# pptest

Phillips-Perron test for one unit root

## Description

example

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

example

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

example

StatTbl = pptest(Tbl) returns the table StatTbl 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 Tbl. To select a different variable in Tbl to test, use the DataVariable name-value argument.

example

[___] = pptest(___,Name=Value) uses additional options specified by one or more name-value arguments, using any input-argument combination in the 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.

example

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

## Examples

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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.

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.

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

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

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.

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.

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.

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.

DataTable.LogGDP = log(DataTable.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(DataTable,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

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

Data Types: double

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.

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

Test NameDescription
"t1"

Modification of the standard t statistic

${t}_{1}=\frac{\stackrel{^}{\varphi }-1}{\text{SE}\left(\stackrel{^}{\varphi }\right)},$

computed using the ordinary least squares (OLS) estimate of the AR(1) coefficient $\stackrel{^}{\varphi }$ and its standard error SE($\stackrel{^}{\varphi }$), in the alternative model.

"t2"

Modification of the unstudentized t statistic

t2 = T(ϕ – 1)

${t}_{2}=\frac{T\left(\stackrel{^}{\varphi }-1\right)}{1-{\stackrel{^}{\beta }}_{1}-\dots -{\stackrel{^}{\beta }}_{p}},$

computed using the OLS estimates of the AR(1) coefficient $\stackrel{^}{\varphi }$ 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 εt.

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

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

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

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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.

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.

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.

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.

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.

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

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### Phillips-Perron Test

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

yt = c + δt + ϕyt – 1 + εt

and

• c is the drift coefficient (see Model).

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

• εt 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 ytyt−1; slow rates of decay require more lags.

The Newey-West estimator is consistent when the number of lags is O(T1/4), 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 yt. 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 yt is defined for t = 1,…,T, the lag k series ytk is defined for t = k+1,…,T. Consequently, the effective sample size of the common time base is Tk.

• To account for serial correlations in the innovations process εt, 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