Timeline for How many lags to use in the Ljung-Box test of a time series?
Current License: CC BY-SA 4.0
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| Nov 16 at 8:56 | comment | added | Carlos Sáez | As mentioned already in the comments, and proved in Box and Pierce paper, their Q statistic with the residuals indeed has an asymptotic chi-squared distribution, just with less degrees of freedom than that computed from the true errors (in the case of an AR(1), one less degree of freedom). So it is a little bit misleading saying that the Box-Pierce or Ljung-Box tests "cannot be said to have an asymptotic chi-squared distribution". | |
| Jun 23 at 12:48 | comment | added | Christoph Hanck | @AlecosPapadopoulos, thank you very much, I somehow overlooked this upon first skimming! | |
| Jun 23 at 11:17 | comment | added | Alecos Papadopoulos | @ChristophHanck This is mentioned/discussed already in the foundational article, Box, G. E., & Pierce, D. A. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American statistical Association, 65(332), 1509-1526. see around eq. (2.38). My understanding is that the result is Theory and not simulation, and it has to do with the fact that the VCV matrix of the regression residuals involves an idempotent matrix that has deficient rank (hence we loose degrees of freedom). | |
| Jun 23 at 10:28 | comment | added | Christoph Hanck | @AlecosPapadopoulos, thank you for the pointer! I am familiar with the formula, and rather seek discussion of where the formula to adjust the degrees of freedom comes from. I.e., is subtracting $p+q$ from the standard d.f. of LB when doing a LB test to residuals from fitting an $ARMA(p,q)$ a rule of thumb supported by simulations or is there some theory for this formula? Wikipedia gives a reference to Davidson Econometric Theory which sounds promising, but I will only have access to it next week in the office. | |
| Jun 22 at 23:18 | comment | added | Alecos Papadopoulos | @ChristophHanck Are you perhaps seeking the formulas appearing here, en.wikipedia.org/wiki/Ljung%E2%80%93Box_test ? | |
| Jun 21 at 8:56 | comment | added | Christoph Hanck |
@AlecosPapadopoulos, e.g., ?Box.test suggests to adjust the d.f. when using residuals from models with only predetermined regressors by the number of paramters fitted in the regression model. Do you happen to have further information on how to derive this adjustment? Box and Pierce already make this point in their seminal paper, but without much discussion.
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| Nov 10, 2020 at 10:19 | history | edited | Alecos Papadopoulos | CC BY-SA 4.0 |
Correct name of test
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| Sep 21, 2020 at 23:01 | comment | added | Michael | This issue seems to have been overlooked in the statistics time series literature---e.g. Shumway and Stoffer discusses applying the Q-statistic to check adequacy of fitted AR models. | |
| Apr 3, 2016 at 19:08 | comment | added | Richard Hardy | @AlecosPapadopoulos, an amazing post!!! Among the few best ones I have encountered here at Cross Validated. I just wish it would not disappear in this long thread and many users would find and benefit from it in the future. | |
| Apr 3, 2016 at 17:39 | comment | added | Alecos Papadopoulos | @Aksakal Thanks for the reference. The point exactly is that without strict exogeneity, the Box-Pierce/Ljung-Box do not have an asymptotic chi-square distribution, this is what the mathematics above show. Weak exogeneity (which holds in the above model) is not enough for them. This is exactly what the presentation you link to says in p. 3/44. | |
| Apr 3, 2016 at 17:14 | comment | added | Aksakal | You wrote "But the second expected value is not, since the dependent variable depends on past errors." That's called strict exogeneity. I agree that it's a strong assumption, and you can build AR(p) framework without it, just by using weak exogeneity. This the reason why Breusch-Godfrey test is better in some sense: if the null is not true, then B-L loses power. B-G is based on weak exogeneity. Both tests are not good for some common econometric, applications, see e.g. this Stata's presentation, p. 4/44. | |
| Apr 3, 2016 at 16:25 | history | answered | Alecos Papadopoulos | CC BY-SA 3.0 |