ECO 5316 Time Series Econometrics Spring 2016, HW-4,Problem 2
March 19, 2016
library("zoo")
library("forecast")
library("tseries")
library("urca")
library ("Quandl")
library("fGarch")
library("rugarch")
library("stargazer")Problem 2
Get data for the Microsoft stock price GOOG/NASDAQ_MSFTand construct the log return \(y_t = \Delta log MSFT_t\) .
Introduction and Data :
In this problem, data is analysed about Microsoft daily stock closing price from 1986 to 2016. The data for this problem is collectd from Quand.
microsoft_close_d<- read.csv(file = "http://www.google.com/finance/historical?q=NASDAQ%3AMSFT&startdate=Jan+1%2C+1990&output=csv", header=TRUE, sep=",")
microsoft_close<- ts(microsoft_close_d[,5], start=c(1986,3))
l.microsoft_close= log(microsoft_close)
str(microsoft_close)## Time-Series [1:4000] from 1988 to 5987: 54.1 53.9 53.5 54.7 54.4 ...head(microsoft_close)## [1] 54.07 53.86 53.49 54.66 54.35 53.59tail(microsoft_close)## [1] 35.28 34.94 36.72 34.88 34.91 34.00tsdisplay(microsoft_close, lag.max = 100, xlab=" Years 1986-2016", ylab= "Stock closing price ", main ="Microsoft daily stock closing price from 1986 to 2016")
tsdisplay(l.microsoft_close, lag.max = 100, xlab=" Years 1986-2016", ylab= "log Stock closing price ", main ="Microsoft daily stock closing price from 1986 to 2016")
The plot above indicate both decreasing adn increae trend of stock price over the time frame. ACF plot is decreasing slowing, while PACK does not have any peak.
As the data is not stationary, let take a first difference to make the data stationary.
dl.microsoft_close<-diff(l.microsoft_close)
tsdisplay(dl.microsoft_close, lag.max = 100, xlab=" Years 1986-2016",ylab = "first difference of Stock closing price ", main ="Microsoft daily stock closing price from 1986 to 2016")
The pot suggests the data is stationaty. ACF and PACF plot has second and fourth peak. Let use ARMA (4,4) for modeling the data.
(a) Estimate an ARMA model: use ACF, PACF, Ljung-Box statistics, to identify and then estimate an appropriate model for log return
arma_4_4<- arima(dl.microsoft_close, order = c(4,0,4))
tsdiag(arma_4_4, gof.lag= 100)
The Ljung-Box statistics suggests model has high precdictive power for the first month only.
(b) Use ACF, PACF, Ljung-Box statistics for squared residuals from (a) to identify and then estimate an ARCH(m) model with normal innovations
par(mfcol=c(2,1))
acf( arma_4_4$residuals^2 )
pacf(arma_4_4$residuals^2 )
par (mfcol=c(1,1)) In the PACF plot, peak are not significant after 8 lag. lets combine the ARCh (8) on the above ARMA(4,4)
arch_8 <-garchFit(~ arma(4,4) + garch(8,0), data = dl.microsoft_close, trace = FALSE)## Warning in arima(.series$x, order = c(u, 0, v), include.mean =
## include.mean): possible convergence problem: optim gave code = 1 summary(arch_8)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(4, 4) + garch(8, 0), data = dl.microsoft_close,
## trace = FALSE)
##
## Mean and Variance Equation:
## data ~ arma(4, 4) + garch(8, 0)
## <environment: 0x0000000011e32b58>
## [data = dl.microsoft_close]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3 ar4
## -4.7101e-04 9.6876e-01 -1.3792e-01 -2.6691e-01 -2.0745e-01
## ma1 ma2 ma3 ma4 omega
## -1.0000e+00 1.6506e-01 3.0327e-01 1.5594e-01 7.7442e-05
## alpha1 alpha2 alpha3 alpha4 alpha5
## 1.9195e-01 1.1775e-01 1.1666e-01 1.4464e-01 7.3773e-02
## alpha6 alpha7 alpha8
## 8.4345e-02 9.6400e-02 6.7803e-02
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -4.710e-04 2.667e-04 -1.766 0.077424 .
## ar1 9.688e-01 2.529e-01 3.831 0.000128 ***
## ar2 -1.379e-01 3.241e-01 -0.426 0.670408
## ar3 -2.669e-01 6.727e-01 -0.397 0.691527
## ar4 -2.075e-01 5.318e-01 -0.390 0.696459
## ma1 -1.000e+00 2.551e-01 -3.920 8.85e-05 ***
## ma2 1.651e-01 3.316e-01 0.498 0.618627
## ma3 3.033e-01 6.850e-01 0.443 0.657980
## ma4 1.559e-01 5.480e-01 0.285 0.775979
## omega 7.744e-05 6.317e-06 12.259 < 2e-16 ***
## alpha1 1.920e-01 2.915e-02 6.586 4.53e-11 ***
## alpha2 1.178e-01 2.515e-02 4.682 2.84e-06 ***
## alpha3 1.167e-01 2.158e-02 5.406 6.44e-08 ***
## alpha4 1.446e-01 2.634e-02 5.490 4.01e-08 ***
## alpha5 7.377e-02 2.443e-02 3.020 0.002528 **
## alpha6 8.434e-02 2.278e-02 3.703 0.000213 ***
## alpha7 9.640e-02 2.267e-02 4.253 2.11e-05 ***
## alpha8 6.780e-02 1.971e-02 3.440 0.000581 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 10671.02 normalized: 2.668422
##
## Description:
## Tue Mar 22 23:37:33 2016 by user: kluitel
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 4480.814 0
## Shapiro-Wilk Test R W 0.9570959 0
## Ljung-Box Test R Q(10) 7.245007 0.70213
## Ljung-Box Test R Q(15) 10.722 0.7720408
## Ljung-Box Test R Q(20) 13.26902 0.8655454
## Ljung-Box Test R^2 Q(10) 10.85226 0.3691439
## Ljung-Box Test R^2 Q(15) 13.70664 0.5478847
## Ljung-Box Test R^2 Q(20) 20.52738 0.4254037
## LM Arch Test R TR^2 13.7713 0.3155506
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -5.327841 -5.299512 -5.327882 -5.317799All the parametes estimated on error aalysis is significant except mu.
(c) Perform model checking for the ARCH model from (b): for both standardized residuals and squared standardized residuals plot ACF, examine Q(10) and Q(20) statistics. Also plot and comment on QQ plot and the results of Jarque-Bera test.
arch_8_res<-arch_8@residuals/arch_8@sigma.t
arch_8_res2<-(arch_8@residuals/arch_8@sigma.t)^2
# standardized residuals
par(mfcol=c(2,2))
acf(arch_8_res, na.action = na.exclude, main="ACF for standardized residuals" )
pacf( arch_8_res, na.action = na.exclude, main="PACF for standardized residuals" )
# squared standardized residuals
acf( arch_8_res2 , na.action = na.exclude, main="ACF for Squared standardized residuals")
pacf( arch_8_res2, na.action = na.exclude, main="PACF for Squared standardized residuals" )
par (mfcol=c(1,1)) In ACF plot there is no peak, while in PACF there is peak at 12 lag in standard residual and at 16 lag in squared residual.
Box.test( (arch_8@residuals/arch_8@sigma.t)^2, lag = 10, type = "Ljung")##
## Box-Ljung test
##
## data: (arch_8@residuals/arch_8@sigma.t)^2
## X-squared = 10.852, df = 10, p-value = 0.3691Box.test( (arch_8@residuals/arch_8@sigma.t)^2, lag = 20, type = "Ljung")##
## Box-Ljung test
##
## data: (arch_8@residuals/arch_8@sigma.t)^2
## X-squared = 20.527, df = 20, p-value = 0.4254The p-value of Ljung test is not significat for lag 10 and lag 20 are 0.3598 and 0.3397, respectively.
plot( arch_8, which= 13)
#qqnorm(arch_8_res) #another method to qq plot
#qqline(arch_8_res)The qq-plot suggests data is not normally distributed.
jarque.bera.test(arch_8_res)##
## Jarque Bera Test
##
## data: arch_8_res
## X-squared = 4480.8, df = 2, p-value < 2.2e-16The p value of Jarque–Bera Test is less than 0.0001.
(d) Estimate a GARCH(m, s) model with Student-t innovations. Perform model checking.
garch11<-garchFit(~arma(4,4)+ garch(1,1), data = dl.microsoft_close, trace=FALSE, cond.dist="sstd")## Warning in arima(.series$x, order = c(u, 0, v), include.mean =
## include.mean): possible convergence problem: optim gave code = 1summary(garch11)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(4, 4) + garch(1, 1), data = dl.microsoft_close,
## cond.dist = "sstd", trace = FALSE)
##
## Mean and Variance Equation:
## data ~ arma(4, 4) + garch(1, 1)
## <environment: 0x00000000468c8900>
## [data = dl.microsoft_close]
##
## Conditional Distribution:
## sstd
##
## Coefficient(s):
## mu ar1 ar2 ar3 ar4
## -5.5383e-04 -6.0244e-01 -3.6384e-01 7.6224e-02 6.3205e-01
## ma1 ma2 ma3 ma4 omega
## 5.7657e-01 3.2724e-01 -1.0526e-01 -6.6193e-01 3.0846e-06
## alpha1 beta1 skew shape
## 7.1937e-02 9.2381e-01 9.2912e-01 4.5226e+00
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -5.538e-04 3.515e-04 -1.576 0.11507
## ar1 -6.024e-01 1.409e-01 -4.277 1.90e-05 ***
## ar2 -3.638e-01 1.606e-01 -2.266 0.02347 *
## ar3 7.622e-02 1.651e-01 0.462 0.64426
## ar4 6.321e-01 1.206e-01 5.242 1.59e-07 ***
## ma1 5.766e-01 1.358e-01 4.247 2.17e-05 ***
## ma2 3.272e-01 1.536e-01 2.130 0.03319 *
## ma3 -1.053e-01 1.579e-01 -0.667 0.50490
## ma4 -6.619e-01 1.161e-01 -5.700 1.20e-08 ***
## omega 3.085e-06 1.174e-06 2.627 0.00861 **
## alpha1 7.194e-02 1.539e-02 4.674 2.95e-06 ***
## beta1 9.238e-01 1.610e-02 57.375 < 2e-16 ***
## skew 9.291e-01 2.001e-02 46.423 < 2e-16 ***
## shape 4.523e+00 3.222e-01 14.035 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 10938.85 normalized: 2.735397
##
## Description:
## Tue Mar 22 23:37:42 2016 by user: kluitel
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 9504.006 0
## Shapiro-Wilk Test R W 0.9426393 0
## Ljung-Box Test R Q(10) 6.680128 0.7552583
## Ljung-Box Test R Q(15) 14.63704 0.4778638
## Ljung-Box Test R Q(20) 16.5612 0.6812552
## Ljung-Box Test R^2 Q(10) 3.245137 0.9750524
## Ljung-Box Test R^2 Q(15) 7.462484 0.9435239
## Ljung-Box Test R^2 Q(20) 8.731585 0.9858127
## LM Arch Test R TR^2 4.260435 0.9782651
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -5.463793 -5.441759 -5.463818 -5.455983garch11_res<-garch11@residuals/garch11@sigma.t
garch11_res2<-(garch11@residuals/garch11@sigma.t)^2
# standardized residuals
par(mfcol=c(2,2))
acf(garch11_res, na.action = na.exclude, main="ACF for standardized residuals" )
pacf( garch11_res, na.action = na.exclude, main="PACF for standardized residuals" )
# squared standardized residuals
acf( garch11_res2 , na.action = na.exclude, main="ACF for Squared standardized residuals")
pacf( garch11_res2, na.action = na.exclude, main="PACF for Squared standardized residuals" )
par (mfcol=c(1,1))
plot(garch11, which= 13)
The Ljung BOx-Test for Q(10) and Q(20) p-value is no significant. The parameter estimates of model are mostly significan. There is no peak in ACF and PACF residual plot except in the PACF plot of standard residual has peak at 12 lag. The qq-plot suggests the data is skewed at the right side.
(e) Estimate a TGARCH(m, s) model, again perform model checking. What does this model tell us about the effects of negative vs positive shocks?
In TGARCH model dealta is equal to 1.
Tgarch11<-garchFit(~arma(4,4)+ garch(1,1), data = dl.microsoft_close, delta = 1, trace=FALSE, leverage = TRUE, cond.dist="sstd")## Warning in arima(.series$x, order = c(u, 0, v), include.mean =
## include.mean): possible convergence problem: optim gave code = 1summary(Tgarch11)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(4, 4) + garch(1, 1), data = dl.microsoft_close,
## delta = 1, cond.dist = "sstd", leverage = TRUE, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ arma(4, 4) + garch(1, 1)
## <environment: 0x0000000011e43580>
## [data = dl.microsoft_close]
##
## Conditional Distribution:
## sstd
##
## Coefficient(s):
## mu ar1 ar2 ar3 ar4
## -7.6564e-05 -3.5884e-01 -4.3180e-02 3.9578e-01 8.5424e-01
## ma1 ma2 ma3 ma4 omega
## 3.5020e-01 3.2243e-02 -4.0004e-01 -8.6685e-01 9.0936e-05
## alpha1 gamma1 beta1 skew shape
## 6.7544e-02 3.0112e-01 9.4621e-01 9.1519e-01 4.8053e+00
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -7.656e-05 8.337e-05 -0.918 0.358420
## ar1 -3.588e-01 9.151e-02 -3.921 8.80e-05 ***
## ar2 -4.318e-02 4.157e-02 -1.039 0.298943
## ar3 3.958e-01 7.076e-02 5.594 2.23e-08 ***
## ar4 8.542e-01 5.908e-02 14.459 < 2e-16 ***
## ma1 3.502e-01 8.775e-02 3.991 6.58e-05 ***
## ma2 3.224e-02 4.054e-02 0.795 0.426376
## ma3 -4.000e-01 6.656e-02 -6.010 1.85e-09 ***
## ma4 -8.668e-01 5.889e-02 -14.720 < 2e-16 ***
## omega 9.094e-05 3.826e-05 2.377 0.017460 *
## alpha1 6.754e-02 9.773e-03 6.912 4.79e-12 ***
## gamma1 3.011e-01 8.910e-02 3.380 0.000726 ***
## beta1 9.462e-01 8.318e-03 113.751 < 2e-16 ***
## skew 9.152e-01 2.025e-02 45.187 < 2e-16 ***
## shape 4.805e+00 3.527e-01 13.623 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 10969.88 normalized: 2.743155
##
## Description:
## Tue Mar 22 23:37:51 2016 by user: kluitel
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 33452.5 0
## Shapiro-Wilk Test R W 0.9133639 0
## Ljung-Box Test R Q(10) 12.09046 0.2790481
## Ljung-Box Test R Q(15) 21.97301 0.1085135
## Ljung-Box Test R Q(20) 24.32253 0.228587
## Ljung-Box Test R^2 Q(10) 172.1581 0
## Ljung-Box Test R^2 Q(15) 174.0975 0
## Ljung-Box Test R^2 Q(20) 174.8971 0
## LM Arch Test R TR^2 4.743178 0.9660115
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -5.478808 -5.455201 -5.478836 -5.470440Tgarch11_res<-Tgarch11@residuals/Tgarch11@sigma.t
Tgarch11_res2<-(Tgarch11@residuals/Tgarch11@sigma.t)^2
# standardized residuals
par(mfcol=c(2,2))
acf(Tgarch11_res, na.action = na.exclude, main="ACF for standardized residuals" )
pacf( Tgarch11_res, na.action = na.exclude, main="PACF for standardized residuals" )
# squared standardized residuals
acf( Tgarch11_res2 , na.action = na.exclude, main="ACF for Squared standardized residuals")
pacf( Tgarch11_res2, na.action = na.exclude, main="PACF for Squared standardized residuals" )
par (mfcol=c(1,1))
plot(Tgarch11, which= 13)
plot(Tgarch11, which= 3)# series y with +/- 2*sigma superimposed
plot(Tgarch11, which= 9)# standardized residuals
The parameter estimates of model are mostly significan at 10% level. The Ljung BOx-Test for square of teh residual test for Q(10), Q(15),and Q(20) p-value is significant. In PACF plot of standard residual there is peak at 11 lag. In lag 4 there is peak in both ACF and PACF of the square residual plot.
The qq-plot suggests the data is more skewed at the right side. The plot of series y with +/- 2*sigma superimposed and standardized residuals suggests the positive and negative shocks have same effects on volatility
(f) Create a summary table using stargazerX showing the models side by side.
stargazer(arma_4_4,arch_8,garch11,Tgarch11,align=TRUE,column.sep.width="0pt", report="vc*", single.row=TRUE, header=FALSE,model.numbers=FALSE,dep.var.caption="", dep.var.labels="Microsoft Daily Returns",column.labels=c("ARMA(1,1)","ARCH(8)","GARCH(1,1)","TGARCH(1,1)"))##
## Call:
## arima(x = dl.microsoft_close, order = c(4, 0, 4))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ma1 ma2 ma3 ma4
## 0.1326 -0.3941 0.2365 -0.0059 -0.1608 0.3564 -0.2493 -0.0538
## s.e. NaN NaN 0.1389 0.0531 NaN NaN 0.0969 0.0491
## intercept
## -1e-04
## s.e. 3e-04
##
## sigma^2 estimated as 0.0003754: log likelihood = 10096.52, aic = -20173.04
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -9.590539e-06 0.01937634 0.0130522 NaN Inf 0.6812664
## ACF1
## Training set -0.001652477
##
## \begin{table}[!htbp] \centering
## \caption{}
## \label{}
## \begin{tabular}{@{\extracolsep{0pt}}lD{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3} }
## \\[-1.8ex]\hline
## \hline \\[-1.8ex]
## \\[-1.8ex] & \multicolumn{4}{c}{Microsoft Daily Returns} \\
## \\[-1.8ex] & \multicolumn{1}{c}{\textit{ARIMA}} & \multicolumn{3}{c}{\textit{GARCH}} \\
## & \multicolumn{1}{c}{ARMA(1,1)} & \multicolumn{1}{c}{ARCH(8)} & \multicolumn{1}{c}{GARCH(1,1)} & \multicolumn{1}{c}{TGARCH(1,1)} \\
## \hline \\[-1.8ex]
## mu & & -0.0005^{*} & -0.001 & -0.0001 \\
## ar1 & 0.133 & 0.969^{***} & -0.602^{***} & -0.359^{***} \\
## ar2 & -0.394 & -0.138 & -0.364^{**} & -0.043 \\
## ar3 & 0.237^{*} & -0.267 & 0.076 & 0.396^{***} \\
## ar4 & -0.006 & -0.207 & 0.632^{***} & 0.854^{***} \\
## ma1 & -0.161 & -1.000^{***} & 0.577^{***} & 0.350^{***} \\
## ma2 & 0.356 & 0.165 & 0.327^{**} & 0.032 \\
## ma3 & -0.249^{**} & 0.303 & -0.105 & -0.400^{***} \\
## ma4 & -0.054 & 0.156 & -0.662^{***} & -0.867^{***} \\
## intercept & -0.0001 & & & \\
## omega & & 0.0001^{***} & 0.00000^{***} & 0.0001^{**} \\
## alpha1 & & 0.192^{***} & 0.072^{***} & 0.068^{***} \\
## alpha2 & & 0.118^{***} & & \\
## alpha3 & & 0.117^{***} & & \\
## alpha4 & & 0.145^{***} & & \\
## alpha5 & & 0.074^{***} & & \\
## alpha6 & & 0.084^{***} & & \\
## alpha7 & & 0.096^{***} & & \\
## alpha8 & & 0.068^{***} & & \\
## gamma1 & & & & 0.301^{***} \\
## beta1 & & & 0.924^{***} & 0.946^{***} \\
## skew & & & 0.929^{***} & 0.915^{***} \\
## shape & & & 4.523^{***} & 4.805^{***} \\
## \hline \\[-1.8ex]
## Observations & \multicolumn{1}{c}{3,999} & \multicolumn{1}{c}{3,999} & \multicolumn{1}{c}{3,999} & \multicolumn{1}{c}{3,999} \\
## Log Likelihood & \multicolumn{1}{c}{10,096.520} & \multicolumn{1}{c}{-10,671.020} & \multicolumn{1}{c}{-10,938.850} & \multicolumn{1}{c}{-10,969.880} \\
## $\sigma^{2}$ & \multicolumn{1}{c}{0.0004} & & & \\
## Akaike Inf. Crit. & \multicolumn{1}{c}{-20,173.040} & \multicolumn{1}{c}{-5.328} & \multicolumn{1}{c}{-5.464} & \multicolumn{1}{c}{-5.479} \\
## Bayesian Inf. Crit. & & \multicolumn{1}{c}{-5.300} & \multicolumn{1}{c}{-5.442} & \multicolumn{1}{c}{-5.455} \\
## \hline
## \hline \\[-1.8ex]
## \textit{Note:} & \multicolumn{4}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\
## \end{tabular}
## \end{table}(g) Reestimate the best model from (a)-(e) using rugarch package. Create and plot 100 rolling 1 day ahead forecasts
re_model_spec <- ugarchspec(mean.model=list(armaOrder=c(4,4), archm=TRUE, archpow=2, include.mean=TRUE),
variance.model=list(model="sGARCH", garchOrder=c(1,1)))
re_model_spec##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : sGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(4,0,4)
## Include Mean : TRUE
## GARCH-in-Mean : TRUE
##
## Conditional Distribution
## ------------------------------------
## Distribution : norm
## Includes Skew : FALSE
## Includes Shape : FALSE
## Includes Lambda : FALSEre_model <- ugarchfit(data=dl.microsoft_close, spec=re_model_spec, out.sample=200)
re_model##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(4,0,4)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001159 0.000331 -3.503513 0.000459
## ar1 0.998218 0.377134 2.646855 0.008124
## ar2 -0.021385 0.384726 -0.055586 0.955672
## ar3 -0.343943 0.363359 -0.946566 0.343860
## ar4 -0.117405 0.350843 -0.334636 0.737900
## ma1 -1.034100 0.378842 -2.729634 0.006340
## ma2 0.054028 0.388679 0.139004 0.889447
## ma3 0.377676 0.363384 1.039329 0.298652
## ma4 0.048616 0.344056 0.141302 0.887631
## archm 2.739474 1.178752 2.324046 0.020123
## omega 0.000012 0.000001 15.908190 0.000000
## alpha1 0.148011 0.006777 21.838754 0.000000
## beta1 0.826811 0.010524 78.563412 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.001159 0.000389 -2.976987 0.002911
## ar1 0.998218 0.624798 1.597666 0.110117
## ar2 -0.021385 0.486650 -0.043944 0.964949
## ar3 -0.343943 0.436681 -0.787631 0.430913
## ar4 -0.117405 0.587893 -0.199704 0.841712
## ma1 -1.034100 0.626498 -1.650605 0.098819
## ma2 0.054028 0.487139 0.110909 0.911689
## ma3 0.377676 0.426412 0.885707 0.375775
## ma4 0.048616 0.574042 0.084690 0.932508
## archm 2.739474 1.128350 2.427858 0.015188
## omega 0.000012 0.000002 6.923694 0.000000
## alpha1 0.148011 0.015099 9.802975 0.000000
## beta1 0.826811 0.016746 49.374869 0.000000
##
## LogLikelihood : 10281.66
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.4060
## Bayes -5.3846
## Shibata -5.4060
## Hannan-Quinn -5.3984
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1438 0.7046
## Lag[2*(p+q)+(p+q)-1][23] 6.2122 1.0000
## Lag[4*(p+q)+(p+q)-1][39] 16.0648 0.8884
## d.o.f=8
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4409 0.5067
## Lag[2*(p+q)+(p+q)-1][5] 2.2596 0.5585
## Lag[4*(p+q)+(p+q)-1][9] 3.3349 0.7026
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1855 0.500 2.000 0.6667
## ARCH Lag[5] 1.1439 1.440 1.667 0.6908
## ARCH Lag[7] 1.5234 2.315 1.543 0.8167
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 20.3802
## Individual Statistics:
## mu 0.09248
## ar1 0.20516
## ar2 0.53657
## ar3 0.33501
## ar4 0.14497
## ma1 0.14367
## ma2 0.51698
## ma3 0.34966
## ma4 0.16415
## archm 0.04318
## omega 3.04198
## alpha1 0.10461
## beta1 0.11251
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.3004 0.7639
## Negative Sign Bias 0.2703 0.7869
## Positive Sign Bias 0.2255 0.8216
## Joint Effect 0.1310 0.9879
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 169.8 3.055e-26
## 2 30 182.2 3.926e-24
## 3 40 190.9 6.742e-22
## 4 50 209.7 9.130e-22
##
##
## Elapsed time : 4.29943re_model.fcst <- ugarchforecast(re_model, n.ahead=10, n.roll=100)
plot(re_model.fcst, which="all")
The trend of forecast is on average same as the real value for the 100 rolling 1 day ahead.