run_autocovar: Calculate the trailing autocovariance of a _time series_ of...

In algoquant/HighFreq: High Frequency Time Series Management

Calculate the trailing autocovariance of a time series of returns using an online recursive formula.

Description

Calculate the trailing autocovariance of a time series of returns using an online recursive formula.

Usage

run_autocovar(timeser, lambdaf, lagg = 1L)

Arguments

timeser	A time series or a matrix with a single column of returns data.
lambdaf	A decay factor which multiplies past estimates.
lagg	An integer equal to the number of periods to lag. (The default is lagg = 1.)

Details

The function run_autocovar() calculates the trailing autocovariance of a streaming time series of returns, by recursively weighting the past autocovariance estimates {cov}_{t-1}, with the products of their returns minus their means, using the decay factor \lambda:

\bar{x}_t = \lambda \bar{x}_{t-1} + (1 - \lambda) x_t

\sigma^2_{t} = \lambda^2 \sigma^2_{t-1} + (1 - \lambda^2) (x_t - \bar{x}_t)^2

{cov}_t = \lambda^2 {cov}_{t-1} + (1 - \lambda^2) (x_t - \bar{x}_t) (x_{t-l} - \bar{x}_{t-l})

Where {cov}_t is the trailing autocovariance estimate at time t, with lagg=l. And \sigma^2_{t} and \bar{x}_t are the trailing variances and means of the streaming data.

The above online recursive formulas are convenient for processing live streaming data because they don't require maintaining a buffer of past data. The formulas are equivalent to a convolution with exponentially decaying weights, but they're much faster to calculate. Using exponentially decaying weights is more natural than using a sliding look-back interval, because it gradually "forgets" about the past data.

Note that the variance and covariance decay as the square of \lambda, while the mean returns decay as \lambda. This is because the variance is proportional to the square of the returns.

The value of the decay factor \lambda must be in the range between 0 and 1. If \lambda is close to 1 then the decay is weak and past values have a greater weight, and the trailing covariance values have a stronger dependence on past data. This is equivalent to a long look-back interval. If \lambda is much less than 1 then the decay is strong and past values have a smaller weight, and the trailing covariance values have a weaker dependence on past data. This is equivalent to a short look-back interval.

The function run_autocovar() returns three columns of data: the trailing autocovariances, the variances, and the mean values of the argument timeser. This allows calculating the trailing autocorrelations.

Value

A matrix with three columns of data: the trailing autocovariances, the variances, and the mean values of the argument timeser.

Examples

## Not run: 
# Calculate historical returns
retp <- zoo::coredata(na.omit(rutils::etfenv$returns$VTI))
# Calculate the trailing autocovariance
lambdaf <- 0.9 # Decay factor
lagg <- 3
covars <- HighFreq::run_autocovar(retp, lambdaf=lambdaf, lagg=lagg)
# Calculate the trailing autocorrelation
correl <- covars[, 1]/covars[, 2]
# Calculate the trailing autocovariance using R code
nrows <- NROW(retp)
retm <- numeric(nrows)
retm[1] <- retp[1, ]
retd <- numeric(nrows)
covarr <- numeric(nrows)
covarr[1] <- retp[1, ]^2
for (it in 2:nrows) {
  retm[it] <- lambdaf*retm[it-1] + (1-lambdaf)*(retp[it])
  retd[it] <- retp[it] - retm[it]
  covarr[it] <- lambdaf*covarr[it-1] + (1-lambdaf)*retd[it]*retd[max(it-lagg, 1)]
} # end for
all.equal(covarr, covars[, 1])

## End(Not run)  # end dontrun

algoquant/HighFreq documentation built on June 10, 2025, 3:54 p.m.