Hello everyone in this election we will
talk about the Ljung-Box Q-statistic. So objective is to define
Ljung-Box Q-statistic. Will learn how their the decision
rule to test an null hypothesis that the several
autocorrelation coefficients are zero. And we'll learn how to test the null
hypothesis that several autocorrelation coefficients are zero using R. Let me tart with Portmanteau
statistic in 1970, Box and Pierce proposed the following
statistic which is just Q*(m). It is the time,t he length of
the time series is multiplied with the sum of the squares of the sample
of the correlation coefficients until m. And they propose this for testing
the null hypothesis that more than one. Let's say m many of the correlation
coefficient zero against the alternative hypothesis that there is some rho i for
i in between one and m that's not zero. So either all of them are zero or against that there's at least one
of these guys that's not zero and they used Q*(m) statistic to
test this null hypothesis. They showed that under some
conditions Q*(m) has asymptotically Chi-Squared distribution,
with degrees of freedom m. Then Ljung and Box in 1978,
they modified this statistics to increase the power of the test for
a finite samples. And there test is that
Q(m) is equal to now. The length of the time series*(the
length of the time series+2)*the sum, this time the sum is
actually divided by T-l. Decision rule is going
to be the following. We're going to look at the Q(m) and if Q(m) is large enough then we reject
the null hypothesis, now how large enough? Well if it is larger than 100(1-
alpha) quantile in the Chi-Squared distribution with m degrees of freedom. Now most packages actually
give you the p-value. So we're going to reject
the null hypothesis if p value is sufficiently small. So if p value is less than some alpha,
let's say alpha is 0.05, our significance level you're going to reject that the idea
that there is no auto-correlation. Now how are are we going to use this? Well, when we start with this time
series's were going to use this q statistic to see if there
is autocorrelation. Then if there is autocorrelation
were going to try to fit our models. And once they fit the model then there
will be residuals hopefully a white nulls. Now then we're going to
look at the residuals, we shouldn't see n alpha correlation. So we're going to use this test for the residuals to see if there's
a autocorrelation or not as well. Now the question is what is this m? How large do we take m? It is usually taking as the lower than ln,
of the length of the time series. And in the R,
we're going to use this Box.test routine to actually carry out
all of our calculations. So what have we learned? We have learned the definition
of the Ljung-Box Q-statistic. We learned the decision rule to
test the null hypothesis that several auto-correlation
coefficient are zero. We also learned to test
the null hypothesis that several auto-correlation
coefficients are zero using R. In the next lecture we'll actually
look at the real life time series. And we're going to use all of the tools
that we have acquired until now. We have a lot of them now. And we're going to try to fit our real
life model into the real-life data set.