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I want to test if some parameters from a regression is different from 0.

HO: alpha = beta = 0

I have obtained the log-likelihood of the restricted model and the log-likelihood of the unrestricted model.

As you can see below, this is my unrestricted model (Original Model):

*---------------------------------*
*          DCC GARCH Fit          *
*---------------------------------*

Distribution         :  mvnorm
Model                :  DCC(1,1)
No. Parameters       :  117
[VAR GARCH DCC UncQ] : [0+70+2+45]
No. Series           :  10
No. Obs.             :  1302
Log-Likelihood       :  43347.8
Av.Log-Likelihood    :  33.29 

Optimal Parameters
-----------------------------------
                  Estimate  Std. Error     t value Pr(>|t|)
[retBTC].mu       0.003046    0.001241    2.453919 0.014131
[retBTC].ar1      0.960899    0.089217   10.770337 0.000000
[retBTC].ma1     -0.940867    0.108801   -8.647617 0.000000
[retBTC].omega    0.000081    0.000046    1.737272 0.082339
[retBTC].alpha1   0.221456    0.030538    7.251806 0.000000
[retBTC].beta1    0.777544    0.043195   18.000703 0.000000
[retBTC].shape    3.170512    0.182315   17.390335 0.000000
[retSP].mu        0.000703    0.000059   11.996822 0.000000
[retSP].ar1       0.930631    0.009309   99.975481 0.000000
[retSP].ma1      -0.971264    0.002851 -340.713672 0.000000
[retSP].omega     0.000003    0.000004    0.788360 0.430486
[retSP].alpha1    0.229322    0.037754    6.074183 0.000000
[retSP].beta1     0.742375    0.072137   10.291153 0.000000
[retSP].shape     4.470668    0.742508    6.021034 0.000000
[retDAX].mu       0.000734    0.000225    3.271370 0.001070
[retDAX].ar1     -0.598341    0.229636   -2.605603 0.009171
[retDAX].ma1      0.608832    0.225663    2.697967 0.006976
[retDAX].omega    0.000002    0.000002    0.695826 0.486538
[retDAX].alpha1   0.101317    0.026963    3.757589 0.000172
[retDAX].beta1    0.897683    0.024746   36.276457 0.000000
[retDAX].shape    4.841694    0.806891    6.000433 0.000000
[retKS11].mu      0.000361    0.000165    2.179652 0.029283
[retKS11].ar1     0.290953    0.249720    1.165114 0.243973
[retKS11].ma1    -0.278195    0.247850   -1.122435 0.261677
[retKS11].omega   0.000004    0.000001    7.429448 0.000000
[retKS11].alpha1  0.078169    0.006439   12.139971 0.000000
[retKS11].beta1   0.846931    0.013745   61.615500 0.000000
[retKS11].shape   4.771639    0.617535    7.726908 0.000000
[retVGLT].mu      0.000109    0.000161    0.677069 0.498362
[retVGLT].ar1     0.755963    0.260954    2.896921 0.003768
[retVGLT].ma1    -0.788746    0.246379   -3.201353 0.001368
[retVGLT].omega   0.000000    0.000000    1.784266 0.074380
[retVGLT].alpha1  0.021034    0.001745   12.050702 0.000000
[retVGLT].beta1   0.971663    0.002363  411.122574 0.000000
[retVGLT].shape  15.183606    6.718171    2.260080 0.023816
[retEUR].mu      -0.000073    0.000114   -0.634293 0.525890
[retEUR].ar1      0.065867    0.843151    0.078121 0.937732
[retEUR].ma1     -0.110990    0.837008   -0.132604 0.894507
[retEUR].omega    0.000000    0.000001    0.079021 0.937016
[retEUR].alpha1   0.034322    0.011046    3.107275 0.001888
[retEUR].beta1    0.963804    0.010703   90.048781 0.000000
[retEUR].shape    6.893396    1.278341    5.392456 0.000000
[retCFH].mu      -0.000108    0.000104   -1.044847 0.296094
[retCFH].ar1      0.924264    0.015728   58.765918 0.000000
[retCFH].ma1     -0.940493    0.010285  -91.438891 0.000000
[retCFH].omega    0.000000    0.000000    1.921093 0.054720
[retCFH].alpha1   0.003871    0.001513    2.558370 0.010516
[retCFH].beta1    0.986317    0.001150  857.870655 0.000000
[retCFH].shape    5.760995    1.572449    3.663708 0.000249
[retYEN].mu      -0.000153    0.000115   -1.328011 0.184175
[retYEN].ar1      0.908598    0.022023   41.256053 0.000000
[retYEN].ma1     -0.916286    0.017632  -51.967086 0.000000
[retYEN].omega    0.000000    0.000000    0.462780 0.643522
[retYEN].alpha1   0.050058    0.007148    7.003523 0.000000
[retYEN].beta1    0.946988    0.007853  120.596116 0.000000
[retYEN].shape    4.720927    0.549495    8.591398 0.000000
[retOil].mu       0.000094    0.000394    0.238040 0.811850
[retOil].ar1     -0.397718    0.674308   -0.589816 0.555314
[retOil].ma1      0.366654    0.682885    0.536919 0.591324
[retOil].omega    0.000001    0.000005    0.274973 0.783337
[retOil].alpha1   0.056285    0.030819    1.826317 0.067803
[retOil].beta1    0.942637    0.031191   30.221770 0.000000
[retOil].shape    7.677219    1.930047    3.977737 0.000070
[retGLD].mu      -0.000022    0.000204   -0.106983 0.914802
[retGLD].ar1     -0.174956    0.218806   -0.799591 0.423948
[retGLD].ma1      0.128977    0.217954    0.591765 0.554008
[retGLD].omega    0.000001    0.000001    1.132616 0.257376
[retGLD].alpha1   0.022788    0.003623    6.289842 0.000000
[retGLD].beta1    0.969654    0.004865  199.330554 0.000000
[retGLD].shape    4.520722    0.436011   10.368365 0.000000
[Joint]dcca1      0.019558    0.004801    4.073449 0.000046
[Joint]dccb1      0.776304    0.180508    4.300672 0.000017

Here is my restricted model (Where the Alphas and betas are set as 0)

*---------------------------------*
*          DCC GARCH Fit          *
*---------------------------------*

Distribution         :  mvnorm
Model                :  DCC(1,1)
No. Parameters       :  97
[VAR GARCH DCC UncQ] : [0+50+2+45]
No. Series           :  10
No. Obs.             :  1302
Log-Likelihood       :  42009.12
Av.Log-Likelihood    :  32.27 

Optimal Parameters
-----------------------------------
                 Estimate  Std. Error     t value Pr(>|t|)
[retBTC].mu      0.006031    0.000404  1.4941e+01 0.000000
[retBTC].ar1     0.980255    0.008860  1.1063e+02 0.000000
[retBTC].ma1    -0.953782    0.002130 -4.4772e+02 0.000000
[retBTC].omega   0.013624    0.001016  1.3414e+01 0.000000
[retBTC].shape   2.100000    0.000713  2.9472e+03 0.000000
[retSP].mu       0.000671    0.000010  6.8354e+01 0.000000
[retSP].ar1      0.913399    0.014899  6.1307e+01 0.000000
[retSP].ma1     -0.957483    0.000670 -1.4284e+03 0.000000
[retSP].omega    0.000071    0.000010  7.4110e+00 0.000000
[retSP].shape    3.015369    0.288986  1.0434e+01 0.000000
[retDAX].mu      0.000586    0.000253  2.3184e+00 0.020426
[retDAX].ar1     0.161144    0.716854  2.2479e-01 0.822140
[retDAX].ma1    -0.163866    0.713626 -2.2962e-01 0.818384
[retDAX].omega   0.000146    0.000014  1.0246e+01 0.000000
[retDAX].shape   3.422244    0.392206  8.7256e+00 0.000000
[retKS11].mu     0.000321    0.000172  1.8659e+00 0.062054
[retKS11].ar1    0.211304    0.249752  8.4606e-01 0.397522
[retKS11].ma1   -0.197172    0.247087 -7.9799e-01 0.424878
[retKS11].omega  0.000053    0.000004  1.3881e+01 0.000000
[retKS11].shape  4.165704    0.543065  7.6707e+00 0.000000
[retVGLT].mu     0.000040    0.000032  1.2334e+00 0.217425
[retVGLT].ar1    0.993468    0.003037  3.2717e+02 0.000000
[retVGLT].ma1   -1.000000    0.000002 -6.4850e+05 0.000000
[retVGLT].omega  0.000050    0.000002  2.3390e+01 0.000000
[retVGLT].shape 13.081433    0.498740  2.6229e+01 0.000000
[retEUR].mu     -0.000058    0.000116 -5.0122e-01 0.616218
[retEUR].ar1     0.757117    0.489581  1.5465e+00 0.121994
[retEUR].ma1    -0.793335    0.457476 -1.7342e+00 0.082890
[retEUR].omega   0.000029    0.000002  1.5047e+01 0.000000
[retEUR].shape   4.783291    0.639763  7.4767e+00 0.000000
[retCFH].mu     -0.000123    0.000135 -9.1077e-01 0.362418
[retCFH].ar1    -0.047935    0.361320 -1.3267e-01 0.894457
[retCFH].ma1     0.034679    0.360080  9.6310e-02 0.923274
[retCFH].omega   0.000032    0.000004  9.0431e+00 0.000000
[retCFH].shape   4.505839    0.902360  4.9934e+00 0.000001
[retYEN].mu     -0.000177    0.000135 -1.3066e+00 0.191336
[retYEN].ar1     0.558427    0.449970  1.2410e+00 0.214594
[retYEN].ma1    -0.580397    0.439878 -1.3194e+00 0.187019
[retYEN].omega   0.000041    0.000004  1.0714e+01 0.000000
[retYEN].shape   3.726435    0.432189  8.6222e+00 0.000000
[retOil].mu     -0.000098    0.000474 -2.0739e-01 0.835704
[retOil].ar1    -0.403995    0.551857 -7.3207e-01 0.464129
[retOil].ma1     0.372228    0.561428  6.6300e-01 0.507330
[retOil].omega   0.000531    0.000055  9.6492e+00 0.000000
[retOil].shape   3.372255    0.345858  9.7504e+00 0.000000
[retGLD].mu     -0.000014    0.000209 -6.8093e-02 0.945711
[retGLD].ar1    -0.118646    0.218152 -5.4387e-01 0.586532
[retGLD].ma1     0.051282    0.216398  2.3698e-01 0.812673
[retGLD].omega   0.000096    0.000009  1.0748e+01 0.000000
[retGLD].shape   3.825087    0.450614  8.4886e+00 0.000000
[Joint]dcca1     0.007401    0.002920  2.5341e+00 0.011274
[Joint]dccb1     0.867725    0.076019  1.1415e+01 0.000000

Then I've calculated the likelihood ratio statistic (LR) to be 0.062

Now the tricky part is how i go from here to say that i can reject my null hypothesis. My LR is very low and the degrees of freedom is 20 (117-97). When i look at the Left tail of a chi^2 table, the low test values like my LR are only for 1 df and not 12.

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It sounds like you are attempting to perform a likelihood-ratio test comparing a broader class of models to a smaller class restricted by your null hypothesis. Since you have already calculated the likelihood-ratio statistic, the next step of is to calculate the p-value of the test by comparing the observed value to its null distribution. This is usually done by calculating the deviance value $D$, which is twice the logarithm of the likelihood-ratio statistic. The asymptotic null distribution that is commonly usually used for the test is the chi-squared distribution:

$$D \equiv 2 (\ell_{\text{A}} - \ell_{0}) \sim \text{Chi-Sq}(df = df_{\text{A}} - df_0).$$

Since higher values of this test-statistic are more conducive to the alternative hypothesis, this means that the p-value of your test is the upper-tail area of the chi-squared distribution:

$$p = \int \limits_{d}^\infty \text{Chi-Sq}(r | df_{\text{A}} - df_0) dr.$$

You should be able to calculate this tail area in any standard mathematical or statistical program. In R you can use 1 - pchisq(0.062, 12) which gives you the value $p = 1$, so there is no evidence to reject the null.

(Also: The title of your question misstates the substance of what you are actually asking, which is about the mechanics of performing the test, not the interpretation of the LR-statistic.)

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  • $\begingroup$ Somehow, the book i used, defined the likelihood ratio statistic (LR) as twice the difference in the log-likelihood. With my LR i actually ment the deviance value D, sorry for the misunderstanding. So in this case my deviance value is 0.062 with a p-value = 1. Does this mean that the alpha and beta is jointly significant? $\endgroup$ – basse Apr 11 '18 at 3:14
  • $\begingroup$ It means that you do not reject the null hypothesis, and hence, there is no evidence that the more complex model is required. To remember this, just remember the mnemonic, "If p is low, the null must go!" $\endgroup$ – Ben Apr 11 '18 at 3:34
  • $\begingroup$ Hi, again. I might have done some calculation error in order to find the right deviance number. When i did my unrestricted and restricted regression i had log-likelihood values of: (Unrestricted = 42009.12 and Restricted = 43347.8) Now, i plotted this in the formula 2(ln(42009.12)-ln(43347.8)) which is 0.062. However, this is the part where i think i made the error. Because, since the two regressions gave me the log-likelihoods for each models the right calculation will be 2(42009.12-43347.8) = 2677. Can anyone confirm this? $\endgroup$ – basse Apr 11 '18 at 11:17
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    $\begingroup$ We can't confirm the numbers you got from your analysis, but we can confirm that you shouldn't be taking logarithms twice. $\endgroup$ – Ben Apr 11 '18 at 12:31
  • $\begingroup$ Looking at your numbers again, they can't be right. You have a higher maximum for the restricted model, which is impossible. $\endgroup$ – Ben Apr 11 '18 at 12:34

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