Analysis

日経平均株価と東京市場ドル円レート(Source:日本銀行,日本経済新聞社)

分析設計

[1] "USDJPY"
         Date   Open   High    Low  Close Center  Index CloseToOpen HighToLow     MA25 DeviationRate Close:Diff(lag=1) Close:Ratio(lag=1)
56 2019-11-25 108.73 108.89 108.65 108.84 108.78 104.77       0.101     0.221 108.7444          0.09              0.23              0.212
57 2019-11-26 109.00 109.21 108.92 108.96 109.05 104.62      -0.037     0.266 108.7568          0.19              0.12              0.110
58 2019-11-27 109.10 109.19 109.03 109.19 109.11 104.19       0.082     0.147 108.7816          0.38              0.23              0.211
59 2019-11-28 109.38 109.56 109.34 109.42 109.39 104.23       0.037     0.201 108.8232          0.55              0.23              0.211
60 2019-11-29 109.52 109.60 109.45 109.50 109.47 104.27      -0.018     0.137 108.8552          0.59              0.08              0.073
61 2019-12-02 109.54 109.73 109.49 109.60 109.66 104.69       0.055     0.219 108.8944          0.65              0.10              0.091
62 2019-12-03 109.02 109.21 108.95 109.10 109.13 105.13       0.073     0.239 108.9088          0.18             -0.50             -0.456
63 2019-12-04 108.68 108.67 108.43 108.48 108.55 104.80      -0.184     0.221 108.8936         -0.38             -0.62             -0.568
64 2019-12-05 108.84 108.93 108.78 108.89 108.86 104.78       0.046     0.138 108.8948          0.00              0.41              0.378
65 2019-12-06 108.73 108.78 108.66 108.67 108.78 104.94      -0.055     0.110 108.8972         -0.21             -0.22             -0.202
66 2019-12-09 108.64 108.66 108.55 108.57 108.58 104.98      -0.064     0.101 108.9216         -0.32             -0.10             -0.092
67 2019-12-10 108.59 108.66 108.57 108.63 108.64 104.75       0.037     0.083 108.9152         -0.26              0.06              0.055
68 2019-12-11 108.78 108.86 108.67 108.73 108.74 104.82      -0.046     0.175 108.9052         -0.16              0.10              0.092
69 2019-12-12 108.56 108.65 108.46 108.65 108.58 103.63       0.083     0.175 108.8928         -0.22             -0.08             -0.074
70 2019-12-13 109.40 109.66 108.84 109.63 109.50     NA       0.210     0.753 108.9064          0.66              0.98              0.902
[1] "NIKKEI"
         Date     Open     High      Low    Close CloseToOpen HighToLow     MA25 DeviationRate Close:Diff(lag=1) Close:Ratio(lag=1)
56 2019-11-25 23292.85 23347.18 23255.39 23292.81       0.000     0.395 23075.06          0.94            179.93              0.778
57 2019-11-26 23451.40 23608.06 23350.10 23373.32      -0.333     1.105 23110.29          1.14             80.51              0.346
58 2019-11-27 23452.85 23507.82 23418.23 23437.77      -0.064     0.383 23145.84          1.26             64.45              0.276
59 2019-11-28 23458.88 23482.32 23367.33 23409.14      -0.212     0.492 23177.19          1.00            -28.63             -0.122
60 2019-11-29 23497.44 23498.77 23273.37 23293.91      -0.866     0.968 23198.92          0.41           -115.23             -0.492
61 2019-12-02 23388.63 23562.05 23378.40 23529.50       0.602     0.786 23228.11          1.30            235.59              1.011
62 2019-12-03 23231.14 23388.18 23186.84 23379.81       0.640     0.868 23248.61          0.56           -149.69             -0.636
63 2019-12-04 23186.74 23203.77 23044.78 23135.23      -0.222     0.690 23255.06         -0.52           -244.58             -1.046
64 2019-12-05 23292.70 23363.44 23259.82 23300.09       0.032     0.445 23273.34          0.11            164.86              0.713
65 2019-12-06 23347.67 23412.48 23338.40 23354.40       0.029     0.317 23290.43          0.27             54.31              0.233
66 2019-12-09 23544.31 23544.31 23360.01 23430.70      -0.483     0.789 23313.63          0.50             76.30              0.327
67 2019-12-10 23372.39 23449.47 23336.93 23410.19       0.162     0.482 23319.96          0.39            -20.51             -0.088
68 2019-12-11 23421.14 23438.43 23333.63 23391.86      -0.125     0.449 23323.48          0.29            -18.33             -0.078
69 2019-12-12 23449.28 23468.15 23360.43 23424.81      -0.104     0.461 23327.26          0.42             32.95              0.141
70 2019-12-13 23810.56 24050.04 23775.73 24023.10       0.893     1.154 23352.51          2.87            598.29              2.554

単位根検定・共和分検定

  • CADFtest {CADFtest}
  • ca.po {urca}
$USDJPY_CloseToOpen

    ADF test

data:  x
ADF(0) = -8.6933, p-value = 0.00000000513
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-1.096216 


$NIKKEI_CloseToOpen

    ADF test

data:  x
ADF(1) = -6.812, p-value = 0.000001879
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-1.360073 

######################################## 
# Phillips and Ouliaris Unit Root Test # 
######################################## 

Test of type Pu 
detrending of series none 


Call:
lm(formula = z[, 1] ~ z[, -1] - 1)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.35048 -0.05973  0.01477  0.09225  0.24228 

Coefficients:
        Estimate Std. Error t value Pr(>|t|)   
z[, -1]  0.10456    0.03607   2.899  0.00525 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1082 on 59 degrees of freedom
Multiple R-squared:  0.1247,    Adjusted R-squared:  0.1099 
F-statistic: 8.406 on 1 and 59 DF,  p-value: 0.005246


Value of test-statistic is: 68.0015 

Critical values of Pu are:
                  10pct    5pct    1pct
critical values 20.3933 25.9711 38.3413

最小二乗法

  • lm {stats}
  • dwtest {lmtest}
  • ks.test {stats}
  • confint {stats}
  • Box.test {stats}
  • 切片項\(\neq0\)
MODEL INFO:
Observations: 60
Dependent Variable: USDJPY_CloseToOpen
Type: OLS linear regression 

MODEL FIT:
F(1,58) = 8.03, p = 0.01
R2 = 0.12
Adj. R2 = 0.11 

Standard errors: OLS
---------------------------------------------------------------
                           Est.    2.5%   97.5%   t val.      p
------------------------ ------ ------- ------- -------- ------
(Intercept)                0.01   -0.02    0.04     0.56   0.58
NIKKEI_CloseToOpen         0.10    0.03    0.18     2.83   0.01
---------------------------------------------------------------

    Durbin-Watson test

data:  OLS_Model
DW = 2.2921, p-value = 0.8726
alternative hypothesis: true autocorrelation is greater than 0

    One-sample Kolmogorov-Smirnov test

data:  ResidualsOLS
D = 0.10009, p-value = 0.551
alternative hypothesis: two-sided
                         2.5 %     97.5 %
(Intercept)        -0.02029313 0.03612537
NIKKEI_CloseToOpen  0.03025021 0.17587693

    Box-Ljung test

data:  ResidualsOLS
X-squared = 12.212, df = 10, p-value = 0.2711
  • 切片項\(=0\)
MODEL INFO:
Observations: 60
Dependent Variable: USDJPY_CloseToOpen
Type: OLS linear regression 

MODEL FIT:
F(1,59) = 8.41, p = 0.01
R2 = 0.12
Adj. R2 = 0.11 

Standard errors: OLS
--------------------------------------------------------------
                           Est.   2.5%   97.5%   t val.      p
------------------------ ------ ------ ------- -------- ------
NIKKEI_CloseToOpen         0.10   0.03    0.18     2.90   0.01
--------------------------------------------------------------

    Durbin-Watson test

data:  OLS_Model_no_intercept
DW = 2.2806, p-value = 0.8898
alternative hypothesis: true autocorrelation is greater than 0

    One-sample Kolmogorov-Smirnov test

data:  ResidualsOLS_no_intercept
D = 0.1286, p-value = 0.2518
alternative hypothesis: two-sided
                       2.5 %   97.5 %
NIKKEI_CloseToOpen 0.0323969 0.176733

    Box-Ljung test

data:  ResidualsOLS_no_intercept
X-squared = 12.28, df = 10, p-value = 0.2667

一般化最小二乗法

  • 切片項\(\neq0\)
Generalized least squares fit by REML
  Model: USDJPY_CloseToOpen ~ NIKKEI_CloseToOpen 
  Data: USDJPY_NIKKEI 
        AIC       BIC   logLik
  -80.60922 -70.30701 45.30461

Correlation Structure: ARMA(1,1)
 Formula: ~1 
 Parameter estimate(s):
      Phi1     Theta1 
-0.8846090  0.7501313 

Coefficients:
                       Value  Std.Error  t-value p-value
(Intercept)        0.0079385 0.01258152 0.630965  0.5305
NIKKEI_CloseToOpen 0.1150429 0.03365376 3.418425  0.0012

 Correlation: 
                   (Intr)
NIKKEI_CloseToOpen -0.073

Standardized residuals:
        Min          Q1         Med          Q3         Max 
-3.28385962 -0.61949713  0.04080907  0.76086081  2.12904950 

Residual standard error: 0.1088565 
Degrees of freedom: 60 total; 58 residual

    One-sample Kolmogorov-Smirnov test

data:  ResidualsGLS
D = 0.095724, p-value = 0.6073
alternative hypothesis: two-sided
                         2.5 %     97.5 %
(Intercept)        -0.01672083 0.03259782
NIKKEI_CloseToOpen  0.04908269 0.18100301

    Box-Ljung test

data:  ResidualsGLS
X-squared = 12.727, df = 10, p-value = 0.2393
  • 切片項\(=0\)
Generalized least squares fit by REML
  Model: USDJPY_CloseToOpen ~ NIKKEI_CloseToOpen - 1 
  Data: USDJPY_NIKKEI 
       AIC       BIC  logLik
  -89.1282 -80.81805 48.5641

Correlation Structure: ARMA(1,1)
 Formula: ~1 
 Parameter estimate(s):
      Phi1     Theta1 
-0.8832993  0.7465474 

Coefficients:
                       Value  Std.Error  t-value p-value
NIKKEI_CloseToOpen 0.1166774 0.03337299 3.496162  0.0009

Standardized residuals:
       Min         Q1        Med         Q3        Max 
-3.2232939 -0.5485805  0.1109947  0.8384689  2.2076837 

Residual standard error: 0.108393 
Degrees of freedom: 60 total; 59 residual

    One-sample Kolmogorov-Smirnov test

data:  ResidualsGLS_no_intercept
D = 0.12087, p-value = 0.3186
alternative hypothesis: two-sided
                        2.5 %    97.5 %
NIKKEI_CloseToOpen 0.05126752 0.1820872

    Box-Ljung test

data:  ResidualsGLS_no_intercept
X-squared = 12.792, df = 10, p-value = 0.2355

散布図・QQプロット・残差の時系列推移

  • (注意)線形回帰の傾き(\(a\))、切片(\(b\))それぞれの検定統計量、p値に関わらず\(y=ax+b\)とした回帰直線やその残差を散布図、QQプロット等にプロットしています。
  • 散布図とQQプロット

  • 残差の自己相関(ACF)

  • 残差の自己相関(PACF)

ダウ平均株価と日経平均株価(Source:Yahoo Finance,FRED,日本経済新聞社)

時系列チャート

  • Source:
  • (注意) 欠損値(休場日)は原系列にスプライン補間を掛けた上で前日比を算出している。
  • 対象期間:2019-08-12~2019-12-13
  • サンプルサイズ:n=90

単位根検定・共和分検定

  • CADFtest {CADFtest}
  • ca.po {urca}
単位根検定
$DOW30

    ADF test

data:  x
ADF(0) = -2.8318, p-value = 0.1902
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-0.159906 


$NIKKEI225

    ADF test

data:  x
ADF(0) = -2.2003, p-value = 0.4832
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-0.108884 


$DOW30_Change

    ADF test

data:  x
ADF(2) = -7.4935, p-value = 0.00000003399
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-1.359004 


$NIKKEI225_Change

    ADF test

data:  x
ADF(0) = -8.6063, p-value = 0.0000000004728
alternative hypothesis: true delta is less than 0
sample estimates:
     delta 
-0.9991729 
共和分検定
[1] "DOW30 × NIKKEI225"

######################################## 
# Phillips and Ouliaris Unit Root Test # 
######################################## 

Test of type Pu 
detrending of series none 


Call:
lm(formula = z[, 1] ~ z[, -1] - 1)

Residuals:
     Min       1Q   Median       3Q      Max 
-1098.49  -584.49   -95.37   506.21  1463.21 

Coefficients:
        Estimate Std. Error t value            Pr(>|t|)    
z[, -1] 1.216907   0.003326   365.9 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 700.5 on 89 degrees of freedom
Multiple R-squared:  0.9993,    Adjusted R-squared:  0.9993 
F-statistic: 1.339e+05 on 1 and 89 DF,  p-value: < 0.00000000000000022


Value of test-statistic is: 4.2058 

Critical values of Pu are:
                  10pct    5pct    1pct
critical values 20.3933 25.9711 38.3413

相互相関関数

  • ggCcf {forecast}

ベクトル自己回帰モデル

  • VARselect {vars}
  • VAR {vars}

VAR Estimation Results:
========================= 
Endogenous variables: DOW30, NIKKEI225 
Deterministic variables: const 
Sample size: 89 
Log Likelihood: -1163.745 
Roots of the characteristic polynomial:
0.9896 0.666
Call:
VAR(y = obj, p = selected_lag, type = "const")


Estimation results for equation DOW30: 
====================================== 
DOW30 = DOW30.l1 + NIKKEI225.l1 + const 

               Estimate Std. Error t value            Pr(>|t|)    
DOW30.l1        0.80811    0.07531  10.730 <0.0000000000000002 ***
NIKKEI225.l1    0.10998    0.05063   2.172              0.0326 *  
const        2770.64717 1100.08744   2.519              0.0136 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 199.5 on 86 degrees of freedom
Multiple R-Squared: 0.9226, Adjusted R-squared: 0.9208 
F-statistic: 512.4 on 2 and 86 DF,  p-value: < 0.00000000000000022 


Estimation results for equation NIKKEI225: 
========================================== 
NIKKEI225 = DOW30.l1 + NIKKEI225.l1 + const 

                Estimate  Std. Error t value             Pr(>|t|)    
DOW30.l1         0.23458     0.05481   4.280            0.0000484 ***
NIKKEI225.l1     0.84753     0.03685  23.002 < 0.0000000000000002 ***
const        -2917.38802   800.64487  -3.644             0.000459 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 145.2 on 86 degrees of freedom
Multiple R-Squared: 0.9816, Adjusted R-squared: 0.9812 
F-statistic:  2294 on 2 and 86 DF,  p-value: < 0.00000000000000022 



Covariance matrix of residuals:
          DOW30 NIKKEI225
DOW30     39813      1839
NIKKEI225  1839     21089

Correlation matrix of residuals:
            DOW30 NIKKEI225
DOW30     1.00000   0.06348
NIKKEI225 0.06348   1.00000

VAR Estimation Results:
========================= 
Endogenous variables: DOW30_Change, NIKKEI225_Change 
Deterministic variables: const 
Sample size: 89 
Log Likelihood: -172.807 
Roots of the characteristic polynomial:
0.1016 0.1016
Call:
VAR(y = obj, p = selected_lag, type = "const")


Estimation results for equation DOW30_Change: 
============================================= 
DOW30_Change = DOW30_Change.l1 + NIKKEI225_Change.l1 + const 

                    Estimate Std. Error t value Pr(>|t|)
DOW30_Change.l1     -0.12461    0.10455  -1.192    0.237
NIKKEI225_Change.l1 -0.02178    0.12114  -0.180    0.858
const                0.10915    0.08428   1.295    0.199


Residual standard error: 0.7739 on 86 degrees of freedom
Multiple R-Squared: 0.01648,    Adjusted R-squared: -0.006391 
F-statistic: 0.7206 on 2 and 86 DF,  p-value: 0.4894 


Estimation results for equation NIKKEI225_Change: 
================================================= 
NIKKEI225_Change = DOW30_Change.l1 + NIKKEI225_Change.l1 + const 

                    Estimate Std. Error t value         Pr(>|t|)    
DOW30_Change.l1      0.60472    0.07376   8.199 0.00000000000209 ***
NIKKEI225_Change.l1  0.02291    0.08546   0.268           0.7893    
const                0.12851    0.05946   2.161           0.0334 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.546 on 86 degrees of freedom
Multiple R-Squared: 0.4387, Adjusted R-squared: 0.4257 
F-statistic: 33.61 on 2 and 86 DF,  p-value: 0.0000000000164 



Covariance matrix of residuals:
                 DOW30_Change NIKKEI225_Change
DOW30_Change          0.59889          0.01169
NIKKEI225_Change      0.01169          0.29807

Correlation matrix of residuals:
                 DOW30_Change NIKKEI225_Change
DOW30_Change          1.00000          0.02768
NIKKEI225_Change      0.02768          1.00000

グレンジャー因果

  • causality {vars}
Dow → Nikkei

    Granger causality H0: DOW30_Change do not Granger-cause NIKKEI225_Change

data:  VAR object var_result
F-Test = 67.221, df1 = 1, df2 = 172, p-value = 0.00000000000005396
Nikkei → Dow

    Granger causality H0: NIKKEI225_Change do not Granger-cause DOW30_Change

data:  VAR object var_result
F-Test = 0.032326, df1 = 1, df2 = 172, p-value = 0.8575

インパルス応答

  • irf {vars}

空売り比率と日経平均株価(Source:日本取引所グループ、日本経済新聞社)

業種別空売り集計

  • 2019年12月13日
  • 「空売り合計:比率」は100から「実注文:比率」を減じた数値としています。

空売り合計:比率:2019年12月13日
N 業種名 空売り合計:比率
1 水産・農林業 35.2
2 鉱業 32.7
3 建設業 38.7
4 食料品 39.4
5 繊維製品 38.3
6 パルプ・紙 42.2
7 化学 38.9
8 医薬品 37.7
9 石油・石炭製品 37.7
10 ゴム製品 35.1
11 ガラス・土石製品 33.5
12 鉄鋼 35.6
13 非鉄金属 35.4
14 金属製品 44
15 機械 40
16 電気機器 38.4
17 輸送用機器 39.1
18 精密機器 38.6
19 その他製品 41
20 電気・ガス業 41.9
21 陸運業 45.6
22 海運業 35.5
23 空運業 37.9
24 倉庫・運輸関連業 38.6
25 情報・通信業 35.9
26 卸売業 36.4
27 小売業 43.4
28 銀行業 35.3
29 証券、商品先物取引業 35.6
30 保険業 44.2
31 その他金融業 40.6
32 不動産業 42.1
33 サービス業 37.9
34 その他(33業種外) 38.4

空売り比率の時系列推移

  • 2019-08-02 ~ 2019-12-13

時系列推移
Date 12-13 12-12 12-11 12-10 12-09 12-06 12-05 12-04
実注文:比率 61.4 61.6 59.4 60 60.5 60.3 59.2 56.5
空売り(価格規制あり):比率 30 31.9 33.2 31.5 32.9 34 35.6 36.1
空売り(価格規制なし):比率 8.6 6.4 7.5 8.5 6.7 5.7 5.2 7.4
空売り合計:比率 38.6 38.4 40.6 40 39.5 39.7 40.8 43.5

日経平均株価と空売り比率

時系列推移

  • 対象期間:2019-08-02 ~ 2019-12-13

単位根検定/共和分検定

  • CADFtest {CADFtest}
  • ca.po {urca}
  • 各系列の“_change“は前営業日との差。
  • 対象期間: 2019-08-02 ~ 2019-12-13,90days
### 単位根検定 ###

$NIKKEI225.close

    ADF test

data:  x
ADF(0) = -2.4001, p-value = 0.377
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-0.124968 


$ShortSalerRatio

    ADF test

data:  x
ADF(0) = -8.3393, p-value = 0.000000001282
alternative hypothesis: true delta is less than 0
sample estimates:
     delta 
-0.9001321 


$NIKKEI225.close_change

    ADF test

data:  x
ADF(0) = -8.7186, p-value = 0.0000000003131
alternative hypothesis: true delta is less than 0
sample estimates:
    delta 
-1.012128 


$ShortSalerRatio_change

    ADF test

data:  x
ADF(2) = -8.509, p-value = 0.0000000006783
alternative hypothesis: true delta is less than 0
sample estimates:
   delta 
-2.28697 
### 共和分検定 ###


######################################## 
# Phillips and Ouliaris Unit Root Test # 
######################################## 

Test of type Pu 
detrending of series none 


Call:
lm(formula = z[, 1] ~ z[, -1] - 1)

Residuals:
   Min     1Q Median     3Q    Max 
 -5433  -1833    687   2305   4421 

Coefficients:
        Estimate Std. Error t value            Pr(>|t|)    
z[, -1]  507.829      6.203   81.87 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2546 on 89 degrees of freedom
Multiple R-squared:  0.9869,    Adjusted R-squared:  0.9867 
F-statistic:  6702 on 1 and 89 DF,  p-value: < 0.00000000000000022


Value of test-statistic is: 0.3668 

Critical values of Pu are:
                  10pct    5pct    1pct
critical values 20.3933 25.9711 38.3413

最小二乗法

  • lm {stats}
  • dwtest {lmtest}
  • ks.test {stats}
  • confint {stats}
  • Box.test {stats}
  • Ljung-Box 検定のラグは15としている。
  • 対象期間: 2019-08-02 ~ 2019-12-13,90days
  • 切片項\(\neq0\)

Call:
lm(formula = NIKKEI225.close_change ~ ShortSalerRatio_change, 
    data = datadf)

Residuals:
    Min      1Q  Median      3Q     Max 
-371.23  -61.03   -5.43   87.26  579.38 

Coefficients:
                       Estimate Std. Error t value    Pr(>|t|)    
(Intercept)              25.178     16.535   1.523       0.131    
ShortSalerRatio_change  -31.316      5.909  -5.299 0.000000851 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 156.8 on 88 degrees of freedom
Multiple R-squared:  0.2419,    Adjusted R-squared:  0.2333 
F-statistic: 28.08 on 1 and 88 DF,  p-value: 0.0000008511

    Durbin-Watson test

data:  OLS_Model
DW = 1.4457, p-value = 0.004184
alternative hypothesis: true autocorrelation is greater than 0

    One-sample Kolmogorov-Smirnov test

data:  ResidualsOLS
D = 0.11211, p-value = 0.1929
alternative hypothesis: two-sided
                            2.5 %    97.5 %
(Intercept)             -7.682342  58.03856
ShortSalerRatio_change -43.059557 -19.57232

    Box-Ljung test

data:  ResidualsOLS
X-squared = 13.385, df = 15, p-value = 0.5726
  • 切片項\(=0\)

Call:
lm(formula = NIKKEI225.close_change ~ ShortSalerRatio_change - 
    1, data = datadf)

Residuals:
    Min      1Q  Median      3Q     Max 
-345.34  -36.18   19.92  113.89  604.60 

Coefficients:
                       Estimate Std. Error t value   Pr(>|t|)    
ShortSalerRatio_change  -31.562      5.951  -5.304 0.00000082 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 158 on 89 degrees of freedom
Multiple R-squared:  0.2402,    Adjusted R-squared:  0.2316 
F-statistic: 28.13 on 1 and 89 DF,  p-value: 0.0000008195

    Durbin-Watson test

data:  OLS_Model_no_intercept
DW = 1.4102, p-value = 0.003377
alternative hypothesis: true autocorrelation is greater than 0

    One-sample Kolmogorov-Smirnov test

data:  ResidualsOLS_no_intercept
D = 0.17402, p-value = 0.007457
alternative hypothesis: two-sided
                           2.5 %    97.5 %
ShortSalerRatio_change -43.38648 -19.73848

    Box-Ljung test

data:  ResidualsOLS_no_intercept
X-squared = 13.342, df = 15, p-value = 0.5759

一般化最小二乗法

  • 切片項\(\neq0\)
Generalized least squares fit by REML
  Model: NIKKEI225.close_change ~ ShortSalerRatio_change 
  Data: datadf 
       AIC      BIC   logLik
  1154.728 1164.637 -573.364

Correlation Structure: AR(1)
 Formula: ~1 
 Parameter estimate(s):
      Phi 
0.2311613 

Coefficients:
                           Value Std.Error   t-value p-value
(Intercept)             26.09793 21.005880  1.242411  0.2174
ShortSalerRatio_change -29.83991  5.157308 -5.785947  0.0000

 Correlation: 
                       (Intr)
ShortSalerRatio_change 0.015 

Standardized residuals:
        Min          Q1         Med          Q3         Max 
-2.38267539 -0.39484827 -0.02644419  0.51725962  3.65953758 

Residual standard error: 157.9872 
Degrees of freedom: 90 total; 88 residual

    One-sample Kolmogorov-Smirnov test

data:  ResidualsGLS
D = 0.10496, p-value = 0.2561
alternative hypothesis: two-sided
                           2.5 %    97.5 %
(Intercept)            -15.07284  67.26870
ShortSalerRatio_change -39.94805 -19.73177

    Box-Ljung test

data:  ResidualsGLS
X-squared = 13.62, df = 15, p-value = 0.5545
  • 切片項\(=0\)
Generalized least squares fit by REML
  Model: NIKKEI225.close_change ~ ShortSalerRatio_change - 1 
  Data: datadf 
       AIC      BIC    logLik
  1162.195 1169.661 -578.0976

Correlation Structure: AR(1)
 Formula: ~1 
 Parameter estimate(s):
      Phi 
0.2366056 

Coefficients:
                           Value Std.Error   t-value p-value
ShortSalerRatio_change -29.90305  5.157079 -5.798447       0

Standardized residuals:
       Min         Q1        Med         Q3        Max 
-2.2067475 -0.2284964  0.1377484  0.6803061  3.8082772 

Residual standard error: 158.673 
Degrees of freedom: 90 total; 89 residual

    One-sample Kolmogorov-Smirnov test

data:  ResidualsGLS_no_intercept
D = 0.16794, p-value = 0.01093
alternative hypothesis: two-sided
                           2.5 %    97.5 %
ShortSalerRatio_change -40.01074 -19.79536

    Box-Ljung test

data:  ResidualsGLS_no_intercept
X-squared = 13.611, df = 15, p-value = 0.5552

残差

  • 時系列推移
  • 自己相関
  • 時系列推移

  • 自己相関

ドル円レートと日経平均株価:ベイズ推定:線形回帰モデル

\[\rm{NIKKEI}\sim\rm{Normal}(\beta_0 + \beta_1 \cdot \rm{USDJPY},\sigma)\]

# 数値はいずれも前月比(%)
head(df)
         Date NIKKEI USDJPY
13 2017-01-01   0.67  -1.05
14 2017-02-01  -0.03  -1.45
15 2017-03-01   0.79  -0.05
16 2017-04-01  -3.12  -2.59
17 2017-05-01   5.29   1.94
18 2017-06-01   1.62  -1.19
tail(df)
         Date NIKKEI USDJPY
43 2019-07-01   2.53   0.15
44 2019-08-01  -4.46  -1.81
45 2019-09-01   4.63   1.06
46 2019-10-01   2.84   0.65
47 2019-11-01   4.87   0.69
48 2019-12-01   0.69   0.02
apply(df[, -1], 2, adf.test)
$NIKKEI

    Augmented Dickey-Fuller Test

data:  newX[, i]
Dickey-Fuller = -3.649, Lag order = 3, p-value = 0.04326
alternative hypothesis: stationary


$USDJPY

    Augmented Dickey-Fuller Test

data:  newX[, i]
Dickey-Fuller = -4.1827, Lag order = 3, p-value = 0.01397
alternative hypothesis: stationary
# 最尤推定
summary(lm(NIKKEI ~ USDJPY), confint = T, ci.width = 0.95)

Call:
lm(formula = NIKKEI ~ USDJPY)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.7694 -1.1114  0.2399  1.2454  6.0705 

Coefficients:
            Estimate Std. Error t value  Pr(>|t|)    
(Intercept)   0.8658     0.4232   2.046    0.0486 *  
USDJPY        1.4564     0.2977   4.892 0.0000237 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.522 on 34 degrees of freedom
Multiple R-squared:  0.4131,    Adjusted R-squared:  0.3958 
F-statistic: 23.93 on 1 and 34 DF,  p-value: 0.00002372
Gaussian <- "
  data{
    int N;
    vector[N] NIKKEI;
    vector[N] USDJPY;
  }
  parameters{
    real beta0;
    real beta1;
    real <lower = 0> sigma;
  }
  model{
    for(i in 1:N)
      NIKKEI[i] ~ normal(beta0 + beta1*USDJPY[i], sigma);
  }
  generated quantities{
    vector[N] pred_NIKKEI;
    real log_lik[N];
    for (i in 1:N){
      pred_NIKKEI[i] = normal_rng(beta0 + beta1*USDJPY[i], sigma);
      log_lik[i] = normal_lpdf(NIKKEI[i] | beta0 + beta1*USDJPY[i], sigma);
    }
  }
"
datalist <- list(N = N, NIKKEI = NIKKEI, USDJPY = USDJPY)
iter <- 1400
warmup <- 400
chains <- 3
fit <- stan(model_code = Gaussian, data = datalist, iter = iter, warmup = warmup, thin = 1, chains = chains)
summary(fit)$summary[c("beta0", "beta1", "sigma"), ]
           mean     se_mean        sd         2.5%       25%       50%      75%    97.5%    n_eff     Rhat
beta0 0.8908768 0.011008357 0.4583084 -0.006569884 0.5839943 0.8893089 1.187002 1.818650 1733.288 1.000400
beta1 1.4705010 0.007251186 0.3150267  0.865748110 1.2584969 1.4683257 1.670457 2.105773 1887.455 1.000086
sigma 2.6356880 0.008491936 0.3484453  2.072707793 2.3993586 2.6031181 2.833059 3.408856 1683.665 1.002696
traceplot(fit) + theme(axis.text.x = element_text(size = 5), axis.text.y = element_text(size = 5), strip.text.x = element_text(size = 5), legend.title = element_text(size = 5), legend.text = element_text(size = 5))

# EAP:事後期待値
df_result <- rstan::extract(fit)$pred_NIKKEI %>% data.frame() %>% gather() %>% dplyr::mutate(id = rep(c(1:N), each = (iter - warmup) * chains)) %>% group_by(id) %>% dplyr::summarize(pred_EAP = mean(value), pred_lower = quantile(value, 0.025), pred_upper = quantile(value, 0.975)) %>% dplyr::ungroup() %>% cbind(data.frame(NIKKEI, USDJPY))
head(df_result)
  id   pred_EAP pred_lower pred_upper NIKKEI USDJPY
1  1 -0.5828066  -5.745521   4.678021   0.67  -1.05
2  2 -1.2837200  -6.650768   4.287679  -0.03  -1.45
3  3  0.8152446  -4.722763   6.061326   0.79  -0.05
4  4 -2.9334397  -8.171576   2.398691  -3.12  -2.59
5  5  3.6976151  -1.739563   9.268521   5.29   1.94
6  6 -0.7826062  -5.981145   4.584849   1.62  -1.19
tail(df_result)
   id   pred_EAP pred_lower pred_upper NIKKEI USDJPY
31 31  1.1389518  -4.123720   6.442027   2.53   0.15
32 32 -1.7560420  -7.085969   3.488656  -4.46  -1.81
33 33  2.3879026  -2.960855   7.723205   4.63   1.06
34 34  1.8619282  -3.553991   7.117496   2.84   0.65
35 35  1.9183019  -3.424474   7.368708   4.87   0.69
36 36  0.8551996  -4.476583   6.247384   0.69   0.02

トランプ大統領の就任前後における日経平均前営業日比ボラティリティの比較

  • 休日、祝日の補間はとっていない。
  • helpより
    • 『garchOrder The ARCH (q) and GARCH (p) orders.』
    • 『external.regressors A matrix object containing the external regressors to include in the variance equation with as many rows as will be included in the data (which is passed in the fit function).』
  • 参照引用Webページ
head(nikkei)
            Date Nikkei_ChangeRate trump
14882 2009-01-20         -2.313958     0
14883 2009-01-21         -2.035139     0
14884 2009-01-22          1.899606     0
14885 2009-01-23         -3.806506     0
14886 2009-01-26         -0.814822     0
14887 2009-01-27          4.932610     0
tail(nikkei)
            Date Nikkei_ChangeRate trump
17547 2019-12-06        0.23308923     1
17548 2019-12-09        0.32670503     1
17549 2019-12-10       -0.08753473     1
17550 2019-12-11       -0.07829924     1
17551 2019-12-12        0.14086097     1
17552 2019-12-13        2.55408688     1
head(nikkei[as.Date("2017-1-17") <= nikkei$Date, ])
            Date Nikkei_ChangeRate trump
16839 2017-01-17        -1.4752891     0
16840 2017-01-18         0.4296908     0
16841 2017-01-19         0.9414445     0
16842 2017-01-20         0.3442698     1
16843 2017-01-23        -1.2900050     1
16844 2017-01-24        -0.5454441     1

nikkei$trump <- as.numeric(nikkei$trump)
datavalue <- nikkei[, 2]
trump <- as.matrix(nikkei$trump)
  • Dummyなし

\[ r_t = \mu + \sqrt{h_t}\epsilon_t,\quad \epsilon_t\sim i.i.d. \textrm{N} (0,1),\quad h_t=\omega+\beta_{1}h_{t-1} + \alpha_{1} r^2_{t-1}\\r_tはt時点の日経平均前営業日比 \]

summary(tseries::garch(x = datavalue, order = c(1, 1)))

 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 

     I     INITIAL X(I)        D(I)

     1     1.627330e+00     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1  2.076e+03
     1    3  2.057e+03  9.19e-03  2.34e-02  3.1e-02  4.9e+03  1.0e-01  5.69e+01
     2    5  2.056e+03  5.57e-04  5.98e-04  2.4e-03  5.1e+02  1.0e-02  3.91e+00
     3    7  2.054e+03  8.95e-04  8.99e-04  5.1e-03  2.0e+00  2.0e-02  1.22e+00
     4    9  2.051e+03  1.24e-03  1.26e-03  1.2e-02  2.1e+00  4.0e-02  1.22e+00
     5   12  2.035e+03  8.12e-03  1.35e-02  2.1e-01  2.4e+00  6.5e-01  1.30e+00
     6   14  1.956e+03  3.86e-02  3.39e-02  4.5e-01  1.9e+00  6.5e-01  2.42e+00
     7   16  1.953e+03  1.77e-03  6.20e-03  6.4e-02  2.0e+00  6.5e-02  7.76e-01
     8   17  1.942e+03  5.31e-03  5.02e-03  6.0e-02  2.0e+00  6.5e-02  1.66e+00
     9   18  1.931e+03  6.12e-03  5.59e-03  8.2e-02  2.0e+00  1.3e-01  1.78e+00
    10   20  1.926e+03  2.11e-03  2.42e-03  1.4e-02  2.0e+00  2.6e-02  8.24e-01
    11   21  1.923e+03  1.82e-03  1.90e-03  3.1e-02  2.0e+00  5.2e-02  2.62e-01
    12   22  1.921e+03  8.54e-04  4.97e-03  6.6e-02  1.8e+00  1.0e-01  4.83e-02
    13   23  1.901e+03  1.05e-02  1.38e-02  6.1e-02  2.0e+00  1.0e-01  2.21e+00
    14   25  1.901e+03  1.88e-04  7.53e-04  7.0e-03  6.7e+00  1.0e-02  2.35e-02
    15   26  1.900e+03  6.77e-04  6.99e-04  6.0e-03  2.0e+00  1.0e-02  1.45e-01
    16   27  1.898e+03  6.82e-04  8.13e-04  1.2e-02  2.0e+00  2.1e-02  1.01e-01
    17   29  1.897e+03  4.08e-04  8.38e-04  9.8e-03  2.0e+00  1.7e-02  2.90e-02
    18   30  1.896e+03  9.84e-04  1.07e-03  8.4e-03  1.8e+00  1.7e-02  8.79e-03
    19   32  1.893e+03  1.26e-03  2.16e-03  3.0e-02  2.6e+00  6.7e-02  1.75e-02
    20   35  1.893e+03  3.41e-04  9.25e-04  2.8e-03  2.8e+00  6.3e-03  3.39e-03
    21   36  1.892e+03  3.96e-04  4.49e-04  2.6e-03  2.0e+00  6.3e-03  2.91e-02
    22   37  1.892e+03  1.33e-04  5.00e-04  7.4e-03  1.9e+00  1.3e-02  1.33e-02
    23   38  1.891e+03  1.41e-04  5.78e-04  6.3e-03  1.7e+00  1.3e-02  1.57e-03
    24   39  1.891e+03  1.72e-04  2.03e-04  6.0e-03  9.7e-01  1.3e-02  2.47e-04
    25   41  1.891e+03  1.05e-05  3.88e-05  8.3e-04  1.7e+00  1.6e-03  4.36e-05
    26   42  1.891e+03  1.11e-05  1.20e-05  7.3e-04  1.5e+00  1.6e-03  1.28e-05
    27   43  1.891e+03  6.47e-09  6.87e-08  7.3e-05  0.0e+00  1.8e-04  6.87e-08
    28   44  1.891e+03  7.28e-10  1.70e-09  2.9e-05  0.0e+00  6.0e-05  1.70e-09
    29   45  1.891e+03  4.10e-09  1.08e-09  1.7e-05  0.0e+00  4.0e-05  1.08e-09
    30   46  1.891e+03 -1.17e-09  6.38e-12  1.6e-06  0.0e+00  3.7e-06  6.38e-12

 ***** RELATIVE FUNCTION CONVERGENCE *****

 FUNCTION     1.890961e+03   RELDX        1.603e-06
 FUNC. EVALS      46         GRAD. EVALS      30
 PRELDF       6.376e-12      NPRELDF      6.376e-12

     I      FINAL X(I)        D(I)          G(I)

     1    6.856020e-02     1.000e+00     4.093e-03
     2    1.274418e-01     1.000e+00     1.315e-02
     3    8.381131e-01     1.000e+00     1.713e-02


Call:
tseries::garch(x = datavalue, order = c(1, 1))

Model:
GARCH(1,1)

Residuals:
     Min       1Q   Median       3Q      Max 
-5.24576 -0.52159  0.06163  0.63817  4.15772 

Coefficient(s):
    Estimate  Std. Error  t value             Pr(>|t|)    
a0  0.068560    0.010411    6.586      0.0000000000453 ***
a1  0.127442    0.009972   12.780 < 0.0000000000000002 ***
b1  0.838113    0.012915   64.893 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Diagnostic Tests:
    Jarque Bera Test

data:  Residuals
X-squared = 358.53, df = 2, p-value < 0.00000000000000022


    Box-Ljung test

data:  Squared.Residuals
X-squared = 2.8441, df = 1, p-value = 0.09171
garchresult <- fGarch::garchFit(formula = ~garch(1, 1), data = datavalue, trace = F)
summary(garchresult)

Title:
 GARCH Modelling 

Call:
 fGarch::garchFit(formula = ~garch(1, 1), data = datavalue, trace = F) 

Mean and Variance Equation:
 data ~ garch(1, 1)
<environment: 0x000000008c17ff08>
 [data = datavalue]

Conditional Distribution:
 norm 

Coefficient(s):
      mu     omega    alpha1     beta1  
0.078889  0.069105  0.131862  0.834027  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value             Pr(>|t|)    
mu       0.07889     0.02137    3.692             0.000223 ***
omega    0.06910     0.01476    4.683           0.00000283 ***
alpha1   0.13186     0.01576    8.368 < 0.0000000000000002 ***
beta1    0.83403     0.01948   42.816 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 -4340.452    normalized:  -1.625029 

Description:
 Mon Dec 16 06:36:22 2019 by user: 20141203 


Standardised Residuals Tests:
                                Statistic p-Value  
 Jarque-Bera Test   R    Chi^2  344.9328  0        
 Shapiro-Wilk Test  R    W      0.9840915 0        
 Ljung-Box Test     R    Q(10)  3.514249  0.9666123
 Ljung-Box Test     R    Q(15)  8.275969  0.9122426
 Ljung-Box Test     R    Q(20)  14.61657  0.7979113
 Ljung-Box Test     R^2  Q(10)  8.527387  0.5774641
 Ljung-Box Test     R^2  Q(15)  13.01353  0.6012528
 Ljung-Box Test     R^2  Q(20)  17.82628  0.5988508
 LM Arch Test       R    TR^2   12.53484  0.403736 

Information Criterion Statistics:
     AIC      BIC      SIC     HQIC 
3.253052 3.261873 3.253048 3.256244 
garchresult@fit$par
        mu      omega     alpha1      beta1 
0.07888931 0.06910475 0.13186152 0.83402657 
  • Dummyあり

\[ r_t = \mu + \sqrt{h_t}\epsilon_t,\quad \epsilon_t\sim i.i.d. \textrm{N} (0,1),\quad h_t=\omega+\beta_{1}h_{t-1} + \alpha_{1} r^2_{t-1} + \delta_{\textrm{dummy}}\\r_tはt時点の日経平均前営業日比 \]

garch_sim <- function(data, v_model, garchorder, armaorder, external_regressors) {
    garch_model <- ugarchspec(variance.model = list(model = v_model, garchOrder = garchorder, external.regressors = external_regressors), mean.model = list(armaOrder = armaorder, include.mean = T))
    garch_result <- ugarchfit(spec = garch_model, data = data)
    return(garch_result)
}
garch_sim(data = datavalue, v_model = "sGARCH", garchorder = c(1, 1), armaorder = c(0, 0), external_regressors = trump)

*---------------------------------*
*          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      0.078877    0.021372   3.6906 0.000224
omega   0.069123    0.015141   4.5653 0.000005
alpha1  0.131748    0.017478   7.5378 0.000000
beta1   0.834051    0.020466  40.7520 0.000000
vxreg1  0.000000    0.012448   0.0000 1.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.078877    0.022695   3.4756 0.000510
omega   0.069123    0.024063   2.8725 0.004072
alpha1  0.131748    0.033317   3.9544 0.000077
beta1   0.834051    0.041182  20.2527 0.000000
vxreg1  0.000000    0.018214   0.0000 1.000000

LogLikelihood : -4340.463 

Information Criteria
------------------------------------
                   
Akaike       3.2538
Bayes        3.2648
Shibata      3.2538
Hannan-Quinn 3.2578

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1169  0.7324
Lag[2*(p+q)+(p+q)-1][2]    0.4528  0.7161
Lag[4*(p+q)+(p+q)-1][5]    0.9484  0.8713
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.711  0.1909
Lag[2*(p+q)+(p+q)-1][5]     3.670  0.2981
Lag[4*(p+q)+(p+q)-1][9]     5.284  0.3889
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.2454 0.500 2.000  0.6204
ARCH Lag[5]    2.1720 1.440 1.667  0.4345
ARCH Lag[7]    2.9727 2.315 1.543  0.5187

Nyblom stability test
------------------------------------
Joint Statistic:  2.2082
Individual Statistics:              
mu     0.02271
omega  0.38688
alpha1 0.29107
beta1  0.37070
vxreg1 0.53095

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           0.7053 0.480662    
Negative Sign Bias  1.8325 0.066991   *
Positive Sign Bias  2.6655 0.007733 ***
Joint Effect       15.6962 0.001309 ***


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic   p-value(g-1)
1    20     75.35 0.000000011612
2    30     97.26 0.000000002676
3    40    107.00 0.000000029860
4    50    122.06 0.000000036318


Elapsed time : 0.441025 
garch_sim(data = datavalue, v_model = "eGARCH", garchorder = c(1, 1), armaorder = c(0, 0), external_regressors = trump)

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

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

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.042082    0.025990   1.6192  0.10541
omega   0.048447    0.009006   5.3792  0.00000
alpha1 -0.130263    0.014699  -8.8622  0.00000
beta1   0.926466    0.011401  81.2620  0.00000
gamma1  0.204882    0.021262   9.6359  0.00000
vxreg1 -0.051263    0.013001  -3.9429  0.00008

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.042082    0.035623   1.1813 0.237480
omega   0.048447    0.014181   3.4164 0.000635
alpha1 -0.130263    0.032092  -4.0591 0.000049
beta1   0.926466    0.019038  48.6643 0.000000
gamma1  0.204882    0.028439   7.2043 0.000000
vxreg1 -0.051263    0.018625  -2.7523 0.005918

LogLikelihood : -4286.183 

Information Criteria
------------------------------------
                   
Akaike       3.2139
Bayes        3.2271
Shibata      3.2139
Hannan-Quinn 3.2187

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1478  0.7007
Lag[2*(p+q)+(p+q)-1][2]    0.4949  0.6960
Lag[4*(p+q)+(p+q)-1][5]    0.8128  0.9004
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                         statistic p-value
Lag[1]                  0.00002303  0.9962
Lag[2*(p+q)+(p+q)-1][5] 0.56777359  0.9467
Lag[4*(p+q)+(p+q)-1][9] 1.55726449  0.9508
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]  0.002464 0.500 2.000  0.9604
ARCH Lag[5]  1.144411 1.440 1.667  0.6906
ARCH Lag[7]  1.349051 2.315 1.543  0.8511

Nyblom stability test
------------------------------------
Joint Statistic:  0.9752
Individual Statistics:              
mu     0.10510
omega  0.07352
alpha1 0.23446
beta1  0.12381
gamma1 0.37863
vxreg1 0.02226

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.49 1.68 2.12
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.4489 0.6535    
Negative Sign Bias  0.1481 0.8823    
Positive Sign Bias  1.4530 0.1463    
Joint Effect        2.4232 0.4893    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     55.87   0.00001711
2    30     72.14   0.00001536
3    40     76.12   0.00034515
4    50    100.04   0.00002338


Elapsed time : 0.6110349