Suppose that a researcher estimates a GARCH(1,1) model and obtains a log likelihood function (LLF) value of 71.22. He is interested in testing whether an ARCH(1) model is a better model at describing volatility. If he estimates a model which imposes the necessary restrictions and obtains an LLF value of 68.21, what would be the conclusion of his likelihood ratio test (assuming a 5% significance level)?
A) Statistical evidence suggesting that ARCH(1) is better than GARCH(1,1)
B) Statistical evidence suggesting that ARCH(1) is not better than GARCH(1,1)
C) Statistical evidence suggesting that GARCH(1,1) is better than ARCH(1)
D) We cannot say because we would need to know the number of observations

Question

Quizplus · Accepted Answer

The likelihood ratio test is calculated as 2*(LLF of unrestricted model - LLF of restricted model) = 2*(71.22 - 68.21) = 6.02. This value would then be compared to a chi-square distribution with degrees of freedom equal to the difference in the number of parameters between the models. Since GARCH(1,1) has more parameters than ARCH(1), the test assesses whether the simpler model (ARCH(1)) significantly reduces the fit of the model. A higher LLF for GARCH(1,1) suggests it is a better fit, and the calculated test statistic would need to be compared to a critical value from the chi-square distribution to make a final decision. However, without knowing the exact critical value (which depends on the degrees of freedom and is not provided), the general principle that a higher LLF indicates a better model leads to the conclusion that there is not statistical evidence suggesting ARCH(1) is better than GARCH(1,1).

Suppose That a Researcher Estimates a GARCH(1,1) Model and Obtains