# #StackBounty: #r #hypothesis-testing #chi-squared #garch #restrictions Restriction test (H0: alpha1+beta1 = 1, H1:alpha1 + beta1 ≠ 1) o…

### Bounty: 50

I am trying to do the restriction test for GARCH model (ugarch from ‘rugarch’ package) using the following hypothesis:

`````` H0: alpha1 + beta1 = 1

H1: alpha1 + beta1 ≠ 1
``````

Testing the sum of GARCH(1,1) parameters

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(Log-likelihood of unrestricted model − Log-likelihood of restricted model) and Obtain the p-value as pchisq(q = LR, df = 1).

I have the following ‘sGARCH’ and ‘iGARCH’ models I am using from ‘rugarch’ package.

(A) sGARCH (unrestricted model):

`````` speccR = ugarchspec(variance.model=list(model = "sGARCH",garchOrder=c(1,1)),mean.model=list(armaOrder=c(0,0), include.mean=TRUE,archm = TRUE, archpow = 1))

ugarchfit(speccR, data=data.matrix(P),fit.control = list(scale = 1))
``````

And the following is this sGARCH output:

``````    *---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model     : sGARCH(1,1)
Mean Model      : ARFIMA(0,0,0)
Distribution    : norm

Optimal Parameters
------------------------------------
Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000355    0.001004 -0.35377 0.723508
archm   0.096364    0.039646  2.43059 0.015074
omega   0.000049    0.000010  4.91096 0.000001
alpha1  0.289964    0.021866 13.26117 0.000000
beta1   0.709036    0.023200 30.56156 0.000000

Robust Standard Errors:
Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000355    0.001580 -0.22482 0.822122
archm   0.096364    0.056352  1.71002 0.087262
omega   0.000049    0.000051  0.96346 0.335316
alpha1  0.289964    0.078078  3.71375 0.000204
beta1   0.709036    0.111629  6.35173 0.000000

LogLikelihood : 5411.828

Information Criteria
------------------------------------

Akaike       -3.9180
Bayes        -3.9073
Shibata      -3.9180
Hannan-Quinn -3.9141

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1]                      233.2       0
Lag[2*(p+q)+(p+q)-1][2]     239.1       0
Lag[4*(p+q)+(p+q)-1][5]     247.4       0
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1]                      4.695 0.03025
Lag[2*(p+q)+(p+q)-1][5]     5.941 0.09286
Lag[4*(p+q)+(p+q)-1][9]     7.865 0.13694
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3]     0.556 0.500 2.000  0.4559
ARCH Lag[5]     1.911 1.440 1.667  0.4914
ARCH Lag[7]     3.532 2.315 1.543  0.4190

Nyblom stability test
------------------------------------
Joint Statistic:  5.5144
Individual Statistics:
mu     0.5318
archm  0.4451
omega  1.3455
alpha1 4.1443
beta1  2.2202

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
t-value   prob sig
Sign Bias           0.2384 0.8116
Negative Sign Bias  1.1799 0.2381
Positive Sign Bias  1.1992 0.2305
Joint Effect        2.9540 0.3988

------------------------------------
group statistic p-value(g-1)
1    20     272.1    9.968e-47
2    30     296.9    3.281e-46
3    40     313.3    1.529e-44
4    50     337.4    1.091e-44

Elapsed time : 0.4910491
``````

(B) iGARCH (restricted model):

`````` speccRR = ugarchspec(variance.model=list(model = "iGARCH",garchOrder=c(1,1)),mean.model=list(armaOrder=c(0,0), include.mean=TRUE,archm = TRUE, archpow = 1))

ugarchfit(speccRR, data=data.matrix(P),fit.control = list(scale = 1))
``````

However, I get the following output of beta1 with N/A in its standard error, t-value, and p-value.

The following is the iGARCH output:

``````    *---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model     : iGARCH(1,1)
Mean Model      : ARFIMA(0,0,0)
Distribution    : norm

Optimal Parameters
------------------------------------
Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000355    0.001001 -0.35485 0.722700
archm   0.096303    0.039514  2.43718 0.014802
omega   0.000049    0.000008  6.42826 0.000000
alpha1  0.290304    0.021314 13.62022 0.000000
beta1   0.709696          NA       NA       NA

Robust Standard Errors:
Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000355    0.001488  -0.2386 0.811415
archm   0.096303    0.054471   1.7680 0.077066
omega   0.000049    0.000032   1.5133 0.130215
alpha1  0.290304    0.091133   3.1855 0.001445
beta1   0.709696          NA       NA       NA

LogLikelihood : 5412.268

Information Criteria
------------------------------------

Akaike       -3.9190
Bayes        -3.9105
Shibata      -3.9190
Hannan-Quinn -3.9159

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1]                      233.2       0
Lag[2*(p+q)+(p+q)-1][2]     239.1       0
Lag[4*(p+q)+(p+q)-1][5]     247.5       0
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1]                      4.674 0.03063
Lag[2*(p+q)+(p+q)-1][5]     5.926 0.09364
Lag[4*(p+q)+(p+q)-1][9]     7.860 0.13725
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3]    0.5613 0.500 2.000  0.4538
ARCH Lag[5]    1.9248 1.440 1.667  0.4881
ARCH Lag[7]    3.5532 2.315 1.543  0.4156

Nyblom stability test
------------------------------------
Joint Statistic:  1.8138
Individual Statistics:
mu     0.5301
archm  0.4444
omega  1.3355
alpha1 0.4610

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
t-value   prob sig
Sign Bias           0.2252 0.8218
Negative Sign Bias  1.1672 0.2432
Positive Sign Bias  1.1966 0.2316
Joint Effect        2.9091 0.4059

------------------------------------
group statistic p-value(g-1)
1    20     273.4    5.443e-47
2    30     300.4    6.873e-47
3    40     313.7    1.312e-44
4    50     337.0    1.275e-44

Elapsed time : 0.365
``````

If I calculate the log-likelihood difference to derive the chi-square value
as suggested I get negative value as such:

`````` 2*(5411.828-5412.268)=-0.88
``````

The Log-likelihood of the restricted model “iGARCH” is 5412.268 which is higher than the Log-likelihood of the unrestricted model “sGARCH” of 5411.828
which should not happen.

The data I use are as follows in time series manner (I am only posting first 100 data due to space limit):

``````   Time      P
1   0.454213593
2   0.10713195
3   -0.106819399
4   -0.101610699
5   -0.094327846
6   -0.037176107
7   -0.101550977
8   -0.016309894
9   -0.041889484
10  0.103384357
11  -0.011746377
12  0.063304432
13  0.059539249
14  -0.049946177
15  -0.023251656
16  0.013989353
17  -0.002815588
18  -0.009678745
19  -0.011139779
20  0.031592303
21  -0.02348106
22  -0.007206591
23  0.077422089
24  0.064632768
25  -0.003396734
26  -0.025524166
27  -0.026632474
28  0.014614485
29  -0.012380888
30  -0.007463018
31  0.022759969
32  0.038667465
33  -0.028619484
34  -0.021995984
35  -0.006162809
36  -0.031187399
37  0.022455611
38  0.011419264
39  -0.005700445
40  -0.010106343
41  -0.004310162
42  0.00513715
43  -0.00498106
44  -0.021382251
45  -0.000694252
46  -0.033326085
47  0.002596086
48  0.011008057
49  -0.004754233
50  0.008969559
51  -0.00354088
52  -0.007213115
53  -0.003101495
54  0.005016228
55  -0.010762641
56  0.030770993
57  -0.015636325
58  0.000875417
59  0.03975863
60  -0.050207219
61  0.011308261
62  -0.021453315
63  -0.003309127
64  0.025687191
65  0.009467306
66  0.005519485
67  -0.011473758
68  0.00223934
69  -0.000913651
70  -0.003055385
71  0.000974694
72  0.000288611
73  -0.002432251
74  -0.0016975
75  -0.001565034
76  0.003332848
77  -0.008007295
78  -0.003086435
79  -0.00160435
80  0.005825885
81  0.020078093
82  0.018055453
83  0.181098137
84  0.102698818
85  0.128786594
86  -0.013587077
87  -0.038429879
88  0.043637258
89  0.042741709
90  0.016384872
91  0.000216317
92  0.009275681
93  -0.008595197
94  -0.016323335
95  -0.024083247
96  0.035922206
97  0.034863621
98  0.032401779
99  0.126333922
100 0.054751935
``````

In order to perform the restriction test from my H0 and H1 hypothesis, may I know how I can fix this problem?

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