GARCH Toolbox™

ppTSTest

Run Phillips-Perron unit root test based on trend stationary AR(1) model

Syntax

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

Description

[H,pValue,TestStat,CriticalValue] = ... ppTSTest(Y,Lags,Alpha,TestType) performs a Phillips-Perron univariate unit root test. This test assumes that the true underlying process is a unit root process with drift. As an alternative, OLS regression estimates a trend stationary first-order autoregressive (AR(1)) model plus additive constant.

Specifically, consider y_t and ε_t to be the time series of observed data and model residuals, respectively. Then under the null hypothesis, ppTSTest assumes that the true underlying process is

for an arbitrary constant C. As an alternative, the estimated OLS regression model is

for some constant C, AR(1) coefficient φ < 1, and trend stationary coefficient δ.

Input Arguments

`Y`	Time-series vector of observed data tested for a unit root. The last element contains the most recent observation. `ppTSTest` 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 autocovariance lags included in the Newey-West estimation of the asymptotic variance of the sample mean of the residuals. `Lags` serves as a correction for serial correlation of residuals. If 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` and `AR`, indicating an OLS t test of the AR coefficient and a test of the unstudentized AR coefficient, respectively. `ppTSTest` 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. `ppTSTest` 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 (`0.001` <= `Alpha` <=`0.999`), a warning appears. `ppTSTest` 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 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 one is specified as a scalar and the other as a vector, ppTSTest performs a scalar expansion to enforce identical-length vectors. If Lags is a scalar or an empty matrix, all outputs are column vectors by default.

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. ppTSTest 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.

ppARTest

price2ret