5-6 Joint Hypothesis Tests We can also test the joint hypothesis that all m of the correlation coefficients are simultaneously equal to zero using the @-statistic developed by box and pierce: Q=7x2 where T= sample size, m= maximum lag length The O-statistic is asymptotically distributed as a x m However, the Box Pierce test has poor small sample properties, so a variant has been developed, called the ljung-Box statistic O*=T(T+2 This statistic is very useful apfa portmanteau(general)test of linear dependence in time series5-6 • We can also test the joint hypothesis that all m of the  k correlation coefficients are simultaneously equal to zero using the Q-statistic developed by Box and Pierce: where T = sample size, m = maximum lag length • The Q-statistic is asymptotically distributed as a . • However, the Box Pierce test has poor small sample properties, so a variant has been developed, called the Ljung-Box statistic: • This statistic is very useful as a portmanteau (general) test of linear dependence in time series. Joint Hypothesis Tests  m 2 = = m k Q T k 1 2  ( ) 2 1 2 2 ~ m m k k T k Q T T   =  − = +