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can someone helps me to solve problem 2 and 3??? thanks a lot...(problem 3 is in the book chapter 3 No.14)

Stat 153 Problem Set 3 (due March 2, 2016)

2

In the following problems, wt ? WN(0, ?w ).

?

1. We saw that xt = j=0 ?j wt?j is the unique stationary solution to the

AR(1) recursion xt ? ?xt?1 = wt for |?| &lt; 1. However, there are many

?

non-stationary solutions. Check that yt = c?t + j=0 ?j wt?j is a solution

to the AR(1) recursion for any c ? R. Show that {yt } is not stationary

for c = 0.

2. Consider the ARMA(2,1) process that satis?es xt ? xt?1 + 1 xt?2 = wt +

2

1

2 wt?1 . Show that there is a causal stationary solution and compute its

ACF.

3. tsa3 3.14

4. Let ? ? Rn be the coe?cients of the best linear forecast of xn+1 given

xn:1 :

P (xn+1 | x1:n ) = ?T xn:1 =

?j xn?j+1 .

j?[n]

Show that the coe?cents of the best linear backcast of x0 given x1:n is

?

? = P ?, where ? ? Rn are the coe?cients of the MMSE linear forecast of

xn+1 given x1:n and P ? Rn×n is the anti-identity matrix. That is, show

that

?

P (x0 | x1:n ) = ?T xn:1 =

?j xj .

j?[n]

5. Download the data on sun spot activity. It consists of n = 285 observations

of sun spot activity from 1700 to 1984. We study the square root of the

data.

? Compute the sample ACF and sample PACF.

? Which of the following models is most appropriate for the data: AR(1),

A(2), MA(1), or MA(2)?

? Use the data to estimate the parameters of the model that you chose.

n

? Using your ?tted model, compute the forecasts Xn+h for h = 1, 2, 3, 4.

Give a 95% prediction interval (assuming the noise is Gaussian)

? The number of sun spots for the years 1985 to 1988 are 17.90, 13.40,

29.20, 100.20 respectively. Plot the data (including the 1985 to 1988

1

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