Midterm Exam
Wayne
2018年1月6日
1 - (a)
ARMA-gjrGARCH-in-mean 在conditional 下的模型假設是 at | F t-1 ~ N(0,σt)。 為了建立合適的ARMA-gjrGARCH-in-mean模型，我先決定ARMA的order，再決定gjrGARCH-in-mean的order。
觀察ACF/PACF
rm(list=ls(all=TRUE))
library(quantmod)
library(rugarch)
library(ggplot2)
library(ggthemes)
start = "2013-01-01"
end = "2015-12-31"
ticker = "^GSPC"
Asset = getSymbols(ticker, from=start, to = end, auto.assign = FALSE)
Return = TTR::ROC(Ad(Asset),type="continuous")
Return = na.omit(Return)
colnames(Return) = "SP500return"
Return1 = Return[1:251]
DataSize = length(Return)
WindowSize = 251
OutSample = DataSize - WindowSize
acf(as.numeric(Return1))
pacf(as.numeric(Return1))
決定ARMA的order
關於決定ARMA的order 我用的方法是先觀察ACF與PACF，視ACF為Cut off 和PACF為Tail off，進而決定ARMA (p , q) ，再進一步確定ARMA的標準化殘差是否通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差為White noice。最後決定ARMA(0,0)。
ARMA_p = 0
ARMA_q = 0
spec1 = arfimaspec(mean.model = list(armaOrder = c(ARMA_p, ARMA_q), include.mean = TRUE),
distribution.model = "norm")
#個人傾向用第二種方法
modelfit1_1 = arfimafit(spec = spec1,
data=Return,
out.sample =OutSample,
solver = "hybrid",
solver.control = list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -896.3238 Pars: 0.0009308 0.0068054
## Iter: 2 fn: -896.3238 Pars: 0.0009308 0.0068054
## solnp--> Completed in 2 iterationsmodelfit1_1##
## *----------------------------------*
## * ARFIMA Model Fit *
## *----------------------------------*
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000931 0.000430 2.167 0.030237
## sigma 0.006805 0.000304 22.408 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000931 0.000380 2.4473 0.014392
## sigma 0.006805 0.000493 13.7930 0.000000
##
## LogLikelihood : 896.3238
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.1261
## Bayes -7.0980
## Shibata -7.1262
## Hannan-Quinn -7.1148
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.125 0.1449
## Lag[2*(p+q)+(p+q)-1][2] 2.144 0.2400
## Lag[4*(p+q)+(p+q)-1][5] 2.590 0.4869
##
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.911 0.088004
## Lag[2*(p+q)+(p+q)-1][2] 6.090 0.020820
## Lag[4*(p+q)+(p+q)-1][5] 10.519 0.006771
##
##
## ARCH LM Tests
## ------------------------------------
## Statistic DoF P-Value
## ARCH Lag[2] 8.255 2 0.01612
## ARCH Lag[5] 10.167 5 0.07065
## ARCH Lag[10] 11.204 10 0.34183
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.2254
## Individual Statistics:
## mu 0.03074
## sigma 0.19826
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.61 0.749 1.07
## Individual Statistic: 0.35 0.47 0.75
##
##
## Elapsed time : 0.8103819決定gjrGARCH-in-mean的order
進一步決定ARMA-gjrGARCH-in-mean的order，我以ARMA(0, 0)為基礎再加上gjrGARCH-in-mean(q, p) 的order，採用方法為窮舉法，gjrGARCH-in-mean (1,0) => gjrGARCH-in-mean (0, 1) =>gjrGARCH-in-mean(1,1) … 直到標準化殘差的平方通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差平方為White noice。雖然gjrGARCH-in-mean(1,0)通過Ljung- Box檢定，但因為不符合第(b)小題探討風險溢酬，於是往下做了幾個模型，最後決定通過Ljung- Box檢定的gjrGARCH-in-mean(1,1)。
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 0
spec2_1 = ugarchspec(mean.model=list(armaOrder=c(ARMA_p,ARMA_q), include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=NULL,
archex=FALSE),
variance.model=list(model="sGARCH", garchOrder=c(GARCH_q,GARCH_p)),
distribution.model="norm")
modelfit2_1 = ugarchfit(spec=spec2_1,
data=Return$SP500return,
out.sample=OutSample,
solver="hybrid",
solver.control=list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -887.9373 Pars: 0.00355699 -0.37053015 0.00002344 0.99899893
## Iter: 2 fn: -887.9373 Pars: 0.00355695 -0.37052415 0.00002344 0.99899893
## solnp--> Completed in 2 iterationsmodelfit2_1 ##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003557 0.001803 1.97230 0.048575
## archm -0.370524 0.311691 -1.18876 0.234536
## omega 0.000023 0.000008 2.97170 0.002962
## alpha1 0.998999 1.050647 0.95084 0.341685
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003557 0.005999 0.59288 0.55326
## archm -0.370524 1.079349 -0.34329 0.73138
## omega 0.000023 0.000029 0.81782 0.41346
## alpha1 0.998999 3.896546 0.25638 0.79766
##
## LogLikelihood : 887.9373
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0433
## Bayes -6.9871
## Shibata -7.0438
## Hannan-Quinn -7.0207
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.520 0.2176
## Lag[2*(p+q)+(p+q)-1][2] 1.604 0.3379
## Lag[4*(p+q)+(p+q)-1][5] 2.507 0.5043
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 4.028 0.04475
## Lag[2*(p+q)+(p+q)-1][2] 4.116 0.07016
## Lag[4*(p+q)+(p+q)-1][5] 5.312 0.12999
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 0.1738 0.500 2.000 0.6768
## ARCH Lag[4] 1.5439 1.397 1.611 0.5503
## ARCH Lag[6] 1.8065 2.222 1.500 0.7237
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.6591
## Individual Statistics:
## mu 0.03541
## archm 0.09270
## omega 0.39048
## alpha1 1.80548
##
## 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 0.290 0.77203
## Negative Sign Bias 1.712 0.08821 *
## Positive Sign Bias 2.012 0.04531 **
## Joint Effect 7.723 0.05209 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.25 0.1523
## 2 30 29.52 0.4383
## 3 40 44.94 0.2372
## 4 50 50.00 0.4336
##
##
## Elapsed time : 1.606733# 利用 spec2_1 的參數估計做為 spec2_2 的 starting value.
spec2_2 = ugarchspec(mean.model=list(armaOrder=c(ARMA_p,ARMA_q), include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=NULL,
archex=FALSE),
variance.model=list(model="gjrGARCH", garchOrder=c(GARCH_q,GARCH_p)),
start.pars=list(mu=0.003557, archm=-0.370524, omega=0.000023, alpha1=0.998999),
distribution.model="norm")
# start.pars
# List of staring parameters for the optimization routine.
# These are not usually required unless the optimization has problems converging.
modelfit2_2 = ugarchfit(spec=spec2_2,
data=Return$SP500return,
out.sample=OutSample,
#solver="hybird",
solver.control=list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -890.5601 Pars: 0.00234929 -0.14946106 0.00002696 0.99899759 -0.42970735
## Iter: 2 fn: -890.5601 Pars: 0.00234929 -0.14946106 0.00002696 0.99899759 -0.42970735
## solnp--> Completed in 2 iterationsmodelfit2_2##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002349 0.001058 2.22131 0.02633
## archm -0.149461 0.116709 -1.28063 0.20033
## omega 0.000027 0.000004 7.15347 0.00000
## alpha1 0.998998 0.615499 1.62307 0.10458
## gamma1 -0.429707 0.628267 -0.68396 0.49400
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002349 0.001407 1.66968 0.094984
## archm -0.149461 0.253561 -0.58945 0.555561
## omega 0.000027 0.000006 4.59788 0.000004
## alpha1 0.998998 0.783324 1.27533 0.202192
## gamma1 -0.429707 0.842029 -0.51032 0.609825
##
## LogLikelihood : 890.5601
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0563
## Bayes -6.9860
## Shibata -7.0570
## Hannan-Quinn -7.0280
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.236 0.2663
## Lag[2*(p+q)+(p+q)-1][2] 1.252 0.4232
## Lag[4*(p+q)+(p+q)-1][5] 2.158 0.5817
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.069 0.07979
## Lag[2*(p+q)+(p+q)-1][2] 3.070 0.13429
## Lag[4*(p+q)+(p+q)-1][5] 4.858 0.16480
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 0.002629 0.500 2.000 0.9591
## ARCH Lag[4] 2.174855 1.397 1.611 0.4028
## ARCH Lag[6] 2.465319 2.222 1.500 0.5784
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.4886
## Individual Statistics:
## mu 0.06202
## archm 0.09937
## omega 0.33431
## alpha1 0.43211
## gamma1 0.04642
##
## 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 1.152 0.25051
## Negative Sign Bias 1.756 0.08033 *
## Positive Sign Bias 1.419 0.15704
## Joint Effect 8.115 0.04369 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.98 0.19692
## 2 30 34.30 0.22849
## 3 40 49.08 0.12931
## 4 50 69.12 0.03063
##
##
## Elapsed time : 0.2535689ARMA_p = 0
ARMA_q = 0
GARCH_q = 0
GARCH_p = 1
spec3 = ugarchspec(mean.model=list(armaOrder=c(ARMA_p,ARMA_q), include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=NULL,
archex=FALSE),
variance.model=list(model="gjrGARCH", garchOrder=c(GARCH_q,GARCH_p)),
distribution.model="norm")
modelfit3 = ugarchfit(spec=spec3,
data=Return$SP500return,
out.sample=OutSample,
solver="hybrid",
solver.control=list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -896.0366 Pars: 4.373e-03 -5.071e-01 9.721e-11 9.982e-01
## Iter: 2 fn: -896.0366 Pars: 4.373e-03 -5.071e-01 9.721e-11 9.982e-01
## solnp--> Completed in 2 iterationsmodelfit3##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(0,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.004373 0.000115 37.929851 0.00000
## archm -0.507096 0.043345 -11.698976 0.00000
## omega 0.000000 0.000001 0.000152 0.99988
## beta1 0.998153 0.000363 2748.374634 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.004373 0.001231 3.553145 0.000381
## archm -0.507096 0.042095 -12.046607 0.000000
## omega 0.000000 0.000011 0.000009 0.999993
## beta1 0.998153 0.001509 661.306877 0.000000
##
## LogLikelihood : 896.0366
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.1079
## Bayes -7.0517
## Shibata -7.1084
## Hannan-Quinn -7.0853
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.484 0.2232
## Lag[2*(p+q)+(p+q)-1][2] 1.509 0.3588
## Lag[4*(p+q)+(p+q)-1][5] 1.971 0.6253
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.194 0.13855
## Lag[2*(p+q)+(p+q)-1][2] 4.623 0.05130
## Lag[4*(p+q)+(p+q)-1][5] 8.552 0.02146
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 4.782 0.500 2.000 0.02876
## ARCH Lag[4] 7.339 1.397 1.611 0.02330
## ARCH Lag[6] 8.649 2.222 1.500 0.02914
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 56.5969
## Individual Statistics:
## mu 0.06838
## archm 0.05339
## omega 37.10437
## beta1 0.21646
##
## 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 0.8021 0.42328
## Negative Sign Bias 1.9191 0.05612 *
## Positive Sign Bias 0.4714 0.63777
## Joint Effect 9.5113 0.02321 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 17.76 0.5382
## 2 30 35.02 0.2040
## 3 40 40.79 0.3916
## 4 50 53.18 0.3163
##
##
## Elapsed time : 0.478941ARMA(0,0)-gjrGARCH(1,1)-in-mean模型
合適的ARMA-gjrGARCH-in-mean模型
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 1
spec4 = ugarchspec(mean.model=list(armaOrder=c(ARMA_p,ARMA_q), include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=NULL,
archex=FALSE),
variance.model=list(model="gjrGARCH", garchOrder=c(GARCH_q,GARCH_p)),
distribution.model="norm")
modelfit4 = ugarchfit(spec=spec4,
data=Return$SP500return,
out.sample=OutSample,
solver="hybrid",
solver.control=list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -895.9343 Pars: -7.798e-04 2.507e-01 5.738e-13 1.141e-02 9.645e-01 4.319e-02
## Iter: 2 fn: -895.9346 Pars: -7.752e-04 2.501e-01 2.915e-13 1.141e-02 9.645e-01 4.319e-02
## solnp--> Completed in 2 iterationsmodelfit4##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000775 0.009385 -0.082604 0.93417
## archm 0.250111 1.381789 0.181005 0.85636
## omega 0.000000 0.000001 0.000000 1.00000
## alpha1 0.011406 0.105506 0.108108 0.91391
## beta1 0.964526 0.030918 31.196764 0.00000
## gamma1 0.043193 0.127350 0.339164 0.73449
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000775 0.055732 -0.013910 0.98890
## archm 0.250111 8.219461 0.030429 0.97572
## omega 0.000000 0.000004 0.000000 1.00000
## alpha1 0.011406 0.606068 0.018820 0.98498
## beta1 0.964526 0.166883 5.779645 0.00000
## gamma1 0.043193 0.767090 0.056307 0.95510
##
## LogLikelihood : 895.9346
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0911
## Bayes -7.0068
## Shibata -7.0922
## Hannan-Quinn -7.0572
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.764 0.05237
## Lag[2*(p+q)+(p+q)-1][2] 3.768 0.08703
## Lag[4*(p+q)+(p+q)-1][5] 4.143 0.23672
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.047 0.3063
## Lag[2*(p+q)+(p+q)-1][5] 5.233 0.1355
## Lag[4*(p+q)+(p+q)-1][9] 7.237 0.1801
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.249 0.500 2.000 0.03927
## ARCH Lag[5] 4.507 1.440 1.667 0.13344
## ARCH Lag[7] 5.669 2.315 1.543 0.16509
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 67.3356
## Individual Statistics:
## mu 0.08733
## archm 0.09145
## omega 22.95788
## alpha1 0.15562
## beta1 0.15308
## gamma1 0.12764
##
## 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 1.0462 0.2965
## Negative Sign Bias 0.8947 0.3718
## Positive Sign Bias 0.5244 0.6005
## Joint Effect 4.5390 0.2088
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 29.88 0.05339
## 2 30 44.10 0.03591
## 3 40 54.18 0.05379
## 4 50 60.35 0.12820
##
##
## Elapsed time : 0.66475011 - (b)
Rolling 過程(暫時省略，因為花時間很久)
#rollingx = ugarchroll(spec=spec4,
# data=Return$SP500return,
# n.ahead=1,
# forecast.length=OutSample,
# refit.every=1,
# refit.window="moving",
# solver="hybrid",
# solver.control=list(tol=1e-5,delta=1e-5,trace=1),
# calculate.VaR = TRUE,
# VaR.alpha = c(0.01,0.05),
# parallel=TRUE,
# parallel.control = list(pkg=c("snowfall"), cores=4),
# keep.coef=TRUE)
#rollingx = resume(rollingx, solver="gosolnp")
#convergence(rollingx)
load("c:/temp/rollingx.RData")
rollingx@model$coef[[1]]$coef## Estimate Std. Error t value Pr(>|t|)
## mu -7.752388e-04 5.573245e-02 -1.391001e-02 9.889018e-01
## archm 2.501112e-01 8.219461e+00 3.042915e-02 9.757248e-01
## omega 2.915273e-13 4.162322e-06 7.003959e-08 9.999999e-01
## alpha1 1.140600e-02 6.060677e-01 1.881968e-02 9.849850e-01
## beta1 9.645262e-01 1.668833e-01 5.779645e+00 7.485849e-09
## gamma1 4.319263e-02 7.670900e-01 5.630713e-02 9.550971e-01Return_b = Return
Return_b$criteria = NA
Return_b$criteria = 0.05
Return_b$model_archm = NA
for (i in WindowSize:(DataSize-1)){
Return_b$model_archm[i] = rollingx@model$coef[[i+1-WindowSize]]$coef[2,1]
}
Return_b$model_archm_pvalue = NA
for (i in WindowSize:(DataSize-1)){
Return_b$model_archm_pvalue[i] = rollingx@model$coef[[i+1-WindowSize]]$coef[2,4]
}
Return_b$model_gamma = NA
for (i in WindowSize:(DataSize-1)){
Return_b$model_gamma[i] = rollingx@model$coef[[i+1-WindowSize]]$coef[6,1]
}
Return_b$model_gamma_pvalue = NA
for (i in WindowSize:(DataSize-1)){
Return_b$model_gamma_pvalue[i] = rollingx@model$coef[[i+1-WindowSize]]$coef[6,4]
}
Return_b$ForecastMu[(WindowSize+1):DataSize]=rollingx@forecast$density$Mu
Return_b$ForecastSigma[(WindowSize+1):DataSize]=rollingx@forecast$density$Sigma
Return_b_1 = data.frame(date = index(Return_b), coredata(Return_b))是否存在風險溢酬?
從「風險溢酬對SP500報酬的影響 與 顯著性(rolling)」圖可以觀察，從2014年初到2015年底的參數估計的p-value幾乎都是高於0.05，意味著風險溢酬對SP500報酬率不顯著影響。由於第一筆估計參數是由前251筆資料估計出來的，我們將推論的時間往前推251天，從2013年初到2015年底對S&P500不存在顯著的風險溢酬影響。
plot1_1 = ggplot(Return_b_1, aes(x=date, y=model_archm)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+ ggtitle("風險溢酬對SP500報酬的影響 與 顯著性_rolling")+
theme_economist() + scale_colour_economist()
plot1_2 = ggplot(Return_b_1, aes(x=date, y=model_archm_pvalue)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
geom_line(aes(y=criteria),colour="red")+
theme_economist(stata=TRUE) + scale_colour_economist(stata=TRUE)
library(gridExtra)
grid.arrange(plot1_1,plot1_2,nrow=2,ncol=1)
是否存在波動不對稱?
從「波動不對稱性對SP500報酬的影響 與 顯著性(rolling)」圖可以觀察，從2014年初到2015年8月底的參數估計的p-value幾乎都是高於0.05，意味著波動不對稱對SP500報酬率不顯著；但從2015年9月初到2015年底的參數估計的p-value幾乎都是低於0.05，意味著波動不對稱對SP500報酬率顯著。由於第一筆估計參數是由前251筆資料估計出來的，我們將推論的時間往前推251天，從2013年初到2014年8月底對S&P500不存在顯著的波動不對稱影響，但從2014年9月初到2015年底對S&P500存在顯著的波動不對稱影響，且影響的程度大約介於0.2與0.5之間，意味著波動不對稱增加1單位對S&P500報酬平均增加0.2到0.5單位之間。
plot2_1 = ggplot(Return_b_1, aes(x=date, y=model_gamma)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+ ggtitle("波動不對稱性對SP500報酬的影響 與 顯著性_rolling")+
theme_economist() + scale_colour_economist()
plot2_2 = ggplot(Return_b_1, aes(x=date, y=model_gamma_pvalue)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
geom_line(aes(y=criteria),colour="red")+
theme_economist(stata=TRUE) + scale_colour_economist(stata=TRUE)
library(gridExtra)
grid.arrange(plot2_1,plot2_2,nrow=2,ncol=1)
S&P500報酬 與 ForecastMu, ForecastSigma關係_rolling
從平均日報酬預測，在2015年7/8月以前日報酬是較平穩的，大約介於-0.001與0.002之間，在2015年7/8月以後日報酬先是較不平穩之後慢慢平緩，大約介於-0.003與0.002之間； 從平均波動預測，在2015年7/8月以前波動是較平穩的，大約介於0.005與0.015之間，在2015年7/8月以後波動先是較不平穩之後慢慢平緩，大約介於0.005與0.03之間。 同時察覺平均日報酬與平均波動預測來看，平均日報酬與平均波動在2015年7/8月以前較穩定的，但在之後先是不穩定再慢慢穩定，意味著其實日報酬變動幅度會受波動影響且是正向影響，波動越大日報酬變動幅度也越大。
plot3_1 = ggplot(Return_b_1, aes(x=date, y=ForecastMu)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
ggtitle("S&P500報酬 與 ForecastMu關係_rolling")
plot3_2 = ggplot(Return_b_1, aes(x=date, y=ForecastSigma)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
ggtitle("S&P500報酬 與 ForecastSigma 關係_rolling")
library(gridExtra)
grid.arrange(plot3_1,plot3_2,nrow=2,ncol=1)
1 - (c)
Return= TTR::ROC(Ad(Asset),type="continuous")
Return_c = cbind(Return, Asset[,5])
colnames(Return_c) = c("SP500return", "Volume")
Return_c$Volume_lag1 = lag(Return_c$Volume,1)
Return_c = na.omit(Return_c)
Regressor.mean = as.matrix(Return_c$Volume_lag1)
Regressor.variance = as.matrix(cbind((Return_c$Volume_lag1)^2))決定ARMA的order
關於決定有解釋變數的ARMA的order 我用的(b)小題的結論ARMA(0,0) ，再進一步確定ARMA的標準化殘差是否通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差為White noice。最後決定ARMA(0,0)。
ARMA_p = 0
ARMA_q = 0
spec5 = arfimaspec(mean.model = list(armaOrder = c(ARMA_p, ARMA_q), include.mean = TRUE,
external.regressors=Regressor.mean),
distribution.model = "norm")
modelfit5 = arfimafit(spec = spec5,
data=Return_c$SP500,
out.sample =OutSample,
solver = "hybrid",
solver.control = list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -896.4561 Pars: -4.564e-04 4.132e-13 6.820e-03
## solnp--> Completed in 1 iterationsmodelfit5##
## *----------------------------------*
## * ARFIMA Model Fit *
## *----------------------------------*
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000456 0.000430 -1.060304 0.28901
## mxreg1 0.000000 0.000000 0.000006 0.99999
## sigma 0.006820 0.000306 22.320396 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000456 0.000382 -1.193294 0.23275
## mxreg1 0.000000 0.000000 0.003457 0.99724
## sigma 0.006820 0.000496 13.757448 0.00000
##
## LogLikelihood : 896.4561
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.1192
## Bayes -7.0770
## Shibata -7.1195
## Hannan-Quinn -7.1022
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.902 0.1678
## Lag[2*(p+q)+(p+q)-1][2] 1.909 0.2783
## Lag[4*(p+q)+(p+q)-1][5] 2.354 0.5375
##
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.383 0.122671
## Lag[2*(p+q)+(p+q)-1][2] 6.074 0.021025
## Lag[4*(p+q)+(p+q)-1][5] 10.786 0.005775
##
##
## ARCH LM Tests
## ------------------------------------
## Statistic DoF P-Value
## ARCH Lag[2] 8.836 2 0.01206
## ARCH Lag[5] 10.722 5 0.05717
## ARCH Lag[10] 12.037 10 0.28262
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.341
## Individual Statistics:
## mu 0.03336
## mxreg1 NaN
## sigma 0.19440
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
##
## Elapsed time : 0.3082778決定gjrGARCH-in-mean的order
進一步決定有解釋變數ARMA-gjrGARCH-in-mean的order，我以ARMA(0, 0)為基礎再加上gjrGARCH-in-mean(q, p) 的order，採用方法為窮舉法，gjrGARCH-in-mean (1,0) => gjrGARCH-in-mean (0, 1) =>gjrGARCH-in-mean(1,1) … 直到標準化殘差的平方通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差平方為White noice。然而在此遇到無法收斂問題的錯誤訊號，故需要面對此問題。
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 0
spec6_1 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="sGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
#modelfit6_1 = ugarchfit(spec=spec6_1, data=Return_c$SP500,
# out.sample=OutSample,
# solver="hybird",
# solver.control=list(tol=1e-5, delta=1e-5, trace=1)
#)
#modelfit6_1
spec6_2 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
start.pars=list(mu=0.005487, archm=0.161375,mxreg1 = 0.000000, omega=0.000000, alpha1=0.050001),
distribution.model="norm")
#modelfit6_2 = ugarchfit(spec=spec6_2, data=Return_c$SP500,
# out.sample=OutSample,
# solver="hybird",
# solver.control=list(tol=1e-5, delta=1e-5, trace=1)
#)
#modelfit6_2解決不收斂問題
由於不收斂 => 正規化Volume ，同除1000000000000
Return= TTR::ROC(Ad(Asset),type="continuous")
Return_c = cbind(Return, Asset[,5])
colnames(Return_c) = c("SP500return", "Volume")
Return_c$Volume_lag1 = lag(Return_c$Volume,1)
Return_c$NewVolume = Return_c$Volume/1000000000000
Return_c$NewVolume_lag1 = lag(Return_c$NewVolume,1)
Return_c = na.omit(Return_c)
Regressor.mean = as.matrix(Return_c$NewVolume_lag1)
Regressor.variance = as.matrix(cbind((Return_c$NewVolume_lag1)^2))重新決定ARMA的order
關於重新決定有解釋變數的ARMA的order 我用的(b)小題的結論ARMA(0,0) ，再進一步確定ARMA的標準化殘差是否通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差為White noice。最後決定ARMA(0,0)。
ARMA_p = 0
ARMA_q = 0
spec7 = arfimaspec(mean.model = list(armaOrder = c(ARMA_p, ARMA_q), include.mean = TRUE,
external.regressors=Regressor.mean),
distribution.model = "norm")
modelfit7 = arfimafit(spec = spec7,
data=Return_c$SP500,
out.sample =OutSample,
solver = "hybrid",
solver.control = list(tol=1e-5, delta=1e-5, trace=1)
)##
## Iter: 1 fn: -896.4561 Pars: -0.0004584 0.4131562 0.0068195
## solnp--> Completed in 1 iterationsmodelfit7##
## *----------------------------------*
## * ARFIMA Model Fit *
## *----------------------------------*
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000458 0.002726 -0.16814 0.86647
## mxreg1 0.413156 0.800412 0.51618 0.60573
## sigma 0.006820 0.000306 22.32087 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000458 0.002643 -0.17341 0.86233
## mxreg1 0.413156 0.772629 0.53474 0.59283
## sigma 0.006820 0.000496 13.75766 0.00000
##
## LogLikelihood : 896.4561
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.1192
## Bayes -7.0770
## Shibata -7.1195
## Hannan-Quinn -7.1022
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.902 0.1678
## Lag[2*(p+q)+(p+q)-1][2] 1.909 0.2783
## Lag[4*(p+q)+(p+q)-1][5] 2.354 0.5375
##
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.383 0.122701
## Lag[2*(p+q)+(p+q)-1][2] 6.073 0.021039
## Lag[4*(p+q)+(p+q)-1][5] 10.785 0.005778
##
##
## ARCH LM Tests
## ------------------------------------
## Statistic DoF P-Value
## ARCH Lag[2] 8.834 2 0.01207
## ARCH Lag[5] 10.722 5 0.05718
## ARCH Lag[10] 12.039 10 0.28247
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.3414
## Individual Statistics:
## mu 0.03284
## mxreg1 0.04096
## sigma 0.19447
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
##
## Elapsed time : 0.008004904重新決定gjrGARCH-in-mean的order
進一步決定ARMA-gjrGARCH-in-mean的order，我以ARMA(0, 0)為基礎再加上gjrGARCH-in-mean(q, p) 的order，採用方法為窮舉法，gjrGARCH-in-mean (1,0) => gjrGARCH-in-mean (0, 1) =>gjrGARCH-in-mean(1,1) … 直到標準化殘差的平方通過Ljung- Box檢定，我以p-value > 0.05為標準，意味著不拒絕Ho，則視為殘差平方為White noice。雖然gjrGARCH-in-mean(1,0)通過Ljung- Box檢定，但因為不符合第(b)小題探討風險溢酬，於是往下做了幾個模型，最後決定通過Ljung- Box檢定的gjrGARCH-in-mean(1,1)。
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 0
spec8_1 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="sGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
modelfit8_1 = ugarchfit(spec=spec8_1, data=Return_c$SP500,
out.sample=OutSample,
solver="hybrid",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Iter: 1 fn: -893.2507 Pars: 0.00287760321 -1.20149266349 1.59093714356 0.00003282769 0.17567117421 0.00000004033
## Iter: 2 fn: -893.2507 Pars: 0.00287760321 -1.20149266349 1.59093714356 0.00003282769 0.17567117421 0.00000004033
## solnp--> Completed in 2 iterationsmodelfit8_1##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002878 0.003203 0.89846 0.368940
## archm -1.201493 0.559058 -2.14914 0.031623
## mxreg1 1.590937 0.259540 6.12985 0.000000
## omega 0.000033 0.000006 5.63068 0.000000
## alpha1 0.175671 0.105981 1.65757 0.097405
## vxreg1 0.000000 0.111496 0.00000 1.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002878 0.00491 0.58606 0.55783
## archm -1.201493 1.14268 -1.05147 0.29304
## mxreg1 1.590937 1.20043 1.32530 0.18507
## omega 0.000033 0.00001 3.23413 0.00122
## alpha1 0.175671 0.16313 1.07686 0.28154
## vxreg1 0.000000 0.99175 0.00000 1.00000
##
## LogLikelihood : 893.2507
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0697
## Bayes -6.9855
## Shibata -7.0708
## Hannan-Quinn -7.0358
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.621 0.2029
## Lag[2*(p+q)+(p+q)-1][2] 1.707 0.3165
## Lag[4*(p+q)+(p+q)-1][5] 2.179 0.5768
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0371 0.8473
## Lag[2*(p+q)+(p+q)-1][2] 1.0782 0.4736
## Lag[4*(p+q)+(p+q)-1][5] 3.9993 0.2542
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 2.049 0.500 2.000 0.1523
## ARCH Lag[4] 4.724 1.397 1.611 0.1027
## ARCH Lag[6] 5.341 2.222 1.500 0.1638
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.5804
## Individual Statistics:
## mu 0.05290
## archm 0.08050
## mxreg1 0.07161
## omega 0.57633
## alpha1 0.17902
## vxreg1 0.96266
##
## 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 0.2459 0.8059
## Negative Sign Bias 0.5662 0.5718
## Positive Sign Bias 0.3870 0.6991
## Joint Effect 1.9015 0.5931
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.30 0.1849
## 2 30 32.39 0.3032
## 3 40 40.79 0.3916
## 4 50 41.63 0.7634
##
##
## Elapsed time : 0.4839029spec8_2 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
start.pars=list(mu=0.035101, archm=-1.471415,mxreg1 = -2.337372, omega=0.000217, alpha1=0.999951, vxreg1=0.00000),
distribution.model="norm")
modelfit8_2 = ugarchfit(spec=spec8_2, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 2.161e-03
## par2 2.573e-01
## par3 -1.226e+00
## par4 2.336e-05
## par5 7.911e-02
## par6 1.107e-01
## par7 3.860e-02
## objf -8.712e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -898.7731 Pars: -0.00083389 0.43502920 -0.26378189 0.00003083 0.08133780 0.36530652 0.03955261
## Iter: 2 fn: -898.7731 Pars: -0.00083389 0.43502920 -0.26378189 0.00003083 0.08133780 0.36530652 0.03955261
## solnp--> Completed in 2 iterationsmodelfit8_2##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000834 0.001171 -0.712153 0.476370
## archm 0.435029 0.250202 1.738711 0.082086
## mxreg1 -0.263782 0.736006 -0.358396 0.720047
## omega 0.000031 0.000007 4.135648 0.000035
## alpha1 0.081338 0.080755 1.007213 0.313832
## gamma1 0.365307 0.166492 2.194143 0.028225
## vxreg1 0.039553 0.648490 0.060992 0.951366
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000834 0.004104 -0.203175 0.838998
## archm 0.435029 0.526292 0.826594 0.408467
## mxreg1 -0.263782 0.802978 -0.328505 0.742530
## omega 0.000031 0.000011 2.717629 0.006575
## alpha1 0.081338 0.137426 0.591868 0.553939
## gamma1 0.365307 0.214219 1.705299 0.088139
## vxreg1 0.039553 0.805081 0.049129 0.960817
##
## LogLikelihood : 898.7731
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.1058
## Bayes -7.0074
## Shibata -7.1073
## Hannan-Quinn -7.0662
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.08915 0.7653
## Lag[2*(p+q)+(p+q)-1][2] 0.09610 0.9224
## Lag[4*(p+q)+(p+q)-1][5] 0.68252 0.9262
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.415 0.12014
## Lag[2*(p+q)+(p+q)-1][2] 4.676 0.04965
## Lag[4*(p+q)+(p+q)-1][5] 8.705 0.01964
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 4.450 0.500 2.000 0.03490
## ARCH Lag[4] 7.558 1.397 1.611 0.02054
## ARCH Lag[6] 8.248 2.222 1.500 0.03624
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.9547
## Individual Statistics:
## mu 0.03430
## archm 0.04816
## mxreg1 0.05558
## omega 0.86328
## alpha1 0.07476
## gamma1 0.06693
## vxreg1 0.93330
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7003 0.4844
## Negative Sign Bias 1.1123 0.2671
## Positive Sign Bias 0.3661 0.7146
## Joint Effect 1.7068 0.6354
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.62 0.17352
## 2 30 44.58 0.03232
## 3 40 54.18 0.05379
## 4 50 70.71 0.02281
##
##
## Elapsed time : 0.459332ARMA_p = 0
ARMA_q = 0
GARCH_q = 0
GARCH_p = 1
spec9 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
modelfit9 = ugarchfit(spec=spec9, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 -4.241e-03
## par2 4.127e-01
## par3 4.554e-01
## par4 4.900e-08
## par5 9.935e-01
## par6 2.405e-08
## objf -8.684e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -896.4210 Pars: -0.00316725967 0.34644606097 0.53731474209 0.00000001293 0.99863889764 0.00000002405
## Iter: 2 fn: -896.5257 Pars: -0.0029425645011 0.3486231456255 0.4417949739786 0.0000000004333 0.9990372517602 0.0000000240473
## Iter: 3 fn: -896.5270 Pars: -0.0029859986814 0.3494561409914 0.4572460817257 0.0000000001714 0.9990449048488 0.0000000240473
## solnp--> Completed in 3 iterationsmodelfit9##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(0,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002986 0.002826 -1.0567e+00 0.290655
## archm 0.349456 0.148139 2.3590e+00 0.018326
## mxreg1 0.457246 0.790131 5.7870e-01 0.562794
## omega 0.000000 0.000001 2.5900e-04 0.999793
## beta1 0.999045 0.000001 1.2137e+06 0.000000
## vxreg1 0.000000 0.000020 1.2020e-03 0.999041
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002986 0.003776 -7.9087e-01 0.42902
## archm 0.349456 0.525107 6.6549e-01 0.50573
## mxreg1 0.457246 0.774706 5.9022e-01 0.55504
## omega 0.000000 0.000012 1.5000e-05 0.99999
## beta1 0.999045 0.000008 1.1849e+05 0.00000
## vxreg1 0.000000 0.000062 3.8600e-04 0.99969
##
## LogLikelihood : 896.527
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0958
## Bayes -7.0116
## Shibata -7.0969
## Hannan-Quinn -7.0619
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.557 0.2121
## Lag[2*(p+q)+(p+q)-1][2] 1.564 0.3465
## Lag[4*(p+q)+(p+q)-1][5] 2.012 0.6157
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.839 0.17502
## Lag[2*(p+q)+(p+q)-1][2] 5.173 0.03657
## Lag[4*(p+q)+(p+q)-1][5] 9.519 0.01222
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 6.561 0.500 2.000 0.010421
## ARCH Lag[4] 9.048 1.397 1.611 0.008651
## ARCH Lag[6] 10.169 2.222 1.500 0.012538
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 58.3619
## Individual Statistics:
## mu 0.09205
## archm 0.04236
## mxreg1 0.10917
## omega 37.42826
## beta1 0.18113
## vxreg1 0.18041
##
## 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 0.5338 0.59393
## Negative Sign Bias 1.8558 0.06467 *
## Positive Sign Bias 0.3297 0.74190
## Joint Effect 7.4033 0.06010 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 20.16 0.3853
## 2 30 27.84 0.5262
## 3 40 40.47 0.4051
## 4 50 60.35 0.1282
##
##
## Elapsed time : 0.853152ARMA(0,0)-gjrGARCH(1,1)-in-mean模型
通過Ljung Box 檢定
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 1
spec10 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
modelfit10 = ugarchfit(spec=spec10, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 -1.145e-03
## par2 1.795e-01
## par3 -1.358e-02
## par4 1.285e-07
## par5 8.145e-02
## par6 8.880e-01
## par7 1.233e-01
## par8 5.839e-09
## objf -8.931e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -895.9041 Pars: -0.001759486585 0.163033275009 0.472861060377 0.000000128374 0.080819830788 0.886318749442 0.086086642955 0.000000005839
## Iter: 2 fn: -895.9041 Pars: -0.001759486585 0.163033275009 0.472861060377 0.000000128374 0.080819830788 0.886318749442 0.086086642955 0.000000005839
## solnp--> Completed in 2 iterationsmodelfit10##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001759 0.003066 -0.57382 0.56609
## archm 0.163033 0.279978 0.58231 0.56036
## mxreg1 0.472861 0.746895 0.63310 0.52667
## omega 0.000000 0.000000 0.27134 0.78613
## alpha1 0.080820 0.140497 0.57524 0.56513
## beta1 0.886319 0.022559 39.28895 0.00000
## gamma1 0.086087 0.173216 0.49699 0.61920
## vxreg1 0.000000 0.109292 0.00000 1.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001759 0.010368 -0.169700 0.86525
## archm 0.163033 1.180984 0.138049 0.89020
## mxreg1 0.472861 0.955682 0.494789 0.62075
## omega 0.000000 0.000003 0.045716 0.96354
## alpha1 0.080820 0.482539 0.167489 0.86699
## beta1 0.886319 0.112937 7.847907 0.00000
## gamma1 0.086087 0.765493 0.112459 0.91046
## vxreg1 0.000000 0.376402 0.000000 1.00000
##
## LogLikelihood : 895.9041
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0749
## Bayes -6.9626
## Shibata -7.0769
## Hannan-Quinn -7.0297
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.231 0.07224
## Lag[2*(p+q)+(p+q)-1][2] 3.278 0.11804
## Lag[4*(p+q)+(p+q)-1][5] 3.575 0.31202
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06199 0.8034
## Lag[2*(p+q)+(p+q)-1][5] 2.53803 0.4978
## Lag[4*(p+q)+(p+q)-1][9] 5.22872 0.3966
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.186 0.500 2.000 0.07429
## ARCH Lag[5] 4.181 1.440 1.667 0.15831
## ARCH Lag[7] 6.306 2.315 1.543 0.12199
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 63.9618
## Individual Statistics:
## mu 0.04005
## archm 0.05571
## mxreg1 0.04366
## omega 6.62034
## alpha1 0.11136
## beta1 0.06196
## gamma1 0.03074
## vxreg1 0.77954
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.51794 0.1303
## Negative Sign Bias 0.48233 0.6300
## Positive Sign Bias 0.05756 0.9541
## Joint Effect 4.14892 0.2458
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.02 0.2363
## 2 30 27.61 0.5391
## 3 40 40.47 0.4051
## 4 50 50.39 0.4181
##
##
## Elapsed time : 0.863651#BIC -6.9637模型精簡 - BIC 判斷
ARMA(0,0)-gjrGARCH(1,1)-in-mean模型通過Ljung Box 檢定，但因為mean.model與variance.model的解釋變數呈現不顯著，在同時滿足Ljung- Box 與題意要求下，運用以下模型精簡手法找到最適合模型，模型精簡方式為mean與variance 模型同時都有解釋變數，或只有variance.model有解釋變數，或只有mean.model有解釋變數，故再比較BIC最小，決定模型精簡的樣貌。最後決定ARMA(0,0)-gjrGARCH(1,1)-in-mean 只有mean.model有解釋變數。
模型精簡1
BIC -6.9637
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 1
spec10 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
modelfit10 = ugarchfit(spec=spec10, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 -7.771e-04
## par2 1.003e-01
## par3 4.892e-01
## par4 8.197e-08
## par5 1.604e-01
## par6 8.772e-01
## par7 -2.149e-02
## par8 2.616e-08
## objf -8.935e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -895.8453 Pars: -0.00297340691 0.33348708298 0.48259328814 0.00000003887 0.05825727578 0.92552029419 0.04245944850 0.00000002616
## Iter: 2 fn: -895.8453 Pars: -0.00297340691 0.33348708298 0.48259328814 0.00000003887 0.05825727578 0.92552029419 0.04245944850 0.00000002616
## solnp--> Completed in 2 iterationsmodelfit10##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002973 0.000812 -3.660692 0.000252
## archm 0.333487 0.320520 1.040457 0.298128
## mxreg1 0.482593 0.789599 0.611188 0.541075
## omega 0.000000 0.000000 0.095612 0.923828
## alpha1 0.058257 0.115024 0.506480 0.612520
## beta1 0.925520 0.012864 71.946398 0.000000
## gamma1 0.042459 0.129571 0.327693 0.743144
## vxreg1 0.000000 0.097605 0.000000 1.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002973 0.006781 -0.438498 0.66103
## archm 0.333487 0.900136 0.370485 0.71102
## mxreg1 0.482593 0.812761 0.593770 0.55267
## omega 0.000000 0.000001 0.028042 0.97763
## alpha1 0.058257 0.259638 0.224379 0.82246
## beta1 0.925520 0.110710 8.359858 0.00000
## gamma1 0.042459 0.296271 0.143313 0.88604
## vxreg1 0.000000 0.250641 0.000000 1.00000
##
## LogLikelihood : 895.8453
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0745
## Bayes -6.9621
## Shibata -7.0764
## Hannan-Quinn -7.0292
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.436 0.06378
## Lag[2*(p+q)+(p+q)-1][2] 3.456 0.10565
## Lag[4*(p+q)+(p+q)-1][5] 3.818 0.27771
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2042 0.6513
## Lag[2*(p+q)+(p+q)-1][5] 3.2708 0.3598
## Lag[4*(p+q)+(p+q)-1][9] 5.8040 0.3214
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.949 0.500 2.000 0.0469
## ARCH Lag[5] 4.578 1.440 1.667 0.1285
## ARCH Lag[7] 6.417 2.315 1.543 0.1156
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 69.7929
## Individual Statistics:
## mu 0.09628
## archm 0.12535
## mxreg1 0.10542
## omega 14.60498
## alpha1 0.07295
## beta1 0.06535
## gamma1 0.05838
## vxreg1 0.37429
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.54363 0.1240
## Negative Sign Bias 0.04838 0.9614
## Positive Sign Bias 0.39528 0.6930
## Joint Effect 4.56056 0.2070
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.89 0.1332
## 2 30 38.36 0.1145
## 3 40 46.53 0.1901
## 4 50 51.19 0.3877
##
##
## Elapsed time : 0.91465模型精簡2
BIC -6.9827
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 1
spec11 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=NULL,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=Regressor.variance,
variance.targeting=FALSE),
distribution.model="norm")
modelfit11 = ugarchfit(spec=spec11, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 6.153e-04
## par2 9.813e-02
## par3 7.049e-08
## par4 1.928e-02
## par5 9.046e-01
## par6 1.429e-01
## par7 2.928e-08
## objf -8.949e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -896.0140 Pars: -0.00059207450 0.22078868107 0.00000004996 0.02015631278 0.94794179118 0.06039498362 0.00000002928
## Iter: 2 fn: -896.0140 Pars: -0.00059207450 0.22078868107 0.00000004996 0.02015631278 0.94794179118 0.06039498362 0.00000002928
## solnp--> Completed in 2 iterationsmodelfit11##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000592 0.001742 -0.339960 0.73389
## archm 0.220789 0.250551 0.881213 0.37820
## omega 0.000000 0.000000 0.132065 0.89493
## alpha1 0.020156 0.053620 0.375912 0.70698
## beta1 0.947942 0.011086 85.511205 0.00000
## gamma1 0.060395 0.063111 0.956959 0.33859
## vxreg1 0.000000 0.014820 0.000002 1.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000592 0.006898 -0.085827 0.93160
## archm 0.220789 1.000087 0.220769 0.82527
## omega 0.000000 0.000002 0.032421 0.97414
## alpha1 0.020156 0.096715 0.208409 0.83491
## beta1 0.947942 0.109656 8.644668 0.00000
## gamma1 0.060395 0.279321 0.216221 0.82882
## vxreg1 0.000000 0.143015 0.000000 1.00000
##
## LogLikelihood : 896.014
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0838
## Bayes -6.9855
## Shibata -7.0853
## Hannan-Quinn -7.0442
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.877 0.04896
## Lag[2*(p+q)+(p+q)-1][2] 3.886 0.08092
## Lag[4*(p+q)+(p+q)-1][5] 4.221 0.22779
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7269 0.3939
## Lag[2*(p+q)+(p+q)-1][5] 4.1716 0.2334
## Lag[4*(p+q)+(p+q)-1][9] 6.3740 0.2575
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.271 0.500 2.000 0.03877
## ARCH Lag[5] 4.614 1.440 1.667 0.12609
## ARCH Lag[7] 6.085 2.315 1.543 0.13563
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 69.5811
## Individual Statistics:
## mu 0.08615
## archm 0.09290
## omega 16.92492
## alpha1 0.12720
## beta1 0.12369
## gamma1 0.08703
## vxreg1 0.45419
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.3105 0.1913
## Negative Sign Bias 0.4619 0.6446
## Positive Sign Bias 0.5167 0.6058
## Joint Effect 4.2945 0.2314
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.50 0.2160
## 2 30 34.54 0.2201
## 3 40 43.66 0.2800
## 4 50 49.60 0.4493
##
##
## Elapsed time : 0.8491809模型精簡3
BIC -6.9876
ARMA_p = 0
ARMA_q = 0
GARCH_q = 1
GARCH_p = 1
spec12 = ugarchspec(mean.model = list(armaOrder=c(ARMA_p,ARMA_q),
include.mean=TRUE,
archm=TRUE,
archpow=1,
external.regressors=Regressor.mean,
archex=FALSE),
variance.model=list(model="gjrGARCH",
garchOrder=c(GARCH_q ,GARCH_p),
submodel=NULL,
external.regressors=NULL,
variance.targeting=FALSE),
distribution.model="norm")
modelfit12 = ugarchfit(spec=spec12, data=Return_c$SP500,
out.sample=OutSample,
solver="gosolnp",
solver.control=list(tol=1e-5, delta=1e-5, trace=1))##
## Calculating Random Initialization Parameters...ok!
##
## Excluding Inequality Violations...
##
## ...Excluded 57/500 Random Sequences
##
## Evaluating Objective Function with Random Sampled Parameters...ok!
##
## Sorting and Choosing Best Candidates for starting Solver...ok!
##
## Starting Parameters and Starting Objective Function:
## [,1]
## par1 -1.824e-03
## par2 3.528e-01
## par3 6.703e-02
## par4 3.806e-08
## par5 2.284e-02
## par6 9.362e-01
## par7 6.782e-02
## objf -8.942e+02
##
## gosolnp-->Starting Solver
##
## Iter: 1 fn: -896.2816 Pars: -0.0022510703276 0.1570532847511 0.6287284415002 0.0000000008651 0.0074647784176 0.9678786031071 0.0437607008518
## Iter: 2 fn: -896.2870 Pars: -0.0022462168936 0.1550614875298 0.6307486993027 0.0000000001592 0.0071698124177 0.9682563524511 0.0436172780079
## solnp--> Completed in 2 iterationsmodelfit12##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002246 0.004032 -0.557081 0.57747
## archm 0.155061 0.853333 0.181713 0.85581
## mxreg1 0.630749 0.676445 0.932446 0.35111
## omega 0.000000 0.000001 0.000249 0.99980
## alpha1 0.007170 0.064478 0.111197 0.91146
## beta1 0.968256 0.012509 77.405426 0.00000
## gamma1 0.043617 0.094807 0.460065 0.64547
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002246 0.026419 -0.085021 0.93224
## archm 0.155061 4.537510 0.034173 0.97274
## mxreg1 0.630749 1.536830 0.410422 0.68150
## omega 0.000000 0.000004 0.000042 0.99997
## alpha1 0.007170 0.306262 0.023411 0.98132
## beta1 0.968256 0.041814 23.156395 0.00000
## gamma1 0.043617 0.539731 0.080813 0.93559
##
## LogLikelihood : 896.287
##
## Information Criteria
## ------------------------------------
##
## Akaike -7.0860
## Bayes -6.9876
## Shibata -7.0875
## Hannan-Quinn -7.0464
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.191 0.07402
## Lag[2*(p+q)+(p+q)-1][2] 3.210 0.12312
## Lag[4*(p+q)+(p+q)-1][5] 3.588 0.31011
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7418 0.3891
## Lag[2*(p+q)+(p+q)-1][5] 5.1112 0.1445
## Lag[4*(p+q)+(p+q)-1][9] 7.2571 0.1786
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.137 0.500 2.000 0.04195
## ARCH Lag[5] 4.437 1.440 1.667 0.13839
## ARCH Lag[7] 5.754 2.315 1.543 0.15865
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 65.851
## Individual Statistics:
## mu 0.06374
## archm 0.07071
## mxreg1 0.06227
## omega 23.57037
## alpha1 0.13992
## beta1 0.14235
## gamma1 0.12448
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.8061 0.4210
## Negative Sign Bias 0.9438 0.3462
## Positive Sign Bias 0.3210 0.7485
## Joint Effect 3.8669 0.2762
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 26.05 0.128752
## 2 30 51.51 0.006179
## 3 40 52.59 0.071791
## 4 50 59.16 0.151704
##
##
## Elapsed time : 1.047289根據BIC最小化選擇模型精簡3
最後決定ARMA(0,0)-gjrGARCH(1,1)-in-mean 只有mean.model有解釋變數
Rolling 過程(暫時省略，因為花時間很久)
#rollingx1 = ugarchroll(spec=spec12,
# data=Return_c$SP500return,
# n.ahead=1,
# forecast.length=OutSample,
# refit.every=1,
# refit.window="moving",
# solver="hybrid",
# solver.control=list(tol=1e-5,delta=1e-5,trace=1),
# calculate.VaR = TRUE,
# VaR.alpha = c(0.01,0.05),
# parallel=TRUE,
# parallel.control = list(pkg=c("snowfall"), cores=4),
# keep.coef=TRUE)
#rollingx1 = resume(rollingx1, solver="gosolnp")
#convergence(rollingx1)
#save(rollingx1, file="c:/temp/rollingx1.RData")
load("c:/temp/rollingx1.RData")
#str(rollingx1)
rollingx1@model$coef[[1]]## $index
## [1] "2013-12-31"
##
## $coef
## Estimate Std. Error t value Pr(>|t|)
## mu -2.252779e-03 2.621327e-02 -8.594041e-02 0.9315138
## archm 1.573776e-01 4.514593e+00 3.485976e-02 0.9721916
## mxreg1 6.299170e-01 1.562441e+00 4.031620e-01 0.6868291
## omega 9.469068e-11 3.752492e-06 2.523408e-05 0.9999799
## alpha1 7.389119e-03 3.084564e-01 2.395515e-02 0.9808884
## beta1 9.679566e-01 4.120387e-02 2.349189e+01 0.0000000
## gamma1 4.366129e-02 5.336000e-01 8.182400e-02 0.9347867Return= na.omit(Return)
Return_c = Return
Return_c$criteria = NA
Return_c$criteria = 0.05
Return_c$model_archm = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_archm[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[2,1]
}
Return_c$model_archm_pvalue = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_archm_pvalue[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[2,4]
}
Return_c$model_gamma = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_gamma[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[7,1]
}
Return_c$model_gamma_pvalue = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_gamma_pvalue[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[7,4]
}
Return_c$model_mxreg1 = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_mxreg1[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[3,1]
}
Return_c$model_mxreg1_pvalue = NA
for (i in WindowSize:(DataSize-1)){
Return_c$model_mxreg1_pvalue[i] = rollingx1@model$coef[[i+1-WindowSize]]$coef[3,4]
}
Return_c$ForecastMu[(WindowSize+1):DataSize]=rollingx1@forecast$density$Mu
Return_c$ForecastSigma[(WindowSize+1):DataSize]=rollingx1@forecast$density$Sigma
Return_c_1 = data.frame(date = index(Return_c), coredata(Return_c))是否存在風險溢酬?
從「風險溢酬對SP500報酬的影響 與 顯著性(rolling)」圖可以觀察，從2014年初到2015年底的參數估計的p-value幾乎都是高於0.05，意味著風險溢酬對SP500報酬率不顯著影響。由於第一筆估計參數是由前251筆資料估計出來的，我們將推論的時間往前推251天，從2013年初到2015年底對S&P500不存在顯著的風險溢酬影響。
plot4_1 = ggplot(Return_c_1, aes(x=date, y=model_archm)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+ ggtitle("風險溢酬對SP500報酬的影響 與 顯著性_rolling")+
theme_economist() + scale_colour_economist()
plot4_2 = ggplot(Return_c_1, aes(x=date, y=model_archm_pvalue)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
geom_line(aes(y=criteria),colour="red")+
theme_economist(stata=TRUE) + scale_colour_economist(stata=TRUE)
library(gridExtra)
grid.arrange(plot4_1,plot4_2,nrow=2,ncol=1)
是否存在波動不對稱性?
從「波動不對稱性對SP500報酬的影響 與 顯著性(rolling)」圖可以觀察，從2014年初到2015年8月底的參數估計的p-value幾乎都是高於0.05，意味著波動不對稱對SP500報酬率不顯著；但從2015年9月初到2015年底的參數估計的p-value幾乎都是低於0.05，意味著波動不對稱對SP500報酬率顯著。由於第一筆估計參數是由前251筆資料估計出來的，我們將推論的時間往前推251天，從2013年初到2014年8月底對S&P500不存在顯著的波動不對稱影響，但從2014年9月初到2015年底對S&P500存在顯著的波動不對稱影響，且影響的程度大約介於0.2與0.5之間，意味著波動不對稱增加1單位對S&P500報酬平均增加0.2到0.5單位之間。
plot5_1 = ggplot(Return_c_1, aes(x=date, y=model_gamma)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+ ggtitle("波動不對稱性對SP500報酬的影響 與 顯著性_rolling")+
theme_economist() + scale_colour_economist()
plot5_2 = ggplot(Return_c_1, aes(x=date, y=model_gamma_pvalue)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
geom_line(aes(y=criteria),colour="red")+
theme_economist(stata=TRUE) + scale_colour_economist(stata=TRUE)
library(gridExtra)
grid.arrange(plot5_1,plot5_2,nrow=2,ncol=1)
是否存在交易量前期的影響
從「解釋變數對SP500報酬的影響 與 顯著性(rolling)」圖可以觀察，從2014年初到2015年底的參數估計的p-value幾乎都是高於0.05，意味著交易量前期對SP500報酬率不顯著影響。由於第一筆估計參數是由前251筆資料估計出來的，我們將推論的時間往前推251天，從2013年初到2015年底對S&P500不存在顯著的交易量前期的影響。
plot6_1 = ggplot(Return_c_1, aes(x=date, y=model_mxreg1)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+ ggtitle("解釋變數對SP500報酬的影響 與 顯著性_rolling")+
theme_economist() + scale_colour_economist()
plot6_2 = ggplot(Return_c_1, aes(x=date, y=model_mxreg1_pvalue)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
geom_line(aes(y=criteria),colour="red")+
theme_economist(stata=TRUE) + scale_colour_economist(stata=TRUE)
library(gridExtra)
grid.arrange(plot6_1,plot6_2,nrow=2,ncol=1)
S&P500報酬 與 ForecastMu, ForecastSigma關係_rolling
從平均日報酬預測，在2015年7/8月以前日報酬是較平穩的，大約介於0.000與0.002之間，在2015年7/8月以後日報酬先是較不平穩之後慢慢平緩，大約介於-0.004與0.002之間； 從平均波動預測，在2015年7/8月以前波動是較平穩的，大約介於0.005與0.015之間，在2015年7/8月以後波動先是較不平穩之後慢慢平緩，大約介於0.005與0.03之間。 同時察覺平均日報酬與平均波動預測來看，平均日報酬與平均波動在2015年7/8月以前較穩定的，但在之後先是不穩定再慢慢穩定，意味著其實日報酬變動幅度會受波動影響且是正向影響，波動越大日報酬變動幅度也越大。
plot7_1 = ggplot(Return_c_1, aes(x=date, y=ForecastMu)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
ggtitle("S&P500報酬 與 ForecastMu關係_rolling")
plot7_2 = ggplot(Return_c_1, aes(x=date, y=ForecastSigma)) +
geom_line() +
xlab("")+
scale_x_date(date_labels = "%Y %b %d")+
ggtitle("S&P500報酬 與 ForecastSigma 關係_rolling")
library(gridExtra)
grid.arrange(plot7_1,plot7_2,nrow=2,ncol=1)
1 - (d)
比較b與c小題
主要的差別是模型是否有解釋變數介入，但同下圖很明白的，解釋變數對S&P500不具影響，故我認為解釋變數介入的模型，與沒有解釋變數介入的模型，這二者之間並無明顯差異。
解釋變數對S&P500不具影響性
grid.arrange(plot6_1,plot6_2,nrow=2,ncol=1)
風險溢酬影響性
介入解釋變數下的風險溢酬差異不大
grid.arrange(plot1,plot4,nrow=1,ncol=2)
波動不對稱性影響性
介入解釋變數下的波動不對稱性差異不大
grid.arrange(plot2,plot5,nrow=1,ncol=2)
平均日報酬與波動預測
介入解釋變數下的平均日報酬與波動預測差異不大
grid.arrange(plot3,plot7,nrow=1,ncol=2)