pptest
PhillipsPerron test for one unit root
Syntax
[h,pValue,stat,cValue,reg] = pptest(y)
[h,pValue,stat,cValue,reg] = pptest(y,'ParameterName',ParameterValue,...)
Description
PhillipsPerron 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 DickeyFuller statistics
(see adftest) to account for serial correlations
in the innovations process e(t).
y 
Vector of timeseries data. The last element is the most recent
observation. NaNs indicating missing values are
removed.

NameValue Pair Arguments
'lags' 
Scalar or vector of nonnegative integers indicating the number
of autocovariance lags to include in the NeweyWest estimator of the
longrun 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 t_{1} = (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 t_{2} = 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 unitroot null in favor of the alternative model.
Values of h equal to 0 indicate
a failure to reject the unitroot null.

pValue 
Vector of pvalues of the test statistics,
with length equal to the number of tests. pvalues
are lefttail 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 lefttail 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 pvalues 
FStat  F statistic and pvalue 
yMu  Mean of the lagadjusted input series 
ySigma  Standard deviation of the lagadjusted input series 
yHat  Fitted values of the lagadjusted input series 
res  Regression residuals 
autoCov  Estimated residual autocovariances 
NWEst  NeweyWest estimator 
DWStat  DurbinWatson 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  HannanQuinn information criterion 

Definitions
The PhillipsPerron 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
expand all
Test GDP data for a unit root using a trendstationary alternative with 0, 1, and 2 lags for the NeweyWest estimator.
Load the GDP data set.
load Data_GDP
logGDP = log(Data);
Perform the PhillipsPerron test including 0, 1, and 2 autocovariance lags in the NeweyWest robust covariance estimator.
h = pptest(logGDP,'model','TS','lags',0:2)
h =
0 0 0
Each test returns h = 0, which means the test fails to reject the unitroot null hypothesis for each set of lags. Therefore, there is not enough evidence to suggest that log GDP is trend stationary.
More About
expand all
pptest performs a leastsquares regression
to estimate coefficients in the null model.
The tests use modified DickeyFuller statistics (see adftest) to account for serial correlations
in the innovations process e(t).
PhillipsPerron 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 pvalues 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  lmctest  vratiotest