\text{t}=1,2,\dots ,n$, has a unit root, the regression model

$$\nabla y_t={\beta }_1{y}_{t-1}+\sum _{i=1}^{p-1}{\delta }_i\nabla y_{t-i}+{ϵ}_t$$

is fitted and the test statistic $\tau$ constructed as

$$\tau =\frac{{\hat{\beta }}_1}{{\sigma }_{11}}$$

where $\nabla$ is the difference operator, with $\nabla y_t=y_t-y_{t-1}$, 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

$$\nabla y_t={\beta }_1{y}_{t-1}+\sum _{i=1}^{p-1}{\delta }_i\nabla y_{t-i}+\alpha +{ϵ}_t$$

is fit and the test statistic ${\tau }_{\mu }$ constructed as

$${\tau }_{\mu }=\frac{{\hat{\beta }}_1}{{\sigma }_{11}}$$

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

$$\nabla y_t={\beta }_1{y}_{t-1}+\sum _{i=1}^{p-1}{\delta }_i\nabla y_{t-i}+\alpha +{\beta }_{2}t+{ϵ}_t$$

is fit and the test statistic ${\tau }_{\tau }$ constructed as

$${\tau }_{\tau }=\frac{{\hat{\beta }}_1}{{\sigma }_{11}}$$

The distributions of the three test statistics; $\tau$, ${\tau }_{\mu }$ and ${\tau }_{\tau }$, are nonstandard. An associated probability can be obtained from stat.prob_dickey_fuller_unit.

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

See also

naginterfaces.library.examples.tsa.uni_dickey_fuller_unit_ex.main()