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. NaNs indicating missing values are removed.

Name-Value 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 Select Appropriate Lag Order. 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 NaNs 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

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

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.

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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

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

 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.

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

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

 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.

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

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

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

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