Solved – How to interpret standardized residuals tests in Ljung-Box Test and LM Arch test

econometricsrresidualstime series

I am new in econometric and I am confused to make conclusion with Ljung-Box test and LM arch test. I used auto.arima and garch to find out relative orders for the TS and got the result as below. It seems all coefficients are fine, however, I cannot conclude whether the modle is fit by analyzing standardised residuals test as they have vary p-value. I would like to learn from you guys what your criteria would be in this case.

Title:              
 GARCH Modelling                
                
Call:               
 garchFit(formula = ~arma(1, 1) + garch(1, 0), data = intc2, 
     trace = FALSE)                 
                
Mean and Variance Equation:             
 data ~ arma(1, 1) + garch(1, 0)                
<environment: 0x00000000045ee8b0>               
 [data = intc2]             
                
Conditional Distribution:               
 norm               
                
Coefficient(s):             
        mu         ar1         ma1       omega      alpha1                  
 0.0060629   0.4602704  -0.5562739   0.0034931   0.1270859                  
                
Std. Errors:                
 based on Hessian               
                
Error Analysis:             
         Estimate  Std. Error  t value Pr(>|t|)                 
mu      0.0060629   0.0019137    3.168  0.00153 **              
ar1     0.4602704   0.1464651    3.143  0.00167 **              
ma1    -0.5562739   0.1419436   -3.919 8.89e-05 (3*)            
omega   0.0034931   0.0002361   14.792  < 2e-16 (3*)            
alpha1  0.1270859   0.0495262    2.566  0.01029 (1*)                
            
Signif. codes:  0 ‘(3*)’ 0.001 ‘(2*)’ 0.01 ‘(1*)’ 0.05 ‘.’ 0.1 ‘ ’ 1                
                
Log Likelihood:             
 1012.501    normalized:  1.350001              
                
Description:                
 Tue Nov 28 00:02:34 2017 by user: ac               
                            
Standardised Residuals Tests:               
                                                    
 Jarque-Bera Test   R    Chi^2  :Statistic: 52.88447  p-Value: 3.283152e-12             
 Shapiro-Wilk Test  R    W      0.9904564 ;8.781227e-05             
 Ljung-Box Test     R    Q(10)  14.35376  ;0.1574584                
 Ljung-Box Test     R    Q(15)  22.47159  ;0.09602165               
 Ljung-Box Test     R    Q(20)  25.98429  ;0.1663314                
 Ljung-Box Test     R^2  Q(10)  16.05112  ;0.09817849               
 Ljung-Box Test     R^2  Q(15)  24.24917  ;0.06098122               
 Ljung-Box Test     R^2  Q(20)  30.81672  ;0.05765708               
 LM Arch Test       R    TR^2   16.61766  ;0.1645545                
                
Information Criterion Statistics:               
      AIC       BIC       SIC      HQIC                 
-10.704096

Best Answer

I think that $R$ in the output stands for residuals and $R^2$ for squared residuals. Given that, the function computes three Ljung-Box p-value for the residuals and three for the squared residuals. As you can see, all the p-values are higher than 0.05 (which is the classical level of significance assumed) therefore you cannot reject the null hypothesis of absence of serial dependence neither in residuals nor in squared residuals. In other terms, your specification is good to capture autocorrelation and time-varying volatility in the data series. By default the function considers lag up to 10, 15 and 20, you can manually specified the test for other lags via the "LjungBoxTest" function on standardized residuals.

Also the LM-ARCH test does not reject the null hypothesis of absence of ARCH effects, i.e. in this model the ARCH effects are caught by the specification.

Your real problem is normality of residuals, from your Jarque-Bera and Shapiro-Wilk tests residuals are clearly not normally distributed.

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