Need help with all of 1 In this part, you may choose any one problem to do. Plea
ID: 3229988 • Letter: N
Question
Need help with all of 1 In this part, you may choose any one problem to do. Please do not do all of the problems. If anyone provides answers to more than one problem, then one answer chosen at random will be graded. Generate a time series {Y_t} with length 2000 from the following model: Y_t = sigma _t u_t, sigma^2 _t = 1 + 0.5Y^2 _t-1, where {u_t} are iid normal with mean 0 and standard deviation one. (i) Plot the time series (Y_t) as well as Y^2 _t. (ii) Plot the PACFs of the two series in (i). (iii) Identify an appropriate model for the sample of and write down your estimated model. (iv) Use ARIMA(.) with Maximum Likelihood Estimation (MLE) to estimate the parameter of the AR(I)-process {Y^2 _t} and report your estimated model. (v) Provide your R codes. Generate a me series {Y_t} with length 2000 from the following model: Y_t = 0.5Y_t-1 + element_t - 0.5 element t-1. where (u_t) are iid normal with mean 0 and standard deviation one. (a) Plot the time series (Y_t) (b) Plot the ACF aand PACF of the series. (c) Identify an appropriate model (AP) for the sample of Y_t. Why do you choose the AP model? (d) Write down your estimated model in (c). (e) Do regression diagnostics for the estimated model in (d) That is, (1) to present Q-Q plot the residuals; 2) to present standard residual plot: 3) to present the ACF of the residuals (f) Provide your R code.
Explanation / Answer
1.
rm(list=ls(all=TRUE))
y=c(); s1=c();
y0=1 ###### start point
s1=1+0.5*(y0^2) ###### generate sigma^2
for( i in 1:2000){
y[i]=sqrt(s1[i])*rnorm(1,0,1); ######## series generate
s1[i+1]=1+0.5*((y[i])^2); ###### generate sigma^2
}
par(mfrow=c(2,1))
plot(y, type="l") ### plot y
plot(y^2,type="l") ### plot y^2
(i)
(ii) R-code
pacf(y)
pacf(y^2)
(iii)
xx=auto.arima(y) ##### find the model
Series: y
ARIMA(1,0,2) with zero mean
Coefficients:
ar1 ma1 ma2
0.6389 -0.5741 -0.0815
s.e. 0.2161 0.2157 0.0225
sigma^2 estimated as 2.169: log likelihood=-3610.55
AIC=7229.1 AICc=7229.12 BIC=7251.5
(iv)
xy=arima(y^2, order=c(1,0,0)) #### estimate AR(1) model
Call:
arima(x = y^2, order = c(1, 0, 0))
Coefficients:
ar1 intercept
0.3809 1.8051
s.e. 0.0207 0.1094
sigma^2 estimated as 9.189: log likelihood = -5055.98, aic = 10115.96