5.56)Use a random number generator and generate 100 observations from the AR(1) model yt = 25 + 0.8yt1 + t. Assume that the errors are normally and independently distributed with mean zero and variance 2 = 1.

a. Verify that your time series is AR(1).

b. Generate 100 observations for a N(0, 1) process and add these random numbers to the 100 AR(1) observations in part a to create a new time series that is the sum of AR(1) and "white noise."

c. Find the sample autocorrelation and partial autocorrelation functions for the new time series created in part b. Can you identify the new time series?

d. Does this give you any insight about how the new time series might arise in practical settings?

5.60 Consider the AR(2) model yt = 25 + 0.6yt1 + 0.25yt2 + t. Assume that the variance of the white noise process is 2 = 1.

a. Sketch the theoretical ACF and PACF for this model.

b. Generate 50 realizations of this AR(1) process and compute the sample ACF and PACF. Compare the sample ACF and the sample PACF to the theoretical ACF and PACF. How similar to the theoretical values are the sample values?

c. Repeat part b using 200 realizations. How has increasing the sample size impacted the agreement between the sample and theoretical ACF and PACF? Does this give you any insight about the sample sizes required for model building, or the reliability of models built to short time series.

Can i please get R code for the above 2 questions?