TS Module 4: Variance of mean homework assignment

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Homework assignment: Variance of mean

An MA(2) process Y_t = e_t – è₁ e_t-1 – è₂ e_t-2 has N observations, with ó²_e = 1, –1 è₁ +1, –1 è₂ +1. What values of è₁ and è₂ maximize the variance of , the average of the Y values?What values of è₁ and è₂ minimizes the variance of , the average of the Y values?Your answer should give a line of values for each part, such as è₁ + è₂ = k.

Jacob: How should we reason through this homework assignment?

Rachel: Write the value of in terms of the å’s: y_j = å_n + (1 – è₁) å_n-1 + (1 – è₁ – è₂) å_n-2 + …

Most of the terms have (1 – è₁ – è₂) å_n-2 ; only the two terms at the beginning and the two terms at the end have fewer è’s. Ignore these beginning and end terms (assuming n is large).

All the å’s are independent. We choose è₁ and è₂ to maximize or minimize (1 – è₁ – è₂) å_n-2, which is easy.

For the homework assignment, ignoring the end terms is fine. If N is small, the answer differs slightly, and the calculations are messy.