Ljung-box test using fixed lag value of 1 in ARIMA model

[chintanr97]

I am using an ARIMA model to predict the time series values. I identify the best fit ARIMA model using the AIC value and it turns out that for all the different orders that I tried, the best AIC is returned for the ARIMA order (4, 0, 1).

As I am using statsmodels.tsa.arima.model.ARIMA package, I could see the description of the fitted model using model.summary() and I could observe the following results for tests:

===================================================================================
Ljung-Box (L1) (Q):                   0.00   Jarque-Bera (JB):              4412.50
Prob(Q):                              0.99   Prob(JB):                         0.00
Heteroskedasticity (H):               5.21   Skew:                            -0.01
Prob(H) (two-sided):                  0.00   Kurtosis:                        12.17
===================================================================================

I also manually ran the test using the statsmodels.stats.diagnostic.acorr_ljungbox and statsmodels.stats.stattools.jarque_bera packages respectively. I am using the following code snippet for the same:

jbTest = jarque_bera(model.resid)
print("JBTest:", jbTest)
lbTest = acorr_ljungbox(model.resid, model_df=p_+q_, lags=[20], boxpierce=False)
print("LBTest:", lbTest)

where, model is my best-fit ARIMA model and p_ and q_ are the respective orders of the model. The output from the snippet:

JBTest: (4412.481863170613, 0.0, -0.008442090306796898, 12.171354626003195)
LBTest: (array([32.85885778]), array([0.00490945]))

I can observe that the Jarque-Bera test output is close to the approximations from the model. However, it looks like the Ljunge-box test output is significantly small and does not match with the output from model. I am not sure if I am mis-interpreting the output from the model description versus the output from the acorr_ljungbox function.

However, when I changed my code to use lbTest = acorr_ljungbox(model.resid, lags=[1], boxpierce=False) instead of above snippet, I got values very close to the estimates from the model: LBTest: (array([0.000103]), array([0.9919025])). I did not understand why we are using the lag=1 as found in the class here.

Also, I am confused about the theoretical understanding of the nobs_diffuse value used in the test_serial_correlation function in same class. Any pointers would be really helpful!

I am using statsmodels version 0.12.2.

[ChadFulton]

I can observe that the Jarque-Bera test output is close to the approximations from the model.

The model is not reporting an approximation. In a state space model, diagnostic tests should be performed on the standardized residuals (model.standardized_forecasts_error), and this is why you are getting a different result when you run the diagnostics on model.resid.

I did not understand why we are using the lag=1

The practical reason is just pragmatic - we have a limited space in the summary output, and we have to show something. But still, this is a good point, and lag=1 is not the best thing to show. Harvey (1989, section 5.4.2) suggests using log(T) or sqrt(T), so maybe we should report lag=int(np.log(self.nobs_effective))

Also, I am confused about the theoretical understanding of the nobs_diffuse value used in the test_serial_correlation function in same class. Any pointers would be really helpful!

Here is an intuitive way to understand nobs_diffuse: when you have an integrated model (i.e. ARIMA(p, d, q) with d > 0), then a typical way to proceed is to difference the data d times. When you do this, you lose the first d observations from the original series, and so the residuals from these observations obviously cannot be included in statistical tests. When put in state space form, the model does not literally difference the data, but the states associated with the first few observations still have a "diffuse" distribution (see Durbin and Koopman [2012], sections 5.1 - 5.2 for rigorous details about the diffuse observations and how nobs_diffuse is computed) and so they should not be included in statistical tests (see ibid. section 7.5 for details of diagnostic tests in state space models). So nobs_diffuse gives the number of observations with diffuse state distributions.

[chintanr97]

Thanks a lot @ChadFulton, this was really helpful! I will read over the referenced articles too!