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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?

Richard Hardy
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user137425
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  • Hi. I am also trying to do the same thing here using R's ugarchfit. So do you mean "sGARCH" and "iGARCH" provide restricted and unrestricted GARCH models? Why is this? – Eric Mar 05 '18 at 20:30
  • I really need to solve this problem. I posted the actual question I wanted to ask directly in the following: https://stats.stackexchange.com/questions/333256/restriction-test-h0-alpha1beta1-1-h1alpha1-beta1-%e2%89%a0-1-on-garch-model-in – Eric Mar 14 '18 at 10:23

1 Answers1

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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:

  1. Estimate the unrestricted GARCH(1,1). Obtain the model log-likelihood $L_{unres.}$.
  2. Estimate the restricted GARCH(1,1) subject to the restriction $\alpha_1+\beta_1=1$. Obtain the model likelihood $L_{res.}$.
  3. 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:

  1. 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.
  2. Specify the restricted model using ugarchspec with option variance.model = list(model = "iGARCH") and estimate it using ugarchfit. Obtain the log-likelihood as above.
  3. 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.

Richard Hardy
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  • Hi. I am also trying to do the same thing here using R's ugarchfit. So do you mean "sGARCH" and "iGARCH" provide restricted and unrestricted GARCH models? Why is this? – Eric Mar 05 '18 at 20:29
  • By the way, if the pchisq function you provided gives out value of zero, does this mean alpha + beta = 1? What if this pchisq function gives out value of 1? – Eric Mar 05 '18 at 23:01
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    Re first comment: Yes, I mean that. Why is this? Because sometimes you know what you want, either an integrated or a stationary equation, and you can specify this by selecting the appropriate option. Re second comment: A low p-value suggests rejection of the null hypothesis while a high one suggests the opposite. The null hypothesis is that the restriction holds. So a low p-value suggests it does not hold, hence, $\alpha_1+\beta_1<0$. – Richard Hardy Mar 06 '18 at 08:10
  • Sorry I mean the other way around. If P-value is zero it means "alpha + beta is not one" while if P-value is one it means "alpha + beta =1". right? – Eric Mar 06 '18 at 10:40
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    @Eric, that's right. – Richard Hardy Mar 06 '18 at 10:52
  • By the way, how to you convert the chi-square probability value from 'pchisq' into the actual chi-square value? Should I use 'qchisq' over it? If so, how? – Eric Mar 08 '18 at 21:42
  • Oh, isn't the q = LR itself is the chi-square value? – Eric Mar 09 '18 at 01:03
  • If I assume my q=LR is the chi-square value, I get a negative chi-square value and the p = 0. Is this possible? Does this mean I strongly reject the null hypothesis? – Eric Mar 09 '18 at 01:20
  • @Eric, LR itself is the test statistic. It should not be possible to get a negative value as the likelihood of the unrestricted model must be higher than that of the restricted model. Perhaps there is some wrong in the functions you are using?.. – Richard Hardy Mar 09 '18 at 07:16
  • Sorry is the “sGARCH” restricted model or unrestricted model? – Eric Mar 09 '18 at 08:43
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    sGARCH is unrestricted because its coefficients do not need to sum to one. iGARCH is restricted because they must (which is a linear restriction: $\alpha_1+\beta_1=1$). – Richard Hardy Mar 09 '18 at 09:17
  • If the pchisq = 0, how shall I interpret it then? Or if it is 1, what should it mean? – Eric Mar 09 '18 at 10:08
  • When I use "iGARCH" on my existing GARCH model, the beta coefficient is alive but its standard error, t-value, and p-value becomes N/A. Does this have to do with this strange symptom I am having? – Eric Mar 09 '18 at 10:20
  • Actually, the loglikelihood of "iGARCH" becomes larger than "sGARCH" when I simply switch the "sGARCH" with "iGARCH". In "iGARCH" however, I get n/a in the beta1's standard error, t-value, and p-value while beta1 itself shows a valid coefficient. – Eric Mar 09 '18 at 10:36
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    Low p-value means low probability to have observed what you have observed given that the null hypothesis is correct, which is a motivation to reject the null hypothesis. Technically the likelihood of sGARCH cannot be lower than that of iGARCH; but there might be some bug in the calculation or oherwise poor definition in the function. – Richard Hardy Mar 09 '18 at 12:37
  • I use the following model but could not find any poor definition in the function you are mentioning. For “sGARCH” model I use I just simply replace the “iGARCH” into “sGARCH”.: ugarchspec=(variance.model=list(model=“iGARCH”, garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0),include.mean=TRUE)) – Eric Mar 09 '18 at 18:24
  • I thought the reason why “iGARCH” was giving larger loglikelihood value than that of “sGARCH” was due to the N/A in beta1’s standard error, p-value, and t-value although the actual beta1 value is available. But this is just my guess. – Eric Mar 09 '18 at 18:30
  • In the "iGARCH" model, if beta1 value is available but its standard error, p-value, and t-value are shown as N/A, is this a credible model? If not, is there a way to fix this? – Eric Mar 09 '18 at 21:00
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    Due to the linear restriction, you are estimating one coefficient less than in an sGARCH. That might be the reason for why you have those NA values. Knowing $\alpha_1$ and the fact that $\alpha_1+\beta_1=1$ you know $\beta_1$, so you are not estimating it. Then you have no standard error etc. – Richard Hardy Mar 10 '18 at 08:02
  • Thank you. As a result, the log-likelihood value for the iGARCH is higher than that of sGARCH. In this case, is there a way to perform restriction test for “alpha1 + beta1 = 1”? – Eric Mar 10 '18 at 09:54
  • I am just interested in testing whether “alpha1 + beta1” equals 1 or not. I am not trying to find which one is greater than the other as you mentioned. – Eric Mar 10 '18 at 10:08
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    @Eric, since the result is theoretically impossible, I do not see a way how it could help solve the problem. Perhaps the calculation of the likelihoods differ by a constant or something like that, making them incomparable. But you could calculate the likelihoods manually based on the standardized residuals and then compare. – Richard Hardy Mar 10 '18 at 10:16
  • I hope you can join the discussion in the link above :) – Eric Mar 10 '18 at 14:27
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    @Eric, Julius might be right. I think I had discovered some issue with the sign of the likelihood a few years ago, but I forgot what it was and how it ended. Sorry. – Richard Hardy Mar 10 '18 at 14:31
  • Is there a clue that I can get from one of your solution in the past as follows? https://stats.stackexchange.com/questions/234738/dcc-garch-how-to-test-restricted-vs-unrestricted-model-with-lr-test – Eric Mar 13 '18 at 22:14
  • Or may I know how I can do this using Wald test instead if this is a GARCH model hopefully using R? – Eric Mar 13 '18 at 22:28
  • Particularly, when I run: nlWaldtest(specc,"alpha1+beta1=1"), I get the error saying: Error in nlWaldtest(specc, "alpha1+beta1=1") : There are no coef() methods for model objects of class "uGARCHspec". Input the 'coeff' parameter. – Eric Mar 13 '18 at 22:32
  • I really need to solve this problem. I posted the actual question I wanted to ask directly in the following: https://stats.stackexchange.com/questions/333256/restriction-test-h0-alpha1beta1-1-h1alpha1-beta1-%e2%89%a0-1-on-garch-model-in – Eric Mar 14 '18 at 10:23
  • Sorry but one more thing..what should be the interpretations of alpha1 + beta1 = 1 and ≠ 1? – Eric Mar 29 '18 at 10:24
  • @Eric, =1 means conditional variance is an integrated process (sort of a random walk) while <1 means it is mean-reverting (to the mean of $\frac{\omega}{1-\alpha-\beta}$) and >1 means it is explosive. – Richard Hardy Mar 29 '18 at 10:52
  • Thank you. So if alpha1+beta1=1, can I say the model is stationary? – Eric Mar 29 '18 at 10:55
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    @Eric, no no, a random walk is not stationary and an integrated cond. variance process is not stationary. – Richard Hardy Mar 29 '18 at 11:07
  • Thank you. Then what does it mean by having "integrated process" in the conditional variance term? If it is a good thing, why is it? If not, why..? – Eric Mar 29 '18 at 11:29
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    @Eric, As I noted above, this means the conditional variance is essentially a random walk. I do not think this should be evaluated as good or bad; it is just a characteristic of a time series. It is less predictable and less stable than a stationary process. Anyway, this is getting quite involved for a comment thread... – Richard Hardy Mar 29 '18 at 11:32