hyavar {yuima}

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.

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 (HY_{ii},HY_{ij},HY_{jj}) is converted to its spectral absolute value. Here, HY_{ij} 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).

Value

A list with components:

covmat

the estimated covariance matrix

cormat

the estimated correlation matrix

avar.cov

the estimated asymptotic variances for covariances

avar.cor

the estimated asymptotic variances for correlations

Note

Construction of kernel-type estimators for off-diagonal components is implemented after pseudo-aggregation described in Bibinger (2011).

Author(s)

Yuta Koike with YUIMA Project Team

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 doi:10.48550/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

## Not run: 
## 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)) 

## End(Not run)