highfrequency (version 1.0.1)

rSemiCov: Realized semicovariance

Description

Calculate the Realized Semicovariances (rSemiCov). Let \( r_{t,i} \) be an intraday \(M x N\) return matrix and \(i=1,...,M\) the number of intraday returns. Then, let \(r_{t,i}^{+} = max(r_{t,i},0)\) and \(r_{t,i}^{-} = min(r_{t,i},0)\).

Then, the realized semicovariance is given by the following three matrices:

$$ \mbox{pos}_t =\sum_{i=1}^{M} r^{+}_{t,i} r^{+'}_{t,i} $$ $$ \mbox{neg}_t =\sum_{i=1}^{M} r^{-}_{t,i} r^{-'}_{t,i} $$ $$ \mbox{mixed}_t =\sum_{i=1}^{M} (r^{+}_{t,i} r^{-'}_{t,i} + r^{-}_{t,i} r^{+'}_{t,i}) $$

The mixed covariance matrix will have 0 on the diagonal. From these three matrices, the realized covariance can be constructed as \(pos + neg + mixed\). The concordant semicovariance matrix is \(pos + neg\). The off-diagonals of the concordant matrix is always positive, while for the mixed matrix, it is always negative.

Usage

rSemiCov(
  rData,
  cor = FALSE,
  alignBy = NULL,
  alignPeriod = NULL,
  makeReturns = FALSE
)

Value

In case the data consists of one day a list of five \(N \times N\) matrices are returned. These matrices are named mixed, positive, negative, concordant, and rCov. The latter matrix corresponds to the realized covariance estimator and is thus named like the function rCov. In case the data spans more than one day, the list for each day will be put into another list named according to the date of the estimates.

Arguments

rData

an xts or data.table object containing returns or prices, possibly for multiple assets over multiple days.

cor

boolean, in case it is TRUE, and the input data is multivariate, the correlation is returned instead of the covariance matrix. FALSE by default.

alignBy

character, indicating the time scale in which alignPeriod is expressed. Possible values are: "ticks", "secs", "seconds", "mins", "minutes", "hours"

alignPeriod

positive numeric, indicating the number of periods to aggregate over. For example, to aggregate based on a 5-minute frequency, set alignPeriod = 5 and alignBy = "minutes".

makeReturns

boolean, should be TRUE when rData contains prices instead of returns. FALSE by default.

Author

Emil Sjoerup.

Details

In the case that cor is TRUE, the mixed matrix will be an \(N \times N\) matrix filled with NA as mapping the mixed covariance matrix into correlation space is impossible due to the 0-diagonal.

References

Bollerslev, T., Li, J., Patton, A. J., and Quaedvlieg, R. (2020). Realized semicovariances. Econometrica, 88, 1515-1551.

See Also

ICov for a list of implemented estimators of the integrated covariance.

Examples

# Realized semi-variance/semi-covariance for prices aligned
# at 5 minutes.

# Univariate:
rSVar = rSemiCov(rData = sampleTData[, list(DT, PRICE)], alignBy = "minutes",
                   alignPeriod = 5, makeReturns = TRUE)
rSVar
if (FALSE) {
library("xts")
# Multivariate multi day:
rSC <- rSemiCov(sampleOneMinuteData, makeReturns = TRUE) # rSC is a list of lists
# We extract the covariance between stock 1 and stock 2 for all three covariances.
mixed <- sapply(rSC, function(x) x[["mixed"]][1,2])
neg <- sapply(rSC, function(x) x[["negative"]][1,2])
pos <- sapply(rSC, function(x) x[["positive"]][1,2])
covariances <- xts(cbind(mixed, neg, pos), as.Date(names(rSC)))
colnames(covariances) <- c("mixed", "neg", "pos")
# We make a quick plot of the different covariances
plot(covariances)
addLegend(lty = 1) # Add legend so we can distinguish the series.
}