MATH1318 Semester 1, 2018
Time Series - Final Project(Competitve)
s3650497- Mohammad,s3689517-Malgorzata Sikora,s3638787 - Ravi Pandey
Importing Data
Bitcoin <- read.csv("Bitcoin_Historical_Price.csv")
head(Bitcoin)## Close
## 1 134.21
## 2 144.54
## 3 139.00
## 4 116.99
## 5 105.21
## 6 97.75Converting to Time series Object
Bitcoin.ts<- ts(Bitcoin,start=c(2013,4,27),frequency=365.25)
class(Bitcoin.ts)## [1] "ts"head(Bitcoin.ts)## Time Series:
## Start = 2013.00821355236
## End = 2013.0219028063
## Frequency = 365.25
## Close
## [1,] 134.21
## [2,] 144.54
## [3,] 139.00
## [4,] 116.99
## [5,] 105.21
## [6,] 97.75plot(Bitcoin.ts,ylab='Price in (USD)',xlab='Year', main = "Time series plot for Bitcoin Price")
Random Walk(Linear Regression Model)
model1 = lm(Bitcoin.ts~time(Bitcoin.ts))
summary(model1)##
## Call:
## lm(formula = Bitcoin.ts ~ time(Bitcoin.ts))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2462.4 -1658.8 -558.4 1049.6 15182.3
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.522e+06 8.502e+04 -29.66 <2e-16 ***
## time(Bitcoin.ts) 1.252e+03 4.218e+01 29.68 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2487 on 1770 degrees of freedom
## Multiple R-squared: 0.3322, Adjusted R-squared: 0.3319
## F-statistic: 880.7 on 1 and 1770 DF, p-value: < 2.2e-16Added the fitted least squares line from model1
plot(Bitcoin.ts, ylab='Price in (USD)', main = "Time series plot for simulated linear series")
abline(model1) 
Quadratic Model
t = time(Bitcoin.ts)
t2 = t^2
model1.1 = lm(Bitcoin.ts~t+t2) # label the model as model1
summary(model1.1)##
## Call:
## lm(formula = Bitcoin.ts ~ t + t2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3041.0 -1317.1 248.7 893.0 12418.7
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.834e+09 1.020e+08 37.59 <2e-16 ***
## t -3.806e+06 1.012e+05 -37.60 <2e-16 ***
## t2 9.444e+02 2.511e+01 37.61 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1854 on 1769 degrees of freedom
## Multiple R-squared: 0.629, Adjusted R-squared: 0.6285
## F-statistic: 1499 on 2 and 1769 DF, p-value: < 2.2e-16plot(ts(fitted(model1.1)), ylim = c(min(c(fitted(model1.1),
as.vector(Bitcoin.ts))), max(c(fitted(model1.1),as.vector(Bitcoin.ts)))),ylab='Price in (USD)' ,
main = "Fitted quadratic curve to linear data")
lines(as.vector(Bitcoin.ts))
Cyclical or Seasonal Trends
Bitcoin.ts1=(ts(Bitcoin.ts, start = 2013,end = 2018, frequency = 12))
month.=season(Bitcoin.ts1)
model2=lm(Bitcoin.ts1~month.-1)
summary(model2)##
## Call:
## lm(formula = Bitcoin.ts1 ~ month. - 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.972 -8.364 -1.266 7.398 23.058
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## month.January 116.492 4.835 24.09 <2e-16 ***
## month.February 121.482 5.296 22.94 <2e-16 ***
## month.March 122.110 5.296 23.06 <2e-16 ***
## month.April 118.470 5.296 22.37 <2e-16 ***
## month.May 113.988 5.296 21.52 <2e-16 ***
## month.June 112.034 5.296 21.15 <2e-16 ***
## month.July 113.952 5.296 21.52 <2e-16 ***
## month.August 117.176 5.296 22.12 <2e-16 ***
## month.September 116.220 5.296 21.94 <2e-16 ***
## month.October 114.676 5.296 21.65 <2e-16 ***
## month.November 114.564 5.296 21.63 <2e-16 ***
## month.December 112.364 5.296 21.22 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.84 on 49 degrees of freedom
## Multiple R-squared: 0.9917, Adjusted R-squared: 0.9897
## F-statistic: 489.2 on 12 and 49 DF, p-value: < 2.2e-16model3=lm(Bitcoin.ts1~month.)
summary(model3)##
## Call:
## lm(formula = Bitcoin.ts1 ~ month.)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.972 -8.364 -1.266 7.398 23.058
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 116.4917 4.8348 24.094 <2e-16 ***
## month.February 4.9903 7.1712 0.696 0.490
## month.March 5.6183 7.1712 0.783 0.437
## month.April 1.9783 7.1712 0.276 0.784
## month.May -2.5037 7.1712 -0.349 0.728
## month.June -4.4577 7.1712 -0.622 0.537
## month.July -2.5397 7.1712 -0.354 0.725
## month.August 0.6843 7.1712 0.095 0.924
## month.September -0.2717 7.1712 -0.038 0.970
## month.October -1.8157 7.1712 -0.253 0.801
## month.November -1.9277 7.1712 -0.269 0.789
## month.December -4.1277 7.1712 -0.576 0.568
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.84 on 49 degrees of freedom
## Multiple R-squared: 0.07783, Adjusted R-squared: -0.1292
## F-statistic: 0.3759 on 11 and 49 DF, p-value: 0.9594Cosine Models(Harmonic Models)
har.=harmonic(Bitcoin.ts,1) # calculate cos(2*pi*t) and sin(2*pi*t)
model4=lm(Bitcoin.ts~har.)
summary(model4)##
## Call:
## lm(formula = Bitcoin.ts ~ har.)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2439.0 -1665.5 -718.9 44.3 16814.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1543.40 69.68 22.149 < 2e-16 ***
## har.cos(2*pi*t) -591.69 99.42 -5.951 3.2e-09 ***
## har.sin(2*pi*t) -985.67 97.64 -10.095 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2931 on 1769 degrees of freedom
## Multiple R-squared: 0.07325, Adjusted R-squared: 0.07221
## F-statistic: 69.92 on 2 and 1769 DF, p-value: < 2.2e-16##Plotting Harmonic models
plot(ts(fitted(model4),freq=365,start=c(2013,1)),ylab='Price in (USD)',type='l',
ylim=range(c(fitted(model4),Bitcoin.ts)),main="Fitted model to average monthly Bitcoin Price series")
points(Bitcoin.ts)
Stationarity Through Differencing(White noise is generated from the standard normal distribution)
plot(Bitcoin.ts,ylab=expression(Y[t]),type='o')
Time series plot of the differences of logarithms of that series
par(mfrow=c(2,1))
par(cex=0.7)
plot(Bitcoin.ts, ylab='Price in (USD)',type='l')
first.diff = diff(log(Bitcoin.ts))*100
plot(first.diff,ylab='Change in Log(Price)',type='l')
abline(h=0)
Natural log transformation
plot(log(Bitcoin.ts),ylab='Log(Bitcoin)',
main = "Time series plot of logarithms of Bitcoin values.")
Differences of the logarithms of the Bitcoin values
plot(diff(log(Bitcoin.ts)), ylab='Difference of Log(Bitcoin)',
main = "Difference of logarithms for Bitcoin series.")
#ADF
adf.test(first.diff)## Warning in adf.test(first.diff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: first.diff
## Dickey-Fuller = -10.167, Lag order = 12, p-value = 0.01
## alternative hypothesis: stationaryARMA
Significant Lags can be seen in ACF with first significant lag in PACF. From Eacf, we got ARMA(1,1), ARMA (1,2)
#Arma (1,1) (1,2)
par(mfrow=c(1,2))
acf(as.vector(log(Bitcoin.ts)),xaxp=c(0,24,12), main="ACF Bitcoin(USD) Price Time Series.")
pacf(as.vector(log(Bitcoin.ts)),xaxp=c(0,24,12), main=" PACF Bitcoin(USD) Price Time Series.")
eacf(first.diff)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 o o o o x x o o o x x o o o
## 1 x o o o o x o o o o x o o o
## 2 x x o o o x o o o o x o o o
## 3 x x x o o x o o o o x o o o
## 4 x x o x o x o o o o x o o o
## 5 x x x x x o o o o o x o o o
## 6 x x o x x x o o o o o o o o
## 7 x x o x x x x o o o o o o oACF and PACF with absolute values and returns
There are significant lags in both ACF and PACF with deteoriating trend . From Eacf we get ARMA(2,2)& ARMA(2,3)
#ARMA(2,2)(2,3)
par(mfrow=c(1,2))
acf(abs(first.diff),main="ACF of the \n
Absolute Daily Bitcoin Price")
pacf(abs(first.diff),main="PACF of the Absolute Daily Bitcoin Price")
eacf(abs(first.diff))## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x x x x x
## 1 x o o x x x o o o x o o o o
## 2 x x o o o o o o o x o o o o
## 3 x x x o o o o o o x o o o o
## 4 x x x x o o o o o x o o o o
## 5 x x x x x o o o o o o o o o
## 6 x x x x x o o o o o o o o o
## 7 x o x x x x x x o o o o o oACF and PACF for the squared absolute values
It was squared and we found out that there were no obvious models due to volitality clustering which is more visible for the squared values there is fuziness due to high variablilty
par(mfrow=c(1,2))
acf(first.diff^2,main="ACF(Squared Daily Bitcoin Price)")
pacf(first.diff^2,main="PACF(Squared Daily Bitcoin Price)")
eacf(first.diff^2)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x x x x x
## 1 x x x x x x o x o x o o x o
## 2 x x o o o x o o o x o o x o
## 3 x x x o o x o o o x o o x o
## 4 x x x o o o o o o o o o x o
## 5 x x x o x x o o o o o o o o
## 6 x x x x x x o o o o o o o o
## 7 x x x o x x x x o o o o o oMcLeod Li Test
McLeod.Li.test(y=first.diff,main="McLeod-Li Test Statistics for Daily Bitcoin Price")
#Shapiro Test
qqnorm(first.diff,main="Q-Q Normal Plot of Daily Bitcoin Price")
qqline(first.diff)
shapiro.test(first.diff)##
## Shapiro-Wilk normality test
##
## data: first.diff
## W = 0.87737, p-value < 2.2e-16Estimation of Parameters
model1 = garch(first.diff,order=c(2,2),trace = FALSE)
summary(model1) # All the coefficients are significant at 5% level of significance.##
## Call:
## garch(x = first.diff, order = c(2, 2), trace = FALSE)
##
## Model:
## GARCH(2,2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.24961 -0.26953 0.04809 0.44019 5.76487
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## a0 1.619e+01 1.322e+00 12.249 < 2e-16 ***
## a1 1.538e-01 1.525e-02 10.084 < 2e-16 ***
## a2 1.125e-01 2.701e-02 4.165 3.11e-05 ***
## b1 2.788e-13 1.150e-01 0.000 1.000
## b2 7.301e-03 5.967e-02 0.122 0.903
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Diagnostic Tests:
## Jarque Bera Test
##
## data: Residuals
## X-squared = 2757.7, df = 2, p-value < 2.2e-16
##
##
## Box-Ljung test
##
## data: Squared.Residuals
## X-squared = 6.9254, df = 1, p-value = 0.008498model1.1 = garchFit(formula = ~garch(2,2), data =first.diff )##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 2)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 2 2
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 1771
## Recursion Init: mci
## Series Scale: 4.499265
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.55869489 0.5586949 0.05586949 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.05000000 TRUE
## alpha2 0.00000001 1.0000000 0.05000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## gamma2 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.40000000 TRUE
## beta2 0.00000001 1.0000000 0.40000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 alpha2 beta1 beta2
## 1 2 3 4 7 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 2276.2288: 0.0558695 0.100000 0.0500000 0.0500000 0.400000 0.400000
## 1: 2247.9835: 0.0558672 0.0773139 0.0552636 0.0495434 0.387486 0.387320
## 2: 2229.9289: 0.0558634 0.0730970 0.0784165 0.0648966 0.393423 0.393248
## 3: 2218.0767: 0.0558605 0.0533761 0.0800700 0.0627795 0.384556 0.384243
## 4: 2205.0845: 0.0558535 0.0502299 0.0981058 0.0738019 0.391533 0.391348
## 5: 2202.4562: 0.0558501 0.0377075 0.100118 0.0728041 0.387744 0.387514
## 6: 2196.6705: 0.0558411 0.0367573 0.109779 0.0765010 0.394045 0.394093
## 7: 2196.0359: 0.0558096 0.0159333 0.124401 0.0725377 0.400714 0.401552
## 8: 2193.7579: 0.0557909 0.0255357 0.133124 0.0688645 0.402487 0.403974
## 9: 2190.5286: 0.0557640 0.0247611 0.141623 0.0609174 0.396879 0.399154
## 10: 2188.3260: 0.0556220 0.0226784 0.164804 0.0241805 0.398597 0.407603
## 11: 2188.2937: 0.0556195 0.0231320 0.165737 0.0245337 0.399285 0.408409
## 12: 2188.1837: 0.0555974 0.0218661 0.165908 0.0242648 0.398880 0.408956
## 13: 2188.1089: 0.0555267 0.0214863 0.167096 0.0243548 0.398238 0.411347
## 14: 2188.0100: 0.0553698 0.0210424 0.169106 0.0247048 0.393661 0.413343
## 15: 2187.9338: 0.0552215 0.0220237 0.171160 0.0252734 0.389174 0.415466
## 16: 2187.3061: 0.0507429 0.0165422 0.166222 0.0128693 0.283248 0.542611
## 17: 2187.0140: 0.0487850 0.0162417 0.171257 0.0107668 0.228997 0.588527
## 18: 2186.7091: 0.0467236 0.0189561 0.178438 0.0112072 0.177608 0.635940
## 19: 2186.2440: 0.0462731 0.0164278 0.183094 0.0149871 0.202007 0.610285
## 20: 2186.1120: 0.0462693 0.0173464 0.182534 0.0139261 0.199742 0.607798
## 21: 2185.9488: 0.0462196 0.0188431 0.182685 0.0141824 0.202405 0.605955
## 22: 2185.8951: 0.0460175 0.0185454 0.182871 0.0140651 0.201961 0.605520
## 23: 2185.8338: 0.0456243 0.0194903 0.183996 0.0143936 0.200350 0.605738
## 24: 2185.7518: 0.0448482 0.0190714 0.185207 0.0143695 0.196300 0.608332
## 25: 2185.5431: 0.0413050 0.0201658 0.186304 0.0130331 0.211125 0.592379
## 26: 2185.4015: 0.0384602 0.0200734 0.192670 0.0142110 0.189568 0.609189
## 27: 2185.3880: 0.0384596 0.0200201 0.192798 0.0143981 0.189140 0.608590
## 28: 2185.3731: 0.0384943 0.0203729 0.192990 0.0146102 0.189093 0.608605
## 29: 2185.3590: 0.0385694 0.0202934 0.193226 0.0148692 0.188580 0.608172
## 30: 2185.3404: 0.0387366 0.0206605 0.193632 0.0152358 0.188179 0.608140
## 31: 2185.3147: 0.0390718 0.0206960 0.194319 0.0157919 0.187089 0.607666
## 32: 2185.2064: 0.0407879 0.0221875 0.198697 0.0189990 0.182030 0.605577
## 33: 2185.1177: 0.0369560 0.0229093 0.198816 0.0229007 0.181939 0.601478
## 34: 2185.1004: 0.0367706 0.0227510 0.202997 0.0186907 0.192939 0.591298
## 35: 2185.0929: 0.0375845 0.0231102 0.203887 0.0237792 0.190509 0.589493
## 36: 2185.0864: 0.0367108 0.0235883 0.204958 0.0232956 0.186372 0.592255
## 37: 2185.0858: 0.0368695 0.0234028 0.204561 0.0224509 0.187816 0.591901
## 38: 2185.0858: 0.0368549 0.0234244 0.204606 0.0225859 0.187632 0.591936
## 39: 2185.0858: 0.0368548 0.0234241 0.204605 0.0225851 0.187631 0.591938
##
## Final Estimate of the Negative LLH:
## LLH: 4848.518 norm LLH: 2.737729
## mu omega alpha1 alpha2 beta1 beta2
## 0.1658196 0.4741829 0.2046053 0.0225851 0.1876306 0.5919379
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 alpha2 beta1
## mu -198.2453653 -3.392661 30.04548 59.84732 11.09326
## omega -3.3926605 -288.583367 -1273.75572 -1384.59887 -2419.61436
## alpha1 30.0454832 -1273.755722 -10320.37133 -10738.82758 -15318.68352
## alpha2 59.8473165 -1384.598874 -10738.82758 -13459.23250 -17398.10908
## beta1 11.0932638 -2419.614362 -15318.68352 -17398.10908 -26558.75189
## beta2 -0.2774075 -2465.219489 -15517.94037 -17222.51636 -26865.79322
## beta2
## mu -2.774075e-01
## omega -2.465219e+03
## alpha1 -1.551794e+04
## alpha2 -1.722252e+04
## beta1 -2.686579e+04
## beta2 -2.749136e+04
## attr(,"time")
## Time difference of 0.05321717 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.2160699 secssummary(model1.1)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(2, 2), data = first.diff)
##
## Mean and Variance Equation:
## data ~ garch(2, 2)
## <environment: 0x7ff4b6e29948>
## [data = first.diff]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 beta1 beta2
## 0.165820 0.474183 0.204605 0.022585 0.187631 0.591938
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.16582 0.07128 2.326 0.02000 *
## omega 0.47418 0.14796 3.205 0.00135 **
## alpha1 0.20461 0.02995 6.832 8.38e-12 ***
## alpha2 0.02259 0.03160 0.715 0.47477
## beta1 0.18763 0.08186 2.292 0.02189 *
## beta2 0.59194 0.06674 8.869 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -4848.518 normalized: -2.737729
##
## Description:
## Tue May 29 00:55:15 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 5385.597 0
## Shapiro-Wilk Test R W 0.9063864 0
## Ljung-Box Test R Q(10) 37.96593 3.847341e-05
## Ljung-Box Test R Q(15) 44.46622 9.292405e-05
## Ljung-Box Test R Q(20) 54.66252 4.611051e-05
## Ljung-Box Test R^2 Q(10) 7.565348 0.6712095
## Ljung-Box Test R^2 Q(15) 10.16227 0.8094103
## Ljung-Box Test R^2 Q(20) 12.73401 0.8885027
## LM Arch Test R TR^2 8.543814 0.7413215
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 5.482233 5.500797 5.482210 5.489091model2 = garch(first.diff,order=c(2,3),trace = FALSE)
summary(model2 )# All the coefficients but aplha_2 are significant at 5% level of significance.##
## Call:
## garch(x = first.diff, order = c(2, 3), trace = FALSE)
##
## Model:
## GARCH(2,3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.33771 -0.27766 0.04911 0.45160 5.94652
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## a0 1.518e+01 2.209e+00 6.873 6.3e-12 ***
## a1 1.459e-01 1.428e-02 10.213 < 2e-16 ***
## a2 9.227e-02 3.074e-02 3.002 0.00269 **
## a3 7.683e-02 2.490e-02 3.085 0.00203 **
## b1 2.133e-13 1.668e-01 0.000 1.00000
## b2 1.179e-02 6.737e-02 0.175 0.86104
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Diagnostic Tests:
## Jarque Bera Test
##
## data: Residuals
## X-squared = 2928.7, df = 2, p-value < 2.2e-16
##
##
## Box-Ljung test
##
## data: Squared.Residuals
## X-squared = 8.2223, df = 1, p-value = 0.004138model2.1 = garchFit(formula = ~garch(3,2), data =first.diff, trace = FALSE )
summary(model2.1)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(3, 2), data = first.diff, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ garch(3, 2)
## <environment: 0x7ff4ba137660>
## [data = first.diff]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 alpha3 beta1
## 0.16218100 0.45510254 0.20771297 0.01217754 0.00000001 0.20654062
## beta2
## 0.57978910
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.622e-01 7.140e-02 2.271 0.0231 *
## omega 4.551e-01 1.837e-01 2.478 0.0132 *
## alpha1 2.077e-01 3.337e-02 6.225 4.82e-10 ***
## alpha2 1.218e-02 3.208e-02 0.380 0.7042
## alpha3 1.000e-08 5.006e-02 0.000 1.0000
## beta1 2.065e-01 8.814e-02 2.343 0.0191 *
## beta2 5.798e-01 7.864e-02 7.373 1.67e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -4848.6 normalized: -2.737775
##
## Description:
## Tue May 29 00:55:16 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 5336.825 0
## Shapiro-Wilk Test R W 0.9065387 0
## Ljung-Box Test R Q(10) 37.71507 4.253901e-05
## Ljung-Box Test R Q(15) 44.20358 0.0001021775
## Ljung-Box Test R Q(20) 54.35771 5.118887e-05
## Ljung-Box Test R^2 Q(10) 7.561231 0.6716092
## Ljung-Box Test R^2 Q(15) 10.07275 0.8151391
## Ljung-Box Test R^2 Q(20) 12.66779 0.8911786
## LM Arch Test R TR^2 8.559354 0.7400351
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 5.483456 5.505113 5.483425 5.491457Residual Analysis
The residual analysis for both the models are almost same, by lookin at the time series of the residual analysis we can say that the series is stochastic and without a trend, the histogram is almost normally distributed and the PACF and ACF plots have significant lags and the qq plot for standardised residuals is normally distributed
residual.analysis(model1,class="GARCH",start=3)##
## Shapiro-Wilk normality test
##
## data: res.model
## W = 0.90436, p-value < 2.2e-16## Warning in (ra^2)/(n - (1:lag.max)): longer object length is not a multiple
## of shorter object length
residual.analysis(model2,class="GARCH",start=4)##
## Shapiro-Wilk normality test
##
## data: res.model
## W = 0.90502, p-value < 2.2e-16## Warning in (ra^2)/(n - (1:lag.max)): longer object length is not a multiple
## of shorter object length
Shapiro Test
shapiro.test(first.diff)##
## Shapiro-Wilk normality test
##
## data: first.diff
## W = 0.87737, p-value < 2.2e-16Forecast
forecast <- fGarch::predict(model2.1,n.ahead=10,trace=FALSE,plot=TRUE)