#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 

So I am trying to follow the advice from
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    


    Adjusted Pearson Goodness-of-Fit Test:
    ------------------------------------
      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    


    Adjusted Pearson Goodness-of-Fit Test:
    ------------------------------------
      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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