highfrequency (version 1.0.1)

leadLag: Lead-Lag estimation

Description

Function that estimates whether one series leads (or lags) another.

Let \(X_{t}\) and \(Y_{t}\) be two observed price over the time interval \([0,1]\).

For every integer \(k \in \cal{Z}\), we form the shifted time series

$$ Y_{\left(k+i\right)/n}, \quad i = 1, 2, \dots $$ \(H=\left(\underline{H},\overline{H}\right]\) is an interval for \(\vartheta\in\Theta\), define the shift interval \(H_{\vartheta}=H+\vartheta=\left(\underline{H}+\vartheta,\overline{H}+\vartheta\right]\) then let

$$ X\left(H\right)_{t}=\int_{0}^{t}1_{H}\left(s\right)\textrm{d}X_{s} $$

Which will be abbreviated: $$ X\left(H\right)=X\left(H\right)_{T+\delta}=\int_{0}^{T+\delta}1_{H}\left(s\right)\textrm{d}X_{s} $$

Then the shifted HY contrast function is: $$ \tilde{\vartheta}\rightarrow U^{n}\left(\tilde{\vartheta}\right)= \\ 1_{\tilde{\vartheta}\geq0}\sum_{I\in{\cal{I}},J\in{\cal{J}},\overline{I}\leq T}X\left(I\right)Y\left(J\right)1_{\left\{ I\cap J_{-\tilde{\vartheta}}\neq\emptyset\right\}} \\ +1_{\tilde{\vartheta}<0}\sum_{I\in{\cal{I}},J\in{\cal{J}},\overline{J}\leq T}X\left(I\right)Y\left(Y\right)1_{\left\{ J\cap I_{\tilde{\vartheta}}\neq\emptyset\right\} } $$ This contrast function is then calculated for all the lags passed in the argument lags

Usage

leadLag(
  price1 = NULL,
  price2 = NULL,
  lags = NULL,
  resolution = "seconds",
  normalize = TRUE,
  parallelize = FALSE,
  nCores = NA
)

Value

A list with class leadLag which contains contrasts, lead-lag-ratio, and lags, denoting the estimated values for each lag calculated, the lead-lag-ratio, and the tested lags respectively.

Arguments

price1

xts or data.table containing prices in levels, in case of data.table, use a column DT to denote the date-time in POSIXct format, and a column PRICE to denote the price

price2

xts or data.table containing prices in levels, in case of data.table, use a column DT to denote the date-time in POSIXct format, and a column PRICE to denote the price

lags

a numeric denoting which lags (in units of resolution) should be tested as leading or lagging

resolution

the resolution at which the lags is measured. The default is "seconds", use this argument to gain 1000 times resolution by setting it to either "ms", "milliseconds", or "milli-seconds".

normalize

logical denoting whether the contrasts should be normalized by the product of the L2 norms of both the prices. Default = TRUE. This does not change the value of the lead-lag-ratio.

parallelize

logical denoting whether to use a parallelized version of the C++ code (parallelized using OPENMP). Default = FALSE

nCores

integer valued numeric denoting how many cores to use for the lead-lag estimation procedure in case parallelize is TRUE. Default is NA, which does not parallelize the code.

Details

The lead-lag-ratio (LLR) can be used to see if one asset leads the other. If LLR < 1, then price1 MAY be leading price2 and vice versa if LLR > 1.

References

Hoffmann, M., Rosenbaum, M., and Yoshida, N. (2013). Estimation of the lead-lag parameter from non-synchronous data. Bernoulli, 19, 1-37.

Examples

if (FALSE) {
# Toy example to show the usage
# Spread prices
spread <- spreadPrices(sampleMultiTradeData[SYMBOL %in% c("ETF", "AAA")])
# Use lead-lag estimator
llEmpirical <- leadLag(spread[!is.na(AAA), list(DT, PRICE = AAA)], 
                       spread[!is.na(ETF), list(DT, PRICE = ETF)], seq(-15,15))
plot(llEmpirical)
}