# Testing multivariate ARCH effects in R

Using the function MarchTest of package "MTS" in R, I am testing whether or not there are multivariate ARCH effects in my time series. I simulated some series without multivarate ARCH effects, so per definition there must be very high p-values. Running the MarchTest, there are high p-values only in small samples but they turn to 0 when the sample becomes bigger (1,000 or more observations). I am sure that there is no mistake in the construction of the time series.
Does anyone have an idea why there are such misleading results?

The underlying parameters are:

p<-3
mu<-c(0.28,0.17,0.34)
phi<-matrix(c(-0.156,0.277,0.259,0.386,0.184,0.269,-0.265,0.191,0.376),3,3)
psi<-matrix(c(-0.500, 0.612, 0.235, 0.597, 0.414, 0.169, 0.333, 0.273, 0.100),3,3)
eps_t <-c(0,0,0)
n<-c(50,100,500,1000)


Then

Y_t<-matrix(0,ncol=3,nrow=n)
Y<-matrix(0,ncol=3,nrow=n)
set.seed(1234)
Y_t<-(VARMAsim(nobs=n, arlags=1, malags=1, cnst=mu, phi=phi, theta=psi, skip=100, sigma=varcov)\$series)
Y<-data.matrix(Y_t)


As you see, it is a pure VARMA process without any ARCH effects by construction.
The output with 1,000 observations is:

> MarchTest(Y)
Q(m) of squared series(LM test):
Test statistic:  35.62607  p-value:  9.758118e-05
Rank-based Test:
Test statistic:  14.08036  p-value:  0.1693574
Q_k(m) of squared series:
Test statistic:  138.9744  p-value:  0.0007122924
Robust Test(5%) :  100.2475  p-value:  0.215975


The output using the same parameters but only 50 observations is:

Q(m) of squared series(LM test):
Test statistic:  14.46804  p-value:  0.152693
Rank-based Test:
Test statistic:  16.10272  p-value:  0.09672955
Q_k(m) of squared series:
Test statistic:  92.35173  p-value:  0.4116147
Robust Test(5%) :  122.0541  p-value:  0.01385632

• There are four tests reported by MTS::MarchTest; which one is giving you the strange results? Could you include the output your are getting (and the code for generating the series, too)? (Here is a related question, although probably too basic to be useful for you.) Apr 11, 2017 at 13:24
• It is probably that there are differences that are only detected when the sample size gets large. Questions about the software should be asked at another site. Apr 11, 2017 at 13:26
• @MichaelChernick, I tend to view this question as of statistical nature rather than programming related. The question is, how to understand the counterintuitive results, not how to code something. But of course we need the question to be expanded to be able to answer it sensibly. Apr 11, 2017 at 13:27
• In large samples, I receive p values of 0 for all of the four tests. Depending on the simulation, this is the case sometimes already at 1000 observations. Apr 11, 2017 at 13:29
• @MichaelChernick as the poramming of the MarcgTest using the package is not difficult, I am not asking about the programming of this but about the conflicting results. Apr 11, 2017 at 13:34