Type:	Package
Title:	Wavelet Routines for Global and Local Multiple Regression and Correlation
Version:	3.1.2
Description:	Wavelet routines that calculate single sets of wavelet multiple regressions and correlations, and cross-regressions and cross-correlations from a multivariate time series. Dynamic versions of the routines allow the wavelet local multiple (cross-)regressions and (cross-)correlations to evolve over time.
License:	GPL-3
Depends:	R (≥ 3.4.0), waveslim (≥ 1.7.5)
Imports:	plot3D, RColorBrewer
Suggests:	covr, knitr, markdown, rmarkdown, testthat
VignetteBuilder:	knitr
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.1.1
NeedsCompilation:	no
Packaged:	2021-09-02 23:07:09 UTC; Javier
Author:	Javier Fernandez-Macho [aut, cre]
Maintainer:	Javier Fernandez-Macho <javier.fernandezmacho@ehu.es>
Repository:	CRAN
Date/Publication:	2021-09-03 12:40:02 UTC

Wavelet Routines for Global and Local Multiple Regression and Correlation

Description

Wavelet routines that calculate single sets of wavelet multiple regressions and correlations, and cross-regressions and cross-correlations from a multivariate time series. Dynamic versions of the routines allow the wavelet local multiple (cross-)regressions and (cross-)correlations to evolve over time.

Details

Wavelet routines that calculate single sets of wavelet multiple regressions and correlations (WMR and WMC), and cross-regressions and cross-correlations (WMCR and WMCC) from a multivariate time series. Dynamic versions of the routines allow the wavelet local multiple (cross-)regressions (WLMR and WLMCR) and (cross-)correlations (WLMC and WLMCC) to evolve over time. The output from these Wavelet statistics can later be plotted in single graphs, as an alternative to trying to make sense out of several sets of wavelet correlations or wavelet cross-correlations. The code is based on the calculation, at each wavelet scale, of the square root of the coefficient of determination in a linear combination of variables for which such coefficient of determination is a maximum. The code provided here is based on the wave.correlation routine in Brandon Whitcher's waveslim R package Version: 1.6.4, which in turn is based on wavelet methodology developed in Percival and Walden (2000), Gençay, Selçuk and Whitcher (2002) and others. Version 2 incorporates wavelet local multiple correlations (WLMC). These are like the previous global WMC but consisting in one single set of multiscale correlations along time. That is, at each time t, they are calculated by letting a window of weighted wavelet coefficients around t move along time. Six weight functions are provided. Namely, the uniform window, Cleveland's tricube window, Epanechnikov's parabolic window, Bartlett's triangular window and Wendland's truncated power window and the Gaussian window. Version 2.2 incorporates auxiliary functions that calculate local multiple correlations and cross-correlations (LMC, LMCC). They are calculated by letting move along time a window of weighted time series values around t. Any of the six weight functions mentioned above can be used. They also feed a new routine to compute wavelet local multiple cross-correlation (WLMCC). Version 3 extends all the previous correlation routines (WMC, WMCC, LMC, WLMC, WLMCC) to handle wavelet regressions (WMR, WMCR, LMR, WLMR, WLMCR) that provide regression coefficients and statistics across wavelet scales. Auxiliary plot_ and heatmap_ routines are also provided to visualize the wavmulcor statistics.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for heatmaping wave local multiple correlations

Description

Produces a heatmap of wave local multiple correlations.

Usage

heatmap_wave.local.multiple.correlation(Lst, xaxt="s", ci=NULL, pdf.write=NULL)

Arguments

Details

The routine produces a time series vs. wavelet periods heatmap of wave local multiple correlations.

Value

Heat map.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for heatmaping wave local multiple cross-correlations

Description

Produces heatmaps of wave local multiple cross-correlations.

Usage

heatmap_wave.local.multiple.cross.correlation(Lst, lmax,
     lag.first=FALSE, xaxt="s", ci=NULL, pdf.write=NULL)

Arguments

Details

The routine produces a set of time series vs. wavelet periods heatmaps of wave local multiple cross-correlations at different lags and leads.

Value

Heat map.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for heatmaping wave multiple cross-correlations

Description

Produces heatmaps of wave multiple cross-correlations.

Usage

heatmap_wave.multiple.cross.correlation(Lst, lmax, by=3, ci=NULL, pdf.write=NULL)

Arguments

Details

The routine produces a set of time series vs. wavelet periods heatmaps of wave local multiple cross-correlations at different lags and leads.

Value

Heat map.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Routine for local multiple correlation

Description

Produces an estimate of local multiple correlations (as defined below) along with approximate confidence intervals.

Usage

local.multiple.correlation(xx, M, window="gauss", p = .975, ymaxr=NULL)

Arguments

Details

The routine calculates a time series of multiple correlations out of n variables. The code is based on the calculation of the square root of the coefficient of determination in that linear combination of locally weighted values for which such coefficient of determination is a maximum.

Value

List of four elements:

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on  Figure 4 showing correlation structural breaks in Fernandez-Macho (2018).

library(wavemulcor)
options(warn = -1)

xrand1 <- wavemulcor::xrand1
xrand2 <- wavemulcor::xrand2
N <- length(xrand1)
b <- trunc(N/3)
t1 <- 1:b
t2 <- (b+1):(2*b)
t3 <- (2*b+1):N

wf <- "d4"
M <- N/2^3 #sharper with N/2^4
window <- "gaussian"

J <- trunc(log2(N))-3

# ---------------------------

cor1 <- cor(xrand1[t1],xrand2[t1])
cor2 <- cor(xrand1[t2],xrand2[t2])
cor3 <- cor(xrand1[t3],xrand2[t3])
cortext <- paste0(round(100*cor1,0),"-",round(100*cor2,0),"-",round(100*cor3,0))

ts.plot(cbind(xrand1,xrand2),col=c("red","blue"),xlab="time")

xx <- data.frame(xrand1,xrand2)

# ---------------------------

xy.mulcor <- local.multiple.correlation(xx, M, window=window)

val <- as.matrix(xy.mulcor$val)
lo  <- as.matrix(xy.mulcor$lo)
up  <- as.matrix(xy.mulcor$up)
YmaxR <- as.matrix(xy.mulcor$YmaxR)

# ---------------------------

old.par <- par()

# ##Producing line plots with CI

title <- paste("Local Multiple Correlation")
sub <- paste("first",b,"obs:",round(100*cor1,1),"% correlation;","middle",b,"obs:",
             round(100*cor2,1),"%","rest:",round(100*cor3,1),"%")
xlab <- "time"
ylab <- "correlation"

matplot(1:N,cbind(val,lo,up),
        main=title, sub=sub,
        xlab=xlab, ylab=ylab, type="l", lty=1, col= c(1,2,2), cex.axis=0.75)
abline(h=0) ##Add Straight horiz and vert Lines to a Plot

#reset graphics parameters
par(old.par)

Routine for local multiple cross-correlation

Description

Produces an estimate of local multiple cross-correlations (as defined below) along with approximate confidence intervals.

Usage

 local.multiple.cross.correlation(xx, M, window="gauss", lag.max=NULL, p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates a set of time series of multiple cross-correlations, one per lag and lead) out of n variables.

Value

List of four elements:

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on  Figure 4 showing correlation structural breaks in Fernandez-Macho (2018).

library(wavemulcor)

data(exchange)
returns <- diff(log(as.matrix(exchange)))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

M <- 30
window <- "gauss"
lmax <- 1

demusd <- returns[,"DEM.USD"]
jpyusd <- returns[,"JPY.USD"]

set.seed(140859)

xrand <- rnorm(N)

xx <- data.frame(demusd, jpyusd, xrand)
##exchange.names <- c(colnames(returns), "RAND")

Lst <- local.multiple.cross.correlation(xx, M, window=window, lag.max=lmax)
val <- Lst$vals
low.ci <- Lst$lower
upp.ci <- Lst$upper
YmaxR <- Lst$YmaxR

# ---------------------------

##Producing correlation plot

colnames(val) <- paste("Lag",-lmax:lmax)
xvar <- seq(1,N,M)
par(mfcol=c(lmax+1,2), las=1, pty="m", mar=c(2,3,1,0)+.1, oma=c(1.2,1.2,0,0))
ymin <- -0.1
if (length(xx)<3) ymin <- -1
for(i in c(-lmax:0,lmax:1)+lmax+1) {
  matplot(1:N,val[,i], type="l", lty=1, ylim=c(ymin,1), #xaxt="n",
          xlab="", ylab="", main=colnames(val)[i])
  # if(i==lmax+1) {axis(side=1, at=seq(0,N+50,50))}
  #axis(side=2, at=c(-.2, 0, .5, 1))
  abline(h=0)              ##Add Straight horiz
  lines(low.ci[,i], lty=1, col=2) ##Add Connected Line Segments to a Plot
  lines(upp.ci[,i], lty=1, col=2)
  text(xvar,1, labels=names(xx)[YmaxR][xvar], adj=0.25, cex=.8)
}
par(las=0)
mtext('time', side=1, outer=TRUE, adj=0.5)
mtext('Local Multiple Cross-Correlation', side=2, outer=TRUE, adj=0.5)

Routine for local multiple cross-regression

Description

Produces an estimate of local multiple cross-regressions (as defined below) along with approximate confidence intervals.

Usage

local.multiple.cross.regression(xx, M, window="gauss", lag.max=NULL, p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates a set of time series of multiple cross-regressions, one per lag and lead) out of n variables.

Value

List of four elements:

• vals: numeric matrix (rows = #observations, cols = #lags and leads) providing the point estimates for the local multiple cross-correlation.

• lower: numeric vmatrix (rows = #observations, cols = #lags and leads) providing the lower bounds from the confidence interval.

• upper: numeric matrix (rows = #observations, cols = #lags and leads) providing the upper bounds from the confidence interval.

• rval: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of local regression estimates.

• rstd: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their standard deviations.

• rlow: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their lower bounds.

• rupp: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their upper bounds.

• rtst: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their t statistic values.

• rord: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their index order when sorted by significance.

• rpva: numeric array (1st_dim = #observations, 2nd_dim = #lags and leads, 3rd_dim = #regressors+1) of their p values.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on Figure 4 showing correlation structural breaks in Fernandez-Macho (2018).

library(wavemulcor)
data(exchange)
returns <- diff(log(as.matrix(exchange)))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

M <- 30
window <- "gauss"
lmax <- 1

demusd <- returns[,"DEM.USD"]
jpyusd <- returns[,"JPY.USD"]

set.seed(140859)

xrand <- rnorm(N)

# ---------------------------

xx <- data.frame(demusd, jpyusd, xrand)
##exchange.names <- c(colnames(returns), "RAND")

Lst <- local.multiple.cross.regression(xx, M, window=window, lag.max=lmax)

# ---------------------------

##Producing correlation plot

plot_local.multiple.cross.correlation(Lst, lmax) #, xaxt="s")

##Producing regression plot

plot_local.multiple.cross.regression(Lst, lmax) #, nsig=2, xaxt="s")

Routine for local multiple regression

Description

Produces an estimate of local multiple regressions (as defined below) along with approximate confidence intervals.

Usage

local.multiple.regression(xx, M, window="gauss", p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates a set of time series of multiple regression coefficients out of n variables.

Value

List of four elements:

• val: numeric vector (rows = #observations) of point estimates for the local multiple correlation.

• lo: numeric vector (rows = #observations) of lower bounds of the confidence interval.

• up: numeric vector (rows = #observations) of upper bounds of the confidence interval.

• rval: numeric matrix (rows = #observations, cols = #regressors+1) of local regression estimates.

• rstd: numeric matrix (rows = #observations, cols = #regressors+1) of their standard deviations.

• rlow: numeric matrix (rows = #observations, cols = #regressors+1) of their lower bounds.

• rupp: numeric matrix (rows = #observations, cols = #regressors+1) of their upper bounds.

• rtst: numeric matrix (rows = #observations, cols = #regressors+1) of their t statistic values.

• rord: numeric matrix (rows = #observations, cols = #regressors+1) of their index order when sorted by significance.

• rpva: numeric matrix (rows = #observations, cols = #regressors+1) of their p values.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on  Figure 4 showing correlation structural breaks in Fernandez-Macho (2018).

library(wavemulcor)
options(warn = -1)

xrand1 <- wavemulcor::xrand1
xrand2 <- wavemulcor::xrand2
N <- length(xrand1)
b <- trunc(N/3)
t1 <- 1:b
t2 <- (b+1):(2*b)
t3 <- (2*b+1):N

wf <- "d4"
M <- N/2^3 #sharper with N/2^4
window <- "gaussian"

J <- trunc(log2(N))-3

# ---------------------------

cor1 <- cor(xrand1[t1],xrand2[t1])
cor2 <- cor(xrand1[t2],xrand2[t2])
cor3 <- cor(xrand1[t3],xrand2[t3])
cortext <- paste0(round(100*cor1,0),"-",round(100*cor2,0),"-",round(100*cor3,0))

ts.plot(cbind(xrand1,xrand2),col=c("red","blue"),xlab="time")

xx <- data.frame(xrand1,xrand2)

# ---------------------------

Lst <- local.multiple.regression(xx, M, window=window) #, ymax=1)

# ---------------------------

##Producing correlation plot
plot_local.multiple.correlation(Lst)

##Producing regression plot
plot_local.multiple.regression(Lst)

Auxiliary routine for plotting local multiple correlations

Description

Produces a plot of local multiple correlations.

Usage

plot_local.multiple.correlation(Lst, xaxt="s")

Arguments

Details

The routine produces a time series plot of local multiple correlations with its confidence interval. Also, at every upturn and downturn, the name of the variable that maximizes its multiple correlation against the rest is shown. Note that the routine is optimize for local.multiple.regression. If you want to use output from the legacy local.multiple.correlation function then you must create an empty list and put that output into a list element named cor like this: Lst <- list(); Lst$cor <- local.multiple.cross.correlation(xx, M, window=window, lag.max=lmax); Lst$YmaxR <- Lst2$cor$YmaxR; Lst$cor$YmaxR <- NULL.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting local multiple cross-correlations

Description

Produces a plot of local multiple cross-correlations.

Usage

plot_local.multiple.cross.correlation(Lst, lmax, xaxt="s")

Arguments

Details

The routine produces a set of time series plots of local multiple cross-correlations, one per lag and lead, each with its confidence interval. Also, at every upturn and downturn, the name of the variable that maximizes its multiple correlation against the rest is shown. Note that the routine is optimize for local.multiple.cross.regression. If you want to use output from local.multiple.cross.correlation function then you must create an empty list and put that output into a list element named cor like this: Lst <- list(); Lst$cor <- local.multiple.cross.correlation(xx, M, window=window, lag.max=lmax); Lst$YmaxR <- Lst2$cor$YmaxR; Lst$cor$YmaxR <- NULL.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting local multiple cross-regressions

Description

Produces a plot of local multiple cross-regressions.

Usage

plot_local.multiple.cross.regression(Lst, lmax, nsig=2, xaxt="s")

Arguments

Details

The routine produces time series plots of local multiple cross-regressions with their confidence interval for every lag and lead. Also, at every upturn and downturn of the corresponding local multiple cross-correlation, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named ordered by significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting local multiple regressions

Description

Produces a plot of local multiple regressions.

Usage

plot_local.multiple.regression(Lst, nsig=2, xaxt="s")

Arguments

Details

The routine produces a time series plot of local multiple regressions with its confidence interval. Also, at every upturn and downturn of the corresponding local multiple correlation, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named ordered by significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave local multiple correlations

Description

Produces a plot of wave local multiple correlations.

Usage

plot_wave.local.multiple.correlation(Lst, xaxt="s")

Arguments

Details

The routine produces time series plots of wave local multiple correlations with their confidence intervals. Also, at every upturn and downturn, the name of the variable that maximizes its multiple correlation against the rest is shown.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave local multiple cross-correlations

Description

Produces a plot of wave local multiple cross-correlations.

Usage

plot_wave.local.multiple.cross.correlation(Lst, lmax,
     lag.first=FALSE, xaxt="s", pdf.write=NULL)

Arguments

Details

The routine produces time series plots of wave local multiple cross-correlations with their confidence intervals. Also, at every upturn and downturn, the name of the variable that maximizes its multiple cross-correlation against the rest is shown.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave local multiple cross-regressions

Description

Produces a set of plots of wave local multiple cross-regressions.

Usage

plot_wave.local.multiple.cross.regression(Lst, lmax, nsig=2, 
     xaxt="s", pdf.write=NULL)

Arguments

Details

The routine produces J+1 pages, one per wavelet level, each with time series plots of wave multiple cross-regressions at different lags and leads, each with their confidence interval. Also, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named with their order of significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave local multiple regressions

Description

Produces a set of plots of wave local multiple regressions.

Usage

plot_wave.local.multiple.regression(Lst,nsig=2, xaxt="s")

Arguments

Details

The routine produces J+1 time series plots of wave multiple regressions, each with their confidence interval. Also, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named with their order of significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave multiple correlations

Description

Produces a plot of wave multiple correlations.

Usage

plot_wave.multiple.correlation(Lst)

Arguments

Details

The routine produces a plot of wave multiple correlations, at each wavelet level, with its confidence interval. Also, at each wavelet level, the name of the variable that maximizes its multiple correlation against the rest is shown.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Auxiliary routine for plotting wave multiple cross-correlations

Description

Produces a plot of wave multiple cross-correlations.

Usage

plot_wave.multiple.cross.correlation(Lst, lmax, by=3)

Arguments

Details

The routine produces a set of plots of wave multiple cross-correlations, one per wavelet level, with their confidence intervals. Also, at each wavelet level, the name of the variable that maximizes its multiple cross-correlation against the rest is shown.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Auxiliary routine for plotting wave multiple cross-regressions

Description

Produces a plot of wave multiple cross.regressions.

Usage

plot_wave.multiple.cross.regression(Lst, lmax, nsig=2, by=3)

Arguments

Details

The routine produces a plot of wave multiple regressions, one per wavelet level, with their confidence intervals. Also, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named with their order of significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Auxiliary routine for plotting wave multiple regressions

Description

Produces a plot of wave multiple regressions.

Usage

plot_wave.multiple.regression(Lst, nsig=2)

Arguments

Details

The routine produces a plot of wave multiple regressions with their confidence interval. Also, the name of the variable that maximizes that multiple correlation against the rest is shown on top. The others are named with their order of significance when they are relevant.

Value

Plot.

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Wavelet routine for local multiple correlation

Description

Produces an estimate of the multiscale local multiple correlation (as defined below) along with approximate confidence intervals.

Usage

wave.local.multiple.correlation(xx, M, window="gauss", p = .975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple correlations out of n variables that can be plotted in in either single heatmap or J line graphs (the former is usually the best graphic option but the latter is useful if confidence intervals are explicitly needed), as an alternative to trying to make sense out of n(n-1)/2 [J \times T] sets of local wavelet correlations. The code is based on the calculation, at each wavelet scale, of the square root of the coefficient of determination in that linear combination of locally weighted wavelet coefficients for which such coefficient of determination is a maximum. The code provided here is based on the wave.multiple.correlation routine in this package which in turn is based on the wave.correlation routine in Brandon Whitcher's waveslim R package Version: 1.6.4, which in turn is based on wavelet methodology developed in Percival and Walden (2000); Gençay, Selçuk and Whitcher (2001) and others.

Value

List of four elements:

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on  Figure 4 showing correlation structural breaks in Fernandez-Macho (2018).

library(wavemulcor)
options(warn = -1)

data(xrand)
N <- length(xrand1)
b <- trunc(N/3)
t1 <- 1:b
t2 <- (b+1):(2*b)
t3 <- (2*b+1):N

wf <- "d4"
M <- N/2^3 #sharper with N/2^4
window <- "gauss"
J <- 3 #trunc(log2(N))-3

# ---------------------------

cor1 <- cor(xrand1[t1],xrand2[t1])
cor2 <- cor(xrand1[t2],xrand2[t2])
cor3 <- cor(xrand1[t3],xrand2[t3])
cortext <- paste0(round(100*cor1,0),"-",round(100*cor2,0),"-",round(100*cor3,0))

ts.plot(cbind(xrand1,xrand2),col=c("red","blue"),xlab="time")

xrand1.modwt <- modwt(xrand1, wf, J)
xrand1.modwt.bw <- brick.wall(xrand1.modwt, wf)

xrand2.modwt <- modwt(xrand2, wf, J)
xrand2.modwt.bw <- brick.wall(xrand2.modwt, wf)

xx <- list(xrand1.modwt.bw,xrand2.modwt.bw)

# ---------------------------

xy.mulcor <- wave.local.multiple.correlation(xx, M, window=window)

val <- as.matrix(xy.mulcor$val)
lo  <- as.matrix(xy.mulcor$lo)
up  <- as.matrix(xy.mulcor$up)
YmaxR <- as.matrix(xy.mulcor$YmaxR)

# ---------------------------

old.par <- par()

# ##Producing heat plot

scale.names <- paste0("(",c("2-4","4-8","8-16","16-32","32-64","64-128","128-256","256-512",
                            "512-1024","1024-2048"),"]")
scale.names <- c(scale.names[1:J],"smooth")

title <- paste("Wavelet Local Multiple Correlation")
sub <- paste("first",b,"obs:",round(100*cor1,1),"% correlation;","middle",b,"obs:",
             round(100*cor2,1),"%","rest:",round(100*cor3,1),"%")
xlab <- "time"
ylab <- "periods"

plot3D::image2D(z=val, x=1:nrow(val), y=1:ncol(val),
        main=title, #sub=sub,
        xlab=xlab, ylab=ylab, axes=FALSE, clab = expression(varphi),
        rasterImage = TRUE, contour = list(lwd = 2, col = plot3D::jet.col(11)))
axis(side=1, at=seq(10,nrow(val),by=10), cex.axis=0.75)
axis(side=2, at=1:ncol(val),labels=scale.names, las=1,cex.axis=0.75)

# ---------------------------

##Producing line plots with confidence intervals

colnames(val)[1:J] <- paste0("level",1:J)
par(mfrow=c(3,2), las=1, pty="m", mar=c(2,3,1,0)+.1, oma=c(1.2,1.2,0,0))
for(i in J:1) {
  matplot(1:N,val[,i], type="l", lty=1, ylim=c(-1,1), xaxt="n",
          xlab="", ylab="", main=colnames(val)[i])
  if(i<3) {axis(side=1, at=seq(10,N,by=10))}
  #axis(side=2, at=c(-.2, 0, .5, 1))
  lines(lo[,i], lty=1, col=2) ##Add Connected Line Segments to a Plot
  lines(up[,i], lty=1, col=2)
  abline(h=0)              ##Add Straight horiz and vert Lines to a Plot
}
par(las=0)
mtext('time', side=1, outer=TRUE, adj=0.5)
mtext('Wavelet Local Multiple Correlation', side=2, outer=TRUE, adj=0.5)

#reset graphics parameters
par(old.par)

Wavelet routine for local multiple cross-correlation

Description

Produces an estimate of the multiscale local multiple cross-correlation (as defined below) along with approximate confidence intervals.

Usage

wave.local.multiple.cross.correlation(xx, M,
     window="gauss", lag.max=NULL, p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates J+1 sets of wavelet multiple cross-correlations,one per wavelet level, out of n variables, that can be plotted each as lags and leads time series plots.

Value

List of four elements:

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on data from Figure 7.9 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.


library(wavemulcor)

data(exchange)
returns <- diff(log(exchange))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]
wf <- "d4"
M <- 30
window <- "gauss"
J <- 3 #trunc(log2(N))-3
lmax <- 2

set.seed(140859)

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

##xx <- list(demusd.modwt.bw, jpyusd.modwt.bw)
xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)
names(xx) <- c("DEM.USD","JPY.USD","rand")

## Not run: 
# Note: WLMCC may take more than 10 seconds of CPU time on some systems

Lst <- wave.local.multiple.cross.correlation(xx, M, window=window, lag.max=lmax)
val <- Lst$val
low.ci <- Lst$lo
upp.ci <- Lst$up
YmaxR <- Lst$YmaxR

# ---------------------------

##Producing cross-correlation plot

xvar <- seq(1,N,M)
level.lab <- c(paste("Level",1:J),paste("Smooth",J))
ymin <- -0.1
if (length(xx)<3) ymin <- -1
for(j in 1:(J+1)) {
  par(mfcol=c(lmax+1,2), las=1, pty="m", mar=c(2,3,1,0)+.1, oma=c(1.2,1.2,1.2,0))
  # xaxt <- c(rep("n",lmax),"s",rep("n",lmax))
  for(i in c(-lmax:0,lmax:1)+lmax+1) {
    matplot(1:N,val[[j]][,i], type="l", lty=1, ylim=c(ymin,1), #xaxt=xaxt[i],
            xlab="", ylab="", main=paste("Lag",i-lmax-1))
    abline(h=0)              ##Add Straight horiz
    lines(low.ci[[j]][,i], lty=1, col=2) ##Add Connected Line Segments to a Plot
    lines(upp.ci[[j]][,i], lty=1, col=2)
    text(xvar,1, labels=names(xx)[YmaxR[[j]]][xvar], adj=0.25, cex=.8)
  }
  par(las=0)
  mtext('time', side=1, outer=TRUE, adj=0.5)
  mtext('Local Multiple Cross-Correlation', side=2, outer=TRUE, adj=0.5)
  mtext(level.lab[j], side=3, outer=TRUE, adj=0.5)
}


## End(Not run)

Wavelet routine for local multiple cross-regression

Description

Produces an estimate of the multiscale local multiple cross-regression (as defined below) along with approximate confidence intervals.

Usage

wave.local.multiple.cross.regression(xx, M,
     window="gauss", lag.max=NULL, p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates J+1 sets of wavelet multiple cross-regressions, one per wavelet level, out of n variables, that can be plotted each as lags and leads time series plots.

Value

List of four elements:

• val: list of J+1 dataframes, each (rows = #observations, columns = #levels) providing the point estimates for the wavelet local multiple correlation.

• lo: list of J+1 dataframes, each (rows = #observations, columns = #lags and leads) providing the lower bounds from the confidence interval.

• up: list of J+1 dataframes, each (rows = #observations, columns = #lags and leads) providing the upper bounds from the confidence interval.

• rval: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of local regression estimates.

• rstd: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their standard deviations.

• rlow: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their lower bounds.

• rupp: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their upper bounds.

• rtst: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their t statistic values.

• rord: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their index order when sorted by significance.

• rpva: numeric array (1st_dim = #observations, 2nd-dim = #lags and leads, 3rd_dim = #regressors+1) of their p values.

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on data from Figure 7.9 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)

data(exchange)
returns <- diff(log(exchange))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]
wf <- "d4"
M <- 30
window <- "gauss"
J <- 3 #trunc(log2(N))-3
lmax <- 2

set.seed(140859)

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)
names(xx) <- c("DEM.USD","JPY.USD","rand")

## Not run: 
# Note: WLMCR may take more than 10 seconds of CPU time on some systems

Lst <- wave.local.multiple.cross.regression(xx, M, window=window, lag.max=lmax) #, ymaxr=1)

# ---------------------------

##Producing cross-correlation plot

plot_wave.local.multiple.cross.correlation(Lst, lmax, lag.first=FALSE) #, xaxt="s")

##Producing cross-regression plot

plot_wave.local.multiple.cross.regression(Lst, lmax, nsig=2) #, xaxt="s")

## End(Not run)

Wavelet routine for local multiple regression

Description

Produces an estimate of the multiscale local multiple regression (as defined below) along with approximate confidence intervals.

Usage

wave.local.multiple.regression(xx, M, window="gauss", p = .975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple regressions out of n variables that can be plotted in in J line graphs with explicit confidence intervals.

Value

List of four elements:

• val: numeric matrix (rows = #observations, columns = #levels in the wavelet transform object) providing the point estimates for the wavelet local multiple correlation.

• lo: numeric matrix (rows = #observations, columns = #levels in the wavelet transform object) providing the lower bounds from the confidence interval.

• up: numeric matrix (rows = #observations, columns = #levels in the wavelet transform object) providing the upper bounds from the confidence interval.

• rval: numeric matrix (rows = #observations, cols = #regressors+1) of local regression estimates.

• rstd: numeric matrix (rows = #observations, cols = #regressors+1) of their standard deviations.

• rlow: numeric matrix (rows = #observations, cols = #regressors+1) of their lower bounds.

• rupp: numeric matrix (rows = #observations, cols = #regressors+1) of their upper bounds.

• rtst: numeric matrix (rows = #observations, cols = #regressors+1) of their t statistic values.

• rord: numeric matrix (rows = #observations, cols = #regressors+1) of their index order when sorted by significance.

• rpva: numeric matrix (rows = #observations, cols = #regressors+1) of their p values.

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on data from Figure 7.8 in Gencay, Selcuk and Whitcher (2001)
## plus two random series.

library(wavemulcor)

data(exchange)
returns <- diff(log(as.matrix(exchange)))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
M <- 30
window <- "gauss"
J <- 3 #trunc(log2(N))-3

set.seed(140859)

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
xrand.modwt <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

# ---------------------------

xx <- list(demusd.modwt, jpyusd.modwt, xrand.modwt)
names(xx) <- c("DEM.USD","JPY.USD","rand")

Lst <- wave.local.multiple.regression(xx, M, window=window) #, ymaxr=1)

# ---------------------------

##Producing line plots with CI

plot_wave.local.multiple.correlation(Lst) #, xaxt="s")

##Producing regression plots

plot_wave.local.multiple.regression(Lst) #, xaxt="s")

Wavelet routine for multiple correlation

Description

Produces an estimate of the multiscale multiple correlation (as defined below) along with approximate confidence intervals.

Usage

wave.multiple.correlation(xx, N, p = 0.975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple correlations out of n variables that can be plotted in a single graph, as an alternative to trying to make sense out of n(n-1)/2 sets of wavelet correlations. The code is based on the calculation, at each wavelet scale, of the square root of the coefficient of determination in the linear combination of variables for which such coefficient of determination is a maximum. The code provided here is based on the wave.correlation routine in Brandon Whitcher's waveslim R package Version: 1.6.4, which in turn is based on wavelet methodology developed in Percival and Walden (2000); Gençay, Selçuk and Whitcher (2001) and others.

Value

List of two elements:

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Examples

## Based on data from Figure 7.8 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)
data(exchange)
returns <- diff(log(as.matrix(exchange)))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
J <- trunc(log2(N))-3

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)

Lst <- wave.multiple.correlation(xx, N = length(xx[[1]][[1]]))
returns.modwt.cor <- Lst$xy.mulcor[1:J,]
YmaxR <- Lst$YmaxR

exchange.names <- c("DEM.USD", "JPY.USD", "RAND")

##Producing plot

par(mfrow=c(1,1), las=0, mar=c(5,4,4,2)+.1)
matplot(2^(0:(J-1)), returns.modwt.cor[-(J+1),], type="b",
  log="x", pch="*LU", xaxt="n", lty=1, col=c(1,4,4),
  xlab="Wavelet Scale", ylab="Wavelet Multiple Correlation")
axis(side=1, at=2^(0:7))
abline(h=0)
text(2^(0:7), min(returns.modwt.cor[-(J+1),])-0.03,
  labels=exchange.names[YmaxR], adj=0.5, cex=.5)

Wavelet routine for multiple cross-correlation

Description

Produces an estimate of the multiscale multiple cross-correlation (as defined below).

Usage

 wave.multiple.cross.correlation(xx, lag.max=NULL, p=.975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple cross-correlations out of n variables that can be plotted as one single set of graphs (one per wavelet level), as an alternative to trying to make sense out of n(n-1)/2 . J sets of wavelet cross-correlations. The code is based on the calculation, at each wavelet scale, of the square root of the coefficient of determination in a linear combination of variables that includes a lagged variable for which such coefficient of determination is a maximum.

Value

List of two elements:

• xy.mulcor: numeric matrix with as many rows as levels in the wavelet transform object. The columns provide the point estimates for the wavelet multiple cross-correlations at different lags (and leads). The central column (lag=0) replicates the wavelet multiple correlations. Columns to the right (lag>0) give wavelet multiple cross-correlations with positive lag, i.e. with y=var[Pimax] lagging behind a linear combination of the rest: x[t]hat –> y[t+j]. Columns to the left (lag<0) give wavelet multiple cross-correlations with negative lag, i.e. with y=var[Pimax] leading a linear combination of the rest: y[t-j] –> x[t]hat.

• ci.mulcor: list of two elements:

  • lower: numeric matrix of the same dimensions as xy.mulcor giving the lower bounds of the corresponding 100(1-2(1-p))\% confidence interval.

  • upper: idem for the upper bounds.

• YmaxR: numeric vector giving, at each wavelet level, the index number of the variable whose correlation is calculated against a linear combination of the rest. By default, wmcc chooses at each wavelet level the variable maximizing the multiple correlation.

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

Examples

## Based on data from Figure 7.9 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)
data(exchange)
returns <- diff(log(exchange))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
J <- trunc(log2(N))-3
lmax <- 36
n <- dim(returns)[1]

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

##xx <- list(demusd.modwt.bw, jpyusd.modwt.bw)
xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)

Lst <- wave.multiple.cross.correlation(xx, lmax)
returns.cross.cor <- Lst$xy.mulcor[1:J,]
returns.lower.ci <- Lst$ci.mulcor$lower[1:J,]
returns.upper.ci <- Lst$ci.mulcor$upper[1:J,]
YmaxR <- Lst$YmaxR

# ---------------------------

##Producing correlation plot

rownames(returns.cross.cor) <- rownames(returns.cross.cor, do.NULL = FALSE, prefix = "Level ")
par(mfrow=c(3,2), las=1, pty="m", mar=c(2,3,1,0)+.1, oma=c(1.2,1.2,0,0))
ymin <- -0.1
if (length(xx)<3) ymin <- -1
for(i in J:1) {
  matplot((1:(2*lmax+1)),returns.cross.cor[i,], type="l", lty=1, ylim=c(ymin,1), xaxt="n",
          xlab="", ylab="", main=rownames(returns.cross.cor)[[i]][1])
  if(i<3) {axis(side=1, at=seq(1, 2*lmax+1, by=12), labels=seq(-lmax, lmax, by=12))}
  #axis(side=2, at=c(-.2, 0, .5, 1))
  abline(h=0,v=lmax+1)              ##Add Straight horiz and vert Lines to a Plot
  lines(returns.lower.ci[i,], lty=1, col=2) ##Add Connected Line Segments to a Plot
  lines(returns.upper.ci[i,], lty=1, col=2)
  text(1,1, labels=names(xx)[YmaxR[i]], adj=0.25, cex=.8)
}
par(las=0)
mtext('Lag (months)', side=1, outer=TRUE, adj=0.5)
mtext('Wavelet Multiple Cross-Correlation', side=2, outer=TRUE, adj=0.5)

Wavelet routine for multiple cross-regression

Description

Produces an estimate of the multiscale multiple cross-regression (as defined below).

Usage

 wave.multiple.cross.regression(xx, lag.max=NULL, p = .975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple cross-regressions out of n variables that can be plotted as one single set of graphs (one per wavelet level).

Value

List of four elements:

• wavemulcor: numeric matrix (rows = #levels, #cols = #lags and leads) with as many rows as levels in the wavelet transform object. The columns provide the point estimates for the wavelet multiple cross-correlations at different lags (and leads). The central column (lag=0) replicates the wavelet multiple correlations. Columns to the right (lag>0) give wavelet multiple cross-correlations with positive lag, i.e. with y=var[Pimax] lagging behind a linear combination of the rest: x[t]hat –> y[t+j]. Columns to the left (lag<0) give wavelet multiple cross-correlations with negative lag, i.e. with y=var[Pimax] leading a linear combination of the rest: y[t-j] –> x[t]hat.

• lower: numeric matrix (rows = #levels, #cols = #lags and leads) of lower bounds of the confidence interval.

• upper: numeric matrix (rows = #levels, #cols = #lags and leads) of upper bounds of the confidence interval.

• rval: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of regression estimates.

• rstd: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their standard deviations.

• rlow: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their lower bounds.

• rupp: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their upper bounds.

• rtst: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their t statistic values.

• rord: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their index order when sorted by significance.

• rpva: numeric array (1stdim = #levels, 2nddim = #lags and leads, 3rddim = #regressors+1) of their p values.

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on data from Figure 7.9 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)

data(exchange)
returns <- diff(log(exchange))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
J <- trunc(log2(N))-3
lmax <- 36

set.seed(140859)

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

# ---------------------------

xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)
names(xx) <- c("DEM.USD","JPY.USD","rand")

Lst <- wave.multiple.cross.regression(xx, lmax)

# ---------------------------

##Producing correlation plot

plot_wave.multiple.cross.correlation(Lst, lmax) #, by=2)

##Producing regression plot

plot_wave.multiple.cross.regression(Lst, lmax) #, by=2)

Wavelet routine for multiple regression

Description

Produces an estimate of the multiscale multiple regression (as defined below) along with approximate confidence intervals.

Usage

wave.multiple.regression(xx, N, p = 0.975, ymaxr=NULL)

Arguments

Details

The routine calculates one single set of wavelet multiple regressions out of n variables that can be plotted in a single graph.

Value

List of four elements:

• rval: numeric matrix (rows = #levels, cols = #regressors+1) of regression estimates.

• rstd: numeric matrix (rows = #levels, cols = #regressors+1) of their standard deviations.

• rlow: numeric matrix (rows = #levels, cols = #regressors+1) of their lower bounds.

• rupp: numeric matrix (rows = #levels, cols = #regressors+1) of their upper bounds.

• rtst: numeric matrix (rows = #levels, cols = #regressors+1) of their t statistic values.

• rord: numeric matrix (rows = #levels, cols = #regressors+1) of their index order when sorted by significance.

• rpva: numeric matrix (rows = #levels, cols = #regressors+1) of their p values.

Note

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Author(s)

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A: Statistical Mechanics, vol. 490, p. 1226–1238. <DOI:10.1016/j.physa.2017.11.050>

Examples

## Based on data from Figure 7.8 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)

data(exchange)
returns <- diff(log(as.matrix(exchange)))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
J <- trunc(log2(N))-3

set.seed(140859)

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
xrand.modwt <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

# ---------------------------

xx <- list(demusd.modwt, jpyusd.modwt, xrand.modwt)
names(xx) <- c("DEM.USD","JPY.USD","rand")

Lst <- wave.multiple.regression(xx)

# ---------------------------

##Producing correlation plot
plot_wave.multiple.correlation(Lst)

##Producing regression plot
plot_wave.multiple.regression(Lst)

Correlation structural breaks data

Description

Simulated data showing correlation structural breaks in Figure 4 of Fernández-Macho (2018).

Usage

data("xrand")

Format

A data frame with 512 observations on the following 2 variables.

xrand1

a numeric vector

xrand2

a numeric vector

Details

xrand1[t] and xrand2[t] are highly correlated at low frequencies (long timescales) but uncorrelated at high frequencies (short timescales). However, during a period of time spanning the second third of the sample (T/3<t<2T/3) that behavior is reversed so that data become highly correlated at short timescales but uncorrelated at low frequencies.

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A, 492, https://doi.org/10.1016/j.physa.2017.11.050

Examples

data(xrand)
## maybe str(xrand) ; plot(xrand) ...

Correlation structural breaks variable 1

Description

Simulated data showing correlation structural breaks in Figure 4 of Fernández-Macho (2018).

Usage

data("xrand")

Format

A data frame with 512 observations on 1 variables.

xrand1

a numeric vector

Details

xrand1[t] and xrand2[t] are highly correlated at low frequencies (long timescales) but uncorrelated at high frequencies (short timescales). However, during a period of time spanning the second third of the sample (T/3<t<2T/3) that behavior is reversed so that data become highly correlated at short timescales but uncorrelated at low frequencies.

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A, 492, https://doi.org/10.1016/j.physa.2017.11.050

Examples

data(xrand)
## maybe str(xrand) ; plot(xrand) ...

Correlation structural breaks variable 2

Description

Simulated data showing correlation structural breaks in Figure 4 of Fernández-Macho (2018).

Usage

data("xrand")

Format

A data frame with 512 observations on 1 variables.

xrand2

a numeric vector

Details

xrand1[t] and xrand2[t] are highly correlated at low frequencies (long timescales) but uncorrelated at high frequencies (short timescales). However, during a period of time spanning the second third of the sample (T/3<t<2T/3) that behavior is reversed so that data become highly correlated at short timescales but uncorrelated at low frequencies.

References

Fernández-Macho, J., 2018. Time-localized wavelet multiple regression and correlation, Physica A, 492, https://doi.org/10.1016/j.physa.2017.11.050

Examples

data(xrand)
## maybe str(xrand) ; plot(xrand) ...