yuima (version 1.1.6)

hyavar:

Asymptotic Variance Estimator for the Hayashi-Yoshida estimator

Description

This function estimates the asymptotic variances of covariance and correlation estimates by the Hayashi-Yoshida estimator.

Usage

hyavar(yuima, bw, nonneg = TRUE, psd = TRUE)

Arguments

yuima

an object of yuima-class or yuima.data-class.

bw

a positive number or a numeric matrix. If it is a matrix, each component indicate the bandwidth parameter for the kernel estimators used to estimate the asymptotic variance of the corresponding component (necessary only for off-diagonal components). If it is a number, it is converted to a matrix as matrix(bw,d,d), where d=dim(x). The default value is the matrix whose $(i,j)$-th component is given by $min(n_i,n_j)^{0.45}$, where $n_i$ denotes the number of the observations for the $i$-th component of the data.

nonneg

logical. If TRUE, the asymptotic variance estimates for correlations are always ensured to be non-negative. See `Details'.

psd

passed to cce.

Value

A list with components: A list with components:

Details

The precise description of the method used to estimate the asymptotic variances is as follows. For diagonal components, they are estimated by the realized quarticity multiplied by 2/3. Its theoretical validity is ensured by Hayashi et al. (2011), for example. For off-diagonal components, they are estimated by the naive kernel approach descrived in Section 8.2 of Hayashi and Yoshida (2011). Note that the asymptotic covariance between a diagonal component and another component, which is necessary to evaluate the asymptotic variances of correlation estimates, is not provided in Hayashi and Yoshida (2011), but it can be derived in a similar manner to that paper.

If nonneg is TRUE, negative values of the asymptotic variances of correlations are avoided in the following way. The computed asymptotic varaince-covariance matrix of the vector $(HYii,HYij,HYjj)$ is converted to its spectral absolute value. Here, $HYij$ denotes the Hayashi-Yohida estimator for the $(i,j)$-th component.

The function also returns the covariance and correlation matrices calculated by the Hayashi-Yoshida estimator (using cce).

References

Barndorff-Nilesen, O. E. and Shephard, N. (2004) Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics, Econometrica, 72, no. 3, 885--925.

Bibinger, M. (2011) Asymptotics of Asynchronicity, technical report, Available at arXiv:1106.4222.

Hayashi, T., Jacod, J. and Yoshida, N. (2011) Irregular sampling and central limit theorems for power variations: The continuous case, Annales de l'Institut Henri Poincare - Probabilites et Statistiques, 47, no. 4, 1197--1218.

Hayashi, T. and Yoshida, N. (2011) Nonsynchronous covariation process and limit theorems, Stochastic processes and their applications, 121, 2416--2454.

See Also

setData, cce

Examples

## Set a model
diff.coef.1 <- function(t, x1 = 0, x2 = 0) sqrt(1+t)
diff.coef.2 <- function(t, x1 = 0, x2 = 0) sqrt(1+t^2)
cor.rho <- function(t, x1 = 0, x2 = 0) sqrt(1/2)
diff.coef.matrix <- matrix(c("diff.coef.1(t,x1,x2)", 
"diff.coef.2(t,x1,x2) * cor.rho(t,x1,x2)", 
"", "diff.coef.2(t,x1,x2) * sqrt(1-cor.rho(t,x1,x2)^2)"), 2, 2) 
cor.mod <- setModel(drift = c("", ""), 
diffusion = diff.coef.matrix,solve.variable = c("x1", "x2")) 

set.seed(111) 

## We use a function poisson.random.sampling to get observation by Poisson sampling.
yuima.samp <- setSampling(Terminal = 1, n = 1200) 
yuima <- setYuima(model = cor.mod, sampling = yuima.samp) 
yuima <- simulate(yuima) 
psample<- poisson.random.sampling(yuima, rate = c(0.2,0.3), n = 1000) 

## Constructing a 95% confidence interval for the quadratic covariation from psample
result <- hyavar(psample)
thetahat <- result$covmat[1,2] # estimate of the quadratic covariation
se <- sqrt(result$avar.cov[1,2]) # estimated standard error
c(lower = thetahat + qnorm(0.025) * se, upper = thetahat + qnorm(0.975) * se)

## True value of the quadratic covariation.
cc.theta <- function(T, sigma1, sigma2, rho) { 
  tmp <- function(t) return(sigma1(t) * sigma2(t) * rho(t)) 
  integrate(tmp, 0, T) 
}

# contained in the constructed confidence interval
cc.theta(T = 1, diff.coef.1, diff.coef.2, cor.rho)$value

# Example. A stochastic differential equation with nonlinear feedback. 

## Set a model
drift.coef.1 <- function(x1,x2) x2
drift.coef.2 <- function(x1,x2) -x1
drift.coef.vector <- c("drift.coef.1","drift.coef.2")
diff.coef.1 <- function(t,x1,x2) sqrt(abs(x1))*sqrt(1+t)
diff.coef.2 <- function(t,x1,x2) sqrt(abs(x2))
cor.rho <- function(t,x1,x2) 1/(1+x1^2)
diff.coef.matrix <- matrix(c("diff.coef.1(t,x1,x2)", 
"diff.coef.2(t,x1,x2) * cor.rho(t,x1,x2)","", 
"diff.coef.2(t,x1,x2) * sqrt(1-cor.rho(t,x1,x2)^2)"), 2, 2) 
cor.mod <- setModel(drift = drift.coef.vector,
 diffusion = diff.coef.matrix,solve.variable = c("x1", "x2"))

## Generate a path of the process
set.seed(111) 
yuima.samp <- setSampling(Terminal = 1, n = 10000) 
yuima <- setYuima(model = cor.mod, sampling = yuima.samp) 
yuima <- simulate(yuima, xinit=c(2,3)) 
plot(yuima)


## The "true" values of the covariance and correlation.
result.full <- cce(yuima)
(cov.true <- result.full$covmat[1,2]) # covariance
(cor.true <- result.full$cormat[1,2]) # correlation

## We use the function poisson.random.sampling to generate nonsynchronous 
## observations by Poisson sampling.
psample<- poisson.random.sampling(yuima, rate = c(0.2,0.3), n = 3000) 

## Constructing 95% confidence intervals for the covariation from psample
result <- hyavar(psample)
cov.est <- result$covmat[1,2] # estimate of covariance
cor.est <- result$cormat[1,2] # estimate of correlation
se.cov <- sqrt(result$avar.cov[1,2]) # estimated standard error of covariance
se.cor <- sqrt(result$avar.cor[1,2]) # estimated standard error of correlation

## 95% confidence interval for covariance
c(lower = cov.est + qnorm(0.025) * se.cov,
 upper = cov.est + qnorm(0.975) * se.cov) # contains cov.true

## 95% confidence interval for correlation
c(lower = cor.est + qnorm(0.025) * se.cor,
 upper = cor.est + qnorm(0.975) * se.cor) # contains cor.true

## We can also use the Fisher z transformation to construct a
## 95% confidence interval for correlation
## It often improves the finite sample behavior of the asymptotic
## theory (cf. Section 4.2.3 of Barndorff-Nielsen and Shephard (2004))
z <- atanh(cor.est) # the Fisher z transformation of the estimated correlation
se.z <- se.cor/(1 - cor.est^2) # standard error for z (calculated by the delta method)
## 95% confidence interval for correlation via the Fisher z transformation
c(lower = tanh(z + qnorm(0.025) * se.z), upper = tanh(z + qnorm(0.975) * se.z))