Consider the ARMA(1, 1) model y t =50- 0.7 y t-1 + 0.5 ε t-1 + ε 1 . Assume that the variance of the white noise process is σ 2 = 2. a. Sketch the theoretical ACF and PACF for this model. b. Generate...

Consider the ARMA(1, 1) model
y_t

=50- 0.7y_t-1

+ 0.5ε_t-1
+
ε

₁
. Assume that the variance of the white noise process is σ²
= 2.

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?