시계열의 Ljung-Box 테스트에 몇 개의 지연이 있습니까?

ARMA 모델이 시계열에 적합하면 Ljung-Box portmanteau 테스트 (다른 ​​테스트 중)를 통해 잔차를 확인하는 것이 일반적입니다. Ljung-Box 테스트는 p 값을 반환합니다. 테스트 할 지연 수인 매개 변수 h 가 있습니다. 일부 텍스트는 h = 20을 사용하는 것이 좋습니다 . 다른 사람들은 h = ln (n); 대부분은 무슨 말을하지 않는 시간을 사용 할 수 있습니다.

h에 단일 값을 사용하는 대신 모든 h <50에 대해 Ljung-Box 테스트를 수행 하고 최소 p 값을 제공하는 h 를 선택 한다고 가정하십시오 . 그 접근법은 합리적입니까? 장점과 단점은 무엇입니까? (한 가지 명백한 단점은 계산 시간이 증가한다는 점이지만 여기서는 문제가되지 않습니다.) 이에 대한 문헌이 있습니까?

약간 정교하게하기 위해 ... 테스트가 모든 h에 대해 p> 0.05를 주면 시계열 (잔여)이 테스트를 통과 한 것입니다. 내 질문은 h의 일부 값에 대해 p <0.05 이고 다른 값이 아닌 경우 테스트를 해석하는 방법에 관한 것 입니다.

대답은 확실히 다음에 달려 있습니다. 테스트를 실제로 사용하려고하는 것은 무엇입니까 ?$Q$

일반적인 이유는 : 더 이하로 자신감을 지연없이 자기 상관 최대의 귀무 가설의 공동 통계적 유의성에 대한 (또는 당신이 뭔가 가까이가 가정 약한 화이트 노이즈를 )하고 구축 할 수 인색 모델을 적게 갖는 가능한 한 많은 매개 변수.$h$

일반적으로 시계열 데이터는 자연적인 계절 패턴을 가지므로 실용적인 경험 규칙은 를이 값의 두 배로 설정하는 것 입니다. 예측 요구에 모델을 사용하는 경우 예측 수평선이 또 하나 있습니다. 마지막으로 후반에 중대한 변화가 발생하면 수정에 대해 생각해보십시오 (일부 계절적 영향으로 인한 것일 수도 있고 데이터가 특이 치에 대해 수정되지 않았을 수도 있음).$h$

h에 단일 값을 사용하는 대신 모든 h <50에 대해 Ljung-Box 테스트를 수행 한 다음 최소 p 값을 제공하는 h를 선택한다고 가정하십시오.

그것은의 공동 유의 의 선택 그렇다면, 시험 데이터 기반은, 왜 내가 미만 어떤 지연에 몇 가지 작은 (가끔?) 출발에 대해 관심을 가져야입니다 시간 이 훨씬 미만이라고 가정하면, n은 물론 (전력 언급 한 테스트 중). 간단하지만 관련성이 높은 모델을 찾기 위해 아래에 설명 된대로 정보 기준을 제안합니다.$h$$h$$n$

내 질문 은 h의 일부 값에 대해 이고 다른 값이 아닌 경우 테스트를 해석하는 방법에 관한 것 입니다.$p<0.05$$h$

따라서 현재와 얼마나 멀리 떨어져 있는지에 달려 있습니다. 원거리 이탈의 단점 : 더 많은 매개 변수 추정, 더 적은 자유도, 모델의 예측력 저하.

출발이 발생하는 지연 시간에 MA 및 / 또는 AR 부품을 포함하여 모델을 추정하고 추가로 정보 기준 중 하나 (샘플 크기에 따라 AIC 또는 BIC)를 확인하면 더 많은 모델에 대한 통찰력을 얻을 수 있습니다. 지극히 검소한. 샘플 외부 예측 연습도 여기에서 환영합니다.

일반적인 속성을 모두 사용하여 간단한 AR (1) 모델을 지정한다고 가정합니다.

$$y_t = \beta y_{t-1} + u_t$$

오차항의 이론적 공분산을 다음과 같이 나타냅니다.

$$\gamma_j \equiv E(u_tu_{t-j})$$

오차항을 관찰 할 수 있으면 오차항의 표본 자기 상관은 다음과 같이 정의됩니다.

$$\tilde \rho_j \equiv \frac {\tilde \gamma_j}{\tilde \gamma_0}$$

어디

$$\tilde\gamma_j \equiv \frac 1n \sum_{t=j+1}^nu_tu_{t-j},\;\;\; j=0,1,2...$$

그러나 실제로는 오류 용어를 관찰하지 않습니다. 따라서 오차 항과 관련된 표본 자기 상관은 다음과 같이 추정의 잔차를 사용하여 추정됩니다.

$$\hat\gamma_j \equiv \frac 1n \sum_{t=j+1}^n\hat u_t\hat u_{t-j},\;\;\; j=0,1,2...$$

Box-Pierce Q- 통계량 (Ljung-Box Q는 무증상 중립 스케일 버전입니다)

$$Q_{BP} = n \sum_{j=1}^p\hat\rho^2_j = \sum_{j=1}^p[\sqrt n\hat\rho_j]^2\xrightarrow{d} \;???\;\chi^2(p)$$

우리의 문제는 정확히 여부 점근이 모델 (오류 용어에없는 autocorellation의 널 아래) 카이 제곱 분포를 갖는다 고 할 수있다. 이런 일이 일어나려면 √의 모든 사람과$Q_{BP}$
점근 적으로 표준 정규해야합니다. 이를 확인하는 방법은$\sqrt n \hat\rho_j$ 동일한 점근 분포$\sqrt n \hat\rho$ (실제 오류를 사용하여 구성되므로 null 아래에서 원하는 점근 적 동작을 갖습니다).$\sqrt n \tilde\rho$

우리는 그것을 가지고

$$\hat u_t = y_t - \hat \beta y_{t-1} = u_t - (\hat \beta - \beta)y_{t-1}$$

여기서 β는 일관된 추정이다. 그래서$\hat \beta$

$$\hat\gamma_j \equiv \frac 1n \sum_{t=j+1}^n[u_t - (\hat \beta - \beta)y_{t-1}][u_{t-j} - (\hat \beta - \beta)y_{t-j-1}]$$

$$=\tilde \gamma _j -\frac 1n \sum_{t=j+1}^n (\hat \beta - \beta)\big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] + \frac 1n \sum_{t=j+1}^n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1}$$

샘플은 정지되고 인체 공학적인 것으로 가정하고 원하는 순서까지 모멘트가 존재한다고 가정합니다. 추정기 때문에 β가 일치하는 두 개의 합계가 제로에 갈 경우,이 충분하다. 그래서 우리는 결론$\hat \beta$

$$\hat \gamma_j \xrightarrow{p} \tilde \gamma_j$$

이것은

$$\hat \rho_j \xrightarrow{p} \tilde \rho_j \xrightarrow{p} \rho_j$$

그러나 이것이 자동으로 √를 보장하지는 않습니다수렴 $\sqrt n \hat \rho_j$$\sqrt n\tilde \rho_j$(분포) (임의 변수에 적용되는 변환이의존하기 때문에 연속 매핑 정리가 여기에 적용되지 않는다고 생각하십시오). 이런 일이 일어나려면$n$

$$\sqrt n \hat \gamma_j \xrightarrow{d} \sqrt n \tilde \gamma_j$$

(분모 물결 또는 모자-는 두 경우 모두 오차항의 분산으로 수렴하므로 우리의 문제에 중립적입니다).$\gamma_0$

우리는

$$\sqrt n \hat \gamma_j =\sqrt n\tilde \gamma _j -\frac 1n \sum_{t=j+1}^n \sqrt n(\hat \beta - \beta)\big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] \\+ \frac 1n \sum_{t=j+1}^n\sqrt n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1}$$

So the question is : do these two sums, multiplied now by $\sqrt n$, go to zero in probability so that we will be left with $\sqrt n \hat \gamma_j =\sqrt n\tilde \gamma _j$ asymptotically?

For the second sum we have

$$\frac 1n \sum_{t=j+1}^n\sqrt n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1} = \frac 1n \sum_{t=j+1}^n\big[\sqrt n(\hat \beta - \beta)][(\hat \beta - \beta)y_{t-1}y_{t-j-1}]$$

Since $[\sqrt n(\hat \beta - \beta)]$ converges to a random variable, and $\hat \beta$ is consistent, this will go to zero.

For the first sum, here too we have that $[\sqrt n(\hat \beta - \beta)]$ converges to a random variable, and so we have that

$$\frac 1n \sum_{t=j+1}^n \big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] \xrightarrow{p} E[u_ty_{t-j-1}] + E[u_{t-j}y_{t-1}]$$

The first expected value, $E[u_ty_{t-j-1}]$ is zero by the assumptions of the standard AR(1) model. But the second expected value is not, since the dependent variable depends on past errors.

So $\sqrt n\hat \rho_j$ won't have the same asymptotic distribution as $\sqrt n\tilde \rho_j$. But the asymptotic distribution of the latter is standard Normal, which is the one leading to a chi-squared distribution when squaring the r.v.

Therefore we conclude, that in a pure time series model, the Box-Pierce Q and the Ljung-Box Q statistic cannot be said to have an asymptotic chi-square distribution, so the test loses its asymptotic justification.

This happens because the right-hand side variable (here the lag of the dependent variable) by design is not strictly exogenous to the error term, and we have found that such strict exogeneity is required for the BP/LB Q-statistic to have the postulated asymptotic distribution.

Here the right-hand-side variable is only "predetermined", and the Breusch-Pagan test is then valid. (for the full set of conditions required for an asymptotically valid test, see Hayashi 2000, p. 146-149).

Before you zero-in on the "right" h (which appears to be more of an opinion than a hard rule), make sure the "lag" is correctly defined.

http://www.stat.pitt.edu/stoffer/tsa2/Rissues.htm

Quoting the section below Issue 4 in the above link:

"....The p-values shown for the Ljung-Box statistic plot are incorrect because the degrees of freedom used to calculate the p-values are lag instead of lag - (p+q). That is, the procedure being used does NOT take into account the fact that the residuals are from a fitted model. And YES, at least one R core developer knows this...."

Edit (01/23/2011): Here's an article by Burns that might help:

http://lib.stat.cmu.edu/S/Spoetry/Working/ljungbox.pdf

The thread "Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey" shows that the Ljung-Box test is essentially inapplicable in the case of an autoregressive model. It also shows that Breusch-Godfrey test should be used instead. That limits the relevance of your question and the answers (although the answers may include some generally good points).

Escanciano and Lobato constructed a portmanteau test with automatic, data-driven lag selection based on the Pierce-Box test and its refinements (which include the Ljung-Box test).

The gist of their approach is to combine the AIC and BIC criteria --- common in the identification and estimation of ARMA models --- to select the optimal number of lags to be used. In the introduction of they suggest that, intuitively, ``test conducted using the BIC criterion are able to properly control for type I error and are more powerful when serial correlation is present in the first order''. Instead, tests based on AIC are more powerful against high order serial correlation. Their procedure thus choses a BIC-type lag selection in the case that autocorrelations seem to be small and present only at low order, and an AIC-type lag section otherwise.

The test is implemented in the R package vrtest (see function Auto.Q).

The two most common settings are $\min(20,T-1)$ and $\ln T$ where $T$ is the length of the series, as you correctly noted.

The first one is supposed to be from the authorative book by Box, Jenkins, and Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.. However, here's all they say about the lags on p.314:

It's not a strong argument or suggestion by any means, yet people keep repeating it from one place to another.

The second setting for a lag is from Tsay, R. S. Analysis of Financial Time Series. 2nd Ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005, here's what he wrote on p.33:

Several values of m are often used. Simulation studies suggest that the choice of m ≈ ln(T ) provides better power performance.

This is a somewhat stronger argument, but there's no description of what kind of study was done. So, I wouldn't take it at a face value. He also warns about seasonality:

This general rule needs modification in analysis of seasonal time series for which autocorrelations with lags at multiples of the seasonality are more important.

Summarizing, if you just need to plug some lag into the test and move on, then you can use either of these setting, and that's fine, because that's what most practitioners do. We're either lazy or, more likely, don't have time for this stuff. Otherwise, you'd have to conduct your own research on the power and properties of the statistics for series that you deal with.

UPDATE.

Here's my answer to Richard Hardy's comment and his answer, which refers to another thread on CV started by him. You can see that the exposition in the accepted (by Richerd Hardy himself) answer in that thread is clearly based on ARMAX model, i.e. the model with exogenous regressors $x_t$:

$$y_t = \mathbf x_t'\beta + \phi(L)y_t + u_t$$

However, OP did not indicate that he's doing ARMAX, to contrary, he explicitly mentions ARMA:

After an ARMA model is fit to a time series, it is common to check the residuals via the Ljung-Box portmanteau test

One of the first papers that pointed to a potential issue with LB test was Dezhbaksh, Hashem (1990). “The Inappropriate Use of Serial Correlation Tests in Dynamic Linear Models,” Review of Economics and Statistics, 72, 126–132. Here's the excerpt from the paper:

As you can see, he doesn't object to using LB test for pure time series models such as ARMA. See also the discussion in the manual to a standard econometrics tool EViews:

If the series represents the residuals from ARIMA estimation, the appropriate degrees of freedom should be adjusted to represent the number of autocorrelations less the number of AR and MA terms previously estimated. Note also that some care should be taken in interpreting the results of a Ljung-Box test applied to the residuals from an ARMAX specification (see Dezhbaksh, 1990, for simulation evidence on the finite sample performance of the test in this setting)

Yes, you have to be careful with ARMAX models and LB test, but you can't make a blanket statement that LB test is always wrong for all autoregressive series.

UPDATE 2

Alecos Papadopoulos's answer shows why Ljung-Box test requires strict exogeneity assumption. He doesn't show it in his post, but Breusch-Gpdfrey test (another alternative test) requires only weak exogeneity, which is better, of course. This what Greene, Econometrics, 7th ed. says on the differences between tests, p.923:

The essential difference between the Godfrey–Breusch and the Box–Pierce tests is the use of partial correlations (controlling for X and the other variables) in the former and simple correlations in the latter. Under the null hypothesis, there is no autocorrelation in εt , and no correlation between $x_t$ and $\varepsilon_s$ in any event, so the two tests are asymptotically equivalent. On the other hand, because it does not condition on $x_t$ , the Box–Pierce test is less powerful than the LM test when the null hypothesis is false, as intuition might suggest.

... h should be as small as possible to preserve whatever power the LB test may have under the circumstances. As h increases the power drops. The LB test is a dreadfully weak test; you must have a lot of samples; n must be ~> 100 to be meaningful. Unfortunately I have never seen a better test. But perhaps one exists. Anyone know of one ?

Paul3nt

There's no correct answer to this that works in all situation for the reasons other have said it will depend on your data.

That said, after trying to figure out to reproduce a result in Stata in R I can tell you that, by default Stata implementation uses: $\mathrm{min}(\frac{n}{2}-2, 40)$. Either half the number of data points minus 2, or 40, whichever is smaller.

All defaults are wrong, of course, and this will definitely be wrong in some situations. In many situations, this might not be a bad place to start.

Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.

Additionally, I have recently found "Thoughts on the Ljung-Box test" which is excellent discussion on the topic from Rob Hyndman.

Update: Considering the alternative discussion in this thread regarding ARMAX, another incentive to look at hwwntest is the availability of a theoretical power function for one of the tests against an alternative hypothesis of ARMA(p,q) model.