Empirical Fluctuation Processes

Description

Computes an empirical fluctuation process according to a specified method from the generalized fluctuation test framework, which includes CUSUM and MOSUM tests based on recursive or OLS residuals, parameter estimates or ML scores (OLS first order conditions).

Usage

efp(formula, data, type = , h = 0.15,
    dynamic = FALSE, rescale = TRUE, lrvar = FALSE, vcov = NULL)

Arguments

formula	a symbolic description for the model to be tested.
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which efp is called from.
type	specifies which type of fluctuation process will be computed, the default is “Rec-CUSUM”. For details see below.
h	a numeric from interval (0,1) specifying the bandwidth. determines the size of the data window relative to sample size (for MOSUM and ME processes only).
dynamic	logical. If TRUE the lagged observations are included as a regressor.
rescale	logical. If TRUE the estimates will be standardized by the regressor matrix of the corresponding subsample according to Kuan & Chen (1994); if FALSE the whole regressor matrix will be used. (only if type is either “RE” or “ME”)
lrvar	logical or character. Should a long-run variance estimator be used for the residuals? By default, the standard OLS variance is employed. Alternatively, lrvar can be used. If lrvar is character (“Andrews” or “Newey-West”), then the corresponding type of long-run variance is used. (The argument is ignored for the score-based tests where gefp should be used instead.)
vcov	a function to extract the covariance matrix for the coefficients of the fitted model (only for “RE” and “ME”).

Details

If type is one of “Rec-CUSUM”, “OLS-CUSUM”, “Rec-MOSUM” or “OLS-MOSUM” the function efp will return a one-dimensional empirical process of sums of residuals. Either it will be based on recursive residuals or on OLS residuals and the process will contain CUmulative SUMs or MOving SUMs of residuals in a certain data window. For the MOSUM and ME processes all estimations are done for the observations in a moving data window, whose size is determined by h and which is shifted over the whole sample.

If type is either “RE” or “ME” a k-dimensional process will be returned, if k is the number of regressors in the model, as it is based on recursive OLS estimates of the regression coefficients or moving OLS estimates respectively. The recursive estimates test is also called fluctuation test, therefore setting type to “fluctuation” was used to specify it in earlier versions of strucchange. It still can be used now, but will be forced to “RE”.

If type is “Score-CUSUM” or “Score-MOSUM” a k+1-dimensional process will be returned, one for each score of the regression coefficients and one for the scores of the variance. The process gives the decorrelated cumulative sums of the ML scores (in a Gaussian model) or first order conditions respectively (in an OLS framework).

If there is a single structural change point \(t^*\), the recursive CUSUM path starts to depart from its mean 0 at \(t^*\). The Brownian bridge type paths will have their respective peaks around \(t^*\). The Brownian bridge increments type paths should have a strong change at \(t^*\).

The function plot has a method to plot the empirical fluctuation process; with sctest the corresponding test on structural change can be performed.

Value

efp returns a list of class “efp” with components including:

process	the fitted empirical fluctuation process of class “ts” or “mts” respectively,
type	a string with the type of the process fitted,
nreg	the number of regressors,
nobs	the number of observations,
par	the bandwidth h used.

References

Brown R.L., Durbin J., Evans J.M. (1975), Techniques for testing constancy of regression relationships over time, Journal of the Royal Statistical Society, B, 37, 149-163.

Chu C.-S., Hornik K., Kuan C.-M. (1995), MOSUM tests for parameter constancy, Biometrika, 82, 603-617.

Chu C.-S., Hornik K., Kuan C.-M. (1995), The moving-estimates test for parameter stability, Econometric Theory, 11, 669-720.

Hansen B. (1992), Testing for Parameter Instability in Linear Models, Journal of Policy Modeling, 14, 517-533.

Hjort N.L., Koning A. (2002), Tests for Constancy of Model Parameters Over Time, Nonparametric Statistics, 14, 113-132.

Krämer W., Ploberger W., Alt R. (1988), Testing for structural change in dynamic models, Econometrica, 56, 1355-1369.

Kuan C.-M., Hornik K. (1995), The generalized fluctuation test: A unifying view, Econometric Reviews, 14, 135 - 161.

Kuan C.-M., Chen (1994), Implementing the fluctuation and moving estimates tests in dynamic econometric models, Economics Letters, 44, 235-239.

Ploberger W., Krämer W. (1992), The CUSUM test with OLS residuals, Econometrica, 60, 271-285.

Zeileis A., Leisch F., Hornik K., Kleiber C. (2002), strucchange: An R Package for Testing for Structural Change in Linear Regression Models, Journal of Statistical Software, 7(2), 1-38. doi:10.18637/jss.v007.i02.

Zeileis A. (2005), A Unified Approach to Structural Change Tests Based on ML Scores, F Statistics, and OLS Residuals. Econometric Reviews, 24, 445–466. doi:10.1080/07474930500406053.

Zeileis A. (2006), Implementing a Class of Structural Change Tests: An Econometric Computing Approach. Computational Statistics & Data Analysis, 50, 2987–3008. doi:10.1016/j.csda.2005.07.001.

Zeileis A., Hornik K. (2007), Generalized M-Fluctuation Tests for Parameter Instability, Statistica Neerlandica, 61, 488–508. doi:10.1111/j.1467-9574.2007.00371.x.

See Also

gefp, plot.efp, print.efp, sctest.efp, boundary.efp

Examples

library("strucchange")

## Nile data with one breakpoint: the annual flows drop in 1898
## because the first Ashwan dam was built
data("Nile")
plot(Nile)

## test the null hypothesis that the annual flow remains constant
## over the years
## compute OLS-based CUSUM process and plot
## with standard and alternative boundaries
ocus.nile <- efp(Nile ~ 1, type = "OLS-CUSUM")
plot(ocus.nile)

plot(ocus.nile, alpha = 0.01, alt.boundary = TRUE)

## calculate corresponding test statistic
sctest(ocus.nile)

    OLS-based CUSUM test

data:  ocus.nile
S0 = 2.9518, p-value = 5.409e-08

## UK Seatbelt data: a SARIMA(1,0,0)(1,0,0)_12 model
## (fitted by OLS) is used and reveals (at least) two
## breakpoints - one in 1973 associated with the oil crisis and
## one in 1983 due to the introduction of compulsory
## wearing of seatbelts in the UK.
data("UKDriverDeaths")
seatbelt <- log10(UKDriverDeaths)
seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12))
colnames(seatbelt) <- c("y", "ylag1", "ylag12")
seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12))
plot(seatbelt[,"y"], ylab = expression(log[10](casualties)))

## use RE process
re.seat <- efp(y ~ ylag1 + ylag12, data = seatbelt, type = "RE")
plot(re.seat)

plot(re.seat, functional = NULL)

sctest(re.seat)

    RE test (recursive estimates test)

data:  re.seat
RE = 1.6311, p-value = 0.02904