# No ARCH effect in the Univariate case but there is an ARCH effect in the Multivariate case?

I am running some simulations, it seems when I consider my data (multivariate residuals of a VAR model) happen to be observing an ARCH effect. However, when I split them into two dataset I failed to prove that either of them has ARCH effect. How come when I put both data set together the multivariate random variable might witness an ARCH effect?

I am using R. My 2 dimension residuals is defined as resi

TEsting for ARCH effect for the first column results in

# Testing ARCH effects for univariate residuals resi
y1=resi[,1]
var=(y1-mean(y1))^2
Box.test(var,lag=26,type='Ljung')


Outcome:

Box-Ljung test

data:  var
X-squared = 31.929, df = 26, p-value = 0.1955


Second column

y2=resi[,2]
var=(y2-mean(y2))^2
Box.test(var,lag=26,type='Ljung')


Outcome:

Box-Ljung test

data:  var
X-squared = 28.001, df = 26, p-value = 0.3584


Testing both together

library("MTS")
MarchTest(resi) # Multivariate ARCH test


results in

Q(m) of squared series(LM test):
Test statistic:  9.761321  p-value:  0.4616771
Rank-based Test:
Test statistic:  69.60412  p-value:  5.286682e-11
Q_k(m) of squared series:
Test statistic:  51.92913  p-value:  0.09796701
Robust Test(5%) :  78.21383  p-value:  0.0002854892


Two differences between univariate and multivariate ARCH tests:

1. Taking each univariate case separately, only the diagonal element of the covariance matrix are considered. Taking all series together, also off-diagonal elements come into play.
2. Similarly to how an $F$-test of two regression coefficients being equal to zero can yield different results than two individual $t$-tests, also here testing the joint hypothesis of no multivariate ARCH pattern (involving the whole covariance matrix) can yield different results than the individual tests for elements of the covariance matrix taken separately.

Also, note that the $Q^*(m)$ and $Q_k^*(m)$ tests do not reject the null at 95% confidence level ($p=0.46$ and $p=0.10$). The other two tests that each reject the null have different null hypotheses (being based on ranks or on outlier-adjusted data) and do not directly correspond to the univariate tests you do. (See here for a brief summary.)