# [R] Switching log(J) to log(J+1) to avoid log(0) in HAR-RVJ model

cursethiscure caolan.harvey6 at mail.dcu.ie
Thu Jul 19 19:23:03 CEST 2012

```I am working with xts dependent data, and my code is as follows (the problem
is explained throughout):

dat <- getdat("prices")
dat <- read.zoo(dat, sep = "",format="%d/%m/%Y %H:%M",
tz="", FUN=NULL, regular=TRUE,
"numeric"))
dat <- as.xts(dat)
## cleaned data here to get
daylist <- lapply(split(dat, "days"), function(x) {
if(NROW(x) >= 10) x
})
do.call(rbind, daylist) -> dat
makeReturns(dat) -> dat

the code I am running to get the HAR regressions (full code for it shown
below) is

x = harModel(dat, periods = c(1,5,22), periodsJ=c(1), RVest =
c("RCov","RBPCov"),
type="HARRVJ", h=5, transform="log") ; # Estimate the HAR model
of type HARRVJ

The three HAR models on paper are:

1.〖RV〗_(t,t+h) = β_0+ β_D 〖RV〗_t+ β_W 〖RV〗_(t-5)+ β_M 〖RV〗_(t-22)  + β_J J_t
+ ε_(t,t+h) # NULL model in code at bottom

2.  (RV〗_(t,t+h) )^(1/2) = β_0+ β_D 〖〖RV〗_t〗^(1/2)+ β_W (〖RV〗_(t-5) )^(1/2)+
β_M (〖RV〗_(t-22) )^(1/2)  + β_J (〖 J〗_t )^(1/2)  + ε_(t,t+h)

3. log(RV〗_(t,t+h) )= β_0+ β_D.log(〖RV〗_t )+ β_W.log(〖RV〗_(t-5) )+
β_M.log(〖RV〗_(t-22) )  + β_J  log(J_t  + 1)  + ε_(t,t+h)

Basically the harModel in the code allows you to transform the regressions
from NULL  to "sqrt" or "log", but when `transform="log"` is chosen it gives
the following error message.

x = harModel(dat, periods = c(1,5,22), periodsJ=c(1), RVest =
c("RCov","RBPCov"),
+              type="HARRVJ", h=22, transform="log") ; # Estimate ....
[TRUNCATED]
Error in lm.fit(x, y, offset = offset, singular.ok = singular.ok, ...) :
NA/NaN/Inf in foreign function call (arg 1)

which is due to it taking the log(0), the actual log model should take
log(J + 1) in case of a 0 value for the J in the time series ( if J does not
occur it should return a 0, otherwise it should return J), but unfortunately
I do not know how to rectify this. I was wondering if any one could tell me
how how I can achieve this as I am very naive with R still.  I have tried
some modifications you will see at the bottom of the question, but they do
not compute the regression correctly though they do allow it to run without
'error'

A sample of the xts returns are

structure(c(0, 0.00163990674377068, -0.000417175726423302,
-0.000536624643948436,
-0.000417573901733803, -0.000656540057603294, -0.000119417243992004,
0.000149269326922941, 0.000865322983560901, 0.000387660473923468,
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1324052400, 1324052700, 1324053000, 1324053300, 1324053600, 1324053900,
1324054200, 1324054500, 1324054800), tzone = "", tclass = c("POSIXct",
"POSIXt")), .Dim = c(1000L, 1L))

The full code is just below this code and what I have tried to do within it
is change the following two lines when running transform="log" but this
gives unrealistic regression output such as values over 3:

if( type == "HARRVJ" ){
J = J[(maxp:(n-h)),];
*x = cbind(x1,J);*              # bind jumps to RV data
if(!is.null(transform)){ y = Ftransform(y); x = Ftransform(x); }
x = cbind(x,rmin);
model = estimhar(y=y,x=x);
model\$transform = transform; model\$h = h; model\$type = "HARRVJ";
model\$dates = alldates[(maxp+h):n];
class(model) = c("harModel","lm");
return( model )
}#End HAR-RV-J if cond

to

if( type == "HARRVJ" ){
J = J[(maxp:(n-h)),];
*x = cbind(x1,J+1); *             # bind jumps to RV data
if(!is.null(transform)){ y = Ftransform(y); x = Ftransform(x); }
x = cbind(x,rmin);
model = estimhar(y=y,x=x);
model\$transform = transform; model\$h = h; model\$type = "HARRVJ";
model\$dates = alldates[(maxp+h):n];
class(model) = c("harModel","lm");
return( model )
}#End HAR-RV-J if cond

and this

if( type == "HARRVCJ" ){
# Are the jumps significant? if not set to zero:
if( jumptest=="ABDJumptest" ){
TQ = apply.daily(data, TQfun);
J = J[,1];
teststats    = ABDJumptest(RV=RM1,BPV=RM2,TQ=TQ );
}else{ jtest = match.fun(jumptest); teststats = jtest(data,...) }
Jindicators  = teststats > qnorm(1-alpha);
*J[!Jindicators] = 0;*
to
if( type == "HARRVCJ" ){
# Are the jumps significant? if not set to zero:
if( jumptest=="ABDJumptest" ){
TQ = apply.daily(data, TQfun);
J = J[,1];
teststats    = ABDJumptest(RV=RM1,BPV=RM2,TQ=TQ );
}else{ jtest = match.fun(jumptest); teststats = jtest(data,...) }
Jindicators  = teststats > qnorm(1-alpha);
*  J[!Jindicators] = 1;*

the full code is (which I take no credit for):

#  START implementation of paper:
#  ROUGHING IT UP: INCLUDING JUMP COMPONENTS IN THE MEASUREMENT, MODELING,
AND FORECASTING OF RETURN VOLATILITY
#  Torben G. Andersen, Tim Bollerslev, and Francis X. Diebold
#  data: a xts object with the intraday data
#  periods: a vector with time periods to aggregate over, expressed in days
#  RVest: estimator for daily realized volatility,
#  in case a vector is supplied, the first estimator is the unrobust
estimator, the second is the robust estimator
#  type: string defining the type of model
#  "HARRV" from "roughing paper"
#  "HARRVJ" from "roughing paper"
#  "HARRVCJ" from "roughing paper"
#  jumptest: function to calculate the jump test statistic which determines
whether the daily jump contribution is significant
#  alpha: a value between zero and one to indicate what
#  h: integer, determining over how many periods the depend variable should
be aggregated. The default is 1, i.e. no aggregation is done, just one day.
#  TODO ADD extra argument: jump-periods??? for aggregated jumps in the
model...

# Helpfunctions:
TQfun = function(rdata){ #Calculate the realized tripower quarticity
returns = as.vector(as.numeric(rdata));
n = length(returns);
mu43 = 0.8308609; #    2^(2/3)*gamma(7/6) *gamma(1/2)^(-1)
tq = n * ((mu43)^(-3)) *  sum( abs(returns[1:(n - 2)])^(4/3)
*abs(returns[2:(n-1)])^(4/3) *abs(returns[3:n])^(4/3) );
return(tq);
}

ABDJumptest = function(RV, BPV, TQ){ # Comput jump detection stat mentioned
in roughing paper
mu1  = sqrt(2/pi);
n = length(RV);
zstat = ((1/n)^(-1/2))*((RV-BPV)/RV)*(  (mu1^(-4) + 2*(mu1^(-2))-5) *
pmax( 1,TQ*(BPV^(-2)) )   )^(-1/2);
return(zstat);
}

harModel = function(data, periods = c(1,5,22), periodsJ = c(1,5,22),
leverage=NULL, RVest = c("RCov","RBPCov"), type="HARRV",
jumptest="ABDJumptest",alpha=0.001,h=1,transform=NULL,
...){
nperiods = length(periods); # Number of periods to aggregate over
nest = length(RVest);       # Number of RV estimators
if( !is.null(transform) ){ Ftransform = match.fun(transform); }
if( !(type %in% c("HARRV","HARRVJ","HARRVCJ"))){ warning("Please provide a
valid argument for type, see documentation.")  }

if( sum(data<0) != 0 ){ #If it are returns as input
# Get the daily RMs (in a non-robust and robust way)
RV1 = match.fun(  RVest[1]);
RM1 = apply.daily( data, RV1 );
# save dates:
alldates = index(RM1)
if( nest == 2 ){
RV2 = match.fun( RVest[2]);
RM2 = apply.daily( data, RV2 ); }
}

if( sum(data<0) == 0 ){ #The input is most likely already realized
measures
dimdata = dim(data)[2];
alldates = index(data);
RM1 = data[,1];
if( dimdata > 1 ){ RM2 = data[,2]; }
if( type != "HARRV" ){ warning("Please provide returns as input for the
type of model you want to estimate. All your returns are positive which is
quite unlikely honestly. Only for the HAR-RV model you can input realized
measures.") }
}

# Get the matrix for estimation of linear model
maxp      = max(periods,periodsJ); #max number of aggregation levels
if(!is.null(leverage)){ maxp = max(maxp,leverage) }
n         = length(RM1);  #Number of Days

# Aggregate RV:
RVmatrix1 = aggRV(RM1,periods);
if( nest==2 ){ RVmatrix2 = aggRV(RM2,periods); }  # In case a jumprobust
estimator is supplied

# Aggregate and subselect y:
y = aggY(RM1,h,maxp);

# Only keep useful parts:
x1 = RVmatrix1[(maxp:(n-h)),];
if( nest==2 ){ x2 = RVmatrix2[(maxp:(n-h)),]; } # In case a jumprobust
estimator is supplied

# Jumps:
if(type!="HARRV"){ # If model type is as such that you need jump component
J = pmax( RM1 - RM2,0 ); # Jump contributions should be positive
J = aggJ(J,periodsJ);
}

if( !is.null(leverage) ){
if( sum(data<0) == 0 ){ warning("You cannot use leverage variables in
the model in case your input consists of Realized Measures") }
# Get close-to-close returns
e = apply.daily(data,sum); #Sum logreturns daily
# Get the rmins:
rmintemp = pmin(e,0);
# Aggregate everything:
rmin = aggRV(rmintemp,periods=leverage,type="Rmin");
# Select:
rmin = rmin[(maxp:(n-h)),];
}else{ rmin = matrix(ncol=0,nrow=dim(x1)[1]) }

###############################
# Estimate the model parameters, according to type of model :
# First model type: traditional HAR-RV:
if( type == "HARRV" ){
if(!is.null(transform)){ y = Ftransform(y); x1 = Ftransform(x1) }
x1 = cbind(x1,rmin);
model     = estimhar(y=y,x=x1);
model\$transform = transform; model\$h = h; model\$type = "HARRV";
model\$dates = alldates[(maxp+h):n];
class(model) = c("harModel","lm");
return( model )
} #End HAR-RV if cond

if( type == "HARRVJ" ){
J = J[(maxp:(n-h)),];
x = cbind(x1,J);              # bind jumps to RV data
if(!is.null(transform)){ y = Ftransform(y); x = Ftransform(x); }
x = cbind(x,rmin);
model = estimhar(y=y,x=x);
model\$transform = transform; model\$h = h; model\$type = "HARRVJ";
model\$dates = alldates[(maxp+h):n];
class(model) = c("harModel","lm");
return( model )
}#End HAR-RV-J if cond

if( type == "HARRVCJ" ){
# Are the jumps significant? if not set to zero:
if( jumptest=="ABDJumptest" ){
TQ = apply.daily(data, TQfun);
J = J[,1];
teststats    = ABDJumptest(RV=RM1,BPV=RM2,TQ=TQ );
}else{ jtest = match.fun(jumptest); teststats = jtest(data,...) }
Jindicators  = teststats > qnorm(1-alpha);
J[!Jindicators] = 0;
# Get continuus components if necessary RV measures if necessary:
Cmatrix = matrix( nrow = dim(RVmatrix1)[1], ncol = 1 );
Cmatrix[Jindicators,]    = RVmatrix2[Jindicators,1];      #Fill with
robust one in case of jump
Cmatrix[(!Jindicators)]  = RVmatrix1[(!Jindicators),1];   #Fill with
non-robust one in case of no-jump
# Aggregate again:
Cmatrix <- aggRV(Cmatrix,periods,type="C");
Jmatrix <- aggJ(J,periodsJ);
# subset again:
Cmatrix <- Cmatrix[(maxp:(n-h)),];
Jmatrix <- Jmatrix[(maxp:(n-h)),];
x = cbind(Cmatrix,Jmatrix);               # bind jumps to RV data
if(!is.null(transform)){ y = Ftransform(y); x = Ftransform(x); }
x = cbind(x,rmin);
model = estimhar( y=y, x=x );
model\$transform = transform; model\$h = h; model\$type = "HARRVCJ";
model\$dates = alldates[(maxp+h):n];
class(model) = c("harModel","lm");
return(model)
}

} #End function harModel
#################################################################
estimhar = function(y, x){ #Potentially add stuff here
colnames(y)="y";
output = lm( formula(y~x), data=cbind(y,x));
}

# Help function to get nicely formatted formula's for print/summary
methods..
getHarmodelformula = function(x){
modelnames = colnames(x\$model\$x);
if(!is.null(x\$transform)){

visual tingie for plotting transformed RV
betas      = paste("beta",(1:length(modelnames)),"",sep="")
betas2     = paste(" + ",betas,"*")
rightside  = paste(betas2, modelnames,collapse="");
h = x\$h;
left = paste("RV",h,sep="");
if(!is.null(x\$transform)){  left = paste(x\$transform,"(",left,")",sep="" )
}
modeldescription = paste(left,"= beta0",rightside);
return(list(modeldescription,betas))
}

aggRV <- function(RM1,periods,type="RV"){
n = length(RM1);
nperiods = length(periods);
RVmatrix1 = matrix(nrow=n,ncol=nperiods);
for(i in 1:nperiods){
if(periods[i]==1){ RVmatrix1[,i] = RM1;
}else{ RVmatrix1[(periods[i]:n),i] =
rollmean(x=RM1,k=periods[i],align="left")  }
} #end loop over periods for standard RV estimator
colnames(RVmatrix1) = paste(type,periods,sep="");
return(RVmatrix1);
}

aggJ <- function( J, periodsJ ){
n = length(J);
nperiods = length(periodsJ);
JM = matrix(nrow=n,ncol=nperiods);
for(i in 1:nperiods){
if(periodsJ[i]==1){ JM[,i] = J;
}else{ JM[(periodsJ[i]:n),i] = rollmean( x=J, k=periodsJ[i],
align="left")  }
} # End loop over periods for standard RV estimator
colnames(JM) = paste("J",periodsJ,sep="");
return(JM)
}

aggY = function(RM1,h,maxp){
n         = length(RM1);
if( h == 1 ){  y  = RM1[(maxp+1):n]; }
if( h != 1 ){
y = matrix( nrow=length(RM1), ncol=1 ); colnames(y) = "y";
y[(h:n),] = rollmean(x=RM1,k=h,align="left");
y = matrix(y[((maxp+h):n),],ncol=1); y=as.data.frame(y) }
return(y);
}

#########################################################################
# Print method for harmodel:
print.harModel = function(x, digits = max(3, getOption("digits") - 3), ...){
formula = getHarmodelformula(x); modeldescription = formula[[1]]; betas =
formula[[2]];

cat("\nModel:\n", paste(modeldescription, sep = "\n", collapse = "\n"),
"\n\n", sep = "")

coefs = coef(x);
names(coefs)  = c("beta0",betas)

if (length(coef(x))){
cat("Coefficients:\n")
print.default(format(coefs, digits = digits), print.gap = 2,quote =
FALSE);
cat("\n\n");
print.default((format(zz,digits=digits) ),print.gap = 2,quote=FALSE)
}
else cat("No coefficients\n")
cat("\n")
invisible(x)
}

summary.harModel = function(object, correlation = FALSE, symbolic.cor =
FALSE,...){
x=object;
dd = summary.lm(x);
formula = getHarmodelformula(x); modeldescription = formula[[1]]; betas =
formula[[2]];
dd\$call = modeldescription;
rownames(dd\$coefficients) = c("beta0",betas);
return(dd)
}

plot.harModel = function(x, which = c(1L:3L, 5L), caption = list("Residuals
vs Fitted",
"Normal
Q-Q", "Scale-Location", "Cook's distance", "Residuals vs Leverage",

expression("Cook's dist vs Leverage  " * h[ii]/(1 - h[ii]))),
panel = if (add.smooth) panel.smooth else points,
sub.caption = NULL,
main = "", ask = prod(par("mfcol")) < length(which)
&& dev.interactive(),
..., id.n = 3, labels.id = names(residuals(x)),
cex.id = 0.75,
qqline = TRUE, cook.levels = c(0.5, 1), add.smooth
label.pos = c(4, 2), cex.caption = 1){
observed = x\$model\$y;
fitted   = x\$fitted.values;
dates    = x\$dates;
dates    = as.POSIXct(dates);
observed = xts(observed, order.by=dates);
fitted   = xts(fitted, order.by=dates);
type     = x\$type;

g_range = range(fitted,observed)
g_range[1] = 0.95*g_range[1]; g_range[2]= 1.05 * g_range[2];
#ind = seq(1,length(fitted),length.out=5);
title = paste("Observed and forecasted RV based on HAR Model:",type);
plot.zoo(observed,col="red",lwd=2,main=title,
ylim=g_range,xlab="Time",ylab="Realized Volatility");
#  axis(1,time(b)[ind], format(time(b)[ind],), las=2, cex.axis=0.8); not
used anymore
#  axis(2);
lines(fitted,col="blue",lwd=2);
legend("topleft", c("Observed RV","Forecasted RV"), cex=1.1,
col=c("red","blue"),lty=1, lwd=2, bty="n");
}

If anyone could help me with as to how J should be modified when the model
is in log form it would be hugely appreciated, Thanks in advance.

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```