# NAG C Library Function Document

## 1Purpose

nag_prob_dickey_fuller_unit (g01ewc) returns the probability associated with the lower tail of the distribution for the Dickey–Fuller unit root test statistic.

## 2Specification

 #include #include
 double nag_prob_dickey_fuller_unit (Nag_TS_URProbMethod method, Nag_TS_URTestType type, Integer n, double ts, Integer nsamp, Integer state[], 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_prob_dickey_fuller_unit (g01ewc) is designed to be called after nag_tsa_dickey_fuller_unit (g13awc) and returns the probability associated with 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. The three types of test statistic are constructed as follows:
1. 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 fit 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.
2. 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
 $τμ = β^1 σ11 .$
3. 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
 $ττ = β^1 σ11 .$
All three test statistics: $\tau$, ${\tau }_{\mu }$ and ${\tau }_{\tau }$ can be calculated using nag_tsa_dickey_fuller_unit (g13awc).
The probability distributions of these statistics are nonstandard and are a function of the length of the series of interest, $n$. The probability associated with a given test statistic, for a given $n$, can therefore only be calculated by simulation as described in Dickey and Fuller (1979). However, such simulations require a significant number of iterations and are therefore prohibitively expensive in terms of the time taken. As such nag_prob_dickey_fuller_unit (g01ewc) also allows the probability to be interpolated from a look-up table. Two such tables are provided, one from Dickey (1976) and one constructed as described in Section 9. The three different methods of obtaining an estimate of the probability can be chosen via the method argument. Unless there is a specific reason for choosing otherwise, ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$ should be used.
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:    $\mathbf{method}$Nag_TS_URProbMethodInput
On entry: the method used to calculate the probability.
${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$
The probability is interpolated from a look-up table, whose values were obtained via simulation.
${\mathbf{method}}=\mathrm{Nag_ViaLookUpOriginal}$
The probability is interpolated from a look-up table, whose values were obtained from Dickey (1976).
${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$
The probability is obtained via simulation.
The probability calculated from the look-up table should give sufficient accuracy for most applications.
Suggested value: ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$.
Constraint: ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$, $\mathrm{Nag_ViaLookUpOriginal}$ or $\mathrm{Nag_ViaSimulation}$.
2:    $\mathbf{type}$Nag_TS_URTestTypeInput
On entry: the type of test statistic, supplied in ts.
Constraint: ${\mathbf{type}}=\mathrm{Nag_UnitRoot}$, $\mathrm{Nag_UnitRootWithDrift}$ or $\mathrm{Nag_UnitRootWithDriftAndTrend}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the length of the time series used to calculate the test statistic.
Constraints:
• if ${\mathbf{method}}\ne \mathrm{Nag_ViaSimulation}$, ${\mathbf{n}}>0$;
• if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRoot}$, ${\mathbf{n}}>2$;
• if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDrift}$, ${\mathbf{n}}>3$;
• if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDriftAndTrend}$, ${\mathbf{n}}>4$.
4:    $\mathbf{ts}$doubleInput
On entry: the Dickey–Fuller test statistic for which the probability is required. If
${\mathbf{type}}=\mathrm{Nag_UnitRoot}$
ts must contain $\tau$.
${\mathbf{type}}=\mathrm{Nag_UnitRootWithDrift}$
ts must contain ${\tau }_{\mu }$.
${\mathbf{type}}=\mathrm{Nag_UnitRootWithDriftAndTrend}$
ts must contain ${\tau }_{\tau }$.
If the test statistic was calculated using nag_tsa_dickey_fuller_unit (g13awc) the value of type and n must not change between calls to nag_prob_dickey_fuller_unit (g01ewc) and nag_tsa_dickey_fuller_unit (g13awc).
5:    $\mathbf{nsamp}$IntegerInput
On entry: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$, the number of samples used in the simulation; otherwise nsamp is not referenced and need not be set.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$, ${\mathbf{nsamp}}>0$.
6:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$, state must contain information on the selected base generator and its current state; otherwise state is not referenced and may be NULL.
On exit: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$, state contains updated information on the state of the generator otherwise a zero length vector is returned.
7:    $\mathbf{fail}$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.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}\ne \mathrm{Nag_ViaSimulation}$, ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRoot}$, ${\mathbf{n}}>2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDriftAndTrend}$, ${\mathbf{n}}>4$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and ${\mathbf{type}}=\mathrm{Nag_UnitRootWithDrift}$, ${\mathbf{n}}>3$.
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_INVALID_STATE
On entry, ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ and the state vector has been corrupted or not initialized.
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_SAMPLE
On entry, ${\mathbf{nsamp}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$, ${\mathbf{nsamp}}>0$.
NW_EXTRAPOLATION
The supplied input values were outside the range of at least one look-up table, therefore extrapolation was used.

## 7Accuracy

When ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$, the probability returned by this function is unlikely to be accurate to more than $4$ or $5$ decimal places, for ${\mathbf{method}}=\mathrm{Nag_ViaLookUpOriginal}$ this accuracy is likely to drop to $2$ or $3$ decimal places (see Section 9 for details on how these probabilities are constructed). In both cases the accuracy of the probability is likely to be lower when extrapolation is used, particularly for small values of n (less than around $15$). When ${\mathbf{method}}=\mathrm{Nag_ViaSimulation}$ the accuracy of the returned probability is controlled by the number of simulations performed (i.e., the value of nsamp used).

## 8Parallelism and Performance

nag_prob_dickey_fuller_unit (g01ewc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_prob_dickey_fuller_unit (g01ewc) 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.

## 9Further Comments

When ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$ or $\mathrm{Nag_ViaLookUpOriginal}$ the probability returned is constructed by interpolating from a series of look-up tables. In the case of ${\mathbf{method}}=\mathrm{Nag_ViaLookUpOriginal}$ the look-up tables are taken directly from Dickey (1976) and the interpolation is carried out using nag_2d_triang_interp (e01sjc) and nag_2d_triang_eval (e01skc). For ${\mathbf{method}}=\mathrm{Nag_ViaLookUp}$ the look-up tables were constructed as follows:
 (i) A sample size, $n$ was chosen. (ii) ${2}^{28}$ simulations were run. (iii) At each simulation, a time series was constructed as described in chapter five of Dickey (1976). The relevant test statistic was then calculated for each of these time series. (iv) A series of quantiles were calculated from the sample of ${2}^{28}$ test statistics. The quantiles were calculated at intervals of $0.0005$ between $0.0005$ and $0.9995$. (v) A spline was fit to the quantiles using nag_1d_spline_fit (e02bec).
This process was repeated for $n=25,50,75,100,150,200,250,300,350,400,450,500,600,700,800,900,1000,1500,2000,2500,5000,10000$, resulting in $22$ splines.
Given the $\mathrm{22}$ splines, and a user-supplied sample size, $n$ and test statistic, $\tau$, an estimated $p$-value is calculated as follows:
 (i) Evaluate each of the $\mathrm{22}$ splines, at $\tau$, using nag_1d_spline_fit (e02bec). If, for a particular spline, the supplied value of $\tau$ lies outside of the range of the simulated data, then a third-order Taylor expansion is used to extrapolate, with the derivatives being calculated using nag_1d_spline_deriv (e02bcc). (ii) Fit a spline through these $22$ points using nag_monotonic_interpolant (e01bec). (iii) Estimate the $p$-value using nag_monotonic_evaluate (e01bfc).

## 10Example

See Section 10 in nag_tsa_dickey_fuller_unit (g13awc).