Testing the sum of GARCH(1,1) parameters How can I test that the sum of the $\alpha_1$ and $\beta_1$ parameters in a GARCH(1,1) model is significantly different from 1?
 A: The restriction you are interested in is a simple linear restriction. The testing principle is the same as testing for a linear restriction in a regression setting, for example. You may use likelihood ratio (LR), Wald or Lagrange multiplier tests.
Consider the LR test:


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*Estimate the unrestricted GARCH(1,1). Obtain the model log-likelihood $L_{unres.}$.

*Estimate the restricted GARCH(1,1) subject to the restriction $\alpha_1+\beta_1=1$. Obtain the model likelihood $L_{res.}$.

*Calculate the likelihood ratio statistic $LR=-2  \operatorname{log} \left( \frac{L_{res.}}{L_{unres.}} \right)$. Under the null hypothesis that the restriction holds in population, the $LR$ statistic will follow a $\chi^2(1)$ distribution (since there is one linear restriction that distinguishes the restricted model from the unrestricted model). Given that, you can obtain the p-value and compare it to the chosen significance level; then you will see whether you can or cannot reject the null hypothesis.


Since you added an R tag, I suppose you are also interested in implementing the test in R. There are three steps corresponding the three steps above:


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*Specify the restricted model using ugarchspec with option variance.model = list(model = "sGARCH") and estimate it using ugarchfit. Obtain the log-likelihood from the slot fit sub-slot likelihood.

*Specify the restricted model using ugarchspec with option variance.model = list(model = "iGARCH") and estimate it using ugarchfit. Obtain the log-likelihood as above.

*Calculate $LR=2(\text{logLik}_{unres.}-\text{logLik}_{res.})$. Obtain the p-value as pchisq(q = LR, df = 1).


On a second thought, the case where $\alpha_1+\beta_1=1$ is a boundary case. When $\alpha_1+\beta_1<1$ the conditional variance is stationary; but when $\alpha_1+\beta_1=1$ the conditional variance becomes integrated. 
Could this issue somehow invalidate the LR test? I am not quite sure. I hope someone could post an argument in the comments or as another answer.
