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
[[alternative HTML version deleted]]
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
Dr. Adrian Trapletti Trapletti Statistical Computing Wildsbergstrasse 31, 8610 Uster Switzerland Phone & Fax : +41 (0) 1 994 5631 Mobile : +41 (0) 76 370 5631 Email : mailto:a.trapletti at bluewin.ch WWW : http://trapletti.homelinux.com