2

我正在尝试使用以下假设对 GARCH 模型（来自“rugarch”包的 ugarch）进行限制测试：

 H0: alpha1 + beta1 = 1

 H1: alpha1 + beta1 ≠ 1

所以我正在尝试遵循 https://stats.stackexchange.com/questions/151573/testing-the-sum-of-garch1-1-parameters/151578?noredirect=1#comment629951_151578的建议

1.使用带有选项 variance.model = list(model = "sGARCH") 的 ugarchspec 指定受限模型，并使用 ugarchfit 对其进行估计。从槽拟合子槽似然中获得对数似然。

2. 使用带有选项 variance.model = list(model = "iGARCH") 的 ugarchspec 指定受限模型，并使用 ugarchfit 对其进行估计。如上所述获得对数似然。

3.计算LR=2（无限制模型的对数似然-受限模型的对数似然）并获得p值作为pchisq（q = LR，df = 1）。

我从“rugarch”包中使用了以下“sGARCH”和“iGARCH”模型。

(A) sGARCH（无限制模型）：

 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))

以下是 sGARCH 输出：

    *---------------------------------*
    *          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（受限模型）：

 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))

但是，我得到了 beta1 的以下输出，其标准错误、t 值和 p 值中包含 N/A。

以下是 iGARCH 输出：

    *---------------------------------*
    *          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

如果我按照建议计算对数似然差以得出卡方值，我会得到负值：

 2*(5411.828-5412.268)=-0.88

受限模型“iGARCH”的对数似然为 5412.268，高于不应该发生的非受限模型“sGARCH”的对数似然 5411.828。

我使用的数据按时间序列方式如下（由于空间限制，我只发布前100个数据）：

   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

为了从我的 H0 和 H1 假设执行限制测试，我可以知道如何解决这个问题吗？

4

2 回答 2

1

估计程序似乎有问题......由于一个模型是另一个模型的受限版本，因此使用 iGARCH 确实应该导致较低的可能性。

使用数据的子集，

fit1 <- ugarchfit(speca, data = data.matrix(P)) 
# [1] 161.7373
fit2 <- ugarchfit(speca2, data = data.matrix(P))
# [1] 165.333

正如我在已删除的帖子中所说，这些数字看起来很可疑，就好像它们是对数似然。然而，从残差中恢复可能性给出

-sum(log(2 * pi * sigma(fit1)^2)) / 2 - sum(residuals(fit1, standardize = TRUE)^2) / 2
# [1] 161.7373
-sum(log(2 * pi * sigma(fit2)^2)) / 2 - sum(residuals(fit2, standardize = TRUE)^2) / 2
# [1] 165.333

这意味着我的怀疑是错误的（必须是密度值> 1）。出于这个原因，我认为没有办法使用当前输出来构造测试。iGARCH 限制非常适合..

然而，一些实验表明，使用

fit.control = list(scale = 1)

改变事物。尤其是，

fit1 <- ugarchfit(speca, data = data.matrix(P), fit.control = list(scale = 1))
likelihood(fit1)
# [1] 161.7373
-sum(log(2 * pi * sigma(fit1)^2)) / 2 - sum(residuals(fit1, standardize = TRUE)^2) / 2
# [1] 161.7373

fit2 <- ugarchfit(speca2, data = data.matrix(P), fit.control = list(scale = 1))
likelihood(fit2)
# [1] 19.5233
-sum(log(2 * pi * sigma(fit2)^2)) / 2 - sum(residuals(fit2, standardize = TRUE)^2) / 2
# [1] 19.5233

考虑到这在某种程度上是有道理的

（第 25 页）“缩放有时有助于估计过程”

（第 46 页） 问：我的模型不收敛，我该怎么办？

“...此外，在拟合例程的 fit.control 列表中，在拟合之前执行数据缩放的选项通常会有所帮助，尽管它在某些设置下不可用...”

然而，第一个模型的可能性保持不变再次令人怀疑。然后我们有

fit1 <- ugarchfit(speca, data = data.matrix(P), fit.control = list(scale = 0), solver.control = list(trace = TRUE))
# 
# Iter: 1 fn: -161.7373  Pars:  -0.0454619  0.0085993  0.0002706  0.0593231  # 0.6898473
# Iter: 2 fn: -161.7373  Pars:  -0.0454619  0.0085993  0.0002706  0.0593231  0.6898473
# solnp--> Completed in 2 iterations
coef(fit1)
#            mu        mxreg1         omega        alpha1         beta1 
# -0.0454619274  0.0085992743  0.0002706018  0.0593231138  0.6898472858 
fit1 <- ugarchfit(speca, data = data.matrix(P), fit.control = list(scale = 1), solver.control = list(trace = TRUE))

# Iter: 1 fn: 114.8143   Pars:  -0.72230  0.13663  0.06830  0.05930  0.68988
# Iter: 2 fn: 114.8143   Pars:  -0.72228  0.13662  0.06830  0.05931  0.68986
# solnp--> Completed in 2 iterations
coef(fit1)
#           mu       mxreg1        omega       alpha1        beta1 
# -0.045463099  0.008599494  0.000270610  0.059310622  0.689858216

和

fit2 <- ugarchfit(speca2, data = data.matrix(P), fit.control = list(scale = 0), solver.control = list(trace = TRUE))

# Iter: 1 fn: -165.3330  Pars:   0.0292439 -0.0051098  0.0002221  0.7495846
# Iter: 2 fn: -165.3330  Pars:   0.0292434 -0.0051097  0.0002221  0.7495853
# solnp--> Completed in 2 iterations
coef(fit2)
#            mu        mxreg1         omega        alpha1         beta1 
#  0.0292434276 -0.0051096984  0.0002221457  0.7495853224  0.2504146776 
fit2 <- ugarchfit(speca2, data = data.matrix(P), fit.control = list(scale = 1), solver.control = list(trace = TRUE))

# Iter: 1 fn: 111.2185   Pars:   0.46462 -0.08118  0.05607  0.74959
# Iter: 2 fn: 111.2185   Pars:   0.46458 -0.08118  0.05607  0.74959
# solnp--> Completed in 2 iterations
coef(fit2)
#          mu      mxreg1       omega      alpha1       beta1 
#  0.46458110 -0.08117626  0.05607215  0.74959242  0.25040758

由于多重不一致，这让事情变得更加奇怪......

于 2018-03-10T18:23:38.213 回答
0

这是我从包作者“Alexios Galanos”那里得到的答案：

问题是 GARCH 模型的平稳性存在限制，这可能会干扰处于平稳边界的模型的求解器收敛性。这是解决方案：

  library(rugarch)
  library(xts)
  dat<-read.table("data.txt",header = TRUE, stringsAsFactors = FALSE)
  dat = xts(dat[,2], as.Date(strptime(dat[,1],"%d/%m/%Y")))

  spec1<-ugarchspec(mean.model=list(armaOrder=c(0,0), archm=TRUE, archpow=1), variance.model=list(model="iGARCH"))
  spec2<-ugarchspec(mean.model=list(armaOrder=c(0,0), archm=TRUE, archpow=1), variance.model=list(model="sGARCH"))
  mod1<-ugarchfit(spec1, dat, solver="solnp")
  mod2<-ugarchfit(spec2,dat)
  persistence(mod2)
  >0.999

 # at the limit of the internal constraint

 mod2<-ugarchfit(spec2, dat, solver="solnp", fit.control = list(stationarity=0))
  likelihood(mod2)
  >5428.871


  likelihood(mod1)

  >5412.268
  persistence(mod2)
  1.08693
  # above the limit

  Here is one solution to change the constraint:

  .garchconbounds2= function(){
    return(list(LB = 1e-12,UB = 0.99999999999))
  }
  assignInNamespace(x = ".garchconbounds", value=.garchconbounds2, ns="rugarch")
  mod2<-ugarchfit(spec2, dat, solver="solnp")

  likelihood(mod2)
  >5412.268

现在该值与约束模型相同（它们都有效集成），但约束模型少了一个要估计的参数。

我什至根本不需要 fit.control=list(scale=1) 。删除这个比例可能更好。

于 2018-03-17T09:56:42.747 回答