GARCH Toolbox™

dfARTest

Run augmented Dickey-Fuller unit root test based on zero drift AR model

Syntax

[H,pValue,TestStat,CriticalValue] = dfARTest(Y,Lags,Alpha,TestType)

Description

[H,pValue,TestStat,CriticalValue] = dfARTest(Y,Lags,Alpha,TestType) performs an augmented Dickey-Fuller univariate unit root test. This test assumes that the true underlying process is a zero drift unit root process. As an alternative, OLS regression estimates a zero drift (P+1)th order autoregressive (AR(P+1)) model.

Specifically, if:

y_t and ε_t are the time series of observed data and model residuals, respectively, and
is the first difference operator

then under the null hypothesis the true underlying process is a zero drift ARIMA(P,1,0) model

Which is equivalent to an integrated AR(P+1) model.

As an alternative, the estimated OLS regression model is

for some AR(1) coefficient φ < 1.

Input Arguments

`Y`	Time-series vector of observed data tested for a unit root. The last element contains the most recent observation. `dfARTest` represents missing values as `NaN`s and removes them, thereby reducing the sample size.
`Lags`	(Optional) Scalar or vector of nonnegative integers. This parameter indicates the number of lagged changes (that is, first differences) of `Y` included in the OLS regression model (see P above). `Lags` serves as a correction for serial correlation of residuals. If `Lags` is empty or missing, the default is `0` (no correction for serial correlation).
`Alpha`	(Optional) Scalar or vector of significance levels of the test. All elements of the input argument must be 0.001 ≤ `Alpha` ≤ 0.999.
`TestType`	(Optional) Character string indicating the type of unit root test. Possible choices are: `t`, indicating an OLS t test of the AR(1) coefficient `AR`, indicating a test of the unstudentized AR(1) coefficient `dfARTest` performs a case-insensitive check of `TestType`. If it is empty or missing, the default is a t test.

Output Arguments

`H`	Logical decision vector. Elements of `H = 0` indicate acceptance of the null hypothesis; elements of `H = 1` indicate rejection of the null hypothesis. Each element of `H` is associated with a particular lag of `Lags` and significance level of `Alpha`.
`pValue`	Vector of p-values (significance levels) associated with the test decision vector `H`. Each element of `pValue` represents the probability of observing a test statistic at least as extreme as that calculated from the OLS regression model when the null hypothesis is true. `dfARTest` obtains p-values by interpolation into the appropriate table of critical values. When a p-value is outside of the range of tabulated significance levels (that is `0.001` <= `Alpha` <=`0.999`), a warning appears. `dfARTest` then sets `pValue` to the appropriate limit (`pValue` = `0.001` or `0.999`).
`TestStat`	Vector of test statistics associated with the decision vector `H`.
`CriticalValue`	Vector of critical values associated with the decision vector `H`.

Notes

You can specify both Lags and Alpha as scalars or vectors. If you specify both as vectors, they must be the same length (that is, they must have the same number of elements). If you specify one as a scalar and the other as a vector, dfARTest performs a scalar expansion to enforce vectors of identical length. If Lags is a scalar or an empty matrix, by default, all outputs are column vectors.

All vector outputs are the same length as vector inputs Alpha and/or Lags. By default all vector outputs are column vectors. If Lags is a row vector, however, all vector outputs are row vectors.

This univariate unit root test is a conventional lower tailed test. dfARTest compares the test statistic with the critical value to determine whether the test is accepted or rejected. If the test statistic is less than the critical value, reject the null hypothesis.

References

Hamilton, J.D., Time Series Analysis, Princeton University Press, Princeton, NJ, 1994.

Greene, W.H., Econometric Analysis, Prentice Hall, Fifth edition, Upper Saddle River, NJ, 2003.

Enders, W., Applied Econometric Time Series, John Wiley & Sons, New York, 1995.

Campbell, J.Y., A.W. Lo, and A.C. MacKinlay, The GARCH of Financial Markets, Princeton University Press, Princeton, NJ, 1997.

dfARDTest

dfTSTest