**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.

**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

**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:

**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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