I got:
A1) 0.0792
A2) 0.0446

A3) 0.02
A4) 0.0054
A5) 0.0008

I have no idea if I'm right or not. I did this work a while ago.

Not sure if this is frowned upon, but I found this section really tough as I didn't know if I had the correct answers or not.

I got:
A1) 0.0792
A2) 0.0196
A3) 0.04
A4) -0.0046
A5) 0.0008

Appreciate someone letting me know if they think I've gone wrong somewhere.

Thanks

In problem A, we can consider for all 5 series a general form:
Y(t) = mu + e(t) + q·e(t-1)

Then we need to calculate:
gamma0 = Var(Y(t))
and
gamma1 = Cov(Y(t),Y(t-1))
rho1 = gamma1/gamma0

The calculations are pretty straightforward.
Once you get the general formulas as functions of parameter q, just plug in the relevant values of q for each of the 5 series.

If Var(Y-bar) = (gamma₀/n)[1+2(1-1/n)(rho₁)], and you got 0.0495 for Var(Y-bar), through some backward calculation this tells me that if you used the same gamma₀ as me then you used 1/2 for rho₁. 

Recall gamma₀ = Var(Y_t) and rho₁ = Cov(Y_t,Y_t-1)/gamma₀ and Cov(a+bY,c+dY) = bdCov(Y_t,Y_t) = bdVar(Y_t).

Double check your rho.

RDH

just want to check, did anyone get .0495 for the second series?

Yes

RDH

Right, so isn't the following true?

Var(Y_t) = Var(mu + e_t+ .5e_t-1) = Var(mu) + Var(e_t) + Var(.5e_t-1) = 0 + Var(e_t) + .25Var(e_t)

Regarding the "pattern" question, if Y(j) is higher than average, I would expect Y(j+1) to be higher than average if they have positive autocorrelation and lower than average if they have negative correlation...

Sorry for being slow, but I still don't get part B, so are we talking about 1. and 5. as first and last series? Are the mean for all five processes mu right? I am not sure if this is right but if it is, that means Y t+1 should also be mu? I think I might be totally confused.

And can someone explain what C is asking?

Thanks!

Happy Monday, everyone!

hs, I believe there is already a post about this somewhere, however I shall restate it.  Page 25 (Appendix A), equation (2.A.6):  Var(a + bX) = b²Var(X)  where a and b are constants and X is a random variable.  So, to answer your question, yes, Var(mu) is in there, but we know the variance of a constant is zero.

Your "pattern" question was also in a previous post, NEAS asks to describe how we would predict Y_t+1 if Y_t was above or below the mean.  This is easily determined from the autocorrelation function, recalling that a positive autocorrelation describes a positive increase over time.  Let me know if you have any further questions.

RDH

What is part B asking for?  What is the first and last time series?