Tutorial 5
Question 1
A volatile model is fitted to a stock return data and produce the following results
(a) Write down the equation to forecast volatility based on the output above.
(b) What can we conclude based on the results in ARCH LM-test?
(c) Are all the coefficients in the model significant? Explain your answer.
(d) Can we conclude that the standardised residuals follow normal distribution? Explain your answer.
Example Answer
(a)
ARMA(1,1) – GARCH(1,0)
Yt = 0.003766 – 0.279625 Yt-1 – 0.689310 εt-1 + εt
εt = Zt √(σt2 )
σt2 = 0.000192 + 0.999 ε2t-1
(b)
According to ARCH – LM test, p-value ≈ 0 is less than 0.05 significant level. Therefore, reject H0. Thus, there are ARCH effect in the series.
(c)
Test significant.
H0 : ϕ1 = 0
H1 : ϕ1 ≠ 0
The p-value of AR1 is less than 0.05 significant level. Therefore, reject H0. Thus AR1 is significant.
H0 : θ1 = 0
H1 : θ1 ≠ 0
The p-value of MA1 is less than 0.05 significant level. Therefore, reject H0. Thus MA1 is significant.
H0 : α1 = 0
H1 : α1 ≠ 0
The p-value of alpha1is less than 0.05 significant level. Therefore, reject H0. Thus alpha1 is significant.
(d)
From Pearson Goodness-of-Fit test, the p-values are greater than 0.05. This indicates the standardized residuals do not follow normal distribution
Question 2
A volatile model is fitted to a stock return data and produce the following results:
(a) Write down the equation to forecast volatility based on the output above.
(b) What can we conclude based on the result in ARCH LM-test?
(c) Identify whether the volatility shock is persistent in the model.
(d) What can we conclude based on the results in weighted Ljung-Box on standardized and standardized squared residuals?
(e) Can we conclude that the standardized residuals follow normal distribution? Explain your answer.
Example Answer
(a)
ARMA(2,1) – GARCH(1,1)
Yt = 0.008808 + 0.865739 Yt-1 – 0.051294 Yt-2 – 0.878041 εt-1 + εt
σt2 = 0.002077 + 0.539066 ε2t-1 + 0.459934 σ2t-1
(b)
According to ARCH – LM test, p-value ≈ 0 is less than 0.05 significant level. Therefore, reject H0. Thus, there are ARCH effect in the series.
(c)
α1 + β1 = 0.54 + 0.46 ≈ 1
Therefore, the volatile shock is persistent.
(d)
According to weighted Ljung-Box test on standardized residuals, the p-value for lag 1 is greater than 0.05 significant level and do not reject H0. However for other lags, the p-values are less than 0.05 significant level and reject H0. Therefore, there is a serial correlation.
According to Ljung-Box test on standardized squared residuals, all p-values are less than 0.05 significant level. Therefore reject H0. Thus the GARCH shock is present.
(e)
From Pearson Goodness-of-Fit test, all p-values are less than 0.05 significant level. Therefore reject H0. Thus, this indicates the standardized residuals do follow normal distribution.
Kim 2 ML 
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