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:

y_t = c + δt + a y_{t –
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'`	String or cell vector of strings indicating the model variant. Values are: `'AR'` (autoregressive) `pptest` tests the null model y_t = y_{t – 1} + e(t). against the alternative model y_t = a y_{t – 1} + e(t). with AR(1) coefficient a < 1. `'ARD'` (autoregressive with drift) `pptest` tests the `'AR'` null model against the alternative model y_t = c + a y_{t – 1} + e(t). with drift coefficient c and AR(1) coefficient a < 1. `'TS'` (trend stationary) `pptest` tests the null model y_t = c + y_{t – 1} + e(t). against the alternative model y_t = c + δ t + a y_{t – 1} + e(t). with drift coefficient c, deterministic trend coefficient δ, 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 modification of the 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 modification of the "unstudentized" t statistic t₂ = 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.

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

Definitions

The Phillips-Perron model is

y_t = c + δt + a y_{t –
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).

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 Data_GDP
y = log(Data);
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.

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.

How To

Unit Root Nonstationarity

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