TS Module 4: Variance of mean homework assignment

(The attached PDF file has better formatting.)

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.

I'm confused - the homework assignment seems to cover things not in the readings for this module.  The all in 1 pdf of homework (which I understand was submitted by a user, not NEAS) also seems to have a different homework assignment than the pdf attached.  The all in 1 pdf HW assignment for module 4 DOES seem to be relevant to the readings though.  

Is
"TS Module 4: Variance of mean homework assignment" the correct assignment? (NEAS: Yes, this is the correct assignment)
or the:
"TS Module 4: Regression methods HW" correct? (which is in the all in 1 pdf, but seems relevant to the readings assigned)?

Is this just taking the derivative (1-theta(1)-theta(2))(epsilon(n-2)) with respect to epsilon(n-2) and setting the equation equal to zero? If not, is there a page reference in the book to assist?

I'm having a really hard time understanding what this question is even asking.. I dnt see anything related to this in the reading. Can someone please help?

This looks different from the homework assignment I printed. The one that I see has exhibit 3.10 and is showing standardized residuals versus Fitted VAlues for the temp. Seasonal means model.

I’m wondering about questions C and D. For C, it asked if toy sales are higher in December (500,000) and low in February (50,000). Would you expect the residuals to have a higher variance in December or February? What’s the intuition behind this? It does not seem obvious to me.

Part D asks why this reasoning does not apply to daily temperature.

Would you please help me understand the idea behind this, or where in the book it explains this reasoning?