Homework2

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1. Consider the U.S. quarterly GDP growth rates download data via: require(quantmod) gdp = getSymbols(‘GDPC96’,src=‘FRED’, auto.assign=F) gdp.df = data.frame(date = time(gdp), coredata(gdp) )

require(quantmod)

载入需要的程辑包：quantmod
载入需要的程辑包：xts
载入需要的程辑包：zoo

载入程辑包：‘zoo’

The following object is masked from ‘package:timeSeries’:

    time<-

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

Registered S3 method overwritten by 'xts':
  method     from
  as.zoo.xts zoo 
载入需要的程辑包：TTR

载入程辑包：‘TTR’

The following object is masked from ‘package:fBasics’:

    volatility

Registered S3 method overwritten by 'quantmod':
  method            from
  as.zoo.data.frame zoo 
Version 0.4-0 included new data defaults. See ?getSymbols.

gdp = getSymbols('GDPC96',src='FRED', auto.assign=F) 

‘getSymbols’ currently uses auto.assign=TRUE by default, but will
use auto.assign=FALSE in 0.5-0. You will still be able to use
‘loadSymbols’ to automatically load data. getOption("getSymbols.env")
and getOption("getSymbols.auto.assign") will still be checked for
alternate defaults.

This message is shown once per session and may be disabled by setting 
options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.

gdp.df = data.frame(date = time(gdp), coredata(gdp) )
head(gdp.df)

1. Build an AR model for the growth rate series. Perform model checking to validate the fitted model. Write down the model.

require(fBasics)
gr=diff((gdp.df[,2]))/gdp.df[-282,2]
require(ggfortify)

载入需要的程辑包：ggfortify
载入需要的程辑包：ggplot2
Registered S3 methods overwritten by 'ggplot2':
  method         from 
  [.quosures     rlang
  c.quosures     rlang
  print.quosures rlang
Registered S3 method overwritten by 'dplyr':
  method           from
  print.rowwise_df     

autoplot(ts(gr), ts.colour = 'blue', ts.geom = 'ribbon', main = "GNP")

qplot(gr[1:281],gr[2:282], xlab = "index", ylab = "return", geom = "point")

par(mfcol=c(2,1)) 
acf(gr)
pacf(gr)

m0 = ar(gr , method="mle")
m0$order  #An AR(3) is selected based on AIC

[1] 3

autoplot(ts(m0$resid),ts.colour = "blue", ts.geom = 'ribbon', main="AR(3) fit for GNP")

t.test(gr)

    One Sample t-test

data:  gr
t = 13.772, df = 280, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 0.006698639 0.008932837
sample estimates:
  mean of x 
0.007815738 

Box.test(m0$resid,lag=10,type="Ljung")

    Box-Ljung test

data:  m0$resid
X-squared = 6.4222, df = 10, p-value = 0.7786

(pv1=1-pchisq(6.4412,10-3)) 

[1] 0.4892768

sprintf(" Since pv1 = %s ,pv1 > 0.05 ,Do not reject null hypothesis of independence, the model AR(3) is adequate",pv1)

[1] " Since pv1 = 0.489276819373675 ,pv1 > 0.05 ,Do not reject null hypothesis of independence, the model AR(3) is adequate"

(m1 <- arima(gr,order=c(3,0,0)))

Call:
arima(x = gr, order = c(3, 0, 0))

Coefficients:
         ar1     ar2      ar3  intercept
      0.3461  0.1255  -0.0930     0.0078
s.e.  0.0593  0.0624   0.0594     0.0008

sigma^2 estimated as 7.629e-05:  log likelihood = 933.26,  aic = -1856.51

tsdiag(m1)

2. Does the model confirm the existence of business cycles? Why? (Hint: use the command polyroot to find roots of a polynomial.)

(p1 <- c(1,-m1$coef[1:3]))

                    ar1         ar2         ar3 
 1.00000000 -0.34608624 -0.12549869  0.09300821 

(roots=polyroot(p1))

[1]  1.824805+1.15939i -2.300281+0.00000i  1.824805-1.15939i

Mod(roots)

[1] 2.161966 2.300281 2.161966

(k=2*pi/acos(1.824805/2.161966))

[1] 11.10089

sprintf("There are complex solutions, hence the business cycles exist, the average length of business cycles is %s",k)

[1] "There are complex solutions, hence the business cycles exist, the average length of business cycles is 11.1008924427193"

3. Obtain 1-step to 8-step ahead point and 95% interval forecasts for the U.S. quarterly GDP growth rate at the forecast origin April 1, 2017 (the last data point).

require(forecast)
pre <- forecast(m1,level = c(95), h = 9) # Prediction 
autoplot(pre)

#dev.off() # make sure no graphics error

4. Build an ARMA model for the log growth rate series. Perform model checking to validate the fitted model. Write down the model.

x <- gdp.df[,2]
log_x = diff(log(x))
#basicStats(log_x)
autoplot(ts(log_x), ts.colour = 'blue', ts.geom = 'ribbon', main = "GNP")

Ignoring unknown parameters: ts.colour, ts.geom

par(mfcol=c(2,1)) 
acf(log_x,lag=12,main="ACF lag=12")
pacf(log_x,lag.max=12,main="PACF lag=12")

auto.arima(log_x)

Series: log_x 
ARIMA(4,1,1) 

Coefficients:
         ar1     ar2      ar3      ar4      ma1
      0.3324  0.1297  -0.0700  -0.0815  -0.9860
s.e.  0.0601  0.0628   0.0628   0.0601   0.0113

sigma^2 estimated as 7.661e-05:  log likelihood=930.43
AIC=-1848.87   AICc=-1848.56   BIC=-1827.06

(m_log <- arima(log_x,order=c(4,1,1)))

Call:
arima(x = log_x, order = c(4, 1, 1))

Coefficients:
         ar1     ar2      ar3      ar4      ma1
      0.3324  0.1297  -0.0700  -0.0815  -0.9860
s.e.  0.0601  0.0628   0.0628   0.0601   0.0113

sigma^2 estimated as 7.525e-05:  log likelihood = 930.43,  aic = -1848.87

tsdiag(m_log)

(ljunbgbox_1 = Box.test(m_log$resid , lag= 10 ,type="Ljung"))

    Box-Ljung test

data:  m_log$resid
X-squared = 6.4484, df = 10, p-value = 0.7763

(pv = 1-pchisq(ljunbgbox_1$statistic, 10 - 4))

X-squared 
0.3748752 

sprintf("p value: %s > 0.05. Do not reject null hypothesis of independence,the model is adequate",pv)

[1] "p value: 0.374875197390212 > 0.05. Do not reject null hypothesis of independence,the model is adequate"

5. Obtain 1-step to 8-step ahead point and 95% interval forecasts for the U.S. quarterly GDP growth rate at the forecast origin April 1, 2017 (the last data point).

require(forecast)
pre2 <- forecast(m_log,level = c(95), h = 9) # Prediction 
autoplot(pre2)

#dev.off() # make sure no graphics error

2. Consider the Decile 10 portfolio of CRSP. The monthly simple returns are available in the file m-dec125910-5112.txt. Obtain the log returns of the Decile 10 portfolio. (Note that the file contains all decile portfolios, and you should only take prtnam = 10.)

require(data.table)

载入需要的程辑包：data.table
Registered S3 method overwritten by 'data.table':
  method           from
  print.data.table     
data.table 1.12.2 using 4 threads (see ?getDTthreads).  Latest news: r-datatable.com

载入程辑包：‘data.table’

The following objects are masked from ‘package:xts’:

    first, last

csrp = fread("/Users/aisling/Documents/daydayup/nus_mqf/Financial Time Series/homework/Homework/HW 2/m-dec125910-5112.txt")
csrp = csrp[csrp$prtnam == 10]
head(csrp)

log_csrp = log(csrp$totret+1)

1. Build a time series model for the log returns. Perform model checking to verify that the model is adequate, and write down the fitted model.

par(mfcol=c(2,1)) 
pacf(log_csrp ) 
acf(log_csrp ) 

(fit<-auto.arima(log_csrp,ic=c('bic')))

Series: log_csrp 
ARIMA(0,0,1) with non-zero mean 

Coefficients:
         ma1    mean
      0.2175  0.0096
s.e.  0.0356  0.0027

sigma^2 estimated as 0.003802:  log likelihood=1018.18
AIC=-2030.36   AICc=-2030.33   BIC=-2016.52

m_csrp1 = arima(log_csrp,order=c(0,0,1))
tsdiag(m_csrp1,gof=24)

(ljunbgbox_csrp1 = Box.test(m_csrp1$resid , lag= 10 ,type="Ljung"))

    Box-Ljung test

data:  m_csrp1$resid
X-squared = 2.7869, df = 10, p-value = 0.986

(pv_csrp1 = 1-pchisq(ljunbgbox_csrp1$statistic, 10 - 0))

X-squared 
0.9860037 

sprintf("p value: %s > 0.05. Do not reject null hypothesis of independence , the model is adequate ",pv_csrp1)

[1] "p value: 0.986003652074264 > 0.05. Do not reject null hypothesis of independence , the model is adequate "

2. Create a January dummy variable by the command Jan = rep(c(1, rep(0, 11)), 62). Use the dummy variable as an explanatory variable and build a regression model with time series error for the log returns. Perform model checking to justify the model and write down the fitted model.

Jan = rep(c(1, rep(0, 11)), 62) 
m_csrp2 = arima(log_csrp,order=c(0,0,1), xreg=Jan)
tsdiag(m_csrp2,gof=24)

(ljunbgbox_csrp2 = Box.test(m_csrp2$resid , lag= 10 ,type="Ljung"))

    Box-Ljung test

data:  m_csrp2$resid
X-squared = 3.3443, df = 10, p-value = 0.9721

(pv_csrp2 = 1-pchisq(ljunbgbox_csrp2$statistic, 10 - 0))

X-squared 
0.9721214 

sprintf("p value: %s > 0.05. Do not reject null hypothesis of independence , the model is adequate",pv_csrp2)

[1] "p value: 0.972121378634896 > 0.05. Do not reject null hypothesis of independence , the model is adequate"

3. Compare the two models in parts (a) and (b) using AIC criterion. Draw your conclusion about the seasonality of part

sprintf("AIC of a model is: %s, while b model is %s , model in part b has smaller AIC .Hence there is indeed some seasonality ", m_csrp1$aic , m_csrp2$aic)

[1] "AIC of a model is: -2030.36065312675, while b model is -2094.91694558133 , model in part b has smaller AIC .Hence there is indeed some seasonality "

3. In this exercise you will learn how to do basic simulations through an example. Please use google to understand the following R-code: at = rnorm(1000) rt1 = rep(0,1000) for (t in 3:1000) { rt1[t]=0.5rt1[t-1]+1+at[t]+2at[t-2] }

1. What model does the generated sequence rt1 simulate? What is its value at time 1 and 2?

at = rnorm(1000)
rt1 = rep(0,1000) 
for (t in 3:1000) {
  rt1[t]=0.5*rt1[t-1]+1+at[t]+2*at[t-2]
}
sprintf("rt1[t]=0.5*rt1[t-1]+1+at[t]+2*at[t-2] is a time series (ARMA(1,2)) model , and its value at time 1 and 2 is zero  ")

[1] "rt1[t]=0.5*rt1[t-1]+1+at[t]+2*at[t-2] is a time series (ARMA(1,2)) model , and its value at time 1 and 2 is zero  "

2. Try to fit the series rt1 with a ARMA(1, 2) model and explain your finding.

m_rt1 = arima(rt1,order=c(1,0,2))
m_rt1  # i find nothing 

Call:
arima(x = rt1, order = c(1, 0, 2))

Coefficients:
        ar1     ma1     ma2  intercept
      0.524  0.0107  0.5230     1.7674
s.e.  0.040  0.0374  0.0325     0.2041

sigma^2 estimated as 4.028:  log likelihood = -2116.17,  aic = 4242.35

tsdiag(m_rt1,gof=24)

3. Replace the number of iteration 1000 to 100 and 10000, and repeat (b), (c)

at = rnorm(100)
rt2 = rep(0,100)
for (t in 3:100){
  rt2[t]=0.5*rt2[t-1]+1+at[t]+2*at[t-2]
}
(m_rt2 <- arima(rt2,order=c(1,0,2)))

Call:
arima(x = rt2, order = c(1, 0, 2))

Coefficients:
         ar1      ma1     ma2  intercept
      0.4119  -0.0258  0.5373     1.5509
s.e.  0.1281   0.1133  0.0999     0.5524

sigma^2 estimated as 4.74:  log likelihood = -220.21,  aic = 450.41

tsdiag(m_rt2,gof=24)

at = rnorm(10000)
rt3 = rep(0,10000)
for (t in 3:10000){
  rt3[t]=0.5*rt3[t-1]+1+at[t]+2*at[t-2]
}
(m_rt3 <- arima(rt3,order=c(1,0,2)))

Call:
arima(x = rt3, order = c(1, 0, 2))

Coefficients:
         ar1     ma1     ma2  intercept
      0.5022  0.0030  0.5016     1.9895
s.e.  0.0130  0.0124  0.0096     0.0605

sigma^2 estimated as 4.001:  log likelihood = -21123.25,  aic = 42256.49

tsdiag(m_rt3,gof=24)

4. Try to simulate another time series rt2 with the same model, using the same shock series at, but with different initial conditions rt2[1] = 20, rt2[2] = 10. What is the difference between rt1[1000] and rt2[1000]? Try to explain your result.

at = rnorm(1000)
rt1 = rep(0,1000)
for (t in 3:1000){
  rt1[t]=0.5*rt1[t-1]+1+at[t]+2*at[t-2]
}
rt2 = rep(0,1000)
rt2[1] = 20
rt2[2] = 10
for (t in 3:1000){
  rt2[t]=0.5*rt2[t-1]+1+at[t]+2*at[t-2]
}
sprintf("rt1[1000] = %s , rt2[1000] = %s ",rt1[1000],rt2[1000])

[1] "rt1[1000] = 1.56121886821331 , rt2[1000] = 1.56121886821331 "

4. The file d-gmsp9908.txt contains the daily simple returns of GM stock and the S&P composite index from 1999 to 2008. It has three columns denoting date, GM return, and S&P return. We focus on the daily returns of the S&P composite index.

gm=fread("/Users/aisling/Documents/daydayup/nus_mqf/Financial Time Series/homework/Homework/HW 2/d-gmsp9908.txt")
head(gm)

1. Compute the daily log returns of the S&P index. Is there any ARCH effect in the log returns?

require(fUnitRoots)

载入需要的程辑包：fUnitRoots

log_gm=log(gm$sp+1)
adftest4 = adfTest(log_gm)

p-value smaller than printed p-value

sprintf("p value: %s < 0.05. Reject null hypothesis of non stationary",adftest4@test$p.value)

[1] "p value: 0.01 < 0.05. Reject null hypothesis of non stationary"

par(mfcol=c(2,1)) 
pacf(log_gm)
acf(log_gm)  

auto.arima(log_gm) 

Series: log_gm 
ARIMA(1,0,5) with zero mean 

Coefficients:
          ar1     ma1      ma2      ma3      ma4      ma5
      -0.9434  0.8693  -0.1694  -0.0506  -0.0039  -0.0737
s.e.   0.0233  0.0304   0.0264   0.0273   0.0264   0.0200

sigma^2 estimated as 0.0001752:  log likelihood=7311.3
AIC=-14608.59   AICc=-14608.55   BIC=-14567.78

m_gm = arima(log_gm, c(1,0,5))
tsdiag(m_gm)

ljungbox_gm1 = Box.test(m_gm$resid , lag= 10 ,type="Ljung")
(pv_gm1 = 1-pchisq(ljungbox_gm1$statistic, 10 - 1))

X-squared 
0.9186324 

sprintf("p value: %s > 0.05. Do not reject null hypothesis of independence",pv_gm1)

[1] "p value: 0.918632387085003 > 0.05. Do not reject null hypothesis of independence"

require(aTSA)

载入需要的程辑包：aTSA

载入程辑包：‘aTSA’

The following object is masked from ‘package:forecast’:

    forecast

The following object is masked from ‘package:graphics’:

    identify

arch.test(arima(m_gm$residuals,c(1,0,5)))

ARCH heteroscedasticity test for residuals 
alternative: heteroscedastic 

Portmanteau-Q test: 
     order   PQ p.value
[1,]     4  986       0
[2,]     8 2012       0
[3,]    12 3005       0
[4,]    16 3514       0
[5,]    20 4082       0
[6,]    24 4534       0
Lagrange-Multiplier test: 
     order   LM  p.value
[1,]     4 1275 0.00e+00
[2,]     8  427 0.00e+00
[3,]    12  255 0.00e+00
[4,]    16  178 0.00e+00
[5,]    20  137 0.00e+00
[6,]    24  108 6.03e-13

[Hint: First fit an ARMA model for the returns, and then check the residuals of the fitted model for ARCH effect.]

2. Fit a Gaussian ARMA-GARCH model for the log return series. Refine the model. Perform model checking, and write down the fitted model.

require(fGarch)

载入需要的程辑包：fGarch

m_gm_garch1=garchFit(~arma(1,5)+garch(1,1),data=log_gm,trace=F)
summary(m_gm_garch1)

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 5) + garch(1, 1), data = log_gm, 
    trace = F) 

Mean and Variance Equation:
 data ~ arma(1, 5) + garch(1, 1)
<environment: 0x7fa48e7507c0>
 [data = log_gm]

Conditional Distribution:
 norm 

Coefficient(s):
         mu          ar1          ma1          ma2          ma3          ma4          ma5        omega  
 1.8798e-04   3.2974e-01  -3.9134e-01  -1.9104e-02   1.1027e-03  -1.1260e-02  -4.8846e-02   1.0075e-06  
     alpha1        beta1  
 7.1139e-02   9.2362e-01  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu      1.880e-04   1.297e-04    1.450 0.147160    
ar1     3.297e-01   3.226e-01    1.022 0.306771    
ma1    -3.913e-01   3.225e-01   -1.213 0.224947    
ma2    -1.910e-02   3.042e-02   -0.628 0.530032    
ma3     1.103e-03   2.590e-02    0.043 0.966041    
ma4    -1.126e-02   2.270e-02   -0.496 0.619836    
ma5    -4.885e-02   2.335e-02   -2.092 0.036432 *  
omega   1.008e-06   2.937e-07    3.430 0.000604 ***
alpha1  7.114e-02   9.068e-03    7.845 4.44e-15 ***
beta1   9.236e-01   9.670e-03   95.513  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 7867.301    normalized:  3.128151 

Description:
 Fri Jul 26 22:37:20 2019 by user:  


Standardised Residuals Tests:
                                Statistic p-Value     
 Jarque-Bera Test   R    Chi^2  250.0153  0           
 Shapiro-Wilk Test  R    W      0.9882654 1.583554e-13
 Ljung-Box Test     R    Q(10)  2.693501  0.9877461   
 Ljung-Box Test     R    Q(15)  11.97112  0.6812141   
 Ljung-Box Test     R    Q(20)  17.75243  0.6037129   
 Ljung-Box Test     R^2  Q(10)  14.36861  0.1568325   
 Ljung-Box Test     R^2  Q(15)  17.8363   0.2713694   
 Ljung-Box Test     R^2  Q(20)  19.5147   0.4886311   
 LM Arch Test       R    TR^2   15.10128  0.2359443   

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
-6.248351 -6.225170 -6.248382 -6.239937 

plot(m_gm_garch1) 

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

13

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

0

m_gm_garch2=garchFit(~arma(1,5)+garch(1,1),data=log_gm,trace=F,cond.dist = 'std')
summary(m_gm_garch2)

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 5) + garch(1, 1), data = log_gm, 
    cond.dist = "std", trace = F) 

Mean and Variance Equation:
 data ~ arma(1, 5) + garch(1, 1)
<environment: 0x7fa48dd1f688>
 [data = log_gm]

Conditional Distribution:
 std 

Coefficient(s):
         mu          ar1          ma1          ma2          ma3          ma4          ma5  
 2.6512e-04   3.2689e-01  -3.9092e-01  -2.8491e-02   2.7998e-03  -1.1520e-02  -4.0589e-02  
      omega       alpha1        beta1        shape  
 6.1675e-07   7.1623e-02   9.2747e-01   9.3629e+00  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu      2.651e-04   1.521e-04    1.744   0.0812 .  
ar1     3.269e-01   3.095e-01    1.056   0.2909    
ma1    -3.909e-01   3.097e-01   -1.262   0.2069    
ma2    -2.849e-02   3.013e-02   -0.946   0.3443    
ma3     2.800e-03   2.700e-02    0.104   0.9174    
ma4    -1.152e-02   2.244e-02   -0.513   0.6077    
ma5    -4.059e-02   2.210e-02   -1.837   0.0663 .  
omega   6.168e-07   2.879e-07    2.142   0.0322 *  
alpha1  7.162e-02   1.034e-02    6.926 4.32e-12 ***
beta1   9.275e-01   1.029e-02   90.114  < 2e-16 ***
shape   9.363e+00   1.632e+00    5.737 9.64e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 7896.908    normalized:  3.139924 

Description:
 Fri Jul 26 22:46:47 2019 by user:  


Standardised Residuals Tests:
                                Statistic p-Value     
 Jarque-Bera Test   R    Chi^2  336.4595  0           
 Shapiro-Wilk Test  R    W      0.9868748 1.804912e-14
 Ljung-Box Test     R    Q(10)  2.829662  0.9851532   
 Ljung-Box Test     R    Q(15)  11.49686  0.7166439   
 Ljung-Box Test     R    Q(20)  17.03243  0.6508669   
 Ljung-Box Test     R^2  Q(10)  12.16103  0.274424    
 Ljung-Box Test     R^2  Q(15)  15.84869  0.3921786   
 Ljung-Box Test     R^2  Q(20)  17.26528  0.6356867   
 LM Arch Test       R    TR^2   12.88408  0.377522    

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
-6.271100 -6.245600 -6.271138 -6.261845 

plot(m_gm_garch2) 

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

13

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

0

m_gm_garch3=garchFit(~arma(1,5)+garch(1,1),data=log_gm,trace=F,cond.dist = 'sstd')
summary(m_gm_garch3)

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~arma(1, 5) + garch(1, 1), data = log_gm, 
    cond.dist = "sstd", trace = F) 

Mean and Variance Equation:
 data ~ arma(1, 5) + garch(1, 1)
<environment: 0x7fa48e7441d8>
 [data = log_gm]

Conditional Distribution:
 sstd 

Coefficient(s):
         mu          ar1          ma1          ma2          ma3          ma4          ma5  
 1.8570e-04   2.9793e-01  -3.7254e-01  -4.2571e-02   2.4321e-03  -1.5038e-02  -4.8347e-02  
      omega       alpha1        beta1         skew        shape  
 6.4686e-07   7.2390e-02   9.2556e-01   8.8944e-01   1.0000e+01  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu      1.857e-04   1.212e-04    1.532   0.1256    
ar1     2.979e-01   3.043e-01    0.979   0.3276    
ma1    -3.725e-01   3.045e-01   -1.223   0.2212    
ma2    -4.257e-02   3.235e-02   -1.316   0.1882    
ma3     2.432e-03   2.966e-02    0.082   0.9346    
ma4    -1.504e-02   2.267e-02   -0.663   0.5071    
ma5    -4.835e-02   2.332e-02   -2.073   0.0382 *  
omega   6.469e-07   2.866e-07    2.257   0.0240 *  
alpha1  7.239e-02   1.007e-02    7.186 6.67e-13 ***
beta1   9.256e-01   1.015e-02   91.189  < 2e-16 ***
skew    8.894e-01   2.492e-02   35.696  < 2e-16 ***
shape   1.000e+01   1.861e+00    5.372 7.77e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Log Likelihood:
 7905.746    normalized:  3.143438 

Description:
 Fri Jul 26 22:48:23 2019 by user:  


Standardised Residuals Tests:
                                Statistic p-Value     
 Jarque-Bera Test   R    Chi^2  337.984   0           
 Shapiro-Wilk Test  R    W      0.986483  1.006707e-14
 Ljung-Box Test     R    Q(10)  5.989618  0.8161347   
 Ljung-Box Test     R    Q(15)  15.06451  0.4467806   
 Ljung-Box Test     R    Q(20)  20.56417  0.4231719   
 Ljung-Box Test     R^2  Q(10)  12.36954  0.2610858   
 Ljung-Box Test     R^2  Q(15)  15.94966  0.385405    
 Ljung-Box Test     R^2  Q(20)  17.52388  0.6187409   
 LM Arch Test       R    TR^2   13.03785  0.3662954   

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
-6.277333 -6.249515 -6.277378 -6.267237 

plot(m_gm_garch3) 

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

13

Make a plot selection (or 0 to exit): 

 1:   Time Series
 2:   Conditional SD
 3:   Series with 2 Conditional SD Superimposed
 4:   ACF of Observations
 5:   ACF of Squared Observations
 6:   Cross Correlation
 7:   Residuals
 8:   Conditional SDs
 9:   Standardized Residuals
10:   ACF of Standardized Residuals
11:   ACF of Squared Standardized Residuals
12:   Cross Correlation between r^2 and r
13:   QQ-Plot of Standardized Residuals

0

sprintf("We should choose the model m_gm_garch3 according to the QQ plot .")

[1] "We should choose the model m_gm_garch3 according to the QQ plot ."

3. Compute 1-step to 4-step ahead forecasts of the log return and its volatility based on the fitted model.

predict(m_gm_garch3,4)