| lbtest {tsBSS} | R Documentation |
Modified Ljung-Box Test and Volatility Clustering Test for Time Series.
Description
Modified Ljung-Box test and volatility clustering test for time series. Time series can be univariate or multivariate. The modified Ljung-Box test checks whether there is linear autocorrelation in the time series. The volatility clustering test checks whether the time series has squared autocorrelation, which would indicate a presence of volatility clustering.
Usage
lbtest(X, k, type = c("squared", "linear"))
## S3 method for class 'lbtest'
print(x, digits = 3, ...)
Arguments
X |
A numeric vector/matrix or a univariate/multivariate time series object of class |
k |
A vector of lags. |
type |
The type of the autocorrelation test. Options are Modified Ljung-Box test ( |
In methods for class 'lbtest' only:
x |
An object of class lbtest |
digits |
The number of digits when printing an object of class lbtest. Default is 3 |
... |
Further arguments to be passed to or from methods. |
Details
Assume all the individual time series X_i in \bf X with T observations are scaled to have variance 1.
Then the modified Ljung-Box test statistic for testing the existence of linear autocorrelation in X_i (option = "linear") is
T \sum_{j \in k} \left(\sum_{t=1}^T (X_{it} X_{i, t + j})/(T - j)\right)^2/V_{j}.
Here
V_{j} = \sum_{t=1}^{n-j}\frac{x_t^2 x_{t+j}^2}{n-j} + 2 \sum_{k=1}^{n-j-1} \frac{n-k}{n} \sum_{s=1}^{n-k-j}\frac{x_s x_{s+j }x_{s+k} x_{s+k+j}}{n-k-j}.
where t = 1, \ldots, n - j, k = 1, \ldots, n - j - 1 and s = 1, \ldots, n - k - j.
The volatility clustering test statistic (option = "squared") is
T \sum_{j \in k} \left(\sum_{t=1}^T (X_{it}^2 X_{i, t + j}^2)/(T - j) - 1\right)^2
Test statistic related to each time series X_i is then compared to \chi^2-distribution with length(k) degrees of freedom, and the corresponding p-values are produced. Small p-value indicates the existence of autocorrelation.
Value
A list of class 'lbtest' containing the following components:
TS |
The values of the test statistic for each component of X as a vector. |
p_val |
The p-values based on the test statistic for each component of X as a vector. |
Xname |
The name of the data used as a character string. |
varnames |
The names of the variables used as a character string vector. |
k |
The lags used for testing the serial autocorrelation as a vector. |
K |
The total number of lags used for testing the serial autocorrelation. |
type |
The type of the autocorrelation test. |
Author(s)
Markus Matilainen, Jari Miettinen
References
Miettinen, M., Matilainen, M., Nordhausen, K. and Taskinen, S. (2020), Extracting Conditionally Heteroskedastic Components Using Independent Component Analysis, Journal of Time Series Analysis, 41, 293–311.
See Also
FixNA, gFOBI, gJADE, vSOBI, gSOBI
Examples
if(require("stochvol")) {
n <- 10000
s1 <- svsim(n, mu = -10, phi = 0.95, sigma = 0.1)$y
s2 <- rnorm(n)
S <- cbind(s1, s2)
lbtest(S, 1:3, type = "squared")
# First p-value should be very close to zero, as there exists stochastic volatility
}