Description Usage Arguments Details Value Author(s) References See Also Examples

This function implements the local method of moments proposed in Bibinger et al. (2014) to estimate the cummulative covariance matrix of a non-synchronously observed multi-dimensional Ito process with noise.

1 2 |

`x` |
an object of |

`block` |
a positive integer indicating the number of the blocks which the observation interval is split into. |

`freq` |
a positive integer indicating the number of the frequencies used to compute the final estimator. |

`freq.p` |
a positive integer indicating the number of the frequencies used to compute the pilot estimator for the spot covariance matrix (corresponding to the number |

`K` |
a positive integer indicating the number of the blocks used to compute the pilot estimator for the spot covariance matrix (corresponding to the number |

`interval` |
a vector indicating the observation interval. The first component represents the initial value and the second component represents the terminal value. |

`Sigma.p` |
a |

`noise.var` |
character string giving the method to estimate the noise variances. There are several options: |

`samp.adj` |
character string giving the method to adjust the effect of the sampling times on the variances of the spectral statistics for the noise part. The default method |

`psd` |
logical. If |

The default implementation is the adaptive version of the local method of moments estimator, which is only based on observation data. It is possible to implement oracle versions of the estimator by setting user-specified `Sigma.p`

and/or `noise.var`

. An example is given below.

An object of class `"yuima.specv"`

, which is a list with the following elements:

`covmat` |
the estimated covariance matrix |

`vcov` |
the estimated variance-covariance matrix of |

`Sigma.p` |
the pilot estimates of the spot covariance matrix |

Yuta Koike with YUIMA Project Team

Ait-Sahalia, Y., Mykland, P. A. and Zhang, L. (2005)
How often to sample a continuous-time process in the presence of market microstructure noise,
*The Review of Financial Studies*, **18**, 351–416.

Altmeyer, R. and Bibinger, M. (2015)
Functional stable limit theorems for quasi-efficient spectral covolatility estimators,
to appear in *Stochastic processes and their applications*, doi:10.1016/j.spa.2015.07.009.

Bandi, F. M. and Russell, J. R. (2006)
Separating microstructure noise from volatility,
*Journal of Financial Economics*, **79**, 655–692.

Bibinger, M., Hautsch, N., Malec, P. and Reiss, M. (2014)
Estimating the quadratic covariation matrix from noisy observations: local method of moments and efficiency,
*Annals of Statistics*, **42**, 80–114.

Gatheral J. and Oomen, R. C. A. (2010)
Zero-intelligence realized variance estimation,
*Finance Stochastics*, **14**, 249–283.

Oomen, R. C. A. (2006)
Properties of realized variance under alternative sampling schemes,
*Journal of Business and Economic Statistics*, **24**, 219–237.

Reiss, M. (2011)
Asymptotic equivalence for inference on the volatility from noisy observations,
*Annals of Statistics*, **39**, 772–802.

Xiu, D. (2010)
Quasi-maximum likelihood estimation of volatility with high frequency data,
*Journal of Econometrics*, **159**, 235–250.

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# Example. One-dimensional and regular sampling case
# Here the simulated model is taken from Reiss (2011)
## Set a model
sigma <- function(t) sqrt(0.02 + 0.2 * (t - 0.5)^4)
modI <- setModel(drift = 0, diffusion = "sigma(t)")
## Generate a path of the process
set.seed(117)
n <- 12000
yuima.samp <- setSampling(Terminal = 1, n = n)
yuima <- setYuima(model = modI, sampling = yuima.samp)
yuima <- simulate(yuima, xinit = 0)
delta <- 0.01 # standard deviation of microstructure noise
yuima <- noisy.sampling(yuima, var.adj = delta^2) # generate noisy observations
plot(yuima)
## Estimation of the integrated volatility
est <- lmm(yuima)
est
## True integrated volatility and theoretical standard error
disc <- seq(0, 1, by = 1/n)
cat("true integrated volatility\n")
print(mean(sigma(disc[-1])^2))
cat("theoretical standard error\n")
print(sqrt(8*delta*mean(sigma(disc[-1])^3))/n^(1/4))
# Plotting the pilot estimate of the spot variance path
block <- 20
G <- seq(0,1,by=1/block)[1:block]
Sigma.p <- sigma(G)^2 # true spot variance
plot(zoo(Sigma.p, G), col = "blue",, xlab = "time",
ylab = expression(sigma(t)^2))
lines(zoo(est$Sigma.p, G))
## "Oracle" implementation
lmm(yuima, block = block, Sigma.p = Sigma.p, noise.var = delta^2)
# Example. Multi-dimensional case
# We simulate noisy observations of a correlated bivariate Brownian motion
# First we examine the regular sampling case since in this situsation the theoretical standard
# error can easily be computed via the formulae given in p.88 of Bibinger et al. (2014)
## Set a model
drift <- c(0,0)
rho <- 0.5 # correlation
diffusion <- matrix(c(1,rho,0,sqrt(1-rho^2)),2,2)
modII <- setModel(drift=drift,diffusion=diffusion,
state.variable=c("x1","x2"),solve.variable=c("x1","x2"))
## Generate a path of the latent process
set.seed(123)
## We regard the unit interval as 6.5 hours and generate the path on it
## with the step size equal to 1 seconds
n <- 8000
yuima.samp <- setSampling(Terminal = 1, n = n)
yuima <- setYuima(model = modII, sampling = yuima.samp)
yuima <- simulate(yuima)
## Generate noisy observations
eta <- 0.05
yuima <- noisy.sampling(yuima, var.adj = diag(eta^2, 2))
plot(yuima)
## Estimation of the integrated covariance matrix
est <- lmm(yuima)
est
## Theoretical standard error
a <- sqrt(4 * eta * (sqrt(1 + rho) + sqrt(1 - rho)))
b <- sqrt(2 * eta * ((1 + rho)^(3/2) + (1 - rho)^(3/2)))
cat("theoretical standard error\n")
print(matrix(c(a,b,b,a),2,2)/n^(1/4))
## "Oracle" implementation
block <- 20
Sigma.p <- matrix(c(1,rho,rho,1),block,4,byrow=TRUE) # true spot covariance matrix
lmm(yuima, block = block, Sigma.p = Sigma.p, noise.var = rep(eta^2,2))
# Next we extract nonsynchronous observations from
# the path generated above by Poisson random sampling
psample <- poisson.random.sampling(yuima, rate = c(1/2,1/2), n = n)
## Estimation of the integrated covariance matrix
lmm(psample)
## "Oracle" implementation
lmm(psample, block = block, Sigma.p = Sigma.p, noise.var = rep(eta^2,2))
## Other choices of tuning parameters (estimated values are not varied so much)
lmm(psample, block = 25)
lmm(psample, freq = 100)
lmm(psample, freq.p = 15)
lmm(psample, K = 8)
``` |

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