Advanced Statistics with R code 90% doen just check it and make a summary

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EXPLORATORY ANALYSIS

nobs 469.000000

NAs 0.000000

Minimum -0.166652

Maximum 0.164808

1. Quartile -0.022391

3. Quartile 0.029000

Mean 0.003843

Median 0.005614

Sum 1.802469

SE Mean 0.002079

LCL Mean -0.000242

UCL Mean 0.007929

Variance 0.002027

Stdev 0.045027

Skewness -0.040829

Kurtosis 1.569306

C:\Users\XXX\Desktop\PLOTS\Rplot.jpeg 

Title:
 Jarque - Bera Normalality Test
Test Results:


  STATISTIC:


    X-squared: 49.4632


  P VALUE:


    Asymptotic p Value: 1.816e-11 





The distribution of log(returns) is not normal distribution, with symmetry close to 0 but kurtosis statistics is 1.5 which is far from normal distribution statistic. The Jarque-Bera test is significant (p-value <  0.05) indicating that hypothesis of normal distribution for log(returns) can be rejected. 





The following plot is time series plot of weekly price of Google shares which shows increasing trend.


 (
Time series Plot of Google Prices
)



 Time plot of returns show that google stock returns shows periods of high volatility at various times. Most returns are between +/- 10%. Sample moments show that distribution of log returns is somewhat symmetric with very thin tails (kurtosis = 1.56). There is various times where the returns produce +/-15% with higher volatility also. Volatility doesn’t seem to die down quickly and it continues for a while after a volatile period.






C:\Users\XXX\Desktop\PLOTS\Rplot03.jpeg






To check null hypothesis of non stationarity:






Dickey Fuller Test is expressed as H0: φ1=1 vs Ha: φ1<>1 where the null hypothesis indicates unit-root non-stationarity. 




The Dickey-Fuller test shows that the earnings time series is unit-root stationary. The test p-values for lags 7 is less than 0.05, therefore the null hypothesis of non stationarity can be rejected (small p-values< 0.05).





Augmented Dickey-Fuller Test


Test Results:


  PARAMETER:


    Lag Order: 7


  STATISTIC:


    Dickey-Fuller: -7.1331


  P VALUE:


    0.01 


alternative hypothesis: stationary


ACF TESTS


The ACF plots below show the Google stock returns are not correlated indicating a constant mean model for rt. Both the squared returns time series and the absolute time series show large autocorrelations. We can conclude that the log returns process has a strong non-linear dependence.



C:\Users\XXX\Desktop\PLOTS\Rplot02.jpeg




Ljung Box testing for ARCH effects for Lags of 6-12-18 for squared returns and absolute returns





Box-Ljung test


data:  coredata(rets^2)


X-squared = 48.3339, df = 6, p-value = 1.013e-08


Box-Ljung test


data:  coredata(rets^2)


X-squared = 71.9976, df = 12, p-value = 1.352e-10


Box-Ljung test


data:  coredata(rets^2)


X-squared = 98.7945, df = 18, p-value = 3.68e-13





The Ljung Box tests confirm that the Ljung Box tests on the squared returns are autocorrelated. The analysis shows a strong ARCH effect. The squared returns are strongly correlated. The Ljung-Box tests on squared residuals are highly significant with p-values < 0.005, and autocorrelation plots shows large autocorrelations for the first 18 lags.



Plot of PACF: There is no correlation till lag 13 which shows there is no AR lag.



C:\Users\XXX\Desktop\PLOTS\Rplot04.jpeg
















MODEL FITTING





*---------------------------------*


*          GARCH Model Fit        *


*---------------------------------*


Conditional Variance Dynamics 	


-----------------------------------


GARCH Model	: sGARCH(1,1)


Mean Model	: ARFIMA(0,0,0)


Distribution	: std 


Optimal Parameters


------------------------------------


        Estimate  Std. Error  t value Pr(>|t|)


mu      0.004432    0.001778   2.4924 0.012689


omega   0.000103    0.000053   1.9574 0.050295


alpha1  0.094169    0.032716   2.8784 0.003997


beta1   0.856326    0.044635  19.1850 0.000000


shape   6.797338    2.000925   3.3971 0.000681





To fit an ARMA(0,0)-GARCH(2,1) model for the log returns using a normal distribution for the error terms:





Fitted Model:


 


 


  with 7 degrees of freedom



Residual Analysis



Q-Statistics on Standardized Residuals


------------------------------------


              statistic p-value


Lag[1]           0.0339  0.8539


Lag[p+q+1][1]    0.0339  0.8539


Lag[p+q+5][5]    2.4766  0.7800


d.o.f=0


H0 : No serial correlation


Q-Statistics on Standardized Squared Residuals


------------------------------------


              statistic p-value


Lag[1]           0.1274  0.7212


Lag[p+q+1][3]    1.4245  0.2327


Lag[p+q+5][7]    3.4346  0.6333


d.o.f=2


Above output shows there is no evidence of autocorrelation in residuals. They behave like white noise. Also, there is no evidence of serial correlation in squared residuals. They also behave like white noise. 


Adjusted Pearson Goodness-of-Fit Test:


------------------------------------


  group statistic p-value(g-1)


1    20      8465            0


2    30     13148            0


3    40     17834            0


4    50     22521            0




Test for Goodness of fit. Normal Distribution rejected. 



To fit ARMA(0,0)-eGARCH(1,1) model with Gaussian distribution:





Conditional Variance Dynamics 	


-----------------------------------


GARCH Model	: eGARCH(1,1)


Mean Model	: ARFIMA(0,0,0)


Distribution	: norm 


Optimal Parameters


------------------------------------


        Estimate  Std. Error  t value Pr(>|t|)


mu      0.004814    0.001837   2.6205 0.008780


omega  -0.427600    0.165861  -2.5781 0.009936


alpha1 -0.086690    0.033886  -2.5583 0.010518


beta1   0.931657    0.026273  35.4604 0.000000


gamma1  0.177187    0.048248   3.6724 0.000240




Fitted Model is:


 


 


 



Residual Analysis:



Q-Statistics on Standardized Residuals


------------------------------------


              statistic p-value


Lag[1]          0.03655  0.8484


Lag[p+q+1][1]   0.03655  0.8484


Lag[p+q+5][5]   2.73813  0.7403


d.o.f=0


H0 : No serial correlation


Q-Statistics on Standardized Squared Residuals


------------------------------------


              statistic p-value


Lag[1]           0.2255  0.6348


Lag[p+q+1][3]    2.6812  0.1015


Lag[p+q+5][7]    4.1362  0.5300


d.o.f=2


ARCH LM Tests


------------------------------------


             Statistic DoF P-Value


ARCH Lag[2]      2.224   2  0.3290


ARCH Lag[5]      4.682   5  0.4559


ARCH Lag[10]     7.053  10  0.7204


Above output shows there is no evidence of autocorrelation in residuals. They behave like white noise. Also, there is no evidence of serial correlation in squared residuals. They also behave like white noise. 


Adjusted Pearson Goodness-of-Fit Test:


------------------------------------


  group statistic p-value(g-1)


1    20     26.65       0.1131


2    30     32.04       0.3179


3    40     43.67       0.2798


4    50     47.74       0.5243


Above output shows that error terms are normally distributed as all the P values are greater than 0.05.



To fit ARMA(0,0)-tGARCH(1,1) model with Gaussian distribution:



Conditional Variance Dynamics 	


-----------------------------------


GARCH Model	: gjrGARCH(2,1)


Mean Model	: ARFIMA(0,0,0)


Distribution	: norm 


Optimal Parameters


------------------------------------


        Estimate  Std. Error   t value Pr(>|t|)


mu      0.004928    0.001860  2.649926 0.008051


omega   0.000120    0.000053  2.257334 0.023987


alpha1  0.000000    0.000877  0.000011 0.999991


alpha2  0.072854    0.037335  1.951388 0.051011


beta1   0.828559    0.053752 15.414339 0.000000


gamma1  0.397469    0.144242  2.755564 0.005859


gamma2 -0.315459    0.131445 -2.399922 0.016399


Fitted Model is:


 


 


 



Residual Analysis:



Q-Statistics on Standardized Residuals


------------------------------------


              statistic p-value


Lag[1]           0.2311  0.6307


Lag[p+q+1][1]    0.2311  0.6307


Lag[p+q+5][5]    2.9412  0.7090


d.o.f=0


H0 : No serial correlation


Q-Statistics on Standardized Squared Residuals


------------------------------------


              statistic p-value


Lag[1]            2.137 0.14380


Lag[p+q+1][4]     4.191 0.04063


Lag[p+q+5][8]     4.680 0.45620


d.o.f=3


ARCH LM Tests


------------------------------------


             Statistic DoF P-Value


ARCH Lag[2]      2.461   2  0.2921


ARCH Lag[5]      4.904   5  0.4277


ARCH Lag[10]     7.140  10  0.7122


Above output shows there is no evidence of autocorrelation in residuals. They behave like white noise. Also, there is no evidence of serial correlation in squared residuals. They also behave like white noise. 


Adjusted Pearson Goodness-of-Fit Test:


------------------------------------


  group statistic p-value(g-1)


1    20     20.00       0.3947


2    30     32.43       0.3014


3    40     30.87       0.8203


4    50     54.56       0.2714




Above output shows that error terms are normally distributed as all the P values are greater than 0.05.




BEST FIT FOR THE MODEL:


               garch11  egarch11 gjrgarch11


Akaike       -3.479720 -3.455538  -3.474405


Bayes        -3.435471 -3.411288  -3.412455


Shibata      -3.479944 -3.455762  -3.474841


Hannan-Quinn -3.462310 -3.438127  -3.450030


From above output it can be said that AR(0)-EGARCH(1,1) have the (smallest) values for all best fit criterion and hence can be considered as best model.






Forecast Analysis: 







RE-FIT MODELS LEAVING 100 OUT-OF-SAMPLE OBSERVATIONS FOR FORECAST and  EVALUATE STATISTICS for 5 Step Ahead Forecast.


  





	

    garch11     egarch11     tgarch21    

	

MSE 0.000950199 0.0009498531 0.0009499473

	

MAE 0.02257449  0.0225407    0.02255243  

	

DAC 0.6         0.6          0.6  










Since MSE is smallest for eGARCH model. We should consider this model for forecasting.







CODE:



#Tomas Georgakopoulos CSC 425 Final Project


setwd("C:/Course/CSC425")




# Analysis of Google weekly returns from


library(forecast)


library(TSA)




# import data in R and compute log returns


# import libraries for TS analysis


library(zoo)  


library(tseries)


myd= read.table('Weekly-GOOG-TSDATA.csv', header=T, sep=',')


pricets = zoo(myd$price, as.Date(as.character(myd$date), format=c("%m/%d/%Y")))




#log return time series


rets = log(pricets/lag(pricets, -1))




# strip off the dates and just create a simple numeric object


ret = coredata(rets);




#compute statistics


library(fBasics) 


basicStats(rets)




#HISTOGRAM


par(mfcol=c(1,2)) 


hist(rets, xlab="Weekly Returns", prob=TRUE, main="Histogram")


#Add approximating normal density curve


xfit<-seq(min(rets,na.rm=TRUE),max(rets,na.rm=TRUE),length=40)


yfit<-dnorm(xfit,mean=mean(rets,na.rm=TRUE),sd=sd(rets,na.rm=TRUE))


lines(xfit, yfit, col="blue", lwd=2) 




#CREATE NORMAL PROBABILITY PLOT


qqnorm(rets)


qqline(rets, col = 'red', lwd=2) 




# creates time plot of log returns


par(mfrow=c(1,1))


plot(rets)




#Perform Jarque-Bera normality test.


normalTest(rets,method=c('jb'))  




#SKEWNESS TEST


skew_test=skewness(rets)/sqrt(6/length(rets))


skew_test


print("P-value = ")


2*(1-pnorm(abs(skew_test)))




#FAT-TAIL TEST


k_stat = kurtosis(rets)/sqrt(24/length(rets))


print("Kurtosis test statistic")


k_stat


print("P-value = ") 


2*(1-pnorm(abs(k_stat)))




#COMPUTE DICKEY-FULLER TEST


library(fUnitRoots) 


adfTest(rets, lags=7, type=c("c"))






# Computes Ljung-Box test on squared returns to test non-linear independence at lag 6 and 12


Box.test(coredata(rets^2),lag=6,type='Ljung')


Box.test(coredata(rets^2),lag=12,type='Ljung')


Box.test(coredata(rets^2),lag=18,type='Ljung')




# Computes Ljung-Box test on absolute returns to test non-linear independence at lag 6 and 12


Box.test(abs(coredata(rets)),lag=6,type='Ljung')


Box.test(abs(coredata(rets)),lag=12,type='Ljung')


Box.test(abs(coredata(rets)),lag=18,type='Ljung')




# Plots ACF function of vector data


par(mfrow=c(3,1))


acf(ret)


# Plot ACF of squared returns to check for ARCH effect


acf(ret^2)


# Plot ACF of absolute returns to check for ARCH effect


acf(abs(ret))






#plot returns, square returns and abs(returns)


# Plots ACF function of vector data


par(mfrow=c(3,1))


plot(rets, type='l')




# Plot ACF of squared returns to check for ARCH effect


plot(rets^2,type='l')




# Plot ACF of absolute returns to check for ARCH effect


plot(abs(rets),type='l')






par(mfrow=c(1,1))


# plots PACF of squared returns to identify order of AR model


pacf(coredata(rets),lag=30)




#GARCH Models


library(rugarch)


#Fit AR(0,0)-GARCH(1,1) model


garch11.spec=ugarchspec(variance.model=list(garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)),distribution.model = "std")


#estimate model


garch11.fit=ugarchfit(spec=garch11.spec, data=rets)


garch11.fit




#Fit AR(0,0)-eGARCH(1,1) model with Gaussian distribution


egarch11.spec=ugarchspec(variance.model=list(model = "eGARCH",garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)))


#estimate model


egarch11.fit=ugarchfit(spec=egarch11.spec, data=rets)


egarch11.fit




#Fit AR(0,0)-TGARCH(1,1) model with norm-distribution


gjrgarch11.spec=ugarchspec(variance.model=list(model = "gjrGARCH",garchOrder=c(2,1)), mean.model=list(armaOrder=c(0,0)), distribution.model = "norm")


#estimate model


gjrgarch11.fit=ugarchfit(spec=gjrgarch11.spec, data=rets)


gjrgarch11.fit




# compare information criteria


model.list = list(garch11 = garch11.fit,


                  egarch11 = egarch11.fit,


                  gjrgarch11 = gjrgarch11.fit)


info.mat = sapply(model.list, infocriteria)


rownames(info.mat) = rownames(infocriteria(garch11.fit))


info.mat




# RE-FIT MODELS LEAVING 100 OUT-OF-SAMPLE OBSERVATIONS FOR FORECAST


# EVALUATION STATISTICS




garch11.fit = ugarchfit(spec=garch11.spec, data=rets, out.sample=100)


egarch11.fit = ugarchfit(egarch11.spec, data=rets, out.sample=100)


tgarch11.fit = ugarchfit(spec=gjrgarch11.spec, data=rets, out.sample=100)




garch11.fcst = ugarchforecast(garch11.fit, n.roll=100, n.ahead=1)


egarch11.fcst = ugarchforecast(egarch11.fit, n.roll=100, n.ahead=1)


tgarch11.fcst = ugarchforecast(tgarch11.fit, n.roll=100, n.ahead=1)




fcst.list = list(garch11=garch11.fcst,


            egarch11=egarch11.fcst,


            tgarch21=tgarch11.fcst)


fpm.mat = sapply(fcst.list, fpm)


fpm.mat