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Using the R script, answer the following questions. Please show all works for full credit. Please install and load the following package as follows. install.packages("astsa") library(astsa) 1. Consider the following ARMA(1, 2) process. =11++11+22 ~ (0,2) , where 1, 1, and 2 are parameters for the model.

a) Using arima.sim() function with the length of output series n = 250, i) Simulate the model above with parameters 1=0.25,1=0.2, and 2=0.8 (You dont need to show the result. Just show the code for simulation.). ii) Assign the name x to the data generated by the simulation. Note that you have to run the following code with your simulation (that is, this code should be followed by your code for simulation and run all together.)

set.seed(150)

b) Plot the series you obtained from the simulation. c) Plot the sample autocorrelation function (ACF) and the sample partial autocorrelation function (PACF), and comment how they look like.

2. Use the data x you have generated in Q #1. Suppose that you fit a model to the data without knowing the fact that the data is generated by ARMA(1, 2) simulation. Now, you try to find the best model for this data. So, after looking at ACF and PACF, you have decided to try to fit AR(1), AR(2), MA(1), MA(2), ARMA(1, 1), ARMA(1, 2), and ARMA(2, 2). a) Fit each model you have decided to run and show AIC and BIC for each model. What model are you going to choose as the best model in terms of AIC and BIC criteria. b) Check the residual ACF of the best model you choose in a) and comment. Are the residuals white noise or not?

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