pptest

Phillips-Perron test for one unit root

Syntax

h = pptest(y)

[h,pValue,stat,cValue] = pptest(y)

StatTbl = pptest(Tbl)

[___] = pptest(___,Name=Value)

[___,reg] = pptest(___)

Description

h = pptest(y) returns the rejection decision from conducting the Phillips-Perron test for a unit root in the input univariate time series.

[h,pValue,stat,cValue] = pptest(y) also returns the p-value pValue, test statistic stat, and critical value cValue of the test.

StatTbl = pptest(Tbl) returns a table containing variables for the test results, statistics, and settings from conducting the Phillips-Perron test on the last variable of the input table or timetable. To select a different variable to test, use the DataVariable name-value argument.

[___] = pptest(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. pptest returns the output argument combination for the corresponding input arguments.

Some options control the number of tests to conduct. The following conditions apply when pptest conducts multiple tests:

• pptest treats each test as separate from all other tests.

• If you specify y, all outputs are vectors.

• If you specify Tbl, each row of StatTbl contains the results of the corresponding test.

For example, pptest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two tests, at a level of significance of 0.025, on the variable GDP of the table Tbl. The first test includes 0 autocovariance lags in the Newey-West covariance estimator and the second test includes 1 autocovariance lags.

[___,reg] = pptest(___) additionally returns a structure of regression statistics for the hypothesis test.

Examples

Conduct Phillips-Perron Test on Vector of Data

Test a time series for a unit root using the default options of pptest. Input the time series data as a numeric vector.

Load the Canadian inflation rate data and extract the CPI-based inflation rate INF_C.

load Data_Canada
y = DataTable.INF_C;

Test the time series for a unit root.

h = pptest(y)

h = logical
   0

The result h = 0 indicates that this test fails to reject the null hypothesis of a unit root against the AR(1) alternative.

Return Test p-Value and Decision Statistics

Load Canadian inflation rate data and extract the CPI-based inflation rate INF_C.

load Data_Canada
y = DataTable.INF_C;

Test the time series for a unit root. Return the test decision, $\mathit{p}$-value, test statistic, and critical value.

[h,pValue,stat,cValue] = pptest(y)

h = logical
   0

pValue = 
0.3255

stat = 
-0.8769

cValue = 
-1.9476

Conduct Phillips-Perron Test on Table Variable

Test a time series, which is one variable in a table, for a unit root using default options.

Load Canadian inflation rate data, which contains yearly measurements on five time series variables in the table DataTable.

load Data_Canada

Test the long-term bond rate series INT_L, the last variable in the table, for a unit root.

StatTbl = pptest(DataTable)

StatTbl=1×8 table
                h      pValue     stat      cValue     Lags    Alpha    Model      Test 
              _____    ______    _______    _______    ____    _____    ______    ______

    Test 1    false    0.7358    0.24601    -1.9476     0      0.05     {'AR'}    {'T1'}

pptest returns test results and settings in the table StatTbl, where variables correspond to test results (h, pValue, stat, and cValue) and settings (Lags, Alpha, Model, and Test), and rows correspond to individual tests (in this case, pptest conducts one test).

By default, pptest tests the last variable in the table. To select a variable from an input table to test, set the DataVariable option.

Assess Stationarity Using Phillips-Perron Test

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 = 1×3 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.

Inspect Regression Statistics

Test a time series for a unit root against trend-stationary alternatives. Inspect the regression statistics corresponding to each of the tests.

Load a US macroeconomic data set Data_USEconModel.mat. Compute the log of the GDP and include the result as a new variable called LogGDP in the data set.

load Data_USEconModel
DataTimeTable.LogGDP = log(DataTimeTable.GDP);

Test for a unit root in the logged GDP series using three different choices for the number of lagged difference terms. Return the regression statistics for each alternative model.

[StatTbl,reg] = pptest(DataTimeTable,DataVariable="LogGDP",Model="TS",Lags=0:2);
StatTbl

StatTbl=3×8 table
                h      pValue      stat      cValue     Lags    Alpha    Model      Test 
              _____    _______    _______    _______    ____    _____    ______    ______

    Test 1    false      0.999     1.0247    -3.4302     0      0.05     {'TS'}    {'T1'}
    Test 2    false      0.999    0.56702    -3.4302     1      0.05     {'TS'}    {'T1'}
    Test 3    false    0.99829    0.31644    -3.4302     2      0.05     {'TS'}    {'T1'}

pptest treats each of the three lag choices as separate tests, and returns results and settings for each test along the rows of the table StatTbl. reg is a 3-by-1 structure array containing regression statistics corresponding to each of the three alternative models.

Display the names of the coefficients, and their estimates and corresponding $\mathit{p}$-values resulting from the regressions of the three alternative models.

test1 = array2table([reg(1).coeff reg(1).tStats.pVal], ...
    RowNames=reg(1).names,VariableNames=["Coeff" "pValue"])

test1=3×2 table
            Coeff         pValue   
         ___________    ___________

    c      -0.014026         0.6654
    d    -0.00013104        0.22383
    a         1.0061    8.5908e-255

test2 = array2table([reg(2).coeff reg(2).tStats.pVal], ...
    RowNames=reg(2).names,VariableNames=["Coeff" "pValue"])

test2=3×2 table
            Coeff         pValue   
         ___________    ___________

    c      -0.014026         0.6654
    d    -0.00013104        0.22383
    a         1.0061    8.5908e-255

test3 = array2table([reg(3).coeff reg(3).tStats.pVal], ...
    RowNames=reg(3).names,VariableNames=["Coeff" "pValue"])

test3=3×2 table
            Coeff         pValue   
         ___________    ___________

    c      -0.014026         0.6654
    d    -0.00013104        0.22383
    a         1.0061    8.5908e-255

The coefficients c and d are the drift and deterministic trend, respectively, in the alternative model. a is the lag 1 AR term in the null and alternative models. Although each test uses a different number of lags, all coefficient estimates are the same. This result occurs because Lags factors into the Newey-West estimate of the long-run variance, not the model itself.

Compare the Newey-West variance estimates among the tests.

reg.NWEst

ans = 
1.2352e-04

ans = 
1.8025e-04

ans = 
2.2355e-04

Input Arguments

y — Univariate time series data

numeric vector

Univariate time series data, specified as a numeric vector. Each element of y represents an observation.

Data Types: double

Tbl — Time series data

table | timetable

Time series data, specified as a table or timetable. Each row of Tbl is an observation.

Specify a single series (variable) to test by using the DataVariable argument. The selected variable must be numeric.

Note

pptest removes missing observations, represented by NaN values, from the input series.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: pptest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two tests, at a level of significance of 0.025, for the presence of a unit root in the variable GDP of the table Tbl. The first test includes 0 autocovariance lags in the Newey-West estimator of the long-run variance and the second test includes 1 autocovariance lag.

Lags — Number of autocovariance lags

0 (default) | nonnegative integer | vector of nonnegative integers

Number of autocovariance lags to include in the Newey-West estimator of the long-run variance, specified as a nonnegative integer or vector of nonnegative integers. If Lags(j) > 0, pptest includes lags 1 through Lags(j) in the estimator for test j.

pptest conducts a separate test for each element in Lags.

Example: Lags=0:2 includes zero lagged autocovariance terms in the Newey-West estimator for the first test, the lag 1 autocovariance term for the second test, and autocovariance lags 1 and 2 in the third test.

Data Types: double

Model — Model variant

"AR" (default) | "ARD" | "TS" | character vector | string vector | cell vector of character vectors

Model variant, specified as a model variant name, or a string vector or cell vector of model names. This table contains the supported model variant names.

Model Variant Name	Description
"AR"	Autoregressive model variant, which specifies a test of the null model yt = yt – 1 + εt against the alternative model yt = ϕyt – 1 + εt with AR(1) coefficient |ϕ| < 1.
"ARD"	Autoregressive model with drift variant, which specifies a test of the null model yt = yt – 1 + εt against the alternative model yt = c + ϕyt – 1 + εt with drift coefficient c and AR(1) coefficient |ϕ| < 1.
"TS"	Trend-stationary model variant, which specifies a test of the null model yt = yt – 1 + εt against the alternative model yt = c + + ϕyt – 1 + εt with drift coefficient c, deterministic trend coefficient δ, and AR(1) coefficient |ϕ| < 1.

pptest conducts a separate test for each model variant name in Model.

Example: Model=["AR" "ARD"] uses the stationary AR model as the alternative hypothesis for the first test, and then uses the stationary AR model with drift as the alternative hypothesis for the second test.

Data Types: char | cell | string

Test — Test statistic

"t1" (default) | "t2" | character vector | string vector | cell vector of character vectors

Test statistic, specified as a test name, or a string vector or cell vector of test names. This table contains the supported test names.

Test Name	Description
"t1"	Modification of the standard t statistic $$t_1=\frac{\widehat{\varphi }-1}{\text{SE}\left(\widehat{\varphi }\right)},$$ computed using the ordinary least squares (OLS) estimate of the AR(1) coefficient $$\widehat{\varphi }$$ and its standard error SE($$\widehat{\varphi }$$), in the alternative model.
"t2"	Modification of the unstudentized t statistic t2 = T(ϕ – 1) $$t_2=\frac{T(\widehat{\varphi }-1)}{1-{\widehat{\beta }}_1-\dots -{\widehat{\beta }}_p},$$ computed using the OLS estimates of the AR(1) coefficient $$\widehat{\varphi }$$ in the alternative model. T is the effective sample size, which is adjusted for lags and missing values.

The test assesses the significance of the restriction $$\varphi -1=0.$$ pptest modifies the test statistics to account for serial correlations in the innovations process ε_t.

pptest conducts a separate test for each test name in Test.

Example: Test="t2" computes the F test statistic for all tests.

Data Types: char | cell | string

Alpha — Nominal significance level

0.05 (default) | numeric scalar | numeric vector

Nominal significance level for the hypothesis test, specified as a numeric scalar between 0.001 and 0.999 or a numeric vector of such values.

pptest conducts a separate test for each value in Alpha.

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

Data Types: double

DataVariable — Variable in Tbl to test

last variable (default) | string scalar | character vector | integer | logical vector

Variable in Tbl to test, specified as a string scalar or character vector containing a variable name in Tbl.Properties.VariableNames, or an integer or logical vector representing the index of a name. The selected variable must be numeric.

Example: DataVariable="GDP"

Example: DataVariable=[false true false false] or DataVariable=2 tests the second table variable.

Data Types: double | logical | char | string

Note

• When pptest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.

• All vector-valued specifications that control the number of tests must have equal length.

• If you specify the vector y and any value is a row vector, all outputs are row vectors.

Output Arguments

h — Test rejection decisions

logical scalar | logical vector

Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests. pptest returns h when you supply the input y.

• Values of 1 indicate rejection of the unit-root null hypothesis in favor of the alternative.

• Values of 0 indicate failure to reject the unit-root null hypothesis.

pValue — Test statistic p-values

numeric scalar | numeric vector

Test statistic p-values, returned as a numeric scalar or vector with length equal to the number of tests. pptest returns pValue when you supply the input y.

The 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 — Test statistics

numeric scalar | numeric vector

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests. pptest returns stat when you supply the input y.

pptest computes test statistics using OLS estimates of the coefficients in the alternative model.

cValue — Critical values

numeric scalar | numeric vector

Critical values, returned as a numeric scalar or vector with length equal to the number of tests. pptest returns cValue when you supply the input y.

The critical values are for left-tail probabilities.

StatTbl — Test summary

table

Test summary, returned as a table with variables for the outputs h, pValue, stat, and cValue, and with a row for each test. pptest returns StatTbl when you supply the input Tbl.

StatTbl contains variables for the test settings specified by Lags, Alpha, Model, and Test.

reg — Regression statistics

structure array

Regression statistics from the OLS estimation of coefficients in the alternative model, returned as a structure array with number of records equal to the number of tests.

Each element of reg has the fields in this table. You can access a field using dot notation, for example, reg(1).coeff contains the coefficient estimates of the first test.

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

More About

Phillips-Perron Test

The Phillips-Perron test assesses the null hypothesis of a unit root in a univariate time series y_t, where

y_t = c + δt + ϕy_{t – 1} + ε_t

and

• c is the drift coefficient (see Model).

• δ is the deterministic trend coefficient (see Model).

• ε_t is a mean zero innovation process.

The null hypothesis of a unit root restricts ϕ = 1. The alternative hypothesis is ϕ < 1. A test that fails to reject the null hypothesis, fails to reject the possibility of a unit root.

Variants of the model allow for different growth characteristics (see Model). The model with δ = 0 has no trend component, and the model with c = 0 and δ = 0 has no drift or trend.

Tips

• To draw valid inferences from a Phillips-Perron test, you must determine a suitable value for the Lags argument. The following methods help determine a suitable value:

  • Begin by setting a small value and then evaluate the sensitivity of the results by adding more lags.

  • Inspect sample autocorrelations of y_t − y_t−1; slow rates of decay require more lags.

  The Newey-West estimator is consistent when the number of lags is O(T^1/4), where T is the effective sample size, adjusted for lag and missing values. For more details, see [9] and [5].

• With a specific testing strategy in mind, determine the value of Model by the growth characteristics of y_t. If you include too many regressors (see Lags), the test loses power; if you include too few regressors, the test is biased towards favoring the null model [2]. In general, if a series grows, the "TS" model (see Model) provides a reasonable trend-stationary alternative to a unit-root process with drift. If a series is does not grow, the "AR" and "ARD" models provide reasonable stationary alternatives to a unit-root process without drift. The "ARD" alternative model has a mean of c/(1 – a); the "AR" alternative model has mean 0.

Algorithms

• In general, when a time series is lagged, the sample size is reduced. Without a presample, if y_t is defined for t = 1,…,T, the lag k series y_t–k is defined for t = k+1,…,T. Consequently, the effective sample size of the common time base is T − k.

• To account for serial correlations in the innovations process ε_t, pptest uses modified Dickey-Fuller statistics (see adftest).

• Phillips-Perron statistics stat follow nonstandard distributions under the null, even asymptotically. pptest uses tabulated critical values, generated by Monte Carlo simulations, for a range of sample sizes and significance levels of the null model with Gaussian innovations and five million replications per sample size. pptest interpolates critical values cValue and p-values pValue from the tables. Tables for tests of Test types "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, James 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.

Version History

Introduced in R2009b