UnitrootTests {fSeries}R Documentation

Unit Root and Cointegration Time Series Tests

Description

A collection and description of functions for unit root testing. The family of tests includes ADF tests based on Banerjee's et al. tables and on J.G. McKinnons' numerical distribution functions. In addition we have included functions from the contributed R package tseries written by Adrian Trapletti and from the package urca written by Bernhard Pfaff.

The functions are:

adfTest Augmented Dickey–Fuller test for unit roots,
unitrootTest the same based on McKinnons's test statistics.

Added functions based on the 'tseries' package include:

tsadfTest Augmented Dickey–Fuller test for unit roots,
tskpssTest KPSS test for level or trend stationarity,
tsppTest Phillips–Perron test for unit roots,
tspoTest Phillips–Ouliaris test for cointegration.

Added functions based on the 'urca' package include:

urersTest Elliott–Rothenberg–Stock test for unit roots,
urkpssTest KPSS unit root test for stationarity,
urppTest Phillips–Perron test for unit roots,
urspTest Schmidt–Phillips test for unit roots,
urzaTest Zivot–Andrews test for unit roots.

NOTE: The contributed R packages tseries and urca are not required, the functions are builtin.

Usage

urTest(x, method = c("unitroot", "adf", "tsadf", "tskpss", "tspp", 
        "urers", "urkpss", "urpp", "ursp", "urza"), title = NULL, 
        description = NULL, ...)

adfTest(x, type = c("nc", "c", "ct"), lags = 1)
unitrootTest(x, trend = c("nc", "c", "ct"), statistic = c("t", "n"), 
        method = "adf", lags = 1)

tsadfTest(x, alternative = c("stationary", "explosive"), 
        k = trunc((length(x)-1)^(1/3)))
tskpssTest(x, nullhyp = c("level", "trend"), lshort = TRUE) 
tsppTest(x, alternative = c("stationary", "explosive"), 
        type = c("Z(alpha)", "Z(t_alpha)"), lshort = TRUE)
tspoTest(x, demean = TRUE, lshort = TRUE)

urersTest(x, type = c("DF-GLS", "P-test"), model = c("constant", "trend"),
        lag.max = 4)
urkpssTest(x, type = c("mu", "tau"), lags = c("short", "long", "nil"),
    use.lag = NULL)
urppTest(x, type = c("Z-alpha", "Z-tau"), model = c("constant", "trend"),
        lags = c("short", "long"))
urspTest(x, type = c("tau", "rho"), pol.deg = c(1, 2, 3, 4),
        signif = c(0.01, 0.05, 0.1))
urzaTest(x, model = c("intercept", "trend", "both"), lag)

## S3 method for class 'fURTEST':
print(x, ...)
## S3 method for class 'fURTEST':
summary(object, ...)

Arguments

alternative [tsadfTest] -
the argument indicates an alternative hypothesis. The choice must be one of the elements specified in the input vector, by default the first entry is selected.
[tsppTest] -
indicates the alternative hypothesis and must be one of "stationary", the default, or "explosive". One can specify just the initial letter.
demean [tspoTest] -
a logical indicating whether an intercept is included in the cointegration regression or not.
description a character string which allows for a brief description.
k [tsadfTest] -
the lag order to calculate the test statistic.
lag.max [urersTest] -
the maximum numbers of lags used for testing of a decent lag truncation for the "P-test", BIC used, or the maximum number of lagged differences to be included in the test regression for "DF-GLS".
lag [urzaTest] -
the highest number of lagged endogenous differenced variables to be included in the test regression.
lags [urkpssTest][urppTest] -
the maximum number of lags used for error term correction.
lshort [tsppTest] -
a logical indicating whether the short or long version of the truncation lag parameter is used.
[tspoTest] -
a logical indicating whether the short or long version of the truncation lag parameter is used.
method [urTest] -
a character string describing the desired method, one of: "unitroot", "adf", "tsadf", "tskpss", "tspp", "urers", "urkpss", "urpp", "ursp", "urza", or "W3SLS".
[unitrootTest] -
a character string specifying the method of the test. Valid choices are "adf" for the augmented Dickey-Fuller test, and "pp" for Phillips-Perron test. The default is "adf".
model [urersTest] -
a character string dennoting the deterministic model used for detrending, either "constant", the default, or "trend".
[urppTest] -
a character string which determines the deterministic part in the test regression, either "constant", the default, or "trend".
[urzaTest] -
a character string specifying if the potential break occured in either the "intercept", the linear "trend" or in "both".
nullhyp [tskpssTest] -
a character string which indicates the null hypothesis and must be one of "level" (default) or "trend". One can specify just the initial letter.
object an object of class "fURTEST".
pol.deg [urspTest] -
the polynomial degree in the test regression.
signif [urspTest] -
the significance level for the critical value of the test statistic.
statistic [unitrootTest] -
a character string specifying the test statistic. Valid choices are "t" for the t-statistic, the default, and "n" for the normalized statistic, sometimes also referred to as the rho-statistic.
title a character string which allows for a project title.
trend [unitrootTest] -
a character string describing the type of the unit root regression. Valid choices are "nc" for a regression with no intercept (constant) nor time trend, and "c" for a regression with an intercept (constant) but no time trend, "ct" for a regression with an intercept (constant) and a time trend. The default is "c".
type [tsppTest] -
indicates which variant of a test will be computed. The choice must be one of the elements specified in the input vector, by default the first entry is selected.
[urkpssTest] -
a character string which denotes the type of deterministic part, either "mu", the default, or "tau". [urppTest] -
a character string which specifies the test type, either "Z-alpha", the default, or "Z-tau". [urspTest] -
a character string which specifies the test type, either "tau", the default, or "rho".
use.lag [urkpssTest] -
a character string specifying the number of lags. Allowed arguments are lags=c("short", "long", "nil"), for more information see the details section.
x a numeric vector or time series object.
... [urTest] -
optional arguments passed to the underlying test functions.

Details

ADF Tests:

The adftest computes test statistics and p values using the implementation from Trapletti's augmented Dickey–Fuller test for unit roots, see below. In contrast to the function tsadfTest three kind of test types can be selected.

Unit Root Tests from Adrian Trapletti's "tseries" Package:

Augmented Dickey–Fuller Test for Unit Roots:
The general regression equation which incorporates a constant and a linear trend is used and the t-statistic for a first order autoregressive coefficient equals one is computed. The number of lags used in the regression is k. The default value of trunc((length(x)-1)^(1/3)) corresponds to the suggested upper bound on the rate at which the number of lags, k, should be made to grow with the sample size for the general ARMA(p,q) setup. Note that for k equals zero the standard Dickey-Fuller test is computed. The p-values are interpolated from Table 4.2, p. 103 of Banerjee et al. (1993). If the computed statistic is outside the table of critical values, then a warning message is generated. Note, that missing values are not allowed.
[tseries:adf.test]

Kwiatkowski–Phillips–Schmidt–Shin Test:
To estimate sigma^2 the Newey-West estimator is used. If lshort is TRUE, then the truncation lag parameter is set to trunc(3*sqrt(n)/13), otherwise trunc(10*sqrt(n)/14) is used. The p-values are interpolated from Table 1 of Kwiatkowski et al. (1992). If the computed statistic is outside the table of critical values, then a warning message is generated.
[tseries:kpss.test]

Phillips–Ouliaris Test:
The poTest Z(alpha) statistic for a unit root in the residuals of the cointegration regression is computed, see also ppTest. The unit root is estimated from a regression of the first variable (column) of x on the remaining variables of x without a constant and a linear trend. To estimate sigma^2 the Newey-West estimator is used. If lshort is TRUE, then the truncation lag parameter is set to trunc(n/100), otherwise trunc(n/30) is used. The p-values are interpolated from Table Ia and Ib, p. 189 of Phillips and Ouliaris (1990). If the computed statistic is outside the table of critical values, then a warning message is generated. The dimension of x is restricted to six variables.
[tseries:po.test]

Phillips–Perron Test:
The general regression equation which incorporates a constant and a linear trend is used and the Z(alpha) or Z(t_alpha) statistic for a first order autoregressive coefficient equals one are computed. To estimate sigma^2 the Newey-West estimator is used. If lshort is TRUE, then the truncation lag parameter is set to trunc(4*(n/100)^0.25), otherwise trunc(12*(n/100)^0.25) is used. The p-values are interpolated from Table 4.1 and 4.2, p. 103 of Banerjee et al. (1993). If the computed statistic is outside the table of critical values, then a warning message is generated.
[tseries:pp.test]

Unit Root Tests from Berhard Pfaff's "urca" Package:

Elliott–Rothenberg–Stock Test for Unit Roots:
To improve the power of the unit root test, Elliot, Rothenberg and Stock proposed a local to unity detrending of the time series. ERS developed a feasible point optimal test, "P-test", which takes serial correlation of the error term into account. The second test type is the "DF-GLS" test, which is an ADF-type test applied to the detrended data without intercept. Critical values for this test are taken from MacKinnon in case of model="constant" and else from Table 1 of Elliot, Rothenberg and Stock.
[urca:ur.ers]

KPSS Test for Unit Roots:
Performs the KPSS unit root test, where the Null hypothesis is stationarity. The test types specify as deterministic component either a constant "mu" or a constant with linear trend "tau". lags="short" sets the number of lags to root 4 of [4 times (n/100), whereas lags="long" sets the number of lags to root 4 of [12 times (n/100)]. If lags="nil" is choosen, then no error correction is made. Furthermore, one can specify a different number of maximum lags by setting use.lag accordingly.
[urca:ur.kpss]

Phillips–Perron Test for Unit Roots:
Performs the Phillips and Perron unit root test. Beside the Z statistics Z-alpha and Z-tau, the Z statistics for the deterministic part of the test regression are computed, too. For correction of the error term a Bartlett window is used.
[urca:ur.pp]

Schmidt–Phillips Test for Unit Roots:
Performs the Schmidt and Phillips unit root test, where under the Null and Alternative Hypothesis the coefficients of the deterministic variables are included. Two test types are available: the "rho-test" and the "tau-test". Both tests are extracted from the LM principle.
[urca:ur.sp]

Zivot–Andrews Test for Unit Roots:
Performs the Zivot and Andrews unit root test, which allows a break at an unknown point in either the intercept, the linear trend or in both. This test is based upon the recursive estimation of a test regression. The test statistic is defined as the minimum t-statistic of the coeffcient of the lagged endogenous variable.
[urca:ur.za]

Value

All tests return an object of class "fURTEST" with the following slots:

@call the function call.
@data a data frame with the input data.
@data.name a character string giving the name of the data.
@test a list object which holds the output of the underlying test function, usually the same entries as an object of class "htest".
@title a character string with the name of the test.
@description a character string with a brief description of the test.
$statistic the value of the test statistic.
$parameter the lag order.
$p.value the p-value of the test.
$method a character string indicating what type of test was performed.
$data.name a character string giving the name of the data.
$alternative a character string describing the alternative hypothesis.

Note

The output of the various hypothesis tests is an object of class fURTEST. The associated print method gives an unique report about the test results for all tests.

Author(s)

Adrian Trapletti for the tests from R's "tseries" package,
Bernhard Pfaff for the tests from R's "urca" package,
Diethelm Wuertz for the Rmetrics R-port.

References

Banerjee A., Dolado J.J., Galbraith J.W., Hendry D.F. (1993); Cointegration, Error Correction, and the Econometric Analysis of Non-Stationary Data, Oxford University Press, Oxford.

Dickey, D.A., Fuller, W.A. (1979); Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association 74, 427–431.

Kwiatkowski D., Phillips P.C.B, Schmidt P., Shin Y. (1992); Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root, Journal of Econometrics 54, 159–178.

MacKinnon, J.G. (1996); Numerical distribution functions for unit root and cointegration tests, Journal of Applied Econometrics 11, 601–618.

Perron P. (1988); Trends and Random Walks in Macroeconomic Time Series, Journal of Economic Dynamics and Control 12, 297–332.

Phillips P.C.B., Ouliaris S. (1990); Asymptotic Properties of Residual Based Tests for Cointegration, Econometrica 58, 165–193.

Phillips P.C.B., Perron P. (1988); Testing for a unit root in time series regression, Biometrika 75, 335–346.

Said S.E., Dickey D.A. (1984); Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order, Biometrika 71, 599–607.

Schwert G.W. (1989); Tests for Unit Roots: A Monte Carlo Investigation, Journal of Business and Economic Statistics 2, 147–159.

See Also

Fur further details we refer to the manual pages of the "urca" package.

classicalTests, tseriesTests, lmTests.

Examples

## Unit Root Tests build on Adrian Trapletti's Implementation:

## tsadfTest - 
   xmpSeries("\nStart: Augmented Dickey-Fuller Test for Unit Roots >")
   # A time series which contains no unit-root:
   x = rnorm(1000)  
   tsadfTest(x)
   # A time series which contains a unit-root:
   y = diffinv(x)
   tsadfTest(y)

## tskpssTest -
   xmpSeries("\nNext: KPSS test for Level or Trend Stationary >")
   # Time Series is Level Stationary:
   x = rnorm(1000)
   tskpssTest(x)
   # Time Series has Unit Root:
   tskpssTest(cumsum(x))
   # Time Series is Trend Stationary:
   x = 0.3*(1:1000) + rnorm(1000)
   tskpssTest(x, nullhyp = "trend")
   
## tspoTest -
   xmpSeries("\nNext: Phillips-Ouliaris Test for Cointegration >")
   # Non-Cointegrated Case:
   x = ts(diffinv(matrix(rnorm(2000), 1000, 2)))
   tspoTest(x)
   # Cointegrated Case:
   x = diffinv(rnorm(1000))
   y = 2.0 - 3.0 * x + rnorm(x, sd = 5)
   z = ts(cbind(x, y))
   tspoTest(z)
   
## tsppTest -
   xmpSeries("\nNext: Phillips-Perron Test for Unit Roots >")
   # The Time Series has no Unit Root:
   x = rnorm(1000)
   tsppTest(x)
   # The Time Series has Unit Root:
   y = cumsum(x)  
   tsppTest(y)     
 
## Unit Root Tests build on Bernhard Pfaff's Implementation:
    
## ur*Test - 
   # Examples can be found in the demo file "xmpTestUnitRoots".  

[Package fSeries version 200.10058 Index]