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Your task is to provide a short report using R Markdown to carry out a small simulation study to investigate

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Your task is to provide a short report using R Markdown to carry out a small simulation study to investigate how quickly the sample acf and the sample pacf converge to their theoretical values for different n for a particular ARIMA(2,0,2) model, where φ1 = 1.9, φ2 = 0.95, θ1 = 0.8, θ2 =  0.15, σz = 1 and the white noise is normally distributed. Try n = 25, 50, 100, 500, 1000, 10000, and briefly comment on how close the sample and theoretical values are. You should make sure that you include suitable plots and numerical output, and very importantly please provide comments on what you observe. 

θ1,   θ2, etc. compared to the parameterization of our notes.  This is a potential cause of confusion! Always remember to change the sign in the MA parameters when using R (but not the AR parameters).

HINTS: The following commands simulate an ARIMA(2,0,1) process of length n = 100 with parameters φ1 = 0.1, φ2 = 0.2, θ1 = 0.3, and then provides plots of the time-series, the acf and the pacf: 

par(mfrow=c(2,2))   # put  four  plots  on  one  page

x<-arima.sim(n=100,model=list(ar=c(0.1,0.2),ma= -c(0.3)))

 # simulates  from  the  ARIMA(2,0,1)  model ts.plot(x) 
# plot the time series acf(x)
# plot the acf a1  <-  acf(x,max.lag=20)$acf  
#store acf  values  for  lags  up  to  20 # and plot the acf   pacf(x)     
# plot the pacf p1  <-  pacf(x,max.lag=20)$acf  
#store pacf  values  for  lags  up  to  20
# and  plot  the  pacf  The theoretical values of an ARMA acf or pacf for lags up to h = 20 can be found from the commands 

ta1 <- ARMAacf(ar=c(0.1,0.2),ma= -c(0.3),lag.max=20)   ta1

#  theoretical  acf

tp1  <-  ARMAacf(ar=c(0.1,0.2),ma=  -c(0.3),lag.max=20,pacf=TRUE)

#  theoretical  pacf

tp2 

Report length. The report should not be too long. You should aim to convey the important details in a way which is easy to follow, but not excessively long. Think about someone reading it through and try to help make it easy for them. Make it clear, without too much repetition and avoid long items of numerical output.

Grading The coursework will be marked out of 10:

•  5 marks for technical content, use of R and appropriate methods

•  5 marks for presentation and interpretation of results

(5/5)
Attachments:
Instructions Files

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