# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materials including this page, select Japan from the country navigator on the bottom of this page.

# pptest

Phillips-Perron test for one unit root

## Syntax

```[h,pValue,stat,cValue,reg] = pptest(`y`)
[h,pValue,stat,cValue,reg] = pptest(`y`,`'ParameterName'`,`ParameterValue`,...)
```

## Description

Phillips-Perron tests assess the null hypothesis of a unit root in a univariate time series `y`. All tests use the model:

yt = c + δt + a yt – 1 + e(t).

The null hypothesis restricts a = 1. Variants of the test, appropriate for series with different growth characteristics, restrict the drift and deterministic trend coefficients, c and δ, respectively, to be 0. The tests use modified Dickey-Fuller statistics (see `adftest`) to account for serial correlations in the innovations process e(t).

## Input Arguments

 `y` Vector of time-series data. The last element is the most recent observation. `NaN`s indicating missing values are removed.

### Name-Value Pair Arguments

 `'lags'` Scalar or vector of nonnegative integers indicating the number of autocovariance lags to include in the Newey-West estimator of the long-run variance. For best results, give a suitable value for `lags`. For information on selecting `lags`, see Determining an Appropriate Number of Lags. Default: `0` `'model'` Character vector, such as `'AR'`, or cell vector of character vectors indicating the model variant. Values are: `'AR'` (autoregressive) `pptest` tests the null modelyt = yt – 1 + e(t).against the alternative modelyt = a yt – 1 + e(t).with AR(1) coefficient a < 1.`'ARD'` (autoregressive with drift) `pptest` tests the `'AR'` null model against the alternative modelyt = c + a yt – 1 + e(t).with drift coefficient c and AR(1) coefficient a < 1.`'TS'` (trend stationary) `pptest` tests the null modelyt = c + yt – 1 + e(t).against the alternative modelyt = c + δ t + a yt – 1 + e(t).with drift coefficient c, deterministic trend coefficient δ, and AR(1) coefficient a < 1. Default: `'AR'` `'test'` Character vector, such as `'t1'`, or cell vector of character vectors indicating the test statistic. Values are: `'t1'``pptest` computes a modification of the standard t statistict1 = (a – l)/sefrom OLS estimates of the AR(1) coefficient and its standard error (se) in the alternative model. The test assesses the significance of the restriction a – 1 = 0.`'t2'` `pptest` computes a modification of the “unstudentized” t statistict2 = T (a – 1)from an OLS estimate of the AR(1) coefficient `a` and the stationary coefficients in the alternative model. T is the effective sample size, adjusted for lag and missing values. The test assesses the significance of the restriction a – 1 = 0. Default: `'t1'` `'alpha'` Scalar or vector of nominal significance levels for the tests. Set values between `0.001` and `0.999`. Default: `0.05`

## Output Arguments

`h`

Vector of Boolean decisions for the tests, with length equal to the number of tests. Values of `h` equal to `1` indicate rejection of the unit-root null in favor of the alternative model. Values of `h` equal to `0` indicate a failure to reject the unit-root null.

`pValue`

Vector of p-values of the test statistics, with length equal to the number of tests. p-values are left-tail probabilities.

When test statistics are outside tabulated critical values, `pptest` returns maximum (`0.999`) or minimum (`0.001`) p-values.

`stat`

Vector of test statistics, with length equal to the number of tests. Statistics are computed using OLS estimates of the coefficients in the alternative model.

`cValue`

Vector of critical values for the tests, with length equal to the number of tests. Values are for left-tail probabilities.

`reg`

Structure of regression statistics for the OLS estimation of coefficients in the alternative model. The number of records equals the number of tests. Each record has the following fields:

 `num` Length of input series with `NaN`s removed `size` Effective sample size, adjusted for lags `names` Regression coefficient names `coeff` Estimated coefficient values `se` Estimated coefficient standard errors `Cov` Estimated coefficient covariance matrix `tStats` t statistics of coefficients and p-values `FStat` F statistic and p-value `yMu` Mean of the lag-adjusted input series `ySigma` Standard deviation of the lag-adjusted input series `yHat` Fitted values of the lag-adjusted input series `res` Regression residuals `autoCov` Estimated residual autocovariances `NWEst` Newey-West estimator `DWStat` Durbin-Watson statistic `SSR` Regression sum of squares `SSE` Error sum of squares `SST` Total sum of squares `MSE` Mean square error `RMSE` Standard error of the regression `RSq` R2 statistic `aRSq` Adjusted R2 statistic `LL` Loglikelihood of data under Gaussian innovations `AIC` Akaike information criterion `BIC` Bayesian (Schwarz) information criterion `HQC` Hannan-Quinn information criterion

## Examples

collapse all

Test GDP data for a unit root using a trend-stationary alternative with 0, 1, and 2 lags for the Newey-West estimator.

Load the GDP data set.

```load Data_GDP logGDP = log(Data); ```

Perform the Phillips-Perron test including 0, 1, and 2 autocovariance lags in the Newey-West robust covariance estimator.

```h = pptest(logGDP,'model','TS','lags',0:2) ```
```h = 1x3 logical array 0 0 0 ```

Each test returns `h` = 0, which means the test fails to reject the unit-root null hypothesis for each set of lags. Therefore, there is not enough evidence to suggest that log GDP is trend stationary.

## More About

collapse all

### Phillips-Perron Test

The Phillips-Perron model is

yt = c + δt + a yt – 1 + e(t).

where e(t) is the innovations process.

The test assesses the null hypothesis under the model variant appropriate for series with different growth characteristics (c = 0 or δ = 0).

## Algorithms

`pptest` performs a least-squares regression to estimate coefficients in the null model.

The tests use modified Dickey-Fuller statistics (see `adftest`) to account for serial correlations in the innovations process e(t). Phillips-Perron statistics follow nonstandard distributions under the null, even asymptotically. Critical values for a range of sample sizes and significance levels have been tabulated using Monte Carlo simulations of the null model with Gaussian innovations and five million replications per sample size. `pptest` interpolates critical values and p-values from the tables. Tables for tests of type `'t1'` and `'t2'` are identical to those for `adftest`.

## References

[1] Davidson, R., and J. G. MacKinnon. Econometric Theory and Methods. Oxford, UK: Oxford University Press, 2004.

[2] Elder, J., and P. E. Kennedy. “Testing for Unit Roots: What Should Students Be Taught?” Journal of Economic Education. Vol. 32, 2001, pp. 137–146.

[3] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[4] Newey, W. K., and K. D. West. “A Simple Positive Semidefinite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica. Vol. 55, 1987, pp. 703–708.

[5] Perron, P. “Trends and Random Walks in Macroeconomic Time Series: Further Evidence from a New Approach.” Journal of Economic Dynamics and Control. Vol. 12, 1988, pp. 297–332.

[6] Phillips, P. “Time Series Regression with a Unit Root.” Econometrica. Vol. 55, 1987, pp. 277–301.

[7] Phillips, P., and P. Perron. “Testing for a Unit Root in Time Series Regression." Biometrika. Vol. 75, 1988, pp. 335–346.

[8] Schwert, W. “Tests for Unit Roots: A Monte Carlo Investigation.” Journal of Business and Economic Statistics. Vol. 7, 1989, pp. 147–159.

[9] White, H., and I. Domowitz. “Nonlinear Regression with Dependent Observations.” Econometrica. Vol. 52, 1984, pp. 143–162.

## See Also

### Topics

#### Introduced in R2009b

Was this topic helpful?

Download ebook