library(readxl)
library(tseries)
library(zoo)
library(forecast)
setwd("C:/Users/User/Google 雲端硬碟/政治大學ECO/時間序列/作業3")
ts<-read_excel("ARCHQ6.xlsx")

6. The second series on the file ARCH.XLS contains 100 observations of a simulated ARCH-M process.

a.

Estimate the {yt} sequence using the Box–Jenkins methodology. Try to improve on the model.

a.ma<-arima(ts$y,order = c(0,0,6),transform.pars = F,fixed = c(0,0,NA,0,0,NA,NA))
acf(a.ma$residuals)

pacf(a.ma$residuals)

acf(ts)

By the acf and pacf of residuals, we find the residual are not very clean.By the ACF of sequence, I think the model MA(||3,6||) not actually catch all imformation of the sequence. We can find the values of lag(9) and lag(13) are over critical values. Maybe we can change model to MA(||3,6,9,13||).

a.ma2<-arima(ts$y,order = c(0,0,13),transform.pars = F,fixed = c(0,0,NA,0,0,NA,0,0,NA,0,0,0,NA,NA))
acf(a.ma2$residuals)

pacf(a.ma2$residuals)

By the new model, the residuals are more clean.

b.

Examine the ACF and the PACF of the residuals from the MA(∥3, 6∥) model above.Why might someone conclude that the residuals appear to be white noise? Now examine the ACF and PACF of the squared residuals.Perform the LM test for ARCH errors.

ma.r2<-a.ma$residuals^2
ma.r2.acf<-acf(ma.r2)

ma.r2.pacf<-pacf(ma.r2)

figure<-matrix(c(ma.r2.acf$acf[2:7],ma.r2.pacf$acf[1:6]),nrow = 2)
row.names(figure)<-c("ACF","PACF")
colnames(figure)<-c(1,2,3,4,5,6)
figure
##              1         2          3           4           5           6
## ACF  0.4980978 0.2921198 0.04671571 0.498097771  0.21432750 -0.03897722
## PACF 0.2555712 0.1625981 0.12106327 0.009934568 -0.08799931  0.10623009
i<-5
LM.test<-lm(ma.r2[i:100]~ma.r2[(i-1):99]+
              ma.r2[(i-2):98]+ma.r2[(i-3):97]+
              ma.r2[(i-4):96])
summary(LM.test)
## 
## Call:
## lm(formula = ma.r2[i:100] ~ ma.r2[(i - 1):99] + ma.r2[(i - 2):98] + 
##     ma.r2[(i - 3):97] + ma.r2[(i - 4):96])
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.88067 -0.15165 -0.08010  0.05116  2.03431 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        0.09505    0.04895   1.942   0.0552 .  
## ma.r2[(i - 1):99]  0.51142    0.10456   4.891  4.3e-06 ***
## ma.r2[(i - 2):98] -0.10674    0.11442  -0.933   0.3533    
## ma.r2[(i - 3):97]  0.26014    0.11443   2.273   0.0254 *  
## ma.r2[(i - 4):96] -0.08829    0.10463  -0.844   0.4010    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3958 on 91 degrees of freedom
## Multiple R-squared:  0.2889, Adjusted R-squared:  0.2576 
## F-statistic: 9.241 on 4 and 91 DF,  p-value: 2.596e-06

We conduct the LM test for ARCH error using four lag. We find the p-values of lag(1) and lag(3) are very low and we reject the null hypothesis. Meanwhile the F-statistic is 9.241 on the DF=(4,91) ,we reject the null hypothesis. There are arch effect.