solution

Plot the autocorrelation of your time series using acf() and comment on whether it behaves as expected

Discuss whether the Holt-Winters scheme is appropriate for this dataset

 

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SKU: assim17

1.This question relates to the air passenger dataset accessible in R by typing data (“Air Passenger”) ; AP <-Air Passengers.

at the command prompt.

a)How many observations are in this data set?

b)What is the time of the first data point?

c)Plot this data set using standard R commands and comment on the trend and cyclic nature of the dataset

d)You will  recall from lectures that Holt-Winters method is implemented in R using commands such as:

AP.hw <-  Holt Winters (AP)

plot (AP.hw)

Discuss whether the Holt-Winters scheme is appropriate for this dataset.

 

e)Using R code such as:

AP.predict <- predict (AP.hw,  n.ahead=10)

ts.plot (AP,  AP . predict,  lty=1:2)

show a prediction for four years after the dataset ended.

 

f)Plot a similar graph but using the multiplicative model.Comment briefly on whether the multiplicative model is better than the additive.

g)Plot the auto correlation function of AP using  acf() and comment on the structure that this reveals.

 

You will be required to use R and are permitted (furthermore encouraged) to use the R help system.

 

2.This question is concerned with specified auto regressive models.

a)Using R commands such as:

n <- 1000

x <- rep (0,n)

for (i in 2:n) {

    x[i]  <- 0.2*x[i-1] + rnorm(1)

}

x <-ts(x)

simulate the time series specified by the recurrence realtion

xt =0.9 * xt -1 + wt

Where wt is Gaussian white noise

 

b)Plot the auto correlation of your time series using acf() and comment on whether it behaves as expected.

 

c)If,instead,we simulate the relation xt =1.8 * xt -1 + wt, say whether this time series is stationary.Provide numerical and theoretical justification for your answer.

 

You will be required to use R and are permitted (furthermore encouraged!) to use the R help system

 

 

3.This question gives you some recurrence relations.Show how to simulated a length-1000 time series for these relations using the arima.sim() command.Remember you may consult the R help page for guidance.

 

a12

d)For the time series in part (c),use arima() to estimate the coefficients and comment on whether the fit is good

 

You will be required to use R and are permitted (furthermore encouraged!) to use the R help system

 

 

4.This question uses the recurrence

a4

a)Use arima.sim() to simulate the recurrence for a length-1000 time series

b)Use arima(…..,order=c(2,0,2)) to estimate the coefficients

c)Construct a 95% confidence interval for the first coefficient of the AR and the MA components

d)State whether your confidence interval includes the true value of the coefficients.

 

You will be required to use R and are permitted (furthermore encouraged!) to use the R help system

 

 

5.(harder).

Consider the following recurrence:

a5

a)Is the stationary?

b)We usually specify that wt is white noise with distribution N(0,1).But mow suppose that noise is absent,that is wt =0 for all t . Given that x0=0 and x1+1,calculate the first few terms  in the time series and briefly discuss whether it exhibiting nonstationarity

 

You will be required to use R and are permitted (furthermore encouraged!) to use the R help system

 

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