mv_ch_tests: Multivariate Conditional Heteroscedasticity (ARCH) Tests
In sebinum/baqgarchutil: Analyse a Fitted baqGARCH Model

Description Usage Arguments Value Details References See Also Examples

Performs tests to check whether conditional heteroscedasticity in a multivariate time series vector is statistically significant. This is a wrapper function for diag_ljung_box and diag_dufour_roy.

1	mv_ch_tests(x, lags = c(8, 10, 12))

`x`	A `matrix` / `data.frame` / `numeric vector` of (multivariate) financial time series. Each column contains a series, each row an observation of the series.
`lags`	The number of lags of cross-correlation matrices used in the tests. Can take multiple values. Defaults to `lags = c(8, 10, 12)`.

Four different test statistics and their p-values to determine multivariate ARCH-effect as a data.frame. For more information see the Details.

The four test statistics are different approaches to detect conditional heteroscedasticity (ARCH-effect) in multivariate time-series as employed by Ruey S. Tsay (2014) in Multivariate Time Series Analysis with R and Financial Applications.

The k-dimensional series a_t can be transformed to a standardized univariate series e_t:

e_t = a'_t * ∑^-1 * a_t - k

where ∑ denotes the unconditional covariance matrix of the k-dimensional series a_t.

1. Q*(m): univariate Ljung-Box Test on the standardized series e_t

The univariate series e_t is the basis for the univariate Ljung-Box Test

Q*(m) = T * (T + 2) * ∑^m_i=1 * ρ²_i / (T - i)

where T stands for the sample size and ρ_i for the lag-i sample autocorrelation of e_t.

The Hypothesis H0 : ρ_1 = ... = ρ_m = 0 is tested against H1 : ρ_i != 0 for i = (1 ≤ i ≤ m). Under the null hypothesis of no conditional heteroscedasticity in a_t, the test statistic Q*(m) is asymptotically distributed as χ²_m.

2. Q_R(m): Rank-Based Test on the the ranked standardized series e_t

Extreme observations in return series (heavy tails) can have pronounced effects on the results of Q*(m). One approach to circumvent the heavy tails problem is the Rank-Based test on the rank series of e_t by Dufour & Roy (1985, 1986). With R_t being the rank of e_t, the lag-l rank autocorrelation of e_t can be defined as

ρ_l = (∑_{t=l+1}^T * (R_t - R) * (R_{t-l} - R)) / (∑_{t=1}^T * (R_t - R)²) for l = 1, 2, ...,

where

R = ∑_{i=1}^T * R_T/ T = (T + 1) / 2,

∑_{t=1}^T * (R_t - R)² = T * (T² - 1) / 12.

It can be shown that

E(ρ_l) = -(T - l) / [T(T - 1)]

Var(ρ_l) = (5T^4 - (5l + 9)T³ + 9 * (l - 2) * T² + 2l * (5l + 8) * T + 16l²) / (5(T - 1)² * T² * (T + 1))

The Test Statistic

Q_R(m) = ∑_{i=1}^m * [([ρ_{i} - E(ρ_i)]²] / Var(ρ_i)]

is distributed as χ²_m asymptotically if e_t shows no signs of serial dependency.

3. Q*_k(m): multivariate Ljung-Box Test on the k-dimensional series a_t

The multivariate Ljung-Box Statistic is:

Q*_k(m) = T² * ∑_{i=1}^m * (1 / (T - i)) * b'_i(ρ_0^-1 \otimes ρ_0^-1) * b_i

where T stands for the sample size, k for the dimension of a_t and b_i = vec(ρ'_i) with ρ_j being the lag-ρ_j cross-correlation matrix of a²_t. As in the univariate case, under the null hypothesis of no conditional heteroscedasticity in a_t, Q*_k(m) is asymptotically distributed as χ²_{k²m}.

4. Q_k^r(m): robust multivariate Ljung-Box Test on the k-dimensional series a_t

Q*_k(m) may not perform very well when a_t has heavy tails. To make the test results more robust, the heavy tails from a_t are trimmed. This is achieved by removing 5% of the observations of a_t corresponding to the upper 5% quantile from the univariate standardized series e_t (see the details to Q*(m)). Q*_k(m) performed on the trimmed upper 5% quantile series a_t is denoted by Q_k^r(m).

Ljung G. & Box G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 66: 67-72.

Dufour, J. M. & Roy R. (1985). The t copula and related copulas. Working Paper. Department of Mathematics, Federal Institute of Technology.

Dufour, J. M. & Roy R. (1986). Generalized portmanteau statistics and tests of randomness. Communications in Statistics-Theory and Methods, 15: 2953-2972.

Li, W. K. (2004). Diagnostic Checks in Time Series. Chapman & Hall / CRC. Boca Raton, FL.

Tsay, R. S. (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.

Tsay, R. S. (2015). MTS: All-Purpose Toolkit for Analyzing Multivariate Time Series (MTS) and Estimating Multivariate Volatility Models. R package version 0.33.

diag_std_et for the transformation of a multivariate financial time series to a standardized scalar series which can be tested for conditional heteroscedasticity, diag_ljung_box for the Ljung-Box Test statistic, diag_dufour_roy for the Rank-Based Test for serial correlation

# create heteroscedastic data
dat <- mgarchBEKK::simulateBEKK(3, 150)
eps <- data.frame(eps1 = dat$eps[[1]], eps2 = dat$eps[[2]],
                  eps3 = dat$eps[[3]])

# perform multivariate arch test
mv_ch_tests(eps)