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pptest - Phillips-Perron unit root test

Syntax

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

Description

h = pptest(y) assesses the null hypothesis of a unit root in a univariate time series y.

h = pptest(y,'ParameterName',ParameterValue,...) accepts optional inputs as one or more comma-separated parameter-value pairs. 'ParameterName' is the name of the parameter inside single quotation marks. ParameterValue is the value corresponding to 'ParameterName'. Specify parameter-value pairs in any order; names are case-insensitive. Perform multiple tests by passing a vector value for any parameter. Multiple tests yield vector results.

[h,pValue] = pptest(...) returns p-values of the test statistics.

[h,pValue,stat] = pptest(...) returns the test statistics.

[h,pValue,stat,cValue] = pptest(...) returns critical values for the tests.

[h,pValue,stat,cValue,reg] = pptest(...) returns a structure of regression statistics.

Inputs

y	Vector of time-series data. The last element is the most recent observation. The test ignores NaN values, which indicate missing entries.

Parameter/Value Pairs

'alpha'	Scalar or vector of nominal significance levels for the tests. Set values between 0.001 and 0.999. Default: 0.05
'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 Determine Appropriate Lags. Default: 0
'model'	String or cell vector of strings indicating the model variant. Values are: 'AR' (autoregressive) pptest tests the null model yt = yt – 1 + e(t). against the alternative model yt = a yt – 1 + e(t). with AR(1) coefficient a < 1. 'ARD' (autoregressive with drift) pptest tests the 'AR' null model against the alternative model yt = c + a yt – 1 + e(t). with drift coefficient c and AR(1) coefficient a < 1. 'TS' (trend stationary) pptest tests the null model yt = c + yt – 1 + e(t). against the alternative model yt = c + d t + a yt – 1 + e(t). with drift coefficient c, deterministic trend coefficient d, and AR(1) coefficient a < 1. Default: 'AR'
'test'	String or cell vector of strings indicating the test statistic. Values are: 't1' pptest computes a standard t statistic t1 = (a – l)/se from 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 lag-adjusted, "unstudentized" t statistic t2 = T (a – 1)/(1– b1– ...– bp) from OLS estimates of the AR(1) coefficient and the stationary coefficients in the alternative model. T is the effective sample size, adjusted for lags and missing values. The test assesses the significance of the restriction a – 1 = 0. Default: 't1'

Outputs

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.
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. When the value of 'test' is 't1' or 't2', critical values are for left-tail probabilities. When the value of 'test' is 'F', critical values are for right-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 R^2 statistic aRSq Adjusted R^2 statistic LL Loglikelihood of data under Gaussian innovations AIC Akaike information criterion BIC Bayesian (Schwarz) information criterion HQC Hannan-Quinn information criterion

Definitions

The Phillips-Perron model is

y_t = c + dt + a y_{t – 1} + e(t).

where e(t) is the innovations process.

The null hypothesis restricts a = 1.

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

Examples

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

load unitRootData
y = log(GDP);
h = pptest(y,'model','TS','lags',0:2)

The result

h =
     0     0     0

means the test fails to reject the unit-root null for each set of lags.

Algorithm

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. For small samples, values are valid only for Gaussian innovations; for large samples, values are also valid for non-Gaussian innovations. 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

adftest | kpsstest | vratiotest

How To

• Unit Root Tests

ppARTest

ppTSTest

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