Fit the ARIMA$(2,1,2)$ model

$(1-{}_{1}B-{}_2{B}^2)(1-B)x_t=c+(1+{}_{1}B+{}_2{B}^2)Z_t$

to the transport index sample data $(x_t{)}_t=1^{60}$ using the Arima function from the

forecast package and write down the fitted equation. Is there any evidence that the Without using specialised $R$ functions such as arima.sim or the like, write code to

simulate a path $(hat(x{)}_t{)}_t=1^{60}$ of the fitted ARIMA model from $($c$).$ Generate the noise as

Gaussian RVs $($with appropriate variance$)$ using R function rnorm and use initial

values hat$(x{)}_1=x_1,$hat$(x{)}_2=x_2,$hat$(x{)}_3=x_3$ from the sample data. Provide a plot of a simulated

Do question e

use this data xt c$(100,108\mathrm{.}4,109\mathrm{.}7,111\mathrm{.}7,115\mathrm{.}9,112\mathrm{.}2,112\mathrm{.}8,116,112\mathrm{.}5,113\mathrm{.}8,113\mathrm{.}9,111\mathrm{.}6,$

$108\mathrm{.}5,108\mathrm{.}7,83\mathrm{.}4,48\mathrm{.}8,58\mathrm{.}2,66\mathrm{.}8,67\mathrm{.}5,63\mathrm{.}8,68\mathrm{.}5,73\mathrm{.}6,83\mathrm{.}1,85\mathrm{.}5,$

$74\mathrm{.}9,82\mathrm{.}1,89\mathrm{.}4,90\mathrm{.}7,93,89\mathrm{.}8,81\mathrm{.}8,74\mathrm{.}9,77\mathrm{.}9,84\mathrm{.}4,95\mathrm{.}7,94\mathrm{.}6,87\mathrm{.}5,$

$107,115\mathrm{.}8,111\mathrm{.}6,119\mathrm{.}9,126\mathrm{.}8,127\mathrm{.}3,127\mathrm{.}9,132\mathrm{.}1,135,138\mathrm{.}7,129\mathrm{.}3,$

$122\mathrm{.}1,132\mathrm{.}1,135\mathrm{.}4)$