# NAG C Library Function Document

## 1Purpose

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

## 2Specification

 #include #include
 double nag_tsa_dickey_fuller_unit (Nag_TS_URTestType type, Integer p, Integer n, const double y[], NagError *fail)

## 3Description

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
 $∇yt = β1 yt-1 + ∑ i=1 p-1 δi ∇ yt-i +εt$
is fitted and the test statistic $\tau$ constructed as
 $τ = β^1 σ11$
where $\nabla$ is the difference operator, with $\nabla {y}_{t}={y}_{t}-{y}_{t-1}$, and where ${\stackrel{^}{\beta }}_{1}$ 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
 $∇yt = β1 yt-1 + ∑ i=1 p-1 δi ∇ yt-i +α +εt$
is fit and the test statistic ${\tau }_{\mu }$ 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 ${\tau }_{\tau }$ constructed as
 $ττ = β^1 σ11$
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).
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

## 5Arguments

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).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint:
• if ${\mathbf{type}}=\mathrm{Nag_UnitRoot}$, ${\mathbf{n}}>2{\mathbf{p}}$;
• if ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDrift}$, ${\mathbf{n}}>2{\mathbf{p}}+1$;
• if ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDriftAndTrend}$, ${\mathbf{n}}>2{\mathbf{p}}+2$.
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, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
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 ${\stackrel{^}{\beta }}_{1}$, a large positive or negative value has been returned.

None.

## 8Parallelism 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.

None.

## 10Example

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.

### 10.1Program Text

Program Text (g13awce.c)

### 10.2Program Data

Program Data (g13awce.d)

### 10.3Program Results

Program Results (g13awce.r)