How can I test if time series residuals' are uncorrelated ?

Ok I made Jarque-Bera test to the residuals (merv.reg$residual)

library(tseries)
jarque.bera.test(merv.reg$residual)
X-squared = 1772.369, df = 2, p-value = < 2.2e-16
And I reject the null hypotesis (H0: merv.reg$residual are normally
distributed)

So I know that:
1 - merv.reg$residual aren't independently distributed (Box-Ljung test)
2 - merv.reg$residual aren't indentically distributed (Breusch-Pagan 
test)
3 - merv.reg$residual aren't normally distributed (Jarque-Bera test)

My questions is:
It is possible merv.reg$residual be uncorrelated ?
cov[residual_t, residual_(t+k)] = 0 ?
Even when residuals are not independent distributed !

Yes. E.g., in an ARCH(1) process, cov[y_t, y_(t+k) ] = 0 (k \neq 0), 
but cov[(y_t)2, (y_(t+k))2 ] \neq 0,

hence no independence (and this is typical for financial time series).

(and we know that they aren't normally distributed and they aren't
indentically distributed )
And how can I tested it ?

Thanks.

Hint, if a ts is normally distributed then independence and

uncorrelatedness

are equivalent, hence you can test for normally distributed errors 
(e.g.
Jarque-Bera-Test).

HTH,
Bernhard

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Typically, financial time series exhibit fat tails, i.e., are not 
normally distributed (and in an ARCH setup, financial time series are 
usually not even conditionally normally distributed. The fat tails are 
fatter than what we would expect from the clustering of volatility).

best
Adrian

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Dr. Adrian Trapletti
Trapletti Statistical Computing
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