Unit Root Tests for Seasonal Time Series

Description

Canova and Hansen (CH) test for seasonal stability and Hylleberg, Engle, Granger and Yoo (HEGY) test for seasonal unit roots.

Details

Version >= 2.0.0 is a revival of the former package uroot. Some of the functions provided in the original versions have been coded from the scratch.

The current version provides the Canova and Hansen (CH) test for seasonal stability and the Hylleberg, Engle, Granger and Yoo (HEGY) test for seasonal unit roots.

New features:

The original functions have been enhanced with the following new features: 1) the tests are now applicable to series of any seasonal periodicity (not only quarterly and monthly data), 2) p-values based on response surface regressions are available, 3) bootstrapped p-values are available for the HEGY test statistics.

Old features not currently supported:

Some of the utilities available in the initial versions of the package are not available now: graphics for seasonal series, graphical user interface, bootstrap versions of the statistics, ADF test and KPSS test. Some of these utilities are available in other packages and some will probably be incorporated in future versions of this package.

System requirements:

Windows systems: GPU parallelization of the bootstrap is not currently operational on windows systems.

Unix systems: GPU parallelization of the bootstrap requires a CUDA capable GPU with compute capability >= 3.0.

Author(s)

Javier López-de-Lacalle javlacalle@yahoo.es.

References

Burridge, P. and Taylor, R. (2004) "Bootstrapping the HEGY seasonal unit root tests." Journal of Econometrics 123(1), pp. 67-87. DOI: doi:10.1016/j.jeconom.2003.10.029.

Canova, F. and Hansen, Bruce E. (1995) "Are seasonal patterns constant over time? A test for seasonal stability". Journal of Business & Economic Statistics, 13(3), pp. 237-252. DOI: doi:10.1080/07350015.1995.10524598.

Díaz-Emparanza, I. (2014) "Numerical distribution functions for seasonal unit root tests"- Computational Statistics and Data Analysis, 76, pp. 237-247. DOI: doi:10.1016/j.csda.2013.03.006.

Díaz-Emparanza, I. and Moral, M. P. (2013). Seasonal stability tests in gretl. An application to international tourism data. Working paper: Biltoki D.T. 2013.03.

Hylleberg, S., Engle, R., Granger, C. and Yoo, B. (1990) "Seasonal integration and cointegration". Journal of Econometrics 44(1), pp. 215-238. DOI: doi:10.1016/0304-4076(90)90080-D.

BGT-data Sample Data Set

Description

UK macroeconomic data. Sample data set employed in Burridge, Gjorstrup and Taylor (2004).

Usage

bgt.data

Format

A list containing time series objects.

References

Burridge, P., Gjorstrup, F. and Taylor, R. (2004) Robust Inference on Seasonal Unit Roots via a Bootstrap Applied to OECD Macroeconomic Series. Department of Economics, City University London. Working Paper 04/08. URL: https://ideas.repec.org/p/cty/dpaper/04-08.html.

CH-data Sample Data Set

Description

U.S. post World War II macroeconomic time series. The data set is described and employed in Canova and Hansen (JBES, 1995, Section 5.1).

Usage

ch.data

Format

A list containing time series objects.

Source

https://www.ssc.wisc.edu/~bhansen/progs/jbes_95.html

P-values for the CH test statistic

Description

Compute p-values for the Canova and Hansen (CH) test statistic based on the response surface regressions approach.

Usage

ch.rs.pvalue(x, type, lag1, S, n, nobsreg, VMdf)

Arguments

Details

Ported from Gretl code provided by Díaz-Emparanza and Moral (2013).

For type="dummy", the p-value for the joint test statistic is not available.

This function is mainly intended to be used internally by ch.test.

Value

A numeric giving the calculated p-value.

References

Díaz-Emparanza, I. and Moral, M. P. (2013) Seasonal Stability Tests in gretl. An Application to International Tourism Data. Working paper: Biltoki D.T. 2013.03. URL: https://addi.ehu.es/handle/10810/10577. Gretl code: https://www.ehu.eus/ignacio.diaz-emparanza/packages/Canova_Hansen.gfn (seems unavailable, so not linked)

See Also

ch.test.

Canova and Hansen Test for Seasonal Stability

Description

Canova and Hansen (CH) test statistic for the null hypothesis of a stable seasonal pattern.

Usage

ch.test(x, type = c("dummy", "trigonometric"), lag1 = FALSE, NW.order = NULL, 
  sid = NULL, xreg = NULL, pvalue = c("RS", "raw"), rs.nobsreg = 13)

Arguments

Details

The seasons or seasonal cycles to be tested can be chosen through an indicator variable defined in the argument sid. By default, all the t-statistics related to each individual dummy or cycle and the joint F-statistic are returned. If type = "dummy", the index of the target seasons can be specified in sid. For example, in a quarterly series: sid=c(2) returns the test statistic to the stability of the second quarter; sid=c(1,3) returns the joint test statistic for the first and third quarters; sid=c(1,2,3,4) returns the joint test statistic for the null of seasonal stability at all seasons. If type = "trigonometric", the indicator vector sid must be of length floor(frequency(x)/2) and will consist of ones and zeros. Each element in sid is related to each seasonal cycle according to the same order in which the seasonal frequencies, w_j, are defined: w_j=2\pi j/S, j=1,...,Sh, where S is the periodicity and Sh is floor(frequency(x)/2). For example, in a monthly series: sid=c(0,0,0,0,0,1) returns the test statistic to the stability of the cycle with frequency w_6=\pi; sid=c(1,0,0,0,0,1) returns the joint test statistic for cycles related to frequencies w_1=\pi/6 and w_6=\pi; sid=c(1,1,1,1,1,1) returns the joint test statistic for the stability of all seasonal cycles. The following keywords are also admitted: sid="all", computes all the test statistic related to each individual season or cycle as well as the joint test statistic for all seasons or cycles; sid="joint" computes the joint test statistic for all seasons or cycles.

Value

A list of class "CHtest" with components:

The method print displays the test statistics and p-values; summary shows the same output and includes the fitted regression model.

Note

When type = "dummy", the p-value for the joint test statistic based on response surface regressions is not available. If pvalue = "RS", the p-value reported for the joint test statistic in the trigonometric version is based on the tables given in the reference paper, Canova and Hansen (1995).

When sid is a numeric (selected combination of dummies or cycles), the reported p-values are based on interpolation in tables; if pvalue = "RS", it is changed to "raw" and a warning is given.

References

Canova, F. and Hansen, Bruce E. (1995) "Are seasonal patterns constant over time? A test for seasonal stability". Journal of Business & Economic Statistics, 13(3), pp. 237-252. DOI: doi:10.1080/07350015.1995.10524598.

Díaz-Emparanza, I. and Moral, M. P. (2013). Seasonal stability tests in gretl. An application to international tourism data. Working paper: Biltoki D.T. 2013.03. URL: https://addi.ehu.es/handle/10810/10577. Gretl code: https://www.ehu.eus/ignacio.diaz-emparanza/packages/Canova_Hansen.gfn (seems unavailable, so not linked)

See Also

ch.rs.pvalue seasonal.cycles, seasonal.dummies, uroot.raw.pvalue.

Examples

library(uroot)
# example for the series "hours" with the same options 
# employed in Canova and Hansen (1995)
data("ch-data")
hours <- diff(log(ch.data$hours))
res1 <- ch.test(x = hours, type = "dummy", lag1 = TRUE, NW.order = 4)
res1
# the auxiliary regression is stored in the element "fitted.model"
summary(res1$fit)

## Not run: 
# this requires tables not included in the current version of the package 
# see note in main documentation file, uroot-package
res2 <- ch.test(x = hours, type = "trigonometric", lag1 = TRUE, NW.order = 4)
res2
summary(res2$fit)
## End(Not run)

Bootstrapped P-Values for the HEGY Test Statistics

Description

Compute p-values for the HEGY test statistics by means of bootstrap.

Usage

hegy.boot.pval(x, model0, stats0, 
  deterministic = c(1,0,0), lag.method = c("fixed", "AIC", "BIC"), maxlag = 0, 
  byseason = FALSE, nb = 500, u = NULL, debug.tid = -1)

Arguments

Details

See hegy.test for further details about the arguments that have the same name in both functions (deterministic, lag.method, maxlag).

Bootstrapped p-values follow the approach described in Burridge and Robert Taylor (2004), except that here, the residuals are resampled regardless of the season they belong to.

Value

A numeric vector containing the p-values of the the test statistics. The vector is named following the same convention as statistics and pvalues returned by hegy.test.

If the number of bootstrap replicates is nb = 1, the resampled series is returned (relevant for inspection of how the resampled series look like and for debugging).

References

Burridge, P. and Taylor, R. (2004) "Bootstrapping the HEGY seasonal unit root tests." Journal of Econometrics 123(1), pp. 67-87. DOI: doi:10.1016/j.jeconom.2003.10.029.

Hylleberg, S., Engle, R., Granger, C. and Yoo, B. (1990) "Seasonal integration and cointegration." Journal of Econometrics 44(1), pp. 215-238. DOI: doi:10.1016/0304-4076(90)90080-D.

See Also

hegy.test.

Examples

## Not run: 
x <- bgt.data[["LCONSEXPCO"]]
# this requires CUDA capable GPU
hegy.test(x, deterministic = c(1,1,1), lag.method = "fixed", maxlag = 1, 
  pvalue = "bootstrap")
# alternatively, full R non-parallel version
res <- hegy.test(x, deterministic = c(1,1,1), lag.method = "fixed", maxlag = 1)
hegy.boot.pval(x, res$fit, res$stat, deterministic = c(1,1,1), 
  lag.method = "fixed", maxlag = 1, nb = 1000)
## End(Not run)

P-values based on response surface regressions for the HEGY test statistics

Description

Compute p-values for the Hylleberg, Engle, Granger and Yoo (HEGY) test statistic by interpolation in precompiled response surfaces.

Usage

hegy.rs.pvalue(x, type = c("zero", "pi", "pair", "seasall", "all"), 
  deterministic = c(1, 0, 0), lag.method = c("fixed", "AIC", "BIC"),
  lag.order, S, n, nobsreg)

Arguments

Details

Ported from Gretl code provided by Díaz-Emparanza (2014). The original source includes tables with coefficients for the calculation of p-values when the HQC lag order selection criterion is used. These tables are not included here. The case with no deterministic terms (deterministic = c(0,0,0)) is not considered.

This function is mainly intended to be used internally by hegy.test.

Value

A numeric, the p-value.

References

Díaz-Emparanza, I. (2014) "Numerical Distribution Functions for Seasonal Unit Root Tests"- Computational Statistics and Data Analysis 76, pp. 237-247. DOI: doi:10.1016/j.csda.2013.03.006. Gretl code: https://www.ehu.eus/ignacio.diaz-emparanza/packages/Canova_Hansen.gfn/ (seems unavailable, so not linked)

See Also

hegy.test.

Hylleberg, Engle, Granger and Yoo Test for Seasonal Unit Roots

Description

Hylleberg, Engle, Granger and Yoo (HEGY) test statistics for the null hypothesis seasonal unit roots.

Usage

hegy.test(x, deterministic = c(1,0,0),
  lag.method = c("fixed", "AIC", "BIC", "AICc"), maxlag = 0,
  pvalue = c("RS", "bootstrap", "raw"), rs.nobsreg = 15, 
  boot.args = list(seed = 123, lag.method = lag.method[1], maxlag = maxlag, 
    byseason = FALSE, nb = 1000, BTdim = c(100, 10), debug.tid = -1))

hegy.regressors(x)

Arguments

Details

The regression equation employed to obtain the tests statistics may include the following deterministic terms: a constant, a linear trend, seasonal dummies. These terms are selected by setting to 1 the corresponding element in the vector deterministic: deterministic = c(0,0,0) no deterministic components, deterministic = c(1,0,0) includes a constant, deterministic = c(1,0,1) a constant and seasonal dummies, deterministic = c(1,1,0) a constant and a trend and deterministic = c(1,1,1) includes the three components.

The regression equation may include lags of the dependent variable. When lag.method = "fixed", the lag order is fixed to maxlag; otherwise, maxlag is the maximum number of lags considered in a lag selection procedure that minimises the lag.method criterion, which can be AIC or BIC or corrected AIC, AICc, obtained as AIC + \frac{2k(k+1)}{n-k-1}, where k is the number of parameters and n is the number of available observations in the model.

Response surface based p-values, pvalue="RS", is not available with option lag.method = "AICc".

P-values based on response surface regressions follow the method described in MacKinnon (1994), Harvey+vanDijk (2006) and Díaz-Emparanza (2014). Gretl code provided by Díaz-Emparanza (2014) has been ported to function hegy.rs.pvalue. Note: the case deterministic = c(0,0,0) is not considered; NAs are returned if p-values based on response surfaces are selected for that case.

Bootstrapped p-values follow the approach described in Burridge and Robert Taylor (2004). The following arguments can be defined in boot.args:

• seed: a numeric, the seed of the random generator employed for resampling the residuals.

• lag.method: a character, the lag order selection method. This is the same as lag.method employed in the original regression, except that here, the AIC, BIC and AICc are obtained upon the residual sums of squares rather than the likelihood value.

• maxlag: a numeric, maximum number of lags. Same behaviour as maxlag explained above for the original regression.

• byseason: logical, should the residuals be resampled by season? If TRUE, the residuals belonging to the same season are resampled (at each time t, the residuals belonging to the corresponding season at time t are resampled); otherwise, the entire series of residuals is resampled regardless of the season they belong to.

• nb: a numeric, the number of bootstrap replicates.

• BTdim: a vector of length two containing the number of blocks and the number of threads per block to be run on parallel on the GPU. The product of these two elements must be equal or greater than nb.

• debug.tid: an integer employed for debugging. Currently ignored.

By default boot.args$lag.method and boot.args$maxlag are set equal to the same options employed for the original data in arguments lag.method and maxlag; if the default options are desired, these values need not be explicitly defined in the list boot.args.

The standard definition of the AIC, BIC and AICc criteria is used for the original series. For the bootstrapped series, these criteria are defined upon the residual sum squares of the model:

\hbox{AIC} = n \log\left(\hbox{RSS}/n\right) + 2 k

\hbox{BIC} = n \log\left(\hbox{RSS}/n\right) + k \log(n)

\hbox{AICc} = \hbox{AIC} + \frac{2k(k+1)}{n-k-1}

where RSS is the residual sum of squares, k is the number of parameters in the model and n is the number of available observations.

Given a maximum lag, maxlag, the first maxlag observations are removed before fitting the models to be compared by the selected criterion. In this way, all the models contain the same number of observations. Once the lag order is selected, the statistics are obtained using the entire sample (except those observations that are missed due to the eventually chosen lags).

The HEGY regressors were originally proposed in Hylleberg etal. (1990) for quarterly data. They are generalized to monthly series in Beaulieu and Miron (1993) and to weekly data in Cáceres (1996). Franses and Hobijn (1997) show tabulated values for bimonthly and biannual data as well as quarterly and monthly data. hegy.rergessors follows the expressions given in Smith etal. (2009), which define the regressors for a general periodicity of the time series.

Value

hegy.test returns a list of class "HEGYtest" with the following components:

The method print displays the test statistics and p-values; summary shows the same output and includes the fitted regression model; residuals returns the residuals from the regression model fitted to the original data.

hegy.regressors returns a matrix containing the HEGY regressors which are used to test null of unit root at different frequencies.

References

Beaulieu, J. J. Miron, J. A. (1993) "Seasonal unit roots in aggregate U.S. data." Journal of Econometrics 55(1-2), pp. 305-328. DOI: doi:10.1016/0304-4076(93)90018-Z.

Burridge, P. and Taylor, R. (2004) "Bootstrapping the HEGY seasonal unit root tests." Journal of Econometrics 123(1), pp. 67-87. DOI: doi:10.1016/j.jeconom.2003.10.029.

Cáceres, J. J. (1996) "Contraste de raíces unitarias en datos semanales." Estadística Española 38(41), pp. 139-159.

Díaz-Emparanza, I. (2014) "Numerical distribution functions for seasonal unit root tests"- Computational Statistics and Data Analysis, 76, pp. 237-247. DOI: doi:10.1016/j.csda.2013.03.006. Gretl code: https://www.ehu.eus/ignacio.diaz-emparanza/packages/GHegy.gfn (seems unavailable, so not linked)

Franses, F. H. (1991) "Seasonality, non-stationarity and the forecasting of monthly time series." International Journal of Forecasting 7(2), pp. 199-208. DOI: doi:10.1016/0169-2070(91)90054-Y.

Franses, P. H. and Hobijn, H. (1997) "Critical values for unit root tests in seasonal time series." Journal of Applied Statistics 24(1), pp. 25-47.

Harvey D. I. and van Dijk D. (2006). "Sample size, lag order and critical values of seasonal unit root tests." Computational Statistics & Data Analysis, 50(10), 2734-2751. DOI: doi:10.1016/j.csda.2005.04.011.

Hylleberg, S., Engle, R., Granger, C. and Yoo, B. (1990) "Seasonal integration and cointegration." Journal of Econometrics 44(1), pp. 215-238. DOI: doi:10.1016/0304-4076(90)90080-D.

MacKinnon J. G. (1994). "Approximate asymptotic distribution functions for unit-root and cointegration tests." Journal of Business and Economic Statistics, 12(2), 167-176. DOI: doi:10.1080/07350015.1994.10510005.

Smith, R. J., Taylor, A. M. R. and del Barrio Castro, T. (2009) "Regression-based seasonal unit roots."Econometric Theory 25(2), pp. 527-560. DOI: doi:10.1017/S0266466608090166.

See Also

hegy.rs.pvalue, uroot.raw.pvalue.

Examples

x <- bgt.data[["LCONSEXPCO"]]
hegy.test(x, deterministic = c(1,1,1), lag.method = "fixed", maxlag = 1)

## Not run: 
# this requires CUDA capable GPU
hegy.test(x, deterministic = c(1,1,1), lag.method = "fixed", maxlag = 1, 
  pvalue = "bootstrap")
## End(Not run)

Seasonal Dummies and Seasonal Cycles

Description

Generate variables of seasonal dummies and seasonal cycles.

Usage

seasonal.dummies(x)
seasonal.cycles(x)

Arguments

Value

A multivariate time series containing the dummies or cycles by columns.

Examples

# In terms of model fitting
# both sets of variables are equivalent
x <- diff(log(AirPassengers))
sd <- seasonal.dummies(x)
fit1 <- lm(x ~ sd[,-1])
summary(fit1)
sc <- seasonal.cycles(x)
fit2 <- lm(x ~ sc)
summary(fit1)
all.equal(fitted(fit1), fitted(fit2))

Original Tables of Critical Values

Description

Compute p-values by interpolation in the tables of critical values provided in the original references given below.

Usage

uroot.raw.pvalue(x, type = c("CH", "HEGY"), v, n, ctd, S, Ftpi)

Arguments

Details

This function is used internally by ch.test and hegy.test.

Value

A numeric giving the calculated p-value.

References

Beaulieu, J. J. Miron, J. A. (1993) "Seasonal Unit Roots in Aggregate U.S. Data". Journal of Econometrics, 55(1-2), pp. 305-328. DOI: doi:10.1016/0304-4076(93)90018-Z.

Canova, F. and Hansen, Bruce E. (1995) "Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability". Journal of Business & Economic Statistics, 13(3), pp. 237-252. DOI: doi:10.1080/07350015.1995.10524598.

Hylleberg, S., Engle, R., Granger, C. and Yoo, B. (1990) "Seasonal integration and cointegration". Journal of Econometrics 44(1), pp. 215-238. DOI: doi:10.1016/0304-4076(90)90080-D.

See Also

ch.test, hegy.test.