Q-test Lung - Boxing

$Q$ ${\ displaystyle Q}$ $Q$ -test Ljung - Box - a statistical criterion designed to find autocorrelation of time series . Instead of testing for randomness of each individual coefficient, it checks for the difference from zero at once several autocorrelation coefficients ^[1] .

Formal Definition

$Q$ ${\ displaystyle Q}$ Test Lung - Boxing can be defined as follows. Two competing hypotheses are put forward:

H_{0}

{\ displaystyle H_ {0}}

: The data is random (i.e. it is white noise ).

H_{a}

{\ displaystyle H_ {a}}

: The data is not random.

A statistical test is being conducted ^[1] :

{\tilde {Q}}=n\left(n+2\right)\sum _{k=1}^{m}{\frac {{\hat {\rho }}_{k}^{2}}{n-k}},

{\ displaystyle {\ tilde {Q}} = n \ left (n + 2 \ right) \ sum _ {k = 1} ^ {m} {\ frac {{\ hat {\ rho}} _ {k} ^ {2}} {nk}},}

Where $n$ ${\ displaystyle n}$ - number of observations ${\hat {\rho }}_{k}$ ${\ displaystyle {\ hat {\ rho}} _ {k}}$ - autocorrelation $k$ ${\ displaystyle k}$ th order, and $m$ ${\ displaystyle m}$ - the number of checked lags. If a

{\tilde {Q}}>\chi _{1-\alpha ,\;m}^{2},

{\ displaystyle {\ tilde {Q}}> \ chi _ {1- \ alpha, \; m} ^ {2},}

Where $\chi _{1-\alpha ,\;m}^{2}$ ${\ displaystyle \ chi _ {1- \ alpha, \; m} ^ {2}}$ - quantiles of chi-square distribution with $m$ ${\ displaystyle m}$ degrees of freedom , the null hypothesis is rejected, and the presence of autocorrelation to $m$ ${\ displaystyle m}$ th order in the time series. $Q$ ${\ displaystyle Q}$ -test Lung - Boxing is based on the statistics of Boxing - Pierce . So, it has the same asymptotic distribution and, with relatively large values of the number of observations, gives similar results ^[2] . But the Lung - Box test distribution is closer to $\chi ^{2}$ ${\ displaystyle \ chi ^ {2}}$ for finite samples ^[3] . In addition, the criterion does not lose its consistency, even if the process does not have a normal distribution (in the presence of finite dispersion ) ^[1] . $Q$ ${\ displaystyle Q}$ Test Ljung - Boxing is usually used in the construction of ARIMA models . It should be borne in mind that this testing is applied to the remnants of the obtained ARIMA model, and not to the original data ^[3] .

Notes

↑ ¹ ² ³ Suslov V.I., Ibragimov N.M., Talysheva L.P., Tsyplakov A.A. Econometrics. - Novosibirsk: SB RAS, 2005 .-- 744 p. - ISBN 5-7692-0755-8 .
↑ Diebold FX Elements of Forecasting. - 4. - South-Western College Pub, 2007. - P. 129. - 384 p. - ISBN 032432359X .
↑ ¹ ² Magnus Ya. R., Katyshev P.K., Peresetsky A.A. Econometrics. Initial course: Textbook. - Moscow: Case, 2004 .-- 576 p. - ISBN 5-7749-0055-X .

[Суслов-1] ¹ ² ³ Suslov V.I., Ibragimov N.M., Talysheva L.P., Tsyplakov A.A. Econometrics. - Novosibirsk: SB RAS, 2005 .-- 744 p. - ISBN 5-7692-0755-8 .

[2] Diebold FX Elements of Forecasting. - 4. - South-Western College Pub, 2007. - P. 129. - 384 p. - ISBN 032432359X .

[Магнус-3] ¹ ² Magnus Ya. R., Katyshev P.K., Peresetsky A.A. Econometrics. Initial course: Textbook. - Moscow: Case, 2004 .-- 576 p. - ISBN 5-7749-0055-X .

Q-test Lung - Boxing

Formal Definition

See also

Notes

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